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Virginia Tech | Figure 1 - Face and Butt Cleats
Face and butt cleats in conjunction with larger scale fractures, joints and faults constitute
effective flow paths for gas in the coalbeds. As it has been postulated and confirmed by many
experimental results, the anisotropic pore structure of coal results in higher permeability values
along the face cleats compared to butt cleats and significantly higher than in the direction normal
to the bedding plane. Indicative permeability values reported by Gash et al. (1992) are 0.6~1.7
mD for face cleats, 0.3~1 mD for butt cleats and only 0.007 mD for the direction normal to the
bedding plane (Gu, 2009). The porous coal matrix permeability, is reported to be substantially
less than that for the cleats system, normally eight times smaller; thus many researchers
disregard coal matrix permeability and accredit the coalbed permeability to its cleats and larger
scale discontinuities (Robertson, 2005; Liu and Rutqvist, 2009).
Porous coal matrix is the primary medium for gas storage in coalbeds and accounts for up to
95-98% storage, (Gray, 1987). Storage in the coal matrix is performed via two primary
mechanisms. More specifically, gas can either be physically adsorbed on the very large (20-200
m2/g) (Patching, 1970) internal surface of the porous coal matrix, or it can be adsorbed within
the molecular structure of the coal matrix. The remaining gas resides as either a free gas or
dissolved in water within the cleats and larger fractures (Marsh, 1987; Shi and Durucan, 2005).
Typically a Langmuir-type adsorption isotherm is employed to describe the adsorption
phenomenon in coals (Shi and Durucan 2005).
In accordance with their basic characteristics, coalbeds can be regarded as dual porosity and
single permeability systems (Harpalani and Schraufnagel, 1990; Lu and Connell, 2007; Warren
and Root, 1963; Liu, 2011).
With natural gas production (CBM) and CO storage in unmineable coalbeds (ECBM),
2
complex interactions with varying stress state and sorption phenomena are triggered that affect
4 |
Virginia Tech | the transport and sorptive properties of coal and consequently affect production and/or injectivity
rates. Therefore, understanding the mechanisms of the dynamic evolution of sorption, flow, coal
deformation, porosity and permeability is of fundamental importance to CBM/ECBM recovery
(Liu 2011).
Coalbed Methane and Enhanced Coalbed Methane Production Mechanisms
Extraction of methane from undersaturated coalbeds activates a sequence of interactions
among the porous coal matrix, fractures and adsorbed and free gas. In primary gas production,
methane is extracted from the coalbed via decreasing reservoir pressure. More specifically, by
dewatering the targeted seams, the pressure of the gas in the reservoir decreases which results in
an increase of the effective stress and consequently closure of fracture apertures and lower
permeability. Furthermore, when the gas pressure falls below the desorption point, methane is
liberated from the matrix into the cleats system and the matrix shrinks. Based on the zero volume
change condition, matrix shrinkage effectively causes the aperture of fractures to increase and
thus fracture permeability to increase. Hence, an initial decrease in fracture permeability
attributed to an increase in effective stress is counteracted by a gradual increase in permeability
attributed to matrix shrinkage. The net permeability depends on the net impact of these two
competing effects (Chen et al., 2008; Connell, 2009; Liu et al., 2010b, c, d; Shi and Durucan,
2004; Gu, 2009; Liu, 2011).
Coalbed methane extraction for saturated coal seams follows the same mechanisms as in the
case for undersaturated coals without the dewatering phase. When pressure starts dropping there
is immediate release of methane from the coal matrix to the fracture system. Finally for
oversaturated coal seams, reduction of reservoir pressures results in concurrent flow of mobile
water and free gas towards the production wells. As soon as the coalbed seam attains a state of
saturation the production mechanism is similar to that for saturated coal seams (Gu 2009).
Carbon dioxide has been shown to have greater affinity to the porous coal matrix in contrast
to methane and thus it is preferentially adsorbed onto the coal when present. According to
laboratory measurements it is indicated that depending on coal rank, the matrix can adsorb
almost twice the amount of carbon dioxide by volume as methane and in some cases of lower
coal rank this ratio may even be as high as 10 to 1 under conditions of normal reservoir pressure
(Stanton et al., 2001; Shi and Durucan, 2005). In ECBM, when carbon dioxide is competitively
5 |
Virginia Tech | adsorbed onto the coal matrix and methane is displaced the coal matrix swells. Coal matrix
swelling combined with the zero volume change condition results in reduced fracture apertures
and decreased fracture permeability (Mavor et al, 2002; Gu, 2009).
Permeability Models
As discussed, permeability is a key parameter in both coalbed and enhanced coalbed
methane production. US experience suggests that an absolute permeability of 1 mD is generally
required to achieve commercial production rates. For these reasons, the complexity of the
physical interactions affecting permeability has been extensively discussed in the literature and a
variety of models have been proposed attempting to simulate the dynamic evolution of this
phenomenon. Many of these models have also been incorporated in numerical simulators in
order to quantify coal-gas interactions (Liu 2011). All of these models are based on numerous
assumptions regarding flow characteristics and boundary conditions as well as the
geomechanical component that may interact with the above parameters. In the following section
the most important models representing the key parameters in multiphase flow problems are
presented and critically evaluated. Also, in order to achieve a better understanding of the
different mechanisms involved as well as the interaction of the various parameters of the model,
formal definitions of the important variables are briefly outlined below.
A porous medium having a bulk volume (V), comprised of solid volume (Vs) and pore
volume (Vp), where V=Vs+Vp, has porosity (φ) which is defined as the ratio of the volume of
the pores to the bulk volume of the medium, φ=Vp/V. The permeability (k) of the porous
medium is the measure of its ability to allow fluids to pass through it and is expressed in units of
area. Compressibility of a solid medium is described as the measure of relative volume change as
a response to a pressure or a mean stress change. Furthermore, volumetric strain of a deformed
body is defined as the ratio of the change in volume of the body to the deformation of its original
volume. Effective stresses (and or strains) can be calculated from total quantities by subtracting
the fraction attributed to pore pressure.
Physical processes are complex phenomena and their formal descriptions, which are also
known as constitutive laws, are usually very complicated. Incases simplifying assumptions are
used to facilitate analyses with a first order approximation. In the following, it will be
demonstrated that researchers may apply different constitutive equations to describe the same
6 |
Virginia Tech | physical process with a varying number of (simplifying) assumptions. The basic principle of
each formulation will be presented along with the underlying assumptions. On many occasions
the governing equations of these physical processes are termed analytical models in contrast to
the numerical models, which include the methodology for achieving a numerical solution to the
analytical equation.
Liu 2011 stipulated that the total effective volumetric strain responsible for changes in coal
porosity and permeability during production of CBM/ECBM as described by a thermo-poro-
elastic constitutive equation is given by:
(1)
It should be noted that the basic assumptions for this formulation are (i) that thermal
contraction of a non-isothermal body is analogous to coalbed matrix shrinkage/swelling (Palmer
and Mansoori, 1996), and (ii) that the coal seam is treated as a non-isothermal medium.
According to Eq. (1) the total effective volumetric strain is comprised of the total volumetric
strain, the coal compressive strain, the gas sorption-induced volumetric strain and the thermal
strain.
This model is considered to include all major terms affecting fluid flow in single and dual
porosity situations affected by the strain regime of the medium.
Porosity and permeability of the coalbed can be both described as a function of the total
effective volumetric strain ( ) through the following general functions:
( )
(2)
( )
(3)
Also, formulations relating porosity and permeability have been reported in the literature for
different conceptual permeability models for coalbeds. Chilingar (1964) regarded coal as a
continuum porous medium and proposed an equation relating permeability to the porosity and
the effective diameter of its grains, Eq. (4).
(4)
( )
7 |
Virginia Tech | By applying Eq. (4) for initial ( ) and current ( , k) permeability and porosity values,
Eq. (5) can be derived. When the porosity is much smaller than 1 the second term of the right-
hand side is close to unity and the term is dropped. Hence, the relationship between permeability
and porosity for the coal matrix is as follows:
( ) ( ) ( ) (5)
Reiss (1980) idealized coal as a collection of matchsticks (Figure 2) and considered
permeability to be related to its porosity and cleat system spacing, Eq. (6).
(6)
By applying Eq. (6) for initial ( ) and current (k) permeability values, Eq. (7) can be
derived.
( ) (7)
Coal
Confining
formation
Figure 2 - The Matchstick Model
In Eq. (7) which is the same as the simplified version of Eq. (5), the value of the exponent is
related to the flow regime, cleat size and roughness and is normally equal to 2 for laminar flow, 3
for transition flow and 4 for turbulence flow. According to the literature, flow through naturally
fractured coalbed reservoirs is in the transition flow regime, and thus the exponent is taken to be
3 (Reiss, 1980; Seidle, 1992; Avraam and Payatakes, 1995; Wang, 2009).
By applying Eq. (7) which relates permeability to porosity (and thus to elemental volume
conditions), permeability-change formulations are acquired which in many cases may include
coal properties as well as related physical quantities. Extended research has been conducted and
8 |
Virginia Tech | many models have been proposed in the past 30 years in order to combine the flow regime to the
geomechanical characteristics of the medium and examine the dynamic evolution of
permeability. Different classifications have been proposed for these models according to their
respective assumptions, encompassed parameters and employed solving techniques.
More specifically, Palmer (2007) grouped permeability models into analytical permeability-
change models and permeability models that couple different processes through a set of flow and
geomechanical equations and solve the system using numerical simulators. Gu and Chalaturnyk
(2005) and Palmer (2007), further divided analytical permeability-change models into either
stress or strain based models, with or without a geomechanical framework. Strain- based,
permeability-change models where defined as the ones where in the case of CBM production
desorption of methane changes the volumetric strain that causes the porosity to change and
finally affects the permeability. In the case of the stress-based models for CBM production,
desorption of methane changes the volumetric strain, which consequently changes the horizontal
stress and ultimately alters the permeability.
Settari and Walters (2001), Rutqvist et al (2002) and Liu (2011) divided permeability-
change models that couple fluid flow and solid deformation into decoupled simulation,
sequential coupled simulations and fully coupled simulations. In fully coupled simulations, both
reservoir flow variables and geomechanical variables are concurrently determined by solving a
system of equations. In sequential coupled simulation, parameters pertinent to flow, such as
pressure and temperatures, and geomechanical variables, such as stresses and displacements, are
solved sequentially first from a reservoir simulator and then from a geomechanical simulator and
coupling parameters, such as permeability and porosity are iterated between the two simulators
(Figure 3). In decoupled or uncoupled simulation the changes of fluid pressures cause the
changes of stresses and strains, but the changes of stresses and strains are assumed not to affect
fluid pressures (Wang, 2000).
9 |
Virginia Tech | Seidle at al., (1992), Pan et al., (2010), proposed a stress based model for permeability
changes, as shown in Eq. (10). In accordance with experimental results, an exponential function
was used for the permeability-stress relationship (Durucan and Edwards, 1986; Gray 1987;
Wang 2009). Further assumptions applied for this model, were the postulation of a matchstick
geometry for the coal matrix, isotropic permeability, uniaxial strain and constant overburden
stress.
[ ( )]
(10)
where cf is the fracture cleat compressibility defined as follows
(11)
This model accounts for grain compaction and the cleats volume change contribution to the
permeability alteration however it does not incorporate volumetric strain due to gas adsorption or
desorption.
Gilman and Beckie (2000) suggested a stress-based relationship for isotropic permeability
under uniaxial strain and constant overburden stress conditions for saturated coals, as shown by
Eq. (12). An exponential function was employed to relate permeability to the other parameters in
the framework of poro-elasticity. A relative regular cleat system was assumed and it was also
considered that each fracture reacts as an elastic body with respect to change in the normal stress
component. In addition, an extremely slow mechanism of methane release from the coal matrix
to the cleats was postulated.
( ) [ ( )] (12)
Seidle and Huitt (1995) described a strain based permeability change model for isotropic
permeability of saturated coals in the framework of poro-elasticity under uniaxial strain and
constant overburden stress reservoir conditions, as shown in Eq. (13). A cubic function was
employed in this case to correlate the pertinent parameters to permeability. It was assumed that
12 |
Virginia Tech | the coal sorption-induced strain is proportional to the amount of gas sorbed and that the sorbed
gas is related to pressure by Langmuir’s equation (Liu, 2011).
[ ( )( )] (13)
Shi and Durucan (2004) presented a stress based permeability model for isotropic
permeability changes in the framework of linear elasticity for saturated coals, as shown in Eq.
(14). An exponential function was employed to relate permeability to cleat compression and
matrix shrinkage. These two terms affect the dynamic permeability evolution in a competitive
way. For this model as in previous cases, uniaxial strain and constant overburden stresses were
assumed.
[ ( ( ) ( ))]
( ) ( )
(14)
Palmer and Mansoori (1996), proposed a strain based, fully geomechanical model for
isotropic permeability under conditions of uniaxial strain and constant overburden stress in the
framework of linear elasticity for small strain changes, Eq. (15).
[ ( ) ( )( )]
(15)
( )( )
In 2007, Palmer and Mansoori modified their model to account for directional (anisotropic)
permeability and modulus changes with depletion. The Palmer and Mansoori (2007)
permeability change model is also applicable to both saturated and undersaturated coals. The
Computer Modelling Group (CMG) implemented this model in their commercial reservoir
simulator, GEM.
Peekot and Reeves (2002), suggested a permeability model in which matrix strain changes
are extracted from a Langmuir curve of strain versus reservoir pressure which is assumed to be
proportional to the adsorbed gas concentration curve (Palmer, 2007; Liu, 2011). This model is
not developed within a geomechanical framework. The matrix shrinkage is proportional to the
adsorbed gas concentration change multiplied by shrinkage compressibility ( ) that is a free
13 |
Virginia Tech | parameter. This model is incorporated in the commercial reservoir simulator package, COMET3
by ARI (Comet, 2003). It has been shown that for saturated coals the latter model and the Palmer
and Mansoori (1996) model are essentially equivalent (Palmer, 2007; Liu 2011).
Cui and Bustin (2005), described a stress dependent permeability model for isotropic
permeability in the poro-elasticity framework under uniaxial strain and constant overburden
stress reservoir conditions, as described by Eq. (16). This model essentially accounts for the
effects of pressure and sorption induced volumetric strain on permeability constrained by the
adsorption isotherm.
( )
{ [ ( ) ( )]}
( ) ( )
(16)
Gu and Chalaturnyk in 2010 proposed an anisotropic permeability-change model. As shown
in Eq. (17), they determined a directional effective strain, comprised of the directional
mechanical deformation due to stress change, the directional mechanical deformation due to
pressure change, the directional matrix shrinkage/swelling due to desorption/sorption and the
directional thermal contract/expansion due to temperature changes. The directional effective
strain was then related to a directional permeability, Eq. (18), with a cubic function and with
assuming a stack of matchsticks configuration for the coalbed.
(17)
( )
(18)
Pan and Connell (2007) and Clarkson et al. (2008), proposed an elastic strain dependent
model for isotropic permeability under uniaxial strain and constant overburden stress conditions,
Eq. (19). This model accounts for sorption strain, Eq. (20), for a single component adsorption.
(19)
[ ( ) ( ) ]
( )( )
( ) ( ) ( )
(20)
Robertson and Christiansen (2006) suggested an isotropic permeability-change model
assuming a cubic configuration for the coalbed under variable stress reservoir conditions and
14 |
Virginia Tech | constant overlying geology, as shown in Eq. (21). In this model, effective porosity of the coal
matrix is considered to be equal to zero and thus only the porosity of the coalbed is attributed to
the fracture system.
[ ( )]
{
(21)
[ ( ) ( ) ]}
Zhang et al. (2008) presented a permeability change model under variable stress conditions
in the poro-elasticity framework, Eq. (22).
( [( ) ( ]) (22)
where S and
(23)
(24)
Connell et al. (2010), Liu et al. (2010) and Liu and Rutqvist (2010), reported that when the
coalbed is represented by a stack of coal matrix matchsticks separated by the cleat system, under
conditions of constant confining stress (as is usually applied in laboratory experiments), then the
effects of coal matrix swelling will not alter the coal permeability. This statement argues that
since for a given pressure, the coal matrix blocks are fully separated by the fractures and no fixed
boundaries are enforced, the swelling will translate in an increase in fracture spacing and not in a
reduction of fracture aperture. Nevertheless, this conclusion does not tie in with laboratory
observations, where matrix swelling causes significant changes in coal permeability under
conditions of constant confining stress. As discussed by Liu (2010a), in order to account for this
interaction many researchers postulated a zero lateral strain condition in the horizontal plane so
that matrix swelling will result in permeability changes. (e.g., Harpalani and Chen, 1995;
Robertson and Christiansen, 2006; Harpalani and Chen, 1997; Pan et al., 2010; Pini et al., 2009).
Liu et al. (2010) suggested a different approach in order to correctly reproduce the
aforementioned laboratory observations, which accounts for interactions among coal matrix and
15 |
Virginia Tech | fractures. More specifically, the cleat system in this model is not considered to be completely
separate neighboring matrix blocks, but assumes that there are solid bridges connecting the
adjacent coal matrix blocks (Figure 6). For this coalbed configuration, coal matrix-swelling
affects permeability in two opposing ways. When the bridging contacts swell, the coal matrix
blocks are driven apart and thus fracture aperture increases and consequently permeability. At
the same time, swelling of the matrix blocks results in reduction of fracture aperture and thus
permeability.
Figure 6 - Matchstick Model with Bridging Between Coal Blocks
Connell and Detournay (2009) suggested a sequential coupled simulation model to describe
dynamic evolution of permeability in an isotropic linear elasticity framework. Their formulation
was based on Shi and Durucan (2004) isotropic permeability-change stress based model.
However, they discussed a modified version in which directional permeability, triaxial strain and
varying overburden stress conditions were accounted for. Permeability changes for both CBM
and ECBM productions cases were examined by employing the SIMED reservoir simulator
(Spencer et al., 1987; Stevenson and Pinczewski, 1995) and the FLAC3D finite difference
geomechanical code. It was shown that especially for the ECBM production cases, simplifying
assumptions such as uniaxial strain and constant overburden stress can lead to large errors.
Gu and Chalaturnyk, 2010 proposed a sequential coupling permeability change model,
where the discontinuous coal masses were considered as an equivalent elastic continuum. Their
explicit-sequential coupled simulation examined pressure depleting CBM reservoirs. GEM
(CMG, 2003) a multidimensional, multiphase, isothermal and compositional reservoir simulator
and the geomechanical code FLAC3D (Itasca, 2002) designed for rock and soil mechanics
analyses were employed to model fluid flow and calculate coalbed deformation respectively. The
simulation process was as follows: (i) Calculate pore pressures, adsorbed gas volumes, water
saturation and well production rates of gas and water with GEM at an arbitrary time step, then
16
C
C
B
o
o
r
n
a
id
f in in
l
g in g
g F o r m a t io n |
Virginia Tech | (ii) insert pore pressures and adsorbed gas volumes in FLAC3D and calculate stresses and linear
strains and finally (iii) calculate cleat permeability based on Eq. (25).
( )
(25)
Discussion on Implementation of Permeability Models for Reservoir
Simulation
Most commercially and in-house developed software for simulating reservoir behavior for
CBM/ECBM production has implemented one or more of the aforementioned analytical
approaches. In order to apply these relationships to specific geological strata sequences with
varying initial and boundary conditions, a numerical approach should be implemented. This
implementation requires discretization of the domain in two or three dimensions (Figure 7) and
the solution of the governing differential equations in space and time. Commonly finite
difference schemes are used to solve the flow problem. In the more advanced, complicated
approaches where the geomechanical component should also be taken into account a coupled
approach should be used. This can be accomplished either using external coupling (i.e. coupling
to software not directly interfaced with the flow package) or internal coupling where flow
equations can be modified directly by the stress strain regime of the reservoir as prescribed by
the analytical model (see also Figure 3).
The analysis provided in this study will be applied for the simulation of CBM and ECBM
production for a small-scale carbon capture, utilization and storage (CCUS) project in southwest
Virginia. More specifically, the injection and storage potential of unmineable coal seams will be
tested by injecting 20,000 tonnes of carbon dioxide into three wells, in Buchanan County,
Virginia, in the central Appalachian basin, over a period of one year (Figure 8).
17 |
Virginia Tech | Figure 9 - Initial Attempts for Coupled Simulation Using an External Geomechanics
Package (Vasilikou et al., 2012)
Conclusions
Coal permeability and permeability change models are probably the most important
considerations when evaluating the potential of a coal bed methane reservoir for enhanced coal
bed methane production. Different analytical models may be applied depending on the available
knowledge and properties of a reservoir. In the simplest considerations, the flow model will be
solved without input from the changing geomechanics regime in the area. However, as
complexity increases, other conditions should be taken into account (e.g., pore pressure changes,
shrinkage, swelling, etc.) and thus porosity and permeability fluctuations need to be incorporated
as dictated by changes in the stress and strain situation. Furthermore, in ECBM reservoirs
additional complications arise due to the presence of two gases in two-phase flow conditions.
In every step of conducting a reservoir simulation, the user should be aware of the net
contribution of each factor in the final result. Components with minimal contribution may be
safely eliminated after consideration. Also, the underlying assumptions as well as application
conditions should be evident for every analytical model, especially when performing history
matching. As already discussed in the literature, improper model hypothesis and model selection
will lead to large discrepancies between predicted and measured data.
The analytical model evaluation presented in this study will be utilized in reservoir
simulations for ECBM production for a small-scale carbon capture, utilization and storage
(CCUS) project in the central Appalachian basin.
19 |
Virginia Tech | p the gas pressure within ε the volumetric strain of
v
the pores coal
p the initial gas pressure ν Poisson's ratio of coal
0
within the pores
P Langmuir pressure ν Poisson's ratio for solid
L s
constant phase
R the gas constant (8.314 J ρ the density for the solid
s
mol− 1 K− 1) phase
s cleat spacing σ the horizontal stress
h
T reservoir temperature σ the initial horizontal stress
h0
α change rate in fracture ϕ coal porosity
c
compressibility
α the coefficient of ϕ initial coal porosity
T 0
volumetric thermal
expansion
γ the volumetric ϕ the cleat porosity
f
swelling/shrinkage
coefficient
References
Advanced Resources International, I. (2003). COMET3.
Avraam, D. G., & Payatakes, A. C. (1995). Flow regimes and relative permeabilities during
steady-state two-phase flow in porous media. Journal of Fluid Mechanics Digital Archive,
293, 207-236.
Ayers, W. B. (2002). Coalbed gas systems, resources, and production and a review of contrasting
cases from the San Juan and Powder River basins. AAPG Bulletin, 86, 1853–1890.
Bodden, W. P., & Ehrlich, R. (1998). Permeability of coals and characteristics of desorption
tests: implications for coalbed methane production. International Journal of Coal Geology,
35, 333–347.
Chen, Z., Liu, J., Connell, L., Pan, Z., & Zhou, L. (2008). Impact of Effective Stress and CH –
4
CO Counter-Diffusion on CO Enhanced Coalbed Methane Recovery. Paper presented at
2 2
the SPE Asia Pacific Oil and Gas Conference and Exhibition, Perth, Australia.
Chilingar, G. V. (1964). Relationship between porosity, permeability, and grain-size distribution
of sands and sandstones. Paper presented at the 6th International Sedimentological
Congress.
21 |
Virginia Tech | EXPERIENCES IN RESERVOIR MODEL CALIBRATION FOR COAL
BED METHANE PRODUCTION IN DEEP COAL SEAMS IN RUSSELL
COUNTY, VIRGINIA2
Foteini Vasilikou, Virginia Center for Coal and Energy Research, Virginia Tech,
Blacksburg, VA
Cigdem Keles, Virginia Center for Coal and Energy Research, Virginia Tech,
Blacksburg, VA
Zach Agioutantis, Virginia Center for Coal and Energy Research, Virginia Tech,
Blacksburg, VA
Nino Ripepi, Virginia Center for Coal and Energy Research, Virginia Tech,
Blacksburg, VA
Michael Karmis, Virginia Center for Coal and Energy Research, Virginia Tech,
Blacksburg, VA
Abstract
Injection and storage of carbon dioxide (CO ) on the coal seam matrix has two benefits:
2
mitigation of greenhouse gas emissions and enhanced recovery of coalbed methane (ECBM).
The theory of ECBM is based on the natural affinity for the porous coal matrix to preferentially
adsorb CO over methane. According to laboratory measurements on bituminous Appalachian
2
coals, the coal matrix can adsorb almost twice the amount of CO by volume as methane at
2
2 Experiences in Reservoir Model Calibration for Coal Bed Methane Production in Deep
Coal Seams in Russell County, Virginia.
F. Vasilikou, C. Keles, Z. Agioutantis, N. Ripepi, M. Karmis. 2013 Symposium on
Environmental Considerations in Energy Production, SME, April 14-18, Charleston, West
Virginia.
Reprinted with permission of SME.
Foteini Vasilikou researched and prepared this manuscript, with Cigdem Keles, Zach
Agioutantis, Nino Ripepi and Michael Karmis providing technical and editorial input.
27 |
Virginia Tech | typical reservoir pressures. Based on the fact that coalbed methane (CBM) is currently
economically produced in many areas in the world, presents the likelihood that CO injected
2
under the appropriate conditions could sequester the CO while displacing and producing CBM.
2
The Coal Seam Group of the Southeast Regional Carbon Sequestration Partnership
(SECARB) has recently completed the injection of 1,000 tons of carbon dioxide into multiple
deep unminable coal seams as part of a field validation test at their Russell County, VA site in
Central Appalachia and is planning the injection of 20,000 tons of CO in a nearby field in
2
Buchanan County, VA. This paper presents the reservoir modeling procedure and the parameters
developed in order to obtain an accurate history match for both gas and water production for a
group of eight mature CBM wells that were utilized in the Russell County field test. Optimal
parameters were determined both for the single well models as well as for the combined eight-
well model. This model will form the base model for simulating CO injection and subsequently
2
predicting post-injection behavior for the upcoming injection test at the Buchanan county site.
Introduction
The Coal Seam Group of the Southeast Regional Carbon Sequestration Partnership
(SECARB) has recently completed the injection of 1,000 tons of carbon dioxide into multiple
deep unminable coal seams as part of a field validation test at their Russell County, VA site in
Central Appalachia (Ripepi et al, 2009).
Figure 10 shows an aerial view of the general area, while Figure 11 shows an aerial view of
the injection well (BD114), two monitoring wells (black dots) and the closest offset producing
CBM wells.
28 |
Virginia Tech | regarding seam geometry, coal properties (gas content, adsorption isotherms, anisotropy in
permeability), butt and face cleat orientation and by considering well interference.
Geology
A carbon dioxide injection site was selected in Russell County, Virginia, where an existing
CBM well was donated by CNX Gas Corporation, a subsidiary of CONSOL Energy Inc. The
injection well may be found referenced by its company well name, BD114, or by the State of
Virginia designation, RU-0084. BD114 is located at an elevation of 2,523 feet in the Honaker
District of Russell County, Virginia.
The Central Appalachian Basin is a northeast-to-southwest trending basin encompassing
approximately 10,000 square miles in southwestern Virginia, southern West Virginia and eastern
Kentucky (Conrad et al., 2006). Production of CBM began in 1988 with the development of the
Nora Field in Dickenson County, Virginia followed by CONSOL Energy developing the
Oakwood Field in Buchanan County, Virginia. Since that time, over 5,600 CBM wells have
been drilled and brought on-line as producing gas wells in southwest Virginia through 2012
(VaDMME, 2013). The coals in the region near the injection site include those of the
Pocahontas Formation and Lee Formation which directly overlies the late Mississippian
Bluestone Formation. Coal seams of the Pocahontas and Lee Formation are medium to low-
volatile bituminous, high rank and high gas content coals that include the Pocahontas No. 1
through Pocahontas No. 9 seams (Pocahontas Formation and the Upper Seaboard, Greasy Creek,
Middle and Lower Seaboard, Upper and Middle Horsepen, War Creek, and Lower Horsepen
coals (Lee Formation).
The Upper Horsepen (UH) 2&3 coal seams are the thickest single completion at 3.7 feet of
net coal at a depth of 1,374 feet, with a 0.2 feet parting between the seams. The Pocahontas No. 3
coal seam is 2,208 feet deep and 2.4 feet thick at the test site and well outside of current deep
mining activities.
The primary confining units include multiple layers of low permeability shale and siltstone
beds that range in thickness from five to 55 feet in the vicinity of the injection well. Permeability
for the shale and siltstone units is expected to range from 0.001 to 0.1 md, with low porosity.
Even the sandstone units are considered to provide confining beds due to the well cemented
nature of these rocks. The Lee and Pocahontas Formation sandstones are known to have low
30 |
Virginia Tech | permeability and porosity values, and do not produce natural gas in this area. The tight sandstone
units range in thickness from five to 60 feet at the injection site. The sandstone units are expected
to have porosity values that range from 1.0 to 3.0 percent and permeability values ranging from
0.1 to 1.0 md based on core testing. All lithologies will encounter some natural fractures;
however, these fractures are likely to be cemented with quartz and calcite and are not expected to
provide permeability pathways based on core analysis, well logging and field testing (Karmis et
al., 2008).
Face and butt cleat planes were measured at two coal seam outcrop locations near the site
and they corresponded to within 7 degrees of the known face and butt cleat directions of N18W
and N167E respectively, for the deep mined Pocahontas No. 3 coal seam in Buchanan County. A
Rose Diagram (Figure 12) was developed to graphically display the cleat directions from the
injection well (VCCER, 2011).
Figure 12 - (A) Rose Diagram Showing Face and Butt Cleat Orientation (B) Rotation by
35o Clockwise
Well Stimulation
Twenty-four separate coal seams, totaling 36.3 feet, between the depths of 1,044 and 2,259
feet were perforated through the casing (Table 1) for well stimulation. The coal seams perforated
for the four stage well stimulation average 1.5 feet thick.
31
2 7 0 o B
2
D -1 1
o3 6 00
M 2
M 1-11
-1 1 44 /R U -8 4
1
2
A v e ra g e F a ce C le a t
oO rie n ta tio n : 1 6 7
o1 8 0180
H
D ireo
rizo
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ctio n o f M a x
on ta l S tre ss: 5 5
A v e ra g e B u tt C leO
rie n ta tio n : 7 4
o909 0
a t
o
M2 0360o
270o
2
BD-114/RU-1-8144
11
M-1
I Direction
Direction of Max
Face Cleats 1 3 Horizontal Stress
Butt Cleats
2
9090o
181080o
J Direction |
Virginia Tech | Production History
BD114 was brought on-line as a CBM producing well in 2002. Water is produced through a
2 7/8-inch string of tubing set below the Pocahontas No. 3 coal seam, and gas is produced
between the casing and the tubing (annulus). Up until injection commenced in 2009, BD114 has
averaged production of 1.19 thousand cubic meters (Mm3) (42 thousand cubic feet (Mcf)) of
natural gas per day and 2.2 barrels of water per day (Figure 13). BD114 produced 2.89 MMm3
(102 MMcf), and 5,360 barrels of water prior to being taken off-line for conversion to an
injection well. Gas production to date is nearly 10 percent of the estimated gas in place (VCCER,
2011). BD114 is a below average gas producer for this gas field. The average production of the
seven offset wells is 1.87 Mm3 (66 Mcf) per day of natural gas and 2.5 barrels per day of water.
Figure 13 presents the gas production history for the injection well and the nearby offset wells
(VCCER, 2011).
Figure 13 - Gas Production History at the Injection and the Offset Wells
34
1
1
1
4
2
0
8
6
4
2
0
0
0
0
0
0
0
0
)
y
a
d
f/
c
M
(
n
o
ti
c
u
d
o
r
P
s
a
G
2 2 3 3 4 4 5 5 6 6 7 7 8 8 9
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
3/ 9/ 3/ 9/ 3/ 9/ 3/ 9/ 3/ 9/ 3/ 9/ 3/ 9/ 3/
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
B D 1 1 4 B E 1 1 3 B E 1 1 4 B C 1 1 4 B D 1 1 3 B C 1 1 5 B D 1 1 5 B E 1 1 5 |
Virginia Tech | Modeling
Description of Production Model
Preliminary reservoir modeling prior to the actual field test where CO was injected into the
2
coal seams was conducted utilizing Advanced Resources’ COMET3 reservoir simulator for
CBM. The ultimate goal of this preliminary modeling was to estimate the size of the CO plume
2
as well as the expected CO injection rate that could be achieved during injection operations. As
2
additional laboratory analysis, field data and results from the CO injection field test became
2
available the need to revise the original reservoir model became apparent. At the same time,
Computer Modeling Group Ltd.’s GEM software, a compositional and unconventional reservoir
simulator, was utilized since it could simulate the effect of fractures. This paper discusses the
setup and parameters for production modeling of the Russell county field test using CMG’s
reservoir simulator. The ultimate goal of the current study will be to obtain a good history match
that will help modeling of Enhanced Coal Bed Methane production after the injection well comes
back on line. The latter work is detailed in Vasilikou et al. (2013).
A dual porosity, single permeability model was setup, since coal matrix permeability is
considered extremely low. As mentioned before, there were four fracture stages including a total
of 24 coal seams. Gas and water rates were available per well and therefore, only aggregate
production was available for all seams. Since coal seams have different characteristics it was
decided to model four equivalent coal seams at the highest elevation of each fracture stage. The
thickness of these modeled coal seams corresponds to the sum of the thicknesses of each seam
within the fracture stage. A grid was generated that covers all wells with nearly the same
distance from the boundaries for all wells.
The orientation of the hydraulic fractures was critical in determining grid generation. As the
model can only allow for hydraulic fractures along either the I (x) or J (y) direction, and
hydraulic fractures were developed along the maximum horizontal stress in the area (Figure 12),
the grid was rotated by 35 degrees clockwise (-35o) to orient the maximum horizontal stress in
the I direction. In addition, it was assumed that the permeability in the I direction is equal to butt
cleat permeability and that in the J direction equal to face cleat permeability.
35 |
Virginia Tech | Results and Discussion
Table 3 shows the optimal parameters determined for the single-well runs. The bottom hole
pressure (assumed to be equal to the reservoir pore pressure during production) also follows an
expected trend, i.e. increased for higher water production, and with a decreasing trend as gas rate
values increase. Note that these models do not include modeling of hydraulic fractures. Models
with hydraulic fractures are discussed in Vasilikou et al. (2013).
Table 3 - Optimal Parameters Determined for the Dingle-Well Runs
Well Fracture Face Cleat Butt Cleat Face to Butt Bottom Water
Porosity Permeability Permeability Cleat Hole Saturation
(mD) (mD) Permeability Pressure in the Cleats
Ratio (kPa)
RU-84 0.0013 18.97 6.32 3 200 1
RU-85 0.001 14.42 4.12 3.5 700 1
RU-112 0.0013 14.42 4.12 3.5 1200 0.9
RU-123 0.001 14.42 4.12 3.5 200 0.8
RU-132 0.001 10.42 3.47 3 200 0.9
RU-210 0.001 10 7.8 1.28 1300 1
RU-211 0.001 10 2 5 1400 1
RU-284 0.001 7.8 7.8 1 1250 1
Figure 18 presents the reservoir simulation history-match results for BD-115 (RU-132) for a
single well run. Gas rate predictions match exactly the measured data, while water rate
predictions are adequate based on the quality of measured water data. The data for water
production by well are estimated by dividing the total water production of a group of wells by
the time the water pump on each well operated, unlike gas production which is measured at each
well. The bottom hole pressure (assumed to be equal to the reservoir pore pressure during
production) also follows an expected trend, i.e. increased for higher water production, and with a
decreasing trend as gas rate values increase.
Figure 19 presents the reservoir simulation history-match results for BD-115 (RU-132) for a
combined run where the single run optimum parameters were utilized as inputs for each well. In
both cases the gas rate match is perfect as the primary simulation constraint comprises matching
the gas rate.
40 |
Virginia Tech | The optimal parameters for the combined model were then determined using the
optimization algorithm (CMOST) built into the CMG/GEM package. The initial range for each
optimized parameter was determined using data ranges that were based on the data in Table 3 for
the single-well runs. The actual data ranges entered into the optimizer are shown in Table 4. The
optimization algorithm was instructed to find optimum parameter combinations in these ranges.
The objective function was set to minimize the error on cumulative water while the operating
constraint was to match the gas produced. The face to butt cleat permeability ratio was kept at 3
for all runs because preliminary runs showed that approximate ratio to be valid for all wells.
Although the algorithm should examine all possible combinations, some parameter combinations
are evidently not tested due to some internal optimization. In total 835 runs were completed
using CMOST. Table 5 shows the results obtained for all wells.
Table 4 - Parameter Range Used as Input in the CMOST Model Runs
Fracture Butt Cleat Permeability Bottom Hole Water Saturation
Porosity (mD) Pressure (kPa) in the Cleats
0.0010 1 200 0.900
0.0015 2 500 0.925
0.0020 3 800 0.950
0.0025 4 1100 0.975
0.0030 5 1400 1.000
6 1700
2000
In addition the optimum relative permeability curves were determined by the CMOST
algorithm. Two of these diagrams are shown in Figure 20. The fact that different optimum curves
are calculated for each well indicates that the relative permeability curves have a significant
impact on the gas and water rate calculations, may differ considerably between wells.
42 |
Virginia Tech | accurately predict the gas production rate for all wells through history matching, while the total
water produced from the reservoir matches reasonably well the measured values.
In addition, the model identifies the most sensitive parameters in the simulation and how
these interact in closely spaced wells. The development of this model along with the optimal
parameters that were calculated will be subsequently used for modeling of CO injection and the
2
corresponding enhanced gas recovery potential in the same group of seams.
Acknowledgments
Financial assistance for this work was provided by the U.S. Department of Energy through
the National Energy Technology Laboratory’s Program under Contract No. DE-FC26-
04NT42590 and DE-FE0006827.
References
Gash, B.W., Volz, R.F., Potter, G., and Corgan, J. M. 1993. The effects of cleat orientation and
confining pressure on cleat porosity, permeability and relative permeability in coal. Paper
9321 in Proceedings of the 1993 International CoalBed Methane Symposium. Tuscaloosa:
University of Alabama.
Mavor, M.J. and Robinson, J.R., 1993, “Analysis of Coal Gas Reservoir Interference and Cavity
Well,” Paper SPE, 25860, presented at the Joint Rocky Mountain Regional and Low
Permeability Reservoirs Symposium, Denver, CO, April 26-28.
Computer Modelling Group Ltd (CMG), C. (2003). GEM. Calgary, Canada.
Ripepi, N., Karmis, M., Miskovic, I., Shea C., and J.M. Conrad. 2009. Results from the Central
Appalachian Basin Field Verification Test in Coal Seams. 26th Annual International
Pittsburgh Coal Conference, Pittsburgh, PA, USA, 2009. CD-ROM of Proceedings, Paper #
33-4.
Karmis, M., Ripepi, N., Miskovic, I., Conrad, M., Miller, M., and C. Shea. 2008. CO
2
Sequestration in Unminable Coal Seams: Characterization, Modeling, Assessment and
Testing of Sinks in Central Appalachia. Twenty-Fifth Annual International Pittsburgh Coal
Conference, Pittsburgh, PA, USA, 2008. CD-ROM of Proceedings, Paper # 29-3.
45 |
Virginia Tech | MODEL VERIFICATION OF CARBON DIOXIDE SEQUESTRATION
IN UNMINEABLE COAL SEAMS WITH ENHANCED COALBED
METHANE RECOVERY3
Foteini Vasilikou, Virginia Center for Coal and Energy Research, Virginia Tech,
Blacksburg, VA
Cigdem Keles, Virginia Center for Coal and Energy Research, Virginia Tech,
Blacksburg, VA
Zach Agioutantis, Department of Mineral Resources Engineering,
Technical University of Crete
Nino Ripepi, Virginia Center for Coal and Energy Research, Virginia Tech,
Blacksburg, VA
Michael Karmis, Virginia Center for Coal and Energy Research, Virginia Tech,
Blacksburg, VA
Abstract
Commercial deployment of Carbon Capture Utilization and Storage (CCUS) requires field
testing of a scale that can stress the geologic reservoirs. One such reservoir of interest consists of
unminable coal seams that exhibit favorable characteristics and depositional environments and
lower pressure and temperature than other, deeper, reservoirs. Such conditions can reduce
compression costs while utilizing the action of adsorption that offers a more effective carbon
dioxide (CO ) bonding than free storage or solution. To ensure the success of such tests, a
2
3 Model Verification of Carbon Dioxide Sequestration in Unmineable Coal Seams with
Enhanced Coalbed Methane Recovery.
F. Vasilikou, C. Keles, Z. Agioutantis, N. Ripepi, M. Karmis 2013 23rd World Mining
Congress, Montreal, Canada.
Used with permission of the Canadian Institute of Mining, Metallurgy and Petroleum.
Foteini Vasilikou researched and prepared this manuscript, with Cigdem Keles, Zach
Agioutantis, Nino Ripepi and Michael Karmis providing technical and editorial input.
47 |
Virginia Tech | number of parameters should be accurately determined, such as the seam geometry and
stratigraphy, coal porosity and permeability parameters as well as optimum injection conditions.
In essence the CO injection model should be calibrated for the reservoir characteristics using all
2
available data. History matching calibrations are based on gas and water production from
existing wells prior to injection. This paper will present reservoir models that were developed for
a pilot--scale 907 tonnes (1000 tons) CO injection test that was performed in 2009 through one
2
legacy coalbed methane production well in Russell County, Virginia, USA. The model
incorporates a number of individual coal seams about 0.3 m (1ft) in thickness located at depths
ranging from 300 to 700 m (1,000 to 2,200 feet). Model calibration was performed through
history matching to prior production data, and subsequently the model was utilized to develop
different injection scenarios. The developed model was used after injection to calculate CO
2
plume distribution patterns that were monitored at the injection test. The paper will present
model data and assumptions with a special emphasis on the effect of hydraulic fractures and the
skin factor to the coalbed methane (CBM) production model.
Introduction
The mitigation of greenhouse gas emissions and enhanced recovery of coalbed methane are
benefits to sequestering CO in coal seams. This is possible because of the affinity of coal to
2
preferentially adsorb CO over methane (Shi and Durucan, 2005). Coalbed methane (CBM) is
2
the most significant natural gas reserve in the Virginia portion of the Central Appalachian Basin,
USA and currently is economically produced in many fields in the Basin from vertical CBM
wells. The recovery factor for vertical CBM well development is estimated that 55% of the gas
in place would be recovered by primary recovery techniques, 20% of the gas in place is
unrecoverable residual gas, and the remaining 25% can be recovered by implementing CO -
2
sequestration operations. Since coal has a greater affinity for CO than for methane gas, the
2
injected CO should preferentially be adsorbed on the surface of the coal, thereby releasing
2
methane gas that would be recovered at offset producing CBM wells. A field verification test
successfully injected 907 tonnes of CO into a mature CBM production well at the Russell
2
County, Virginia test site hosted by CNX Gas. The injection commenced on January 9, 2009 and
was completed on February 10, 2009. The maximum daily injection rate was over 50 tonnes of
CO per day, with an average injection rate above 36 tonnes per day.
2
48 |
Virginia Tech | Field Description
The Central Appalachian Basin is a northeast-to-southwest trending basin encompassing
approximately 25,900 km2 in southwestern Virginia, southern West Virginia and eastern
Kentucky (Conrad, 2006). Production of CBM began in 1988 with the development of the Nora
Field in Dickenson County, Virginia followed by CONSOL Energy / CNX Gas developing the
Oakwood Field in Buchanan County, Virginia. Since that time, over 5,600 CBM wells have
been drilled and brought on-line as producing gas wells in southwest Virginia through 2012
(VaDMME, 2013). As of year-end 2010, the coal seams in the Central Appalachian Basin had
produced over 28 km3 of CBM (VaDMME, 2013). Virginia is the primary producer of CBM in
the basin accounting for over 90% of the production. In 2010, Virginia produced a record 3.4
km3 of CBM which accounted for nearly 80% of the natural gas produced in the Commonwealth
(VaDMME, 2013). The U.S. Department of Energy (2013) stated that Virginia accounted for
5.1% of the CBM production in the U.S. in 2010 and accounts for 10% of the CBM reserves,
nearly 57 km3. The majority of CBM development is in areas where gas contents range between
12.5 – 18.7 cubic meters of gas per tonne of coal (Conrad, 2006), making these seams some of
the gassiest in the country. The CBM productivity of the basin indicates that coal permeability
should allow for carbon dioxide injection and storage (Karmis, 2008).
The coals in the region include those of the Lee Formation and Pocahontas Formation and
are medium to low-volatile bituminous, high rank and high gas content coals. In the area of the
injection, there are multiple thin unmineable coal seams (up to 24 separate coals) with net coal
thickness of up to 11 m. The coals average less than 0.6 m in thickness and are deposited over a
large range in depth, 250 to 700 m.
Based on geologic characterization and production studies, wells in the South Oakwood
CBM Field were deemed appropriate for the injection test. These CBM wells were developed on
242,800 m2 grid spacing from 2002 to 2005. Stimulation occurs through perforations in the well
casing into coals greater than 0.15 m thick with a multiple-stage nitrogen foam hydraulic
fracturing treatment (VaDMME, 2002). The purpose of hydraulic fracturing is to breakdown the
coal and create fractures that allow gas to flow through the coal matrix to the fractures and then
to the wellbore. In order to keep these fractures open for gas to flow to the well bore, a proppant
49 |
Virginia Tech | of coarse sand was injected with the high pressure nitrogen foam. Water is produced through a
73.025 mm string of tubing set below the deepest coal seam, and gas is produced between the
casing and the tubing (annulus). The water is removed from the formation to decrease the
pressure of the bed and to allow for the methane to be desorbed and be produced through the
well (Holman, 1996).
CNX Gas donated an existing CBM well in the South Oakwood Field in Russell County,
Virginia for the carbon dioxide injection field test. The injection well is referenced by its well
name, BD114, or by its designation assigned by the State of Virginia, RU-84. Prior to injection,
BD114 averaged production of 1189 m3 of natural gas per day and 0.26 m3 of water per day
since 2002 and is a slightly below average gas producer for this gas field.
Field Site Layout
A coal field cleat examination was completed to verify the cleat direction of coals at the
injection site verses known face and butt cleat directions of N18W and N67E respectively, for
the deep mined Pocahontas No. 3 coal seam in Buchanan County. Face and butt cleat planes
were measured at two coal seam outcrop locations near the site and they corresponded to within
7 degrees of the known values. A Rose Diagram (Figure 22) was developed to graphically
display the cleat directions from the injection well (VCCER, 2011).
Prior to injecting carbon dioxide, two monitor wells were drilled in close proximity to the
injection well (BD-114), one 41.2 m away (M1) and the other 87.5 m away (M2). The two
monitor wells were drilled at roughly 90 degree offsets from the injection well through the
deepest coal seam that had been perforated in the injection well and exposed each coal seam
being injected into using a packer and tubing. These monitor wells were used to monitor the
pressure of plume, as well as composition of gas at each well. This study affirms that the
monitoring wells for the test site are arranged in both the hydraulic fracture (M1) and face cleat
(M2) directions (Figure 22).
CO Injection
2
The field test successfully injected 907 tonnes of CO at the Russell County, Virginia test
2
site hosted by CNX Gas. The injection commenced on January 9, 2009 and was completed on
February 10, 2009. The data was gathered on a minute basis throughout the one-month injection
50 |
Virginia Tech | and then averaged on an hourly basis to smooth out the curves. Throughout the injection, the
temperature of CO injection was maintained close to 100 degrees Fahrenheit, the pressure was
2
held below 6895 kPa (1,000 psia) and the flow rate varied according to CO delivery availability
2
and operating below the maximum pressure. The maximum daily injection rate was over 50
tonnes of CO per day, with an average injection rate above 36 tonnes per day. During the last
2
three days of injection, at close to maximum pressure, the injection rate declined to a low of 15.4
tonnes per day. The decrease in the injection rate could be attributed to either the pressure of the
reservoir pushing back or swelling of the coals due to adsorption of CO at the higher pressures.
2
Results from the injection provided essential data for establishing the conditions under which
CO can be injected into underlying coal seams, ultimately to establish a reasonable estimate of
2
the volume of CO that can be sequestered in Central Appalachian coal seams through predictive
2
reservoir modeling.
Figure 22 - (A) Rose Diagram Showing Face and Butt Cleat Orientation (B) Rotation by
35o degrees clockwise
Injection Logging
While the CO was being injected at a defined temperature, pressure, and rate, a spinner
2
survey was run downhole to establish the quantity of CO being injected into each coal seam. As
2
part of the spinner survey, temperature and pressure were logged via a wireline downhole to help
profile the injection operations. This survey encountered problems in that the spinner ran into
liquid CO at 506 m deep in the wellbore. The temperature log shows the sudden decrease in
2
51
o270 B
2
D
o3600
M 2
M 1-11
-114-114/R U -84
1
2
A v e ra g e Fa ce C le a t
oO rie n ta tio n : 1 6 7
o180180
H
D ireo
rizo
3
ctio n o f M a x
on ta l S tre ss: 5 5
A v e ra g e B u tt C le a t
oO rie n ta tio n : 7 4
o9090
0
11 2 -114
-
1 3
2
90
180
Fa ce C le a ts
M 1
B u
D ire ctio n o f M a x
H o rizo n ta l Stre ss
tt C le a ts
M2 360o
270o
BD-114/RU-84
90o
180o
J D ire ctio n
I D ire ctio n |
Virginia Tech | temperature which happened when the CO changed phases from gas to liquid. The results of the
2
temperature log show significant changes in temperature occurring at each open perforation
where CO was injected into coal seams. The change in temperature was plotted and shows that
2
the greatest change occurred at the shallowest seams that were receiving CO . The change in
2
temperature most likely can be correlated with flow rate (VCCER, 2012).
Monitoring Well Results
As the CO injection commenced, it was obvious that there was a direct connection through
2
existing hydraulic fractures to the closest monitoring well, M1. Within 30 minutes of starting the
injection operation, the pressure in M1 unexpectedly raised rapidly to 3,447 kPa (500 psia) and
CO content increased to greater than 95%. The profile of the pressure in M1 followed closely
2
with the pressure at the injection well, lagging about 690 kPa (100 psia) through the one month
injection. At the end of the injection, the pressure in M2 mirrored M1 and the injection well. It is
unknown if the rise in pressure in M2 was as quick as M1, but the results indicate that the M2 is
also interconnected in a fracture network with the injection well, BD114 as the CO content in
2
M2 also reached greater than 95%.
Reservoir Modeling
The CBM production data between 2002 and 2009 was used to develop a reservoir model of
the area. The original model was developed utilizing Advanced Resources’ COMET3 reservoir
simulator for CBM (Ripepi et al., 2009), but in 2013 a new model was developed using
Computer Modeling Group Ltd.’s GEM software, a compositional and unconventional reservoir
simulator (Vasilikou et al., 2013), so that the effect of fractures could be incorporated. The 2013
study utilized a different geometry and reservoir parameters resulting in a better history match
than before, while obtaining insight regarding the specific reservoir characteristics and behavior.
This paper complements existing work by introducing the effect of hydraulic fractures to the
CBM production model by employing two methods: using the “Skin Factor” (SF) approach and
using the “Hydraulic Fracture” (HF) approach. Subsequently the central production well is
converted to an injector and CO is injected into the reservoir for a month according to the
2
injection history presented in the previous section.
52 |
Virginia Tech | Theoretical Considerations for the Skin Factor and the Hydraulic Fracture Approach
The permeability near the wellbore is different from the permeability deeper in the
formation. It is either reduced as affected by drilling activities or it is enhanced due to well
stimulation. This reduced or enhanced permeability zone, termed as “skin” zone, causes an
additional pressure change when compared to the pressure drop due to the original permeability
of the formation. According to Hawkins (1956), the additional pressure change across this zone
can be approximated by Darcy’s equation by introducing a skin factor term, defined as follows:
[ ] ( ) (26)
When the permeability around the wellbore is higher than the formation permeability, i.e.,
due to well stimulation, , the first term if eq (26) is negative. Since the ( ) term
is always positive, in cases of stimulated wells the skin factor is always negative. In GEM, the
skin factor is incorporated in the molar flow rate equation of a particular layer into the well. The
extent of the effective radius ( ) for each well can be calculated using the following equation
(Peaceman, 1983). For the case of the SF Model, a skin factor of -4 was used for all the wells in
the modeling area.
√
(27)
√ √
Darcy’s equation is employed in order to simulate laminar flow of fluid through porous
media. However, as it has been reported in literature, for hydraulically fractured wells, gas flow
velocities near the wellbore and the pressure drop, are not proportionally increased. For this
reason in order to account for both laminar and turbulent phenomena the following equation
suggested by Forchheimer (1914) is used:
(28)
where is the pressure gradient, μ the viscosity, k the formation’s permeability, ρ the fluid’s
density, and β the non-Darcy flow coefficient, a factor characteristic of the porous medium. The
β factor is dependent on the gas saturation in the hydraulic fracture, the relative permeability of
gas flow and the fracture porosity. Several authors have published different correlations of the β
factor. In GEM, the following models can be applied in order to determine the β factor: The
Geertsma, Frederick and Graves I or II or a general non-Darcy correlation model.
53 |
Virginia Tech | Another important aspect of accurately simulating a non-Darcy flow response in the
hydraulic fractures is the use of a pseudo fracture. More specifically, in order to model an actual
fracture width of 0.0058 meters, as in the case of the HF model, very high fracture permeability
(i.e. 60,000md) is assigned to that narrow fracture. This is subsequently mapped to the modeled
fracture width, i.e. 0.4m and converted to effective permeability by observing the equation k x
width = constant.
The Importance of Relative Permeability
The relative permeability (RP) is defined as the ratio of the effective permeability (EP) to a
given fluid at a definite saturation to the permeability at 100% saturation (k). Since k is a
constant for a given porous material, the RP varies with the fluid saturation in the same fashion
as does the EP. The RP to a fluid will vary from a value of zero at some low saturation of that
fluid to a value of 1.0 at 100% saturation of that fluid. In this case study, only two phase flow in
the cleats needs to be considered: flow of water and flow of gas (CH ). In the current case study,
4
the initial water saturation in the cleats is assumed high and numerically is assigned between 0.9
and 1.0. The RP curves are then optimized to work with the permeability of the strata to yield the
appropriate water and gas rates. Figure 23 presents a typical example of the RP curves for two of
the wells in the modelled area. In addition, Figure 24 compares the RP curves used in the SF and
HF approaches.
Figure 23 - Typical Relative Permeability Curves for Two Wells
54 |
Virginia Tech | Figure 24 - Relative Permeability Curves for the HF and SF Models
History Match
A dual porosity, single permeability model was setup, since coal matrix permeability is
considered extremely low. The 24 coal seams encountered at the test site were grouped in four
flat zones set at the average elevation of each group (Vasilikou et al., 2013). The generated grid
encompasses all production wells allowing for a distance of 150-200 m between each well and
the grid boundary. The grid size was set to 25mx25m, with the grid extending to 84 blocks in the
I (x) direction and 56 blocks in the J (y) direction.
The orientation of the hydraulic fractures was critical in determining grid generation. As the
GEM model can only allow for hydraulic fractures along either the I (x) or J (y) direction, and
hydraulic fractures were developed along the maximum horizontal stress in the area, the grid was
rotated by 35 degrees clockwise to orient the maximum horizontal stress in the I direction
(Vasilikou et al., 2013). In addition, it was assumed that the permeability in the I direction is
equal to the butt cleat permeability and that the permeability in the J direction is equal to the face
cleat permeability. Furthermore, the modeled group of eight wells was considered independent of
other wells surrounding this grid. The resulting grid for all layers including the eight wells is
shown in 3D view in Figure 25. The face cleat permeability distribution for all layers varies
between 8 and 12 md variation. Similar results are obtained for the SF and HF models, where the
face cleat permeability values are in the same range as before, but different since these reservoir
models exhibit different flow characteristics.
55 |
Virginia Tech | Figure 26 - Typical HF
Figure 25 - Face Cleat Permeability Variation for the Discretization for All
Base Model Wells
Figure 26 presents a detail of the model at the central production / injection well where the
hydraulic fracture is shown as an area with finer grid discretization. The effective permeability
for that area is in the order of 57,000 mD. It should be noted that these are optimum permeability
values as determined by optimization of the reservoir properties. The RP curves varied per well
for the base model, while they were kept constant for the SF and HF models since the later allow
for higher permeability values in the near well bore areas. Adsorption isotherms for CO and
2
CH were developed by a commercial laboratory, Pine Crest Technology, on three coal seams
4
that are representative of coals developed for CBM in the South Oakwood CBM field.
In all above runs, the primary operating constraint was to meet the gas rate, while the
secondary operating constraint was that the bottom hole pressure should not fall below a
minimum value of 200 kPa. The objective function for the optimization was to minimize the
error for the cumulative water that is recorded per well. It should be noted that the water output
logged contains water introduced into the well and the formation during the hydraulic fracturing
procedure.
Results and Discussion
Figure 27 compares the cumulative water produced at well BC-115 (RU-123) over a six year
period. The dots represent the measured values, while the dashed lines represent the calculated
values by the three models discussed previously. In this case all models over-predict the water
56 |
Virginia Tech | produced. Although only one such diagram is presented in this paper, such charts can be
composed for all wells. For other wells, the calculated cumulative water does not follow the
same pattern and some models under predict while others over predict the measured values. It
should be emphasized, that a complex problem like water and gas production from multiple
wells may have more than one solution depending on the parameters selected. In fact, the
solution space can be considered practically infinite. In addition, the HF model requires more
computer resources to complete than the other models, while it is also the most sensitive of the
three when input parameters vary. In all cases the models matched the historic data for the gas
rate very well.
Figure 27 - Cumulative Water Production vs. Measured Data for Well BC-115 (RU-123)
for All Models
Following the production simulation, and the respective history match, CO was injected
2
into well BD 114 (RU 84) for a period of about one month. The actual injection data were
available on a daily basis, while the production data were available on a monthly basis. Figure 28
presents the injected gas rate (surface conditions, m3/day) and the injected cumulative gas
(surface conditions, m3) for the injection period. Following injection, the injector well was shut
in for a few months.
57 |
Virginia Tech | Conclusions
Due to the complexity, non-linearity and parameter variability of CBM reservoir modeling
multiple solutions may be obtained that satisfy input parameters, boundary conditions, solution
constraints and the objective function. A successful model should predict future production after
being calibrated using historic data. In this work, two advanced CBM production models (the
Skin Factor model and the Hydraulic Fracture model) were developed by incorporating the
effects of permeability changes due to hydraulic fracturing. The models are compared to each
other and also compared to a base model that does not incorporate any effective permeability
changes due to fracturing.
The advantage of the SF model is that it can run in much less time than the HF model. The
skin factor approach however induces a symmetric effective permeability change around each
well since the skin factor is not directional. This is also evident in the plume representations for
the SF models. In contrast the HF models correctly account for preferential effective
permeability changes around the fractured area, but take a lot more computer resources to
complete. Although all models converge and meet the specified gas rate and cumulative gas
production values, none of the models provides a unique match to water production data for all
wells.
These preliminary results verify that the injection process characterized by sorption and the
process of flowback followed by CH and/or CO desorption is mainly driven by the effective
4 2
permeability of the formation around each well. The skin factor approach may be applicable for
large scale simulations where well interference is not an issue, while the hydraulic fracture
simulation should be used when more accuracy in the formation and migration of injected and/or
produced gases is needed.
Acknowledgments
Financial assistance for this work was provided by the U.S. Department of Energy through
the National Energy Technology Laboratory’s Program under Contract No. DE-FC26-
04NT42590 and DE-FE0006827.
59 |
Virginia Tech | RESERVOIR SIMULATIONS FOR COAL BED METHANE (CH )
4
PRODUCTION AND CARBON DIOXIDE (CO ) INJECTION IN DEEP
2
COAL SEAMS IN BUCHANAN COUNTY, VIRGINIA4
Foteini Vasilikou, Virginia Center for Coal and Energy Research, Virginia Tech,
Blacksburg, VA
Zach Agioutantis, Department of Mineral Resources Engineering,
Technical University of Crete
Nino Ripepi, Virginia Center for Coal and Energy Research, Virginia Tech,
Blacksburg, VA
Michael Karmis, Virginia Center for Coal and Energy Research, Virginia Tech,
Blacksburg, VA
Reservoir Model Calibration for Coal Bed Methane Production from Deep Coal
Seams in Buchanan County, Virginia
Abstract
The potential of commercial deployment of CO sequestration in unmineable coal seams, as
2
a measure to mitigate the greenhouse gas effect, is being tested through a series of small- to
medium-scale injection projects in the Appalachian Basin. Indispensable tools for these projects
are reservoir simulations in which the involved processes, both prior to and post injection field
tests are modeled to enhance understanding and to be used as a decision tool. Due to the large
number of modeling input parameters and the high uncertainty in their values determination, for
development of representative models initial model sensitivity analysis of the key input
parameters is required. In this paper, preliminary single-well reservoir parametric simulations are
conducted to identify how the variation of selected reservoir parameters and properties affect
model results. More specifically, this sensitivity analysis focused on three areas: a) the effect of
4 This paper is intended for publication.
Foteini Vasilikou collected the data, researched, and wrote this manuscript with technical
input from Zach Agioutantis, Nino Ripepi and Michael Karmis.
62 |
Virginia Tech | selected initial conditions (pressure gradient, adsorption isotherm, and porosity), b) the influence
of the production mechanism (permeability, relative permeability, and compressibility), and c)
the contribution of the well characteristics (hydraulically stimulated well, skin factor). This
modeling work forms the base for calibration of reservoir models for simulating CO injection
2
and subsequent prediction of post-injection behavior for an upcoming injection test in Buchanan
County, VA.
Introduction
The Coal Seam Group, led by researchers at the Virginia Center for Coal and Energy
Research (VCCER), part of the Southeast Regional Carbon Sequestration Partnership
(SECARB), completed a small-scale validation field test in Russell County, Virginia, during
2009, where 1,000 tons of CO were injected into one vertical coalbed methane well over a
2
period of one month (Ripepi et al., 2009). Prior to conducting the injection test, preliminary
reservoir simulation models were developed to predict the extent of the CO plumes and estimate
2
injection pressures (VCCER, 2011). After the injection test was completed the original reservoir
models were updated to account for post injection data and were refined based on the knowledge
gained (Vasilikou et al., 2013).
In 2014, the VCCER will be performing a larger scale validation test in Buchanan County,
Virginia, approximately seven miles to the north west of the Russell County site. Twenty
thousand (20,000) tons of CO will be injected into three vertical coalbed methane wells over a
2
period of one year. The enhanced understanding of the production and injection mechanisms into
stacked geologic systems gained from the small injection test in Russell County, along with
updated reservoir software capabilities, will be used to develop improved simulation predictions
for the CO injection test in Buchanan County.
2
Geology
CNX Gas Corporation, a subsidiary of CONSOL Energy Inc., has donated three vertical
coalbed methane production (CBM) wells in Buchanan County, Virginia within the Central
Appalachian Basin region. These three production wells will be shut in and converted into
injection wells a month prior to the field injection test start date. As recorded by the State of
Virginia the candidate injection wells, BU 1923, BU 3337, BU 1998 or alternatively as
63 |
Virginia Tech | referenced by their company name, DD7, DD7A, DD8, are at an elevation of 2286.6 feet, 1932.2
feet, and 1876.2 feet, respectively (DMME, 2011).
The Central Appalachian Basin is a northeast-to-southwest trending basin encompassing
approximately 10,000 square miles in southwestern Virginia, southern West Virginia and eastern
Kentucky (Conrad et al., 2006). Production of CBM began in 1988 with the development of the
Nora Field in Dickenson County, Virginia, followed by CONSOL Energy developing the
Oakwood Field in Buchanan County, Virginia. The coals in the region near the injection site
include those of the Pocahontas Formation and Lee Formation, which directly overlie the late
Mississippian Bluestone Formation. Coal seams of the Pocahontas and Lee Formation are
medium to low-volatile bituminous, high rank and high gas content coals that include the
Pocahontas No. 1 through Pocahontas No. 11 seams (Pocahontas Formation and the Greasy
Creek, Middle and Lower Seaboard and Upper Horsepen coals) (Vasilikou et al, 2013).
Cardno MM&A, a research partner on this project, created a database for the coal seams in
the study area by integrating data from donated sources with geophysical well log data from the
Virginia Division of Gas and Oil and the West Virginia Geological and Economic Survey
(VCCER, 2014). This database was used to create elevation structure maps and net thickness
isopachs for both the Lee and Pocahontas Formation coals in southwestern Virginia and southern
West Virginia, encompassing the Central Appalachian Basin. This assessment accounts for the
coal seams that were stimulated for coalbed methane development.
Natural occurring fracture networks in coalbeds and the face and butt cleat planes were
measured at two coal seam outcrop locations near the site and they corresponded to within 7
degrees of the published face and butt cleat directions of N18W and N167E respectively, for the
deep mined Pocahontas No. 3 coal seam in Buchanan County (McCulloch, 1974).
The primary confining units in the study area include multiple layers of low permeability
shale and siltstone beds. Permeability for the shale and siltstone units is expected to range from
0.001 to 0.1 mD, with low porosity (Grimm, 2010). The Lee and Pocahontas Formation
sandstones are known to have low permeability and porosity values, and do not produce natural
gas in this area (VCCER, 2011). Based on well log interpretations, the sandstone units are
expected to have porosity values that range from 1.0 to 3.0 percent and permeability values
ranging from 0.1 to 1.0 mD. All lithologies will encounter some natural fractures; however, these
fractures are likely to be cemented with quartz and calcite and are not expected to provide
64 |
Virginia Tech | permeability pathways based on core analysis, well logging and field testing (Karmis et al.,
2008).
Well Stimulation
All wells within the study area have been perforated and hydraulically fractured using a
nitrogen foam hydraulic fracturing treatment. The wells selected for injection, DD7 (BU 1923),
DD7A (BU 3337), DD8 (BU 1998) were stimulated in November 2000, March 2007 and May
2001, respectively. DD7 was completed in four zones ranging in depth over 1216-1385 feet,
1531-1570 feet, 1674-1866 feet and 1997-2133 feet. DD7A was stimulated in three stages at
856-975.5 feet, 1203-1420 feet and 1599-1803.5 feet. DD8 was completed in three zones ranging
in depth over 871-1205 feet, 1311-1677.5 feet and 1710.5-1802 feet (DMME, 2011).
In each well, the main hydraulic fractures at each perforation are oriented at approximately
N57E (VCCER, 2011). Although in some cases fracturing may extend up to 700-800 feet away
from the borehole, in this analysis a more conservative approach was used where the fracture
half-length is assumed to be 350 feet. In addition, after personal communication with the
operator of the wells, it was suggested that only 40 percent of the coal seams per stage were
successfully completed (CNX, 2013).
Production History
The three injectors, DD7, DD7A, and DD8, were brought on-line as coalbed methane
production wells in 2000, 2007 and 2001, respectively. To date, DD7, DD7A, and DD8 have
averaged a production of 82.52 Mcf, 41.25 Mcf and 47.88 Mcf of natural gas per day and 1.42
barrels, 1.11 barrels and 1.74 barrels of water per day (DMME, 2013). The cumulative gas
production of these three wells to date is estimated to account for 58 percent, 56 percent and 33
percent, respectively, of their expected ultimate gas recovery (VCCER, 2014).
The gas production data are measured at each well, whereas the water production data are
estimated by dividing the total production of a group of wells by the total time the water pump
on each well was operating. For this reason, there is a higher degree of confidence in the
accuracy of the gas production data versus the reported water production of the wells.
65 |
Virginia Tech | Figure 31 - Daily Water Production Rate of the Injection Wells
Description of Reservoir Model
Reservoir simulations prior to commencing with the CO injection field test were performed
2
to estimate the extent of the CO plumes, the storage capacity of unmineable coal seams and the
2
potential of enhanced gas recovery of the field. The reservoir simulator employed in this study is
the compositional and unconventional simulator GEM by the Computer Modeling Group Ltd.
This paper focuses on studying the interaction of key properties which affect the initial gas
and water volumetrics in coalbeds and the production mechanism from multiple thin coal seams.
In addition, well characteristics relevant to how the well was drilled and perforated, such as the
frequency of hydraulic fractures, the fracture lengths and widths, are examined with respect to
ultimate well productivity. Due to the nature of the multi-seam configuration of the study area
and the degree of interference between input parameters, this paper focuses on part of the study
area that includes a single well, DD7, in order to investigate commingled primary production for
different scenarios. Simulation results of the whole study area with CO injection in three of the
2
wells are detailed below.
67 |
Virginia Tech | The geologic structure employed for the single well simulations in this study is an 18-layer
model (Figure 32A) proposed by Cardno MM&A (2013), a project subcontractor. Table 6 shows
the top elevations and thicknesses for each simulated coal seam. Numbering of the layers starts
with the shallowest seams and increases towards the deepest seams. The drainage area for the
single well simulations is approximately 42 acres.
In addition, based on a slightly different interpretation of the aforementioned geologic
structure proposed by Cardno MM&A, a second structure where 5 zones of coal seams are
assumed is also investigated in order to examine sensitivity of model response to variation of
number of layers in the model (Figure 32B).
Figure 32 - (A) 18-Layer Model, (B) 5-Zone Model
18- Layer Models
The key parameters investigated in this analysis are: (i) pore pressure gradient, (ii) coal and
cleat porosity, (iii) Langmuir constants, (iv) water saturation in cleats, (v) cleat permeability, (vi)
relative permeability curves to water and gas, (vii) methane desorption time from the coal matrix,
(viii) skin factor and (ix) hydraulic fractures for well stimulation. The primary objective in each
model was to match the historic gas and water production up to year 2013 for the selected well
(Figure 33).
68
( A ) ( B ) |
Virginia Tech | Figure 33 - History Matching of Monthly Gas Rate and Cumulative Gas for Well DD7 Up
to Year 2013
The first set of input parameters that were varied in the simulations is as follows: pore
pressure gradient, coal and cleat porosity, Langmuir volume and Langmuir pressure constants,
and the initial water saturation in the cleats, shown in Table 7. The remaining parameters used in
the first set of models (C) are kept constant as shown in Table 6. Cases C1 through C11
investigate the effect of the aforementioned parameters on the initial gas and water in place of
the models. In cases C1, C2 and C3 the variation of the original gas and water in place is
examined in relationship to the variation of the Langmuir volume constant, V ; in case C4, the
L
water saturation in the cleats is reduced from 100 to 90 percent and all the other parameters are
the same as in case C1; in case C5 the Langmuir pressure constant, P , is reduced from 0.01 to
L
0.003 and the rest of the assigned properties are the same as in case C3; in cases C6, C7, C8 all
the parameters are kept the same as in cases C1, C2, C3 respectively, but the pore pressure
gradient assigned in the fractures is increased from 0.315 psi/ft to 0.36 psi/ft; in cases C9, C10,
C11 the coal and cleat porosity are reduced compared to the respective values in cases C1, C2,
C3 from 1 percent (Pashin, 2014) to 0.15 percent (Liu et al., 2012) and from 0.1 to 0.015 percent.
Figure 34 shows the effect of the variation of the Langmuir constants to the amount of adsorbed
69 |
Virginia Tech | The second input parameter investigated was cleat permeability. The sensitivity of the 18-
layer model to variation of cleat permeability was examined. Three models - P1, P2 and P3 -
with an isotropic cleat permeability in space and in depth of 20 mD, 30 mD and 45 mD (same as
in cases C1 through C11) were studied. The rest of the parameters were kept constant as shown
in Table 6.
Two different relative permeability curves for water and two for gas were considered in this
analysis as shown in Figure 35. All other input parameters for the models were as shown in
Table 6.
Desorption time from the coal matrix to the cleat system was also varied. In the
investigation of cases C1 to C11 desorption time was set to 20 days. Desorption times of 5 and
50 days was also examined. The rest of the parameters were kept constant as shown in Table 6.
Figure 35 - Different Scenarios for the Relative Permeability Curves to Water and Gas
Because it was hydraulically stimulated, to account for enhanced flow properties around
DD7 three scenarios were studied (S1, S2 and S3). In the first scenario (S1) a skin equal to zero
was assigned; in the second (S2) a negative skin of -2 and in the third (S3) a negative skin of -4
72 |
Virginia Tech | was used. The minimum value of skin factor that could be assigned in order to maintain a
positive production index for the well was -4.
Another way to account for hydraulically stimulated wells in reservoir simulations is to
explicitly model hydraulic fractures. This approach is more computationally intensive compared
to assigning a negative skin factor. For the purposes of this investigation four scenarios with
hydraulic fractures were studied. In all four scenarios (H1, H2, H3, H4) only 40 percent of the
seams per stage, as suggested by the operator of the wells, were assumed to be successfully
fractured. For scenarios H1, H2 and H3 the thickest seams per stage were selected to be fractured
(Pashin, 2014). In scenario H4 the seams per stage selected to be hydraulically fractured were the
ones that initially had a higher amount of gas in place. The properties of the hydraulic fractures
assumed in the modeling work and the layer numbers selected to be fractured per scenario are
shown in Table 8.
Table 8 - Input Parameters for Modeling Hydraulic Fractures in Scenarios H1, H2, H3, H4
H1 H2 H3 H4
Layer Number 4, 6, 8, 12, 14, 4, 6, 8, 12, 14, 4, 6, 8, 12, 14, 4, 5, 9, 13, 15,
with HF 16, 18 16, 18 16, 18 16, 18
0.00833 0.01042 0.12500 0.00833
Primary Fracture
Width (ft)
(GEM, 2003) (EPA, 2004) (EPA, 2004) (GEM, 2003)
6,000 6,000 6,000 6,000
Primary Fracture
Permeability (mD)
(GEM, 2003) (GEM, 2003) (GEM, 2003) (GEM, 2003)
Fracture Half
300 300 300 300
Length (ft)
Grid Refinement I: 9, J: 9, K: 1 I: 9, J: 9, K: 1 I: 9, J: 9, K: 1 I: 9, J: 9, K: 1
Effective Fracture
55 61 405 55
Permeability (mD)
5-Zone Model
Finally, due to restricted computational capabilities with respect to model size, it is common
practice in reservoir modeling when simulating complex geologic structures to aggregate
73 |
Virginia Tech | and C3 was not affected by the variation of the Langmuir constant volume and it was the same
for all cases.
In case C4 the initial water saturation in the cleat system was reduced from 100 to 90
percent. A negligible increase in the original gas in place was noted of approximately 0.42
percent in case C4 compared to case C1. The small increase was due to the increase of free gas in
the cleat system. The original water in place was affected the most by the reduction in the initial
water cleat saturation. In case C4 the original water in place reduced by 10 percent compared to
case C1 (Figure 37).
In cases C5 and C3 all input parameters are the same with the exception of the Langmuir
pressure constant, which in case C5 is reduced from 0.01 1/psi to 0.003 1/psi. The change in the
Langmuir pressure constant had an effect on the curvature of the adsorption isotherm as shown
in Figure 34 and, for the given pore pressure range in the reservoir, it resulted in a reduction of
the original gas in place in the system of approximately 28.22 percent (Figure 36). As expected,
the original water in place of the system was not affected by the reduction in the Langmuir
pressure constant.
In cases C6, C7, C8 a higher pore pressure gradient of 0.36 psi/ft compared to 0.315 psi/ft of
cases C1, C2, C3 is assigned in the models and all other input parameters are assumed to be the
same. The increased pore pressure gradient used in cases C6, C7 and C8 resulted in an increase
by 2.22 percent of the original gas in place compared to cases C1, C2 and C3. There was no
change in the original water in place because of the increase in pore pressure.
In cases C9, C10 and C11, both the coal and cleat porosity of the models were reduced to
0.15 percent and to 0.015 percent respectively, compared to cases C1, C2, C3. Reduction of the
coal porosity results in a negligible increase of the original gas in place (Figure 36). Reduction in
the cleat porosity has a significant effect to the original water in place (Figure 37). It results in a
decrease of 85 percent of the overall water in the system from cases C1, C2, and C3 to C9, C10
and C11.
In all cases aforementioned more than 99 percent of the original gas in place is stored in an
adsorbed state on the coal matrix. Only in case C4, where an initial cleat water saturation less
than 100 percent was assigned, there was more than 1 percent free gas in the cleat system. In the
models it is assumed that no methane is dissolved in the water.
75 |
Virginia Tech | For the coal seams in the study area in Buchanan County, VA, the pore pressure gradient is
assumed to range between 0.315 and 0.36 psi/ft (VCCER, 2013). By employing the 18-layer
geologic structure proposed, the pressure profile per depth per coal seam for the two different
pore pressure gradients of 0.315 psi/ft and 0.36 psi/ft is developed as shown in Figure 39. For the
0.315 psi/ft pressure gradient assumed, pore pressures in the model range between 239- 692 psi.
For the 0.36 psi/ft gradient, pore pressures in the 18-layer model vary between 274-791 psi.
Figure 40 - Initially Adsorbed Gas per Layer for Cases C3, C5 and C8
The initial coal matrix volume per seam (Figure 38) with specific adsorption properties
assigned and the initial pore pressures per layer (Figure 39) shape the initially adsorbed gas
profile in depth for the 18-layer model. Figure 40 presents the initially adsorbed gas per coal
seam for cases C3, C5 and C8. Initially adsorbed gas for case C8 is higher than in case C5 and
significantly higher compared to case C3. In case C8, the lower Langmuir pressure constant used
shifts the adsorption isotherm towards higher gas contents per coal mass and, with the pore
pressures range shifted to the right as shown in Figure 34, overall higher originally adsorbed gas
in the model compared to cases C5 and C3 (Figure 40) was expected. Layers 9 and 15 have the
highest initially adsorbed gas for all cases. It must be noted that coal matrix volume is
78 |
Virginia Tech | significantly higher for layers 9 and 15 compared to the other layers. For layers 4 and 5, while
coal matrix volumetrics for layer 4 is greater than layer 5, layer 5 is a deeper seam with a higher
pore pressure and thus has a greater initial amount of adsorbed gas. This is also the case for
layers 2 and 13 with similar coal matrix volumetrics, but due to the significant difference in pore
pressures the deepest seam has higher initial adsorbed gas content.
Production Mechanism
In the analysis of the (C) models, variation of the input parameters under consideration
affected the initial volumetrics of the models. Parameters such as cleat permeability, relative
permeability curves to water and gas, and desorption time have an effect on the production
mechanism of coalbed wells.
Three different scenarios for cleat permeability were examined. In scenarios P1, P2 and P3,
cleat permeability of 20, 30 and 45 mD was respectively assigned, with a primary operational
constraint to meet the historic water rate and with a secondary constraint of minimum bottom
hole pressure. The results showed that the percentage of gas produced over the initial gas in
place for scenarios P1, P2 and P3 was 35.40, 35.40 and 39.84 percent, respectively. There is
essentially no change in the recovery factor between scenarios P1 and P2, whereas an increase of
12.32 percent in gas production is noted for scenario P3. The layers in order of highest to lowest
production were as follows: layer 9, 15, 5, 4, 13, 2, 11, 16, 3, 8, 17, 14, 7, 6, 18, 1, 12, and 10. As
expected, the layers with the highest initial gas content are also the ones producing the most gas.
Two different curves for relative permeability to water and two curves for relative
permeability to gas were studied (Figure 35). In coalbed reservoirs primary dewatering of the
seams is important for gas production. Hence, the magnitude of the relative permeability to water
for high water saturations is important for gas production. When water and gas start to co-exist in
the cleats, the shape of the relative permeability curves has a significant effect on the production
profile. In this analysis, when relative permeability to water 1 was changed to relative
permeability to water 2 (Figure 35), more water was initially produced but gas production did not
start until lower water saturations and ultimately gas deliverability of the system were reduced.
When relative permeability curve to gas was changed from (1) to (2), as shown in Figure 35,
initial water deliverability was unaffected and gas production increased.
Regarding variations in desorption time of 5, 20 and 50 days, the results show that the
percentage of gas recovery over the initial gas in place are 39.94, 39.84 and 39.64 percent,
79 |
Virginia Tech | respectively. It was expected that the lower and hence faster desorption time used in the
modeling work would yield higher gas production, but it is not significant.
From the investigation of the (S) models where a skin factor was used to account for
enhanced flow around the well, it was shown that recovery factors of the original gas in place for
scenarios S1 (zero skin), S2 (negative 2) and S3 (negative 4) were 38.82, 41.59 and 45.77,
respectively. The recovery factors in the (S) models were higher compared to the ones of the (H)
models where hydraulic fractures were explicitly simulated. For the H models, recovery factors
for scenarios H1, H2, H3 and H4 were 38.00, 38.68, 40.50 and 38.38 percent, respectively. It
must be noted that even scenario S1, where a zero skin factor is used, has a higher recovery
factor than most of the H models. This is most likely due to the different flow properties assumed
for the H models versus the S models. For the H models, it must be noted that increase in the
primary fracture width (Table 8) yields an increased effective permeability and ultimately a
higher recovery factor. Also, in scenario H4 where hydraulic fractures per stage were selected
based on higher initial gas content, the recovery factor was slightly higher compared to the
recovery of scenario H1 where stimulated seams were selected based on the thickest seams per
stage criterion.
Figure 41 - Originally Adsorbed Gas per Layer for the 5-Zone Model
80 |
Virginia Tech | CO Injection Model for Enhanced Coalbed Methane Recovery in Deep Coal
2
Seams in Buchanan County, VA
Abstract
This paper presents preliminary reservoir simulations which were conducted to predict the
amount of carbon dioxide that can be stored in multiple thin coal seams following injection, the
extent of CO plumes by seam, the potential for CO breakthrough at offset wells, and the
2 2
amount of CO stored if the injection wells are flowed back. Finally, the paper will present an
2
assessment of the potential of enhanced gas recovery for all wells within the study area.
Development of Reservoir Models
Reservoir simulations prior to the commence of the CO injection field test were performed
2
to estimate the extent of the CO plumes and identify suitable locations for monitoring wells, the
2
storage capacity of unmineable coal seams and the potential of enhanced gas recovery of the
field. The simulation covers a drainage area of approximately 1552 acres and incorporates the
three selected injection wells and seventeen offset CBM wells. Table 10 shows the designations
of these 20 wells as well as the date that production started at each well. The date shows that the
wells in the simulated area were brought on-line at different times within a time span of sixteen
years; the first well started production in 1993 and the latest producer started in 2009.
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Virginia Tech | Coal (matrix) porosity and cleat (fracture) porosity are key input parameters determining the
initial volumetrics of the models and consequently affecting the ultimate methane recovery of the
field. High values of matrix porosity allow for less coal volume for gas adsorption on the coal
surface and thus translate to lower values of initial gas in place. High fracture porosity values
suggest higher initial water in the system and higher pressure. Due to a lack of specific
information on reservoirs conditions regarding both the matrix and the fracture porosity, porosity
values were selected for the reservoir models in order to match the volume of the historic gas
production of the field to date and to be in accordance with the estimated ultimate recovery
predicted through decline curve analysis (VCCER, 2013). Cleat (fracture) porosity was assumed
to be 10 percent of the coal (matrix) porosity. After personal communication with Pashin (2013),
it was suggested that there is a linear decrease of matrix and fracture porosity values with depth.
For this reason a coal matrix porosity of 2 percent was assumed for the first 6 shallower seams in
both the combined and four wells models and 1 percent for the deeper seams.
Langmuir type isotherms were employed in the models to represent methane and carbon
dioxide adsorption isotherms for coalbeds. For the specific injection site in Buchanan County,
there are currently no data available on the adsorptive properties of coals. For the Russell County
injection site, which is seven miles to the south-east of the current site, coal samples were sent to
Pine Crest Technologies and the methane and carbon dioxide adsorption isotherms for three coal
seams, Pocahontas No.3, Pocahontas No.7 and Pocahontas No.11, were determined (VCCER,
2011). Due to the close proximity of the two sites, an assumption was made that the coals would
have similar properties, and by taking into account that the coals in Buchanan County have
slightly higher gas contents, uniform per layer adsorption isotherms for methane and carbon
dioxide were assigned to the models.
Pore pressure is a critical parameter in modeling CBM production because it essentially
drives production. An initial pore pressure gradient of 0.315 psi/ft was used based on historical
field data (VCCER, 2013) to reflect under-saturated conditions in the study area. The
corresponding pressure values vary between 173-717 psi, depending on the depth of the coal
seam in the reservoir.
As already noted, a single cleat permeability model is employed during the simulation of
coalbed reservoirs (CMG, 2003). The stress field in the study area suggests similar fracture
permeability values along the face and butt cleats (Pashin, 2013), and therefore isotropic cleat
87 |
Virginia Tech | permeability was assumed in the models. Cleat permeability values were determined through
history matching the gas production of the field since no other laboratory or field data for the
Buchanan injection site were available. It is believed that cleat permeability declines
exponentially with depth; as the depths in the study area vary between approximately 1000 and
2000 feet it is assumed that cleat permeability varies between 70 mD and 10 mD (McKee et al.,
1988). In the models, cleat permeability towards the lower end of the suggested range for
moderate analysis was selected; 30 mD was assigned to the first 10 seams and it was reduced to
10 mD for the 8 deepest seams.
Gas and water relative permeability curves were unavailable. Relative permeability was
assumed to be similar to that used by Mavor and Robinson (1993) when evaluating pressure
transient data for the San Juan coals based on initially developed gas and water relative
permeability curves by Gash for Fruitland coals (Gash et al., 1993). The relative permeability
curves were adjusted for the specific study area through history matching gas and water well data
(Figure 43).
Operational constraints are also an important modeling parameter. Initially the models were
set to primarily match the historic gas rate up to year 2013, while having a secondary constraint
of a minimum bottom-hole pressure of 28 psi. Even though dewatering of the coal seams is the
driving force for primary methane production from coalbed wells, as previously mentioned the
quality of the reported water data per well is unknown and, therefore, water production was
taken into account as a secondary check point for the history matching exercise. It should be also
noted that historical production data for gas and water were aggregated on a monthly basis.
88 |
Virginia Tech | Figure 43 - Relative Permeability Curves to Water and Gas as Determined Through
History Matching Gas Production for All the Wells in the Study Area
Results and Discussion
Two different types of models are investigated in this paper. A larger model (L) representing
the whole study area incorporating 20 wells and a smaller four-well model (S) focusing on the
area around the three selected injection wells and the closest offset well, DD8A.
The main objective of the reservoir simulation efforts is to model all phases throughout the
estimated life of the wells within the study area and assess the storage capacity of coalbeds and
the extent of CO plumes. More specifically, the first step is to model the initial period up to year
2
2013, utilizing available historical gas and water production data for all wells in order to
calibrate the models. The next step is to model the one-year injection period during which it is
planned to inject 20,000 tons of carbon dioxide into the three selected wells. Finally, the model
will be used to forecast the behavior of all wells in the study area post injection until the end of
their estimated life, year 2052.
89 |
Virginia Tech | Figure 44 - Gas Rate at Surface Conditions for the Three Injectors, DD7, DD7A, and DD8
throughout History Matching, Injection and Forecasting to Year 2023
S Models
The small models (S) incorporated the three selected injectors: DD7, DD7A and DD8 and
the offset well DD8A. The drainage area of the 18-seam geologic structure for the (S) models
was 167.9 acres and, based on the input parameters, the original gas in place (OGIP) was
estimated to be 1.84e6 standard Mcf and the original water in place 2.64e4 standard bbl.
Three different scenarios with regard to well characteristics were examined (S1, S2, S3). In
scenario S1, all 18 coal seams were perforated but not hydraulically stimulated; in scenario S2 a
negative skin was assigned to all four-wells to account for enhanced flow around the wellbore;
and in scenario S3, the hydraulic fractures were explicitly modeled.
It is common practice in reservoir modeling to assign a negative skin factor to a well in
order to account for its hydraulic fracture stimulation and, at the same time, avoid explicit
modeling of the fractures, which is very computationally intensive. Therefore, to calculate the
skin factor in this modeling work, equation (29) was initially used (Karacan, 2013).
90
H M I N J F O R |
Virginia Tech | (29)
( )
where
α: Function of dimensionless fracture conductivity (FCD)
xf: Fracture half-length (ft)
rw: Well radius (ft)
“α” in the equation is a function of dimensionless fracture conductivity (FCD). NSI
Technologies (2001) and Cunningham et al. (2003) reported that optimum well productivity
occurs when FCD is around 2, and is calculated by using the expression below:
(30)
( )
where
kf: Fracture permeability (mD)
wf: Fracture width (m)
k: Permeability (mD)
xf: Fracture half-length (m)
“α” was determined as 0.3 from the dimensionless fracture conductivity versus alpha plot
(Meyer, 2012) by assuming FCD as 2. Skin factor was calculated as -6.68 by assuming a 350 feet
half-fracture length. However, when a skin factor of -6.68 is used in the GEM/CMG simulator, a
negative well productivity index is calculated. The well productivity index for a phase should be
positive and it is a function of fracture permeability, well drainage effective radius, well radius
and skin factor (GEM, 2013). For the specific reservoir properties assumed in the modeling, the
maximum skin factor that can be assigned to sustain a positive productivity index is -4.
Regarding hydraulic fractures, it was suggested by the operator of the wells that only 40
percent of the coal seams per completion stage have been successfully hydraulically fractured
(CNX, 2013) and that the thickest seams are expected to have received most of the fractures
(Pashin, 2013). For this reason, model S3, where hydraulic fractures are explicitly simulated,
only includes layers 4, 6, 8, 12, 14, 16 and 18, which represent the thickest seams and account
for 38 percent of the stimulated seams. A primary fracture width of 0.01042 feet (EPA, 2004)
with a primary fracture permeability of 10,000 mD and 350 feet of half-length is assigned in the
models. These input parameters for the hydraulic fractures translate into pseudo-fractures with
91 |
Virginia Tech | proper grid refinement that account for non-Darcy flow with an effective permeability of 82.1
mD. Hydraulic fractures were developed along the maximum horizontal stress direction in the
area and thus the modeling grid was rotated to orient the maximum horizontal stress in the I
direction. In addition, it was assumed that the permeability in the I direction is equal to butt cleat
permeability and that in the J direction equal to face cleat permeability (Vasilikou et al., 2013).
Initially, for all three investigated scenarios S1, S2, S3 (base, skin, hydraulic fractures),
history matching of the gas rate was achieved up to year 2013. Cumulative gas production of the
field was similar for all scenarios at approximately 719.65 standard MMcf, accounting for almost
40 percent primary recovery of the original gas in place. The main coal seams contributing to gas
commingle production in all three scenarios are listed below in order (from the largest
contribution to the least): seams number 9, 15, 4, 16, 13, 8, 11, 7, and 14.
The second phase of the modeling was the injection period. The injection start date was set
for the 1st of May 2013 with injection on-going until the 1st of May 2014. Wells DD7, DD7A
and DD8 were shut in for one-month prior to commence of the injection and then turned into
injectors. The same amount of carbon dioxide was injected into DD7, DD7A and DD8,
approximately 6,667 tons per well, at a constant daily rate of 18.26 tons/day over a 1-year period.
After injection of 20,000 tons of CO was completed the three injectors remained shut in for one
2
year until the 1st of May of 2015. Gas production for all four wells in the study area was
projected to year 2023.
From the simulation results it is shown that for the three different scenarios where CO is
2
primarily injected into the coal seams there is a higher differential between the injection pressure
at the wellhead and the flowing pressure yielded at the end of the history matching period at the
specific coal seam depth. As shown in Figure 45 for scenario S3, more CO is injected in coal
2
layer 9 and that was also the seam most depleted during the history matching exercise. The
second most depleted seam during history matching was layer 15. However, during injection
layer 4, which is at a shallower depth and thus has a lower pore pressure, the seam takes up more
of the injected CO (Figure 45).
2
92 |
Virginia Tech | coal seams are hydraulically stimulated the highest levels of CO breakthrough at the offset well,
2
DD8A, are exhibited in scenario S2 (skin scenario), where an enhanced flow zone is assigned
along the wellbore for all the coal seams in the model.
Figure 46 - CO Adsorption Profile in gmole/ft3 at Layer 4 for Scenarios (A) S1 Base, (B) S2
2
Skin and (C) S3 Hydraulic Fractures Scenario
As shown in Table 13, the results of the S model simulations show that the injection wells
DD7 and DD7A produce more CO during flowback when brought back on-line a year after
2
injection is completed in scenario S2 compared to scenarios S1 (base model) and S3 (hydraulic
fractures model). The third injector, DD8, has a higher CO flowback in scenario S3 (hydraulic
2
fractures model), Table 13. In all three scenarios, injector DD7 has a higher CO flowback
2
amount compared to DD7A and DD8; DD8 produces the least amount of CO .
2
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( a )
( c )
( b ) |
Virginia Tech | According to the modeling results of the (S) models, only 11, 6, and 10 percent of the total
20,000 tons of CO injected in the study area is successfully (permanently) stored up to year
2
2023 for scenarios S1, S2 and S3, respectively. The skin model, scenario S2, which has been
assigned the highest flow enhancement around the wellbore, is the one allowing for the least CO
2
storage and maximum flowback.
As far as ultimate field methane recovery is concerned for the (S) models, 89, 94 and 90
percent (for scenarios S1, S2 and S3, respectively) of the initial gas in place is achieved. These
results are higher but still comparable to the assumption that up to more than 75 percent recovery
of methane in place can be achieved with CO injection in coalbeds (VCCER, 2011).
2
Table 13 - CO Flowback at the Injectors and CO Breakthrough at the Offset Well for the
2 2
(S) Models for Projection Time to Year 2023
CO Flowback at Injection Wells (Tons)
2
Wells Base Model (S1) Skin Model (S2) Hydraulically
Fractured Model (S3)
DD7 6,349 6,565 6,384
DD7A 5,381 5,791 5,526
DD8 5,093 4,929 5,175
CO Breakthrough at Offset Well (Tons)
2
DD8A 963 1,462 905
CO Total Field
2
17,786 18,747 17,990
Production
(Tons)
Table 14 - Cumulative CH Production for the (S) Models for the Base (S1), Skin (S2) and
4
Hydraulic Fractures (S3) Scenarios for Projection Time to Year 2023
Cumulative CH Production (Std Mcf)
4
Wells Base Model Skin Model HF Model
(S1) (S2) (S3)
DD7 578,811 639,177 584,588
DD7A 231,607 262,101 237,485
DD8 395,162 422,485 401,668
DD8A 206,529 257,726 204,616
95 |
Virginia Tech | L Models
The larger models (L) incorporated all wells within the study area. The drainage area of the
16-seam geologic structure for the (L) models was 1552 acres. Based on the input parameters,
the original gas in place (OGIP) was estimated to be 1.77e7 standard Mcf and the original water
in place to be 2.50e5 standard bbl.
The L models were first calibrated to match gas and water production data up to 2013. Then
two different injection scenarios were applied in the simulations in order to achieve injection of
20,000 tons of CO in wells DD7, DD7A and DD8. More specifically, in the first injection
2
scenario (L1), 20,000 tons of CO were equally distributed among the three injectors
2
(approximately 6,667 tons per well) and CO was injected at a constant daily rate of 18.26
2
tons/day for 365 days, in the same manner as in the (S) models. In the second injection scenario
(L2), the same amount of CO was injected per well as in scenario L1, but at a higher daily rate
2
of 27.78 tons/day for the first 20 days of each month for a year; the injection wells were shut in
during the final days of each month during injection. Figure 47 shows the CO mass injection
2
rate at surface conditions for injection scenarios L1 and L2 over time.
After the one-year injection period was completed, different cases were examined where the
shut in period of the three injectors varied in time. The performance of all the wells within the
study area was projected to year 2052.
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Virginia Tech | Figure 47 - Gas Mass Rate of Injected CO at Surface Conditions Versus Time for the
2
Injection Scenarios L1 and L2
Four cases with varying injection scenarios (L1, L2) and shut in periods post injection for
wells DD7, DD7A and DD8 were examined. It must be noted that in all cases the well
characteristics were kept constant and no skin factor or explicit simulation of hydraulic fractures
was employed in the (L) models. In case L1a, injection scenario L1 was applied and the three
injectors were shut in for 1 year post injection; in case L1b, injection scenario L1 was used and
the injectors were shut in for 4 years; in case L2a, injection scenario L2 was employed and the
selected injection wells were shut in for 1 year post injection; and in case L1c injection scenario
L1 was applied and the injectors were shut in for 36 years post injection, which means they were
shut in throughout the projected time of the simulation.
During injection, for both injection scenarios L1 and L2, the bottom hole pressures in the
wells did not exceed the maximum allowable pressure as indicated by a U.S. EPA class II
UIC permit (VCCER, 2013). For injection scenario L1, the maximum bottom hole
pressures for the three injectors DD7, DD7A and DD8 were approximately 423, 484 and
416 psi respectively; for injection scenario L2, the maximum bottom hole pressures were
482, 564 and 483 psi respectively. Also, under the specific pressure and temperature
97 |
Virginia Tech | Figure 49 - Cumulative Gas Production of All the Wells in the Study Area for All Cases
Examined for the (L) Models
The main objective of the modeling work as mentioned previously is to assess CO storage
2
capacity of coalbeds, examine the extent of CO plumes and investigate the potential of
2
enhanced gas recovery at the injectors and offset wells.
The cumulative gas production per well in the study area for all different cases (L1a, L1b,
L2a, L1c) is presented in Figure 49. To estimate enhanced gas recovery of the wells, their
cumulative gas production determined by the simulations is compared to the estimated ultimate
recovery based on decline curve analysis. In more detail, Cardno MM&A (2013) forecasted the
ultimate primary gas production of the wells within the study area without taking into account
CO injection, based solely on exponential decline curve fitting of historic gas production data
2
(VCCER, 2013). As shown in Figure 49, for cases L1a, L1b and L2a cumulative gas production
of wells CC6, CC6A, CC7, DD5, DD6, DD6A, DD8A, EE6 and EE7 is higher compared to the
estimated gas production via decline curve analysis by approximately 2, 22, 53, 33, 17, 35, 15,
49 and 10 percent, respectively. For case L1c, where all three injectors remain shut in post
injection until the end of the forecasted period, the cumulative gas production at wells CC7,
DD5, DD6, DD6A, DD8A, EE6 and EE7 is lower than in cases L1a, L1b and L2a but it is still
99 |
Virginia Tech | enhanced in comparison to the predicted ultimate recovery of these wells without CO injection.
2
It must be noted that in all cases, L1a, L1b, L2a, L1c, cumulative gas production for wells CC8,
CC9B and EE9 is significantly higher than what is predicted through decline curve analysis. This
can be partially explained because the average pore pressure for these wells is higher due to a
higher depth to the gas producing seams. In addition, the proximity of these wells to the no-flow
boundary of the model should be further investigated to determine whether it contributes to the
higher gas production.
Regarding CO storage in the coal seams, it has been determined by the modeling work for
2
cases L1a, L1b, L2a, L1c, as shown in Table 16, that 8.15, 18.2, 8.6, 81.64 percent of the total
20,000 tons were successfully sequestered up to year 2052. Comparing the two injection
scenarios that have the same shut in periods for the three injectors post injection, it results that
performing a type of “huff and puff” injection (scenario L2) yields slightly higher storage of
CO . With respect to case L1b, where the continuous injection plan L1 was applied and the three
2
injectors were shut in for 4 years, less CO as compared to cases L1a and L2a has flowed back
2
and broken through in total. More specifically, approximately 16,354 tons of CO versus 18,371
2
and 18,359 tons of CO were produced in cases L1a, L1b and L2a, respectively. Out of all the
2
cases investigated, the most successful regarding CO storage - but the one with the least
2
enhanced gas recovery - is case L1c, where the three injectors remain shut-in throughout the
projected period up to year 2052.
Finally, results of the (L) models showed that ultimate field methane recovery is about 73
percent of the initial gas in place for cases L1a, L1b, L2a. For case L1c it was approximately 67
percent of the original gas in place, six percent lower than what the other models predicted.
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Virginia Tech | Conclusions
In this paper the modeling procedure for coal bed methane production and carbon dioxide
injection from and into multiple seams through selected wells in Buchanan County, VA, is
discussed in detail. Initially, well performance is modeled using three different modeling
approaches. Results show that when a negative skin factor is assigned to the production wells,
primary production during history matching, flowback of CO at the injectors and post injection
2
breakthrough at offset wells is possibly overestimated. When explicitly simulating hydraulic
fractures, modeling is more computationally intensive but results appear more moderate
compared to the case of the negative skin factor. Furthermore, when modeling the behavior of
hydraulically fractured seams, the extent of the CO plumes is more representative of actual field
2
conditions. Nevertheless, explicit representation of hydraulic fractures introduces further
modeling uncertainties on fracture parameters such as the actual width, length, effective
permeability and flow properties.
The process of CO injection into coal seams with the objective of assessing the potential
2
enhanced gas recovery and the permanent storage of CO was also examined.
2
Two different injection scenarios were examined. In the first scenario CO was injected at a
2
constant rate throughout the entire injection period and in the second scenario a small huff and
puff injection test was performed instead, but at a higher injection rate. Results of the second
scenario show that permanent CO storage is better.
2
Furthermore, it was concluded CO permanent storage is proportional to the time interval for
2
which the injection wells are shut in post injection. Maximum CO storage occurred when the
2
wells were not returned to production. As a result, however, enhanced gas recovery of the field is
reduced for wells with higher post injection shut in times.
Acknowledgements
Financial assistance for this work was provided by the U.S. Department of Energy through
the National Energy Technology Laboratory’s Program under Contract No. DE-FC26-
04NT42590 and DE-FE0006827.
103 |
Virginia Tech | SUMMARY AND CONCLUSIONS
Sequestration of carbon dioxide (CO ) into unmineable coal seams has been underway for
2
several years as a way to mitigate the greenhouse effect with a potential of economic prosperity
related to enhanced gas recovery. The Virginia Center for Coal and Energy Research (VCCER),
part of the Coal Seam Group of the Southeast Regional Carbon Sequestration Partnership
(SECARB), successfully completed a small scale injection project in 2009 in Russell County,
VA, where 1000 tons of CO were injected into multiple thin coal seams through one vertical
2
well over a period of one month. In 2014, a larger scale sequestration project is scheduled, where
20,000 tons of CO will be injected into three vertical coalbed methane wells in a coal field in
2
Buchanan County, VA, over a one year period. The main objectives for these injection tests are
to assess storage capacity of “stacked” coal seams, enhance understanding of the physical and
mechanical processes taking place and examine the potential of enhanced gas recovery.
During primary coalbed methane production and enhanced production through CO
2
injection, a series of complex physical and mechanical phenomena occur. The ability to represent
the behavior of a coalbed reservoir as accurately as possible via computer simulations yields
insight into the processes taking place and is an indispensable tool for the decision process of
future operations. The economic viability of projects can be assessed by predicting production,
well performance can be maximized, drilling patterns can be optimized and, most importantly,
associated risks with operations can be accounted for and potentially avoided.
Simulations require a large number of input parameters and high computational capabilities
in order to accurately predict the behavior of the reservoir. Therefore, in current modeling
practices many simplifying assumptions need to be employed. The shortcomings of the modeling
approaches specific to coalbed reservoirs are:
1. In areas with complex geologic structures, it is common practice to use a simplified
approach of aggregating the coal seams into zones to reduce the complexity of the
model and thus the required computational intensity. However, important
information with respect to the commingled production and the injection mechanism
is not accounted for in this method.
107 |
Virginia Tech | 2. Often to reduce modeling time in the simulation of large multi-well areas, separate
single-well models are investigated, disregarding well interference of the field - one
of the key parameters in coalbeds.
3. In order to minimize the computational intensity, well stimulation is either not
accounted for or not representative of the in situ conditions for complex reservoirs.
4. There is an infinite solution space for history matching exercises due to the large
number of unknown vs. known input parameters. Often, the calibration of the models
is not constrained within an acceptable range.
In this dissertation the aforementioned shortcomings were addressed and the initial reservoir
model for the Russell County site was updated. Subsequently the reservoir model for the
Buchanan County test site was constructed and the preliminary simulations for CO sequestration
2
and enhanced gas recovery were conducted. The following were accomplished in this work:
1. Sensitivity analysis was conducted for a number of model input parameters and the
key parameters and their effect identified.
2. The dynamic evolution of permeability during primary and enhanced recovery from
coalbeds, which is extensively referred to in the literature, was investigated. Coupled
flow and geomechanical simulations were developed to assess the significance of
implementing permeability changes into full field scale simulations.
3. Well stimulation approaches, including a negative skin factor and explicit simulation
of hydraulic fractures, were considered and compared.
4. Different CO injection scenarios into multiple seams for a multi-well field area were
2
modeled and the potential of enhanced gas recovery was assessed.
Analytical and numerical models proposed in the literature to address the phenomenon of
dynamic evolution of permeability during primary and enhanced recovery in coalbeds were
critically assessed. These models have been only used for single seams, single well and small
area representations. An algorithm was developed to couple a reservoir simulator with a
geomechanical code to examine permeability changes during methane production and CO
2
injection for a single coal seam. It was concluded that for relatively flat seams, where there are
no areas of large stresses and strains localization, permeability changes are not significant and
thus it was decided to not be considered in the full-field scale simulations.
108 |
Virginia Tech | Different modeling approaches to account for well stimulation were implemented and
compared. From the analyses it was shown that assigning a negative skin factor, which is
determined through history matching, often overestimates the enhanced flow properties around
the wellbore and facilitates post injection flowback of CO at the injector. Explicit simulation of
2
hydraulic fractures can be controlled to be more representative of in situ field conditions,
although more unknowns are introduced in the simulation: primary width, primary permeability,
effective half-length, fracture orientation and flow regime properties. Not accounting for well
stimulation can lead to unrealistic modeling results. For example, such models could not predict
that during injection in the Russell County test site there would be CO breakthrough at the
2
monitoring well closest to the injector within hours of starting injection. More realistic modeling
results were obtained where hydraulic stimulation of the injection well was included in the
simulation model.
Two CO injection scenarios for the Buchanan County, VA, site were examined. In the first
2
scenario approximately 6,667 tons of CO were injected into 18 coal seams at each injector at a
2
constant rate for a year; in the second scenario a “huff and puff” type of CO injection at a higher
2
rate for the first twenty days per month for a year was applied. It was concluded that for the first
scenario there was slightly higher CO breakthrough at the injectors compared to the second
2
scenario. In the second scenario there were intermediate time intervals during the one-year
injection, allowing CO to set in. From the analysis it was also shown that the time the injectors
2
are shut in post injection is critical to the percentage of CO successfully stored. The longer the
2
wells are shut in the less CO flows back at the injectors. Maximum CO storage can be achieved
2 2
when the injector well are not returned to production. CO breakthrough at offset wells for all
2
injection scenarios was also noted, but it was significantly less compared to flowback.
Enhanced gas recovery at the injectors and offset wells was noted in the modeling work. It
was concluded that since the majority of the injected CO flowed back, the primary mechanism
2
for enhanced recovery for the coals in the study area is not due to the CO preferential adsorption
2
by the coal matrix and CH displacement, but is because of the “renewal” of the pressures in the
4
reservoir.
The most important conclusion for the reservoir simulation work is that it is not a
straightforward process. Even though the models can provide reasonable solutions based on the
109 |
Virginia Tech | available data, it is imperative that current models are updated with new data from laboratory
testing and/or field work monitoring to ensure that they remain stable and robust.
Detailed results of the aforementioned are presented in the papers included in the main body
of this work.
Further Work
As discussed, the main problem in reservoir modeling is the uncertainty in the magnitude
and range of several input parameters. Following extensive laboratory testing related to coalbed
parameters over a number of years, reservoir model input parameters have been established;
however, there are still gaps in the understanding of physical processes. For improved
predictions through reservoir simulations, field parameters need to be determined more
accurately and at different times throughout the lifetime of a reservoir.
For instance, it is necessary to achieve better monitoring of the quantity of the water that is
produced from each well during production. This would result in better history matching of the
gas produced. In addition, production pressures at the well head and/or the bottom of the well
should be accurately recorded.
Also, in order to better understand and “decode” commingled production from multiple
seams there is a need to perform field tests where a seam or a group of seams are isolated and
properties such as permeability can be determined via transient pressure testing.
Camera logging of the injectors is proposed to obtain more information regarding hydraulic
fracture simulation of the well and create more representative models. In this way the modeling
efforts will better capture CO flow and potential breakthrough and storage processes.
2
There is a constant need to use reservoir simulators with enhanced capabilities so that less
simplifying assumptions are required for the models and more details are incorporated. The next
step would be to use a higher end simulator such as Eclipse by Schlumberger, where ability to
model more geologic details and create larger element models is provided.
Finally, incorporating a temperature gradient in the models to better account for potential
phase changes of CO during injection and storage will be an important consideration.
2
110 |
Virginia Tech | Recommendations for Transferring Lessons Learned from Coalbed Methane
Modeling to Shale Gas Modeling
Research on primary production from coalbed reservoirs has been underway for several
decades. The best practices to estimate the original gas in place are established and there is
agreement within the research community on the basic methane production mechanism. Yet
there are still questions which need to be answered, as previously mentioned, to further
understand the production mechanism from multiple thin unmineable coal seams and the
interaction with CO injection. Currently, there is an interest in the industry to transfer the
2
experiences and expertise from coalbed reservoir to “unlock” the great energy potential of the
shale reservoirs.
In both coalbed and shale reservoirs a significant portion of the gas in place is stored via
adsorption on the rock matrix. This is their main - and should be considered to be their only -
similarity. There are significant differences between the geologic properties, such as initial pore
pressure, porosity and adsorption isotherms that result in different initial volumetrics and
production mechanisms. In addition, shale gas reservoirs are usually much deeper than coalbed
methane reservoirs. The decades of coalbed methane research poses the right questions to be
answered in order to identify the critical unknown parameters in hydrocarbon exploration from
shale reservoirs. However, there should not be a direct transfer of properties from coalbeds to
shale reservoirs since it is likely to lead to the wrong conclusions. Regarding primary production
from shales, the following critical questions need to be addressed:
1. What is the effective volume activated through hydraulic stimulation, depending on
the stimulation technology employed, particularly with regard to the frac-fluid
composition? Further field characterization and monitoring is required.
2. What are the different flow regimes for the (i) primary artificial channels developed
with hydraulic stimulation, (ii) the activated dendritic pattern perpendicular to the
primary fractures, and (iii) within the shale matrix? Field determination through
pressure transient testing and further laboratory-testing accounting for in situ
conditions needs to be conducted to determine permeability ranges for each case.
3. What happens to the water used for hydraulic treatment? In the case where it is
produced, how are the relative permeability curves affected? If it stays in the
111 |
Virginia Tech | reservoir how are the pressure profile, and the consequent gas release from the shale
formation, shaped?
4. What percentage of the original estimated gas in place is recovered in relation to the
well characteristics, such as lateral length, orientation to in situ stresses, proximity to
other wells, frequency and intensity of hydraulic stages?
5. Could the use of CO for well stimulation, pressure management and/or hydraulic
2
fracturing benefit the production of both gas and condensate, specifically the ability
of CO to change the viscosity of immobile condensates and allow for their
2
production?
There are a number of issues that need to be resolved if shale reservoirs are to be used for
permanent CO storage in a similar manner to deep coal seams. It has been determined that there
2
is a larger affinity of CO to the coal and the shale matrix compared to CH , especially in shale
2 4
when there is an increase of clay content. The in situ behavior of CO and the coal matrix is still
2
under investigation through a series of small and medium CO sequestration projects in
2
unmineable coal seams. The same aspects should be examined for the case of CO sequestration
2
in shale reservoirs with the investigation of CO properties under the different depth and pressure
2
conditions of shales. In addition, there is less interference between wells and different well
development patterns when exploiting shale reservoirs; the potential of enhanced gas recovery at
offset wells should also be examined and potentially could be negligible depending on well
drilling patterns and spacing
Historic production data for coalbeds are available for in excess of 20 to 30 years, a large
portion of the assumed life of a 50-year well. Type curve fitting of the historic points and
development of decline curve analysis to estimate the behavior of new coalbed methane wells is
well established. From the curve fitting it is concluded that for production of coalbeds there is an
initial period of dewatering and increase of the gas rate, then a plateau in production due to
commingle production from multi-seams and at the end a decline period which is best fitted with
an exponential segment. The historic production data available for horizontal shale wells in most
basins are limited to production of approximately seven years. For initial production, a sharp
decline is best fitted via a hyperbolic decline segment for production of free gas in the system
and for the rest of the well life an exponential decline is assumed given that desorption from the
112 |
Virginia Tech | Development of a Flotation Rate Equation from First Principles
under Turbulent Flow Conditions
Ian Michael Sherrell
(ABSTRACT)
A flotation model has been proposed that is applicable in a turbulent environment.
It is the first turbulent model that takes into account hydrodynamics of the flotation cell
as well as all relevant surface forces (van der Waals, electrostatic, and hydrophobic) by
use of the Extended DLVO theory. The model includes probabilities for attachment,
detachment, and froth recovery as well as a collision frequency. A review of the effects
fluids have on the flotation process has also been given. This includes collision
frequencies, attachment and detachment energies, and how the energies of the turbulent
system relate to them. Flotation experiments have been conducted to verify this model.
Model predictions were comparable to experimental results with similar trends.
Simulations were also run that show trends and values seen in industrial flotation
systems. These simulations show the many uses of the model and how it can benefit the
industries that use flotation. |
Virginia Tech | ACKNOWLEDGEMENTS
The author would like to express his deepest appreciation to Dr. Roe-Hoan Yoon.
His guidance and support throughout this work were of the utmost importance. Also, the
support of Dr. Demetri Telionis and Dr. Pavlos Vlachos were immeasurable. With their
significant and incisive advice, this project was able to succeed. Great appreciation and
thanks is extended towards them. The author would also like to thank Dr. Gerald Luttrell
for his words of wisdom, inspiration, and support. The author also thanks Dr. Greg T.
Adel for his timely and insightful advice.
The author would like to thank the Center for Advanced Separation Technology
as well as the Department of Energy for their financial support.
Sincere appreciations are extended to Hubert Schimann, Emilio Lobato,
Selahattin Baris Yazgan, and Mariano Velázquez. With their friendship, support, and
guidance, research went smoothly, but more importantly was enjoyable. Their
friendships will be treasured.
The author is also grateful to all of his other friends for their love and support. A
special thank you to David Gray. He provided great understanding, sympathy, and
encouragement, as well as a lasting friendship.
Lastly, the author would like to express his deepest gratitude and appreciation to
his family. Particular thanks are expressed to his wife, Cam, for her love, support,
understanding, and tremendous patience.
II I |
Virginia Tech | 1 Introduction
Background
Flotation is widely used throughout the mining industry as well as the chemical,
and petroleum industries. It can be a highly efficient process for solid-solid separation of
minerals. It is now more diverse in its application, with uses such as separation of ink
from paper, plastics from each other, radioactive contaminants from soil, and carbon
from fly ash. The entire industry is growing along with knowledge of the process and
sub-processes. With this increased knowledge, a more reliable flotation model can be
derived from first principles. This results in a general flotation model, and allows its use
in the mining industry, regardless of machine type and material being recovered.
Generally, flotation is a three-phase process, which uses a medium of water
(liquid) to separate various particles (solid) by the addition of air bubbles (gas).
Hydrophobic particles attach to the bubbles and rise to the top of the flotation cell where
they are extracted while hydrophilic (or less hydrophobic) particles remain in the slurry.
The attachment of particles to bubbles is the most important sub process in flotation.
Without a selective attachment, no separation would be possible. To enhance this
selective attachment, surfactants may be added to alter surface properties. The
surfactants control the surface tension and contact angles of the particles in the flotation
process.
Flotation occurs within a turbulent environment. Turbulence within a
conventional flotation cell is produced by the action of the impeller, which is used for
mixing purposes, while turbulence within a column flotation cell is induced by rising air
bubbles and settling particles. The combination of the hydrodynamic forces in a
turbulent environment and the surface forces controlled by the addition of surfactants
makes modeling of the entire flotation process very complex.
Modeling
Modeling of flotation has two major benefits. The main benefit being the control
and improvement of the flotation process within an industrial situation. The model will
be able to predict a recovery from certain known inputs. If possible, the controller, either
1 |
Virginia Tech | human or computer, may be able to improve the recovery by modifying those inputs.
The model has the benefit of instantly knowing what that modification will do. The
model can also find the maximum recovery within certain input ranges. The second
benefit is that process design can be more easily accomplished. For a typical flotation
circuit design within a processing plant, many lab tests are run and scale up of those tests
are then performed. A flotation model can bypass the inaccuracy of the scale up process
and do away with many of the flotation lab tests. As long as certain input variables are
known the flotation recovery can be calculated.
A flotation model is similar to a chemical kinetics model, with one form of that
being shown in Equation 1.
dN
1 = f (k,N )=−k Nf −k Ng 1
dt i 1 1 2 2
The model directly predicts the change in particle concentration, N , with respect to time,
1
t, as a function of a certain concentration(s), N, and rate constant(s), k. The negative
i i
sign indicates that the concentration is diminishing due to the loss of particles being
floated. The exponents f and g signify the order of the process. Most researchers believe
that flotation is a first order process and a function of only the particle concentration and
a rate constant (Kelsall 1961; Arbiter and Harris 1962; Mao and Yoon 1997).
dN
1 =−kN 2
dt 1
The rate constant, k, within this equation conveys how rapidly one species floats.
A high rate constant indicates that certain species floats quickly while a low rate constant
indicates slow flotation. Knowing the rate constants of two (or more) species within a
separation process reveals the efficiency of the process. The greater the difference
between the two rate constants, the better the separation is. The recoveries of each
individual species, R, can also be calculated knowing the rate constant as well as the
residence time within the cell, τ.
kτ
R= 3
1+kτ
Since recoveries are the desired output from modeling flotation, the rate constant is the
useful component of Equation 2. Throughout flotation modeling history, the attempt has
been made to produce a general flotation rate constant equation.
2 |
Virginia Tech | The most recent general turbulent flotation rate model was given by Pyke,
Fornasiero, and Ralston (2003).
23
G ⎛0.33ε49d79 ⎞⎛∆ρ ⎞
k =−2.39 fr ⎜ d 2 ⎟⎜ 2 ⎟ E E E 4
d V u ν13 ρ C A S
⎝ ⎠⎝ ⎠
2 cell 2 3
Rate constants are usually modeled as a function of a collision frequency, and
probabilities of attachment and detachment (Schulze 1993; Yoon and Mao 1996; Mao
and Yoon 1997; Lu 2000; Bloom and Heindel 2002; Pyke, Fornasiero et al. 2003). In
Equation 4, the attachment efficiency, E , is taken to be the probability of attachment
A
while the stability efficiency, E , is an inverse probability of detachment. There is also a
S
collision efficiency, E , which takes hydrodynamic effects into account during the
C
collisions of particles and bubbles. The remainder of the equation is the collision
frequency. The true number of collisions, that may or may not become attached, results
from the combination of the collision efficiency and collision frequency. The collision
frequency shown in Equation 4 is a modified equation given by Abrahamson (1975) that
is divided by the number density of particles. The turbulent velocity used within
Abrahamson’s model is given by Liepe and Mockel (1976).
The collision efficiency within Equation 4 takes into account the fact that particles
may deviate from the fluid flow and may not collide due to this deviation. Pyke,
Fornasiero, and Ralston (2003) use a solution of the BBO equation referred to as the
Generalized Sutherland Equation. This takes into account interception and inertial
forces.
The attachment efficiency makes use of the angle of collision, which results in a
certain maximum sliding time, and relates that to the amount of time needed for the
bubble and particle to become attached once collision occurs. When collisions occur, a
certain amount of time, referred to as the induction time, is required for the liquid film to
be drained between the bubble and particle as well as the three-phase-contact line to
spread and become stable. Enough time must be available during the collision process
for the attachment to take place.
The stability efficiency is a relationship between the attachment and detachment
forces. The attachment forces include capillary and hydrostatic forces. The detachment
3 |
Virginia Tech | forces include buoyancy, gravitational, machine acceleration, and capillary force.
Knowing these, the stability of the aggregate can be determined.
Equation 4 provides a model of the flotation process based upon turbulent
characteristics of the flow as well as hydrodynamic forces. What the model does not
account for is the effects of surface forces. Surface forces are known to have an effect on
the outcome of flotation as shown by Mao and Yoon (1997). Without the inclusion of
surface forces a model can predict only a certain percentage of cases and is not general in
nature. A flotation model has been proposed that accounts for surface forces in a
quiescent environment (Yoon and Mao 1996).
2
1 ⎡3 4Re0.72⎤⎛ R ⎞ ⎛ E ⎞⎡ ⎛ W +E ⎞⎤
k = S ⎢ + ⎥⎜ 1 ⎟ exp⎜− 1 ⎟⎢1−exp⎜− a 1 ⎟⎥ 5
4 b ⎣2 15 ⎦⎝ R
2
⎠ ⎝ E
k
⎠⎢⎣ ⎝ E k' ⎠⎥⎦
This model is derived from first principles and is the most rigorous flotation
model, to date, dealing with all dominant surface forces found in flotation. These surface
forces are the electrostatic, van der Waals, and hydrophobic forces and are modeled
based upon the extended DLVO theory. The surface forces are used in calculating the
probability of attachment (first exponential term) and the probability of detachment
(second exponential term). The energies that must be overcome, by the kinetic energies
of the particles and bubbles, for the attachment and detachment processes to occur are
related to the surface forces.
The first half of Equation 5 is a combination of the collision frequency (¼S ) and
b
collision efficiency. The collision frequency is derived from the number of particles one
bubble would collide with assuming it traveled vertically the entire length of the cell and
particles were completely stationary and evenly distributed throughout the cell. This
becomes a function of the surface area rate, S , of the air. Since particles are not
b
stationary, a collision efficiency is used to take into account hydrodynamic effects. This
assumes that particles follow the fluid completely (no inertial effects) and that there is a
governing stream function that takes into account the Reynolds number of the bubble.
The problem with this model is that it was derived for a quiescent environment.
Due to this, no turbulent effects are present. Since turbulence is found in all flotation
situations, this model can not be applied for flotation purposes. It does, on the other
4 |
Virginia Tech | 80
60
40
20
0
-20
-40
-60
-80
0 50 100 150 200
Separation Distance, H (nm)
hand, provide a valuable relationship between the hydrodynamics of the system and the
surface forces.
Surface Forces (Energies)
Surface forces are interactions between surfaces usually on a scale of less then
100 nanometers. The forces can be converted into energies of interaction based upon the
radii of the two interacting surfaces. Surface force modeling in flotation employs the
extended DLVO theory. This theory combines the van der Waals (dispersion) energy
(force), V , electrostatic energy (force), V , as well as the hydrophobic energy (force),
D E
V , into one total surface energy (force), V (see Figure 1). These surface energies are
H T
additive as shown in Equation 6.
V =V +V +V 6
T E D H
All three forces (energies) are either known or shown to have an effect on interactions
between particles and bubbles in a water medium (Yoon and Mao 1996; Yoon 2000).
The electrostatic energy is given by Equation 7 (Shaw 1992).
5
)J(
7101x
V
V
E
V
T
E
1
V
D
H
C
W
A
V
H
Figure 1. Surface energy vs. distance of separation between two particles (i.e.
particle-bubble) |
Virginia Tech | πεεRR ( Ψ2 +Ψ2) ⎡ 2Ψ Ψ ⎛1+e−κH ⎞ ⎤
V = 0 1 2 1 2 ⎢ 1 2 ln⎜ ⎟+ln( 1−e−2κH) ⎥ 7
E (R +R ) ⎣Ψ2 +Ψ2 ⎝1−e−κH ⎠ ⎦
1 2 1 2
The Stern potential for the particle and bubble, Ψ, is substituted with the zeta-potential,
i
ζ, which can be measured.
i
The dispersion energy is given by Equation 8 (Rabinovich and Churaev 1979)
A RR ⎡ 1+2bl ⎤
V =− 132 1 2 1− 8
⎢ ⎥
D 6H(R +R ) ⎣ 1+bc H ⎦
1 2
where A is the Hamaker constant for particle 1 and particle 2 (a bubble in flotation)
132
interacting in a medium (3). The last half of the equation is a correction factor for the
retardation effect, where b is a parameter characterizing materials of interacting particles
(3x10-17 s for most materials), l is a parameter characterizing the medium (3.3x1015 s-1 for
water) and c is the speed of light (Yoon and Mao 1996). The retardation effect can
usually be omitted due to the small effect that it has on the overall energy of interaction.
The equation for hydrophobic energy is an empirical formula as opposed to
Equations 7 and 8 which are theoretical in nature. The most recent proposed form is
similar to the dispersion energy equation (Yoon and Mao 1996).
RR K
V =− 1 2 132 9
H 6(R +R )H
1 2
The Hamaker constant, A , is replaced by the hydrophobic force constant, K , and
132 132
there is no retardation effect. The hydrophobic force constant can be found by using
Equation 10, which includes the interactions between similar particles in a medium
(Yoon, Flinn et al. 1997).
K = K K 10
132 131 232
K and K are given by Equations 11 & 12, where [F] is the dodecylammonium
131 232
hydrochloride concentration.
K =aexp(bθ) 11
131 k
for θ<86.89° a = 2.732E-21, b = 0.04136
k
for 86.89°<θ<92.28° a = 4.888E-44, b = 0.6441
k
for θ>92.28° a = 6.327E-27, b = 0.2172
k
6 |
Virginia Tech | ( )
K =exp d +e [F] 12
232
for all [F] d = -39.67, e = -117.7
Equation 11 was derived from data obtained from Yoon and Ravishankar (1994; 1996),
Rabinovich and Yoon (1994), Vivek (1998), and Pazhianur (1999). Equation 12 was
derived from data obtained by Yoon and Aksoy (1999).
There is an ongoing debate as to the validity and origin of the hydrophobic force.
There has been a tremendous amount of research on this subject, with only a few
examples included here. Most, but not all, researchers believe that the force can be
measured at long ranges (up to 200 nanometers). Of those, there are varying explanations
as to the origin of the force. Attard (1989) proposed that the hydrophobic force is
actually a part of the van der Waals force. This can be discredited in flotation due to the
fact that the van der Waals force is repulsive between a particle and air bubble in a water
medium. This would result in no attractive forces seen in flotation. Another explanation
must be given in view of the fact that an attractive force is still seen under these
circumstances (Ducker, Xu et al. 1994; Lu 2000; Yoon 2000). The force also might
originate from the ordering of water molecules away from the hydrophobic surface
(Eriksson, Ljunggren et al. 1989) or from a hydrodynamic fluctuation at the hydrophobic
surface and liquid interface that might produce an attractive force (Ruckenstein and
Churaev 1991). The most recent explanation is the formation of a microscopic bridging
bubble between the two surfaces. The surface tension along the liquid-vapor interface
creates the long range hydrophobic force. The bridging bubble can be formed by
cavitation of the liquid when the two surfaces approach each other (Berard, Attard et al.
1993; Parker and Claesson 1994) or by stable nano-bubbles that have previously adhered
to the hydrophobic surface (Carambassis, Jonker et al. 1998; Ishida, Sakamoto et al.
2000; Ederth, Tamada et al. 2001; Sakamoto, Kanda et al. 2002). Arguments against
microscopic bridging bubbles state that they are thermodynamically unstable (Eriksson
and Ljunggren 1995; Eriksson and Ljunggren 1999) and too short lived to have a
noticeable effect on experimental measurements (Ljunggren and Eriksson 1997). The
debate as to the cause of the hydrophobic force will continue until a theoretical model for
the interaction can be proposed and verified. Regardless of the origin of the hydrophobic
7 |
Virginia Tech | force, a long range attraction exists between hydrophobic surfaces. Therefore, it is
appropriate to use Equation 9 to quantify that attractive force.
Knowing the total surface force energy (Figure 1), there exists a maximum
repulsive (positive) energy that must be overcome, E . This maximum energy occurs at
1
the critical rupture distance, H . At distances smaller than this the free energy
c
continuously drops. This means there is nothing preventing the particle and bubble from
coming together. Once this maximum is reached, and overcome, the particle and bubble
will spontaneously adhere to each other, and a three-phase contact will occur. Due to
vibrations along the liquid-vapor interface, the average critical rupture thickness is
greater than the instantaneous critical rupture thickness (Yoon 2000). The vibrations
cause the instantaneous distance between the bubble and particle, which is smaller than
the average distance, to be equal to the critical rupture thickness. This smaller distance
overcomes the energy barrier. These vibrations will result in a higher average critical
rupture thickness and a lower energy barrier.
Objectives
The objective of the present research is to derive a general flotation model that
takes into account turbulence within a flotation cell as well as all applicable surface
forces. The main focus of the model is within the slurry portion of the flotation cell.
This model is to be derived, as much as possible, from first principles that relate turbulent
parameters of the fluid to physical and chemical properties of the particles, bubbles and
fluid. With the addition of a froth recovery model, an entire flotation model will be
available.
To accomplish this, a review of flotation fluid dynamics will be performed.
Experiments will also be undertaken to verify the model. Simulations using the model
will show the applicability to industrial systems. It is hoped that the proposed model will
be able to improve upon these systems and be a benefit to the mineral processing
industry.
Organization
8 |
Virginia Tech | The body of this thesis has been presented as three independent papers. Paper 1 is
titled “Fluid Dynamics of Bubbles and Particles under Turbulent Flotation Conditions: A
Review”. This paper presents turbulence effects upon flotation. Included within this
paper is a review of collision frequencies as well as the proposed attachment and
detachment energies imparted by the turbulence. Paper 2 is titled “Developing a
Turbulent Flotation Model from First Principles”. This paper presents the flotation
model that has been derived as well as simulations using this model. Paper 3 is titled “A
Comprehensive Model for Flotation under Turbulent Flow Conditions: Verification”.
This paper demonstrates the validity of the proposed flotation model by experimental
verification. The summary of the entire research is presented following these three
papers. Recommendations for future work follow the summary.
Nomenclature
1 subscript – refers to particle
2 subscript – refers to bubble
3 subscript – refers to liquid
a K parameter [-]
131
b material parameter (V ) [s]
D
b K parameter [-]
k 131
c speed of light [m·s-1]
d K parameter [-]
232
d diameter of i [m]
i
e K parameter [-]
232
k rate constant [s-1]
l medium parameter (V ) [s-1]
D
u bubble velocity [m]
b
A Hamaker constant for 1 interacting with 2 in 3 [-]
132
E surface energy barrier [J]
1
E attachment efficiency [-]
A
E collision efficiency [-]
C
E stability efficiency [-]
S
9 |
Virginia Tech | Fluid Dynamics of Bubbles and Particles under Turbulent Flotation
Conditions: a Review
I. M. Sherrell
Abstract
A review and assessment of the current knowledge of the effects of turbulent flow on the
flotation process has been undertaken. This includes a review of probabilistic models of
collision frequencies with all underlying assumptions. Although there is no model that takes into
account all turbulent effects and conditions for the collisions of particles and bubbles, a model
proposed by Abrahamson (1975) is the most applicable for flotation purposes. Our review
revealed that there are more appropriate models in the literature but modifications are needed to
take into account bubble and particle density effects in a liquid. Attachment and detachment
energies are also described. Attachment energies are related to the Kolmogorov length scale and
the length scales of the particles and bubbles. Detachment energies are related to the system’s
largest length scale (i.e. the impeller).
Introduction
Fluid effects are seen in all flotation processes. Modeling of the process, therefore, must
take this fact into account. Flotation modeling aims at obtaining a rate constant for different
components (e.g. minerals, surface chemistry types, particle sizes) of a feed to a flotation circuit.
These rate constants are usually modeled as a function of collision frequencies, and probabilities
of attachment and detachment (Schulze 1993; Yoon and Mao 1996; Mao and Yoon 1997; Lu
2000; Bloom and Heindel 2002; Pyke, Fornasiero et al. 2003). Since fluid effects are seen in all
aspects of flotation, they must be reflected in the collision frequency, attachment and detachment
model sections.
To model these processes, assumptions must be made. There are many simplifications in
flotation modeling including particle and bubble effects. Shape is assumed to be spherical due to
the great difficulty in accounting for non-spherical particles. Particle surface chemistry is
assumed to be uniform across the entire surface of the particle. Particle composition (e.g.
hydrophobicity) is also assumed to be uniform either throughout the entire flotation cell or
component being modeled or divisions of that cell or component. These are only a few of the
simplifications encountered in flotation modeling, but the greatest simplification comes from
modeling the fluid itself.
Turbulence has always been an extremely complicated subject, and the only way to
model it, except under very special circumstances, is to assume that the turbulence is locally
homogeneous and isotropic. Although, for certain flow fields, this is not as far-fetched as it may
seem, the large scales encountered in flotation modeling do not satisfy this condition and the
assumption is needed. This allows statistical modeling of turbulence to be adopted, based mostly
13 |
Virginia Tech | Particle
collision with
bubble
Particle
Bubble
deviation
around bubble
Stream line
Figure 1. Effects of streamlines for particle
collisions with a rising bubble. The assumption is
made that particles follow the fluid completely.
on the Kolmogorov theory, which in turn allows the establishment of a relationship between
turbulent data such as energy dissipation and root-mean-squared (rms) slip velocity.
Kolmogorov theory is a way of relating scales of turbulence (and in flotation modeling,
what those scales are affecting) to their respective energies. Energy is added at the integral scale
and from there cascades down through the inertial scales to the dissipative scales where it is lost
as heat due to viscous effects. The assumption is that energy is not lost at any scale but the
dissipative scale and is only transferred through the intermediate scales. In the flotation process,
energy is added by the impeller (integral scale), acts upon the particles and bubbles (inertial
scale) and is dissipated from the system at the Kolmogorov microscale (dissipative). Statistical
simplification, such as the Kolmogorov theory, allows modeling of turbulent processes, such as
flotation, to take place.
Collision Frequencies
Although the Kolmogorov theory is not explicitly used in collision frequency modeling,
the assumption that the turbulence is homogeneous and isotropic is still required. Mao and Yoon
bypassed this assumption by modeling flotation in a quiescent environment (1996; 1997). This
required the use of streamlines and the assumption that particles followed the fluid flow
completely.
The total number of collisions was found by multiplying the total possible collisions, Z ,
C
by a collision efficiency, P .
C
Z = Z P [1]
12 C C
The total possible number of collisions was found by multiplying the volume swept by one
bubble, assuming that the bubble rises straight up through the entire cell, and the number density
of particles to get the number of particle collisions with one bubble. Knowing this and the
number of bubbles per unit time through the cell results in the total possible collisions (Equation
[2]).
3V 1
Z = g N = S N [2]
C 4R 1 4 b 1
2
14 |
Virginia Tech | Figure 2. Collision due to shear mechanism.
The collision efficiency, P , was obtained by knowing the stream function for the flow
C
around the bubble (Figure 1). Assuming that the particles completely followed the fluid flow (no
Brownian or inertial effects) the collision efficiency would then be a function of that stream
function and the particle and bubble radii (Luttrell and Yoon 1992; Yoon and Mao 1996; Dai,
Fornasiero et al. 2000). There are forms of collision efficiencies that can also take into account,
among other things, the inertial effects of particles (Dai, Fornasiero et al. 2000).
When turbulence is encountered, an analysis such as this can not be used. For one,
bubbles take meandering paths through the liquid and a total volume that they travel through can
not be accounted for. Also, streamlines are not stationary in turbulence. They are constantly
changing throughout time and space. A more statistical approach must be taken.
Collision frequencies are all based around one simple model
Z =βN N [3]
12 1 2
where β, the collision kernel, is
β= f (C,d ,w) [4]
12
a function of some constant, C, the average diameter of collision, d = (R +R ), and the relative
12 1 2
velocities between the colliding particles, w. A form similar to this was first used by von
Smoluchowski (1917) to model coagulation kinetics.
4
Z = N N d3G [5]
12 3 1 2 12
G in Equation [5] is the velocity gradient perpendicular to the direction of particle motion. Only
laminar flows with collisions occurring due to shear were considered. In shear flows, particles
are able to interact and collide with each other the same way fluid particles can collide with other
fluid particles (Figure 2). There is no deviation from the fluid path with this assumption. Camp
and Stein (1943) were the first to apply this concept to turbulent collisions.
4 ε
Z = N N d3 [6]
12 3 1 2 12 ν
They related G to turbulent fluid parameters, ε and ν. In their assumptions, they
considered only collisions due to shear fields produced by large eddies. Again, there was the
15 |
Virginia Tech | Figure 3. Collision due to acceleration
mechanism. Heavy particle deviates from
streamline to collide with inertia-less particle.
assumption that particles did not deviate from the fluid path. Saffman and Turner (1956) were
later able to refine Camp and Stein’s model by a slight modification of the constant C.
8π ε
Z = N N d3 [7]
12 15 1 2 12 ν
For Equation [7] to be valid the collision diameter (d ) must be small compared to the
12
smallest eddies and the particles and bubbles must follow the fluid completely. In Saffman and
Turner’s case, Equation [7] was used to model droplet collisions in cloud formations, which
followed these assumptions. In flotation these assumptions can not be followed, but Saffman
and Turner also produced another model with the addition of an accelerative mechanism of
collision.
12
⎡ ⎛ ρ ⎞2 ⎛ ⎛ Du⎞2 1 ⎞ 1 ε⎤
Z = 8πN N d2 ⎢⎜1− p ⎟ (τ −τ )2⎜ ⎜ ⎟ + g2⎟+ d2 ⎥ [8]
12 1 2 12 ⎢⎜ ρ ⎟ 1 2 ⎜⎝ Dt ⎠ 3 ⎟ 9 12ν⎥
⎣⎝ f ⎠ ⎝ ⎠ ⎦
The accelerative mechanism accounts for inertial effects in turbulent collisions (Figure
3). This indicates that particles do not have to follow the fluid flow and can, therefore, be larger
than the smallest eddies. For Equation [8] to be valid particles and bubbles must already be close
together (within the same eddy) and must have very similar particle radii (1≤R /R ≤2).
1 2
Abrahamson (1975) then proposed a model for high turbulent energy dissipation, ε. This
is shown in equation [9]
Z = 8πN N d2 U2 +U2 [9]
12 1 2 12 1 2
where the mean squared particle velocity
U2
U2 = f [10]
i 1.5τε
1+ i
U2
f
16 |
Virginia Tech | comes from a simplification of a model for particle relative velocity in an oscillating (sinusoidal)
fluid given by Levins and Glastonbury (1972). Only the assumptions that very energetic
isotropic turbulence was being considered, and that particle velocities were independent and
followed some distribution, needed to be made.
Particle size is not an issue in this model. As long as the particle stokes number is high,
the model is valid. The Stokes number is a ratio of the particle relaxation (response) time to the
smallest fluid relaxation time (for fully developed turbulent flow, this timescale will correspond
to the Kolmogorov timescale).
1
τ ⎛ν⎞ 2
St = i =τ [11]
⎜ ⎟
τ i ⎝ε⎠
η
It represents how well a particle follows the fluid flow and is a way to measure this deviation. If
a particle’s relaxation time is less than the Kolmogorov timescale then the lag time between the
fluid movement and the particle movement will be small and the particle will follow the fluid.
For particles to accurately follow the flow their Stokes number should be much less than one.
Any particles with relaxation times above the Kolmogorov timescale will have some lag in their
response to flow fluctuations. Particle and bubble relaxation times are given by various authors
throughout the literature (Govan 1989; Ceylan, Altunbas et al. 2001; Bourloutski and
Sommerfeld 2002). With a high Stokes number assumption, the retarding effect due to
lubrication theory, for particles nearly in contact, can be ignored (Reade and Collins 1998).
Therefore, no collision efficiency is needed while using Abrahamson’s collision frequency
model.
Sundaram and Collins (1997) compared Abrahamson’s model with Saffman and Turners’
on the basis of particle Stokes numbers. As the Stokes number approached zero, the numerical
simulation approached Saffman and Turner’s prediction. As the Stokes number increased,
Abrahamson’s model more closely predicted the true collision frequency. At a Stokes number
of, roughly, 1, both models had equal error, with Saffman and Turner under predicting and
Abrahamson over predicting. With many flotation particles having Stokes numbers above 1 and
with less assumptions being violated, Abrahamson’s model is more representative of the
flotation process, and, as a result, most flotation models use Equation [9] to determine collision
frequency (Schubert 1999; Bloom and Heindel 2002; Pyke, Fornasiero et al. 2003).
Despite its widespread acceptance and use, Abrahamson’s model is not the most
appropriate for flotation purposes. This stems from the fact that a number of significant
assumptions are violated. The greatest assumption being violated is that all colliding particles
have an infinite Stokes number. This is not the case in flotation. When the particle Stokes
number is above 10, the collision prediction by Equation [9] is fairly accurate, but for Stokes
numbers below 10 it will over predict the true number of collisions taking place within the
flotation cell.
A more comprehensive model would be one that would account for the full range of
Stokes numbers and include both the shear and accelerative mechanisms of collision. Several
authors have proposed models for just this scenario (Yuu 1984; Kruis and Kusters 1997). These
models also include an added mass term for particle flows in liquid environments. This is very
important for flotation modeling since all processes occur within a liquid environment.
One problem with the model proposed by Yuu is its applicability to very small particles
(Kruis and Kusters 1997). This is not the case with Kruis and Kusters’ model (Equations [12]-
[14]).
17 |
Virginia Tech | 1E+15
1E+14
1E+13
1E+12
1E+11
0.1 1 10 100 1000
Stokes number
8π
Z = N N d2 w2 +w2 [12]
12 3 1 2 12 accel shear
w2 ⎛ U2 θ U2 θ θθ U2U2 ⎞
shear =0.238b⎜ 1 1 + 2 2 + 1 2 1 2 ⎟ [13]
U2 ⎜U2 C U2 C C C U2 ⎟
f ⎝ f c,1 f c,2 c,1 c,2 f ⎠
⎡ 1+θ+θ ⎤
⎢ 4θθ 1 2 ⎥
w a2 ccel =3(1−b)2 γ ⎢(θ+θ)− 1 2 (1+θ 1)(1+θ 2) ⎥
U2 γ−1⎢ 1 2 (θ+θ) ⎥
f ⎢ 1 2 ⎥ [14]
⎢⎣ ⎥⎦
⎛ 1 1 ⎞
⎜ − ⎟
⎜(1+θ)(1+θ) (1+γθ)(1+γθ)⎟
⎝ 1 2 1 2 ⎠
In Equations [12] - [14] w is the relative velocity between two particles due to either shear or
acceleration, U is the root-mean-squared fluid velocity, U is the root-mean-squared particle
f i
velocity, b is the added mass coefficient, γ is the turbulence constant, and θ is the dimensionless
i
relaxation time (particle relaxation time with respect to the Lagrangian timescale).
By accounting for intermediate Stokes numbers, Equations [12] - [14] bridge the gap
between Equation [7] and Equation [9]. This is shown in Figure 4. Collisions are calculated
with one particle type’s size being steady while the other varies. Abrahamson’s equation over
predicts the true collision rate while Saffman and Turner’s equation under predicts, with a
18
)1-sm3-(
ycneuqerF
noisilloC
Abrahamson
Kruis and Kusters
Saffman and
Turner
Figure 4. Comparison of collision frequency models. Model predictions are
calculated with one colliding particle type’s diameter steady at 100 microns,
with the other particle type’s diameter varying. Both colliding species have the
same density. |
Virginia Tech | difference of approximately 2 orders of magnitude between the two equations. By accounting
for intermediate Stokes numbers, as suggested by Kruis and Kusters, an intermediate collision
frequency can be calculated between the maximum and minimum predictions given by
Abrahamson and Saffman and Turner. Some variations can be noted between Kruis and Kusters
and the other two models due to the inclusion of the added mass term. The dip in Kruis and
Kusters’ prediction, at approximately a Stokes number of 10, is due to the difference in the
relaxation time between one species and another. The dip corresponds to the Stokes number of
the constant particle size species. When the two relaxation times of the colliding particles are
equal, the collisions are minimized. Collisions at this point mostly occur due to the shear
mechanism. When the difference between the relaxation times increases, collisions increase.
This is shown on either side of the dip in Figure 4.
The current problem with this model is that only one particle density can be given.
Flotation collisions, on the other hand, need to account for two different particle densities. This
should be taken into account with the added mass coefficient, but Equations [12] - [14] can not
properly do this. Since these models can not currently account for bubble-particle collisions, the
best model, to date, that can account for different collision densities is Abrahamson’s (Equation
[9]).
To use Equation [9], the particle and bubble root-mean-squared velocity must be known.
This can be determined using Equation [10], with prior knowledge of the fluid rms velocity.
Since this velocity is not known in flotation practice, a more suitable equation must be used.
Liepe and Moeckel (1976) derived an expression for the slip particle rms velocity from previous
researchers experimental data.
2
ε49d79 ⎛ρ −ρ ⎞ 3
U2 =0.4 1 ⎜ p f ⎟ [15]
1 ν13 ⎜ ρ ⎟
⎝ f ⎠
This equation ([15]) has recently been verified using a digital particle image velocimetry (DPIV)
technique (Brady, Telionis et al. 2004). Although this equation is being used in current flotation
modeling for bubble velocities (Pyke, Fornasiero et al. 2003) it was never derived for bubbles
and should not be used without any experimental verification. For bubbles, a more appropriate
velocity is given by Lee et al. (1987) in equation [16].
U2 =C (εd )23 [16]
2 0 2
C is given by Batchelor (1951) as 2.
0
Attachment and Detachment Energies
For flotation modeling, the attachment and detachment processes are calculated as
probabilities, similar in form to the Arrhenius equation.
⎛ E ⎞
P =exp⎜ B ⎟ [17]
A/D E
⎝ ⎠
K
A certain probability is given for attachment (P ) and detachment (P ) dependent on the energies
A D
needed to be overcome, E , and the energies available in the system, E , to overcome that energy
B K
barrier.
The attachment energy barrier, E , comes from surface forces, which are modeled using
1
the Extended DLVO theory (see Figure 5) (Yoon and Mao 1996). There are three main surface
forces found in flotation; the electrostatic, V , dispersion (van der Waals), V , and hydrophobic,
E D
19 |
Virginia Tech | 80
60
40
20
0
-20
-40
-60
-80
0 50 100 150 200
Separation Distance, H (nm)
V , forces. These are additive and combine to produce the total surface force, V . These surface
H T
forces can easily be converted into energies. When two particles (e.g. particle/bubble) approach
each other, they are initially repulsed (positive surface energy). They must overcome this
repulsion to become attached. The repulsion increases until the energy barrier is overcome.
Below that separation distance, also called the critical rupture distance, H , there is a continuous
C
drop in surface energy. Once this occurs, there is nothing to stop the attachment of the particle
and bubble.
The detachment barrier, W , comes from the change in surface energies when the
A
detachment process occurs. The loss or gain in surface area of the solid, liquid, or vapor state
during the detachment process is multiplied by the respective surface tension to obtain the
required energy. The initial state in this process is the combined bubble and particle while the
final stage is the separated bubble and particle. Both the attachment and detachment barriers are
shown in Figure 5.
To overcome these barriers, a certain amount of energy is required. This energy is
provided by the turbulent environment encountered within a flotation cell. The driving force
within the cell is the motion of the impeller. The impeller produces the turbulent environment by
the creation of eddies. The largest eddy is created by the impeller and is roughly equal to the
impeller size. This eddy is where the energy is input into the turbulent environment. According
to the Kolmogorov theory, this energy is then transferred through intermediate scales of
turbulence to the smallest turbulent scale, the Kolmogorov microscale. This is shown in Figure
6.
The wave number (κ=2π/d) equivalent to the largest eddy size (impeller) corresponds to a
kinetic energy equal to the tip-velocity of the impeller squared.
U2 =( R ω)2 [18]
T−large Imp
20
)J(
7101x
V
V
E
V
T
E
1
V
D
H
C
W
V A
H
Figure 5. Surface energy vs. distance of separation between two
particles (i.e. particle-bubble). |
Virginia Tech | Wave number (κ)
The cascade of energy on a log-log scale, as stated by the Kolmogorov theory, is represented by
a straight line with a slope of -5/3. This is true for a pure liquid system, but with the introduction
of bubbles the slope becomes -8/3 (Wang, Lee et al. 1990). With particles the slope is also
decreased (Buurman 1990). It is assumed that with a combination of all three phases the slope
will follow the -8/3 two-phase prediction. In these dispersed phase systems, the energy is
dissipated much more quickly, due to the fact that a portion of the energy is transferred to the
particles and bubbles (Gore and Crowe 1991). Because of this, at each wave number there is less
energy with particles and bubbles than without. The energy spectrum is shown in Figure 6.
For the attachment and detachment processes, a certain range of these eddy sizes will
have an effect on the particles and bubbles and give them their turbulent kinetic energies. For
attachment to occur there must be a certain differential movement between the fluid and particle.
This allows the particles and bubbles to collide. The attachment energy must, then, overcome
the energy barrier, E , during this collision. Eddy sizes that allow this differential movement,
1
correspond to the particle/bubble size through the Kolmogorov microscale. The fluid within this
range has a different relaxation time than the particles and bubbles, as opposed to large eddies,
where bubbles and particles follow their movement. This out of phase motion allows the
particles and bubbles to move independent of the fluid and each other. It is assumed that the
average amount of this energy over the associated wave numbers (κ to κ) directly
P/B K
corresponds to the particle and bubble attachment energy, U2 .
T-A
1
E = (m +m )U2 [19]
k−A 2 1 2 T−A
For the detachment process, bubbles and particles are already combined and do not need
a differential relaxation time as they do in the attachment process. It is assumed that detachment
comes about from the centrifugal motion of large eddies. When bubbles and particles are
subjected to these eddies, they produce differing behavior. Bubbles tend to travel towards the
21
)2U(
ygrene
citenik
tnelubruT
Integral Inertial Dissipative
D-T2U
A-T2U
-5/3 slope
-8/3 slope
κ I κ P/B κ avg κ K
Figure 6. Turbulent kinetic energy spectrum showing
attachment and detachment energies. |
Virginia Tech | center of vortices, while particles travel outward (Chahine 1995; Crowe and Trout 1995). If a
bubble-particle aggregate were subjected to this, it would tend to pull the aggregate in two which
would lead to the detachment of the particle from the bubble. Given this scenario and the fact
that all aggregates are subjected to the largest eddy produced by the impeller, which contains the
largest available energy within the system, leads to the conclusion that the largest eddy provides
the energy for detachment. The turbulent energy corresponding to the largest eddy, U2 , is,
T−large
therefore, equal to the turbulent detachment energy, U2 .
T−D
1
E = (m +m )U2 [20]
k−D 2 1 2 T−D
Summary
There are many different collision frequency models in the literature to date. All of these
models have varying assumptions, with each model having some violation under flotation
conditions. The best model for use in flotation modeling is one proposed by Abrahamson
(1975). It is assumed that most particles in flotation follow the assumptions used in
Abrahamson’s derivation of the collision frequency model, which includes infinite (very high)
Stokes numbers, independent particle velocities, a particle velocity distribution, and isotropic
turbulence.
Newer models that account for the shear and accelerative mechanisms can also account
for the entire range of Stokes numbers. These would be better suited for flotation purposes,
except for the fact that they can not account for two different particle densities. Therefore, a
particle-bubble collision can not be calculated using these models. When these models are
updated to allow the use of greater and lesser than the surrounding medium densities, they will
be able to account for all particles and bubbles, and their effects, in a turbulent liquid
environment within a flotation cell.
Once particles collide, they have a probability of attaching, and once attached, of
detaching. The forms of these probabilities are similar to the Arrhenius equation, with energy
barriers being overcome by kinetic energies.
For the attachment process, the energy barrier is based upon the Extended DLVO theory
and accounts for the electrostatic, dispersion, and hydrophobic surface forces. The energy
needed to overcome this barrier is provided by the turbulent environment. For the attachment
process to take place, some deviation from the fluid flow is required. The scales of turbulence
that allow this are between the Kolmogorov microscale and the particle/bubble scale. It is
assumed that the average wave number between these two scales accounts for the attachment
energy that is needed to overcome the energy barrier. The Kolmogorov theory is used to
calculate this energy. A modification of the theory is needed to account for particle and bubble
effects on turbulence. This modification changes the slope of the energy spectrum from -5/3 to -
8/3.
For the detachment process, the energy barrier is determined from the change in surface
area of the different phases of the particle-bubble aggregate (i.e. solid, liquid, vapor), and the
respective surface tensions of those phases. The largest scale of turbulence within the system,
the impeller, is used to overcome this barrier. No deviation from the fluid is required for the
detachment process. The detachment process follows from the differing effects vortices have on
22 |
Virginia Tech | Developing a Turbulent Flotation Model from First Principles
I. M. Sherrell
Abstract
A new flotation model has been developed for use in a turbulent environment and is the first
proposed to take into account all three surface forces (electrostatic, van der Waals, and
hydrophobic). Previous models that account for turbulence do not take into account all surface
forces while the one model that does take into account all surface forces (Yoon and Mao 1996) is
only applicable in quiescent conditions. Since most flotation occurs in a turbulent environment,
these previous models cannot be applied. The new model includes attachment, detachment, and
froth recovery probabilities. The effects of each individual process are shown. Simulations
using this model show trends seen in industrial flotation systems.
Introduction
Flotation is widely used throughout the mining industry as well as the chemical,
petroleum, and recycling industries. It is the most efficient process, to date, for solid-solid
separation of minerals. It is now more diverse in its application, with uses such as separation of
ink from paper, plastics separation in recycling, soil contamination separation, and separation of
carbon from fly ash. The industry is growing along with knowledge of the process and sub-
processes. With this increased knowledge, a more reliable flotation model has been derived from
first principles. This results in a general flotation model, and allows its use in the mining
industry, regardless of machine type and material being recovered.
Flotation is a three-phase process, which uses a medium of water (liquid) to separate
various particles (solid) by the addition of air bubbles (gas). Hydrophobic particles attach to the
bubbles and rise to the top of the flotation cell where they are extracted while hydrophilic (or less
hydrophobic) particles remain in the slurry. All three phases are found in flotation machines,
which produce a turbulent environment. To enhance the separation, surfactants may be added
that alter surface properties. The combination of a three-phase turbulent environment with a
modified surface chemistry makes modeling of the process very complex.
Surface forces play a crucial role in the attachment and detachment processes between a
particle and bubble. Proper modeling of these forces is vital to having a general flotation model.
The DLVO theory models some of the surface forces seen in flotation. This theory combines the
van der Waals force and the electrostatic force into a total surface force. The main problem with
the DLVO theory is the lack of any hydrophobic force parameter, which is known to be a major
contributor to surface forces between particles and bubbles in a water medium (Yoon and Mao
1996; Yoon 2000). The extended DLVO theory incorporates this third force (hydrophobic) into
the DLVO theory.
The most rigorous flotation model, to date, dealing with all three surface forces
(electrostatic, van der Waals, and hydrophobic) was proposed by Mao and Yoon (1997).
26 |
Virginia Tech | 2
1 ⎡3 4Re0.72⎤⎛ r ⎞ ⎛ E ⎞⎡ ⎛ W +E ⎞⎤
k = S ⎢ + b ⎥⎜ 1 ⎟ exp⎜− 1 ⎟⎢1−exp⎜− A 1 ⎟⎥ [1]
4 b ⎣2 15 ⎦⎝r
2
⎠ ⎝ E
k−A
⎠⎢⎣ ⎝ E
k−D
⎠⎥⎦
The model is based upon first principles in a quiescent environment and agrees well with
experimental data. The problem of this model is its applicability to industrial applications.
Turbulence is encountered in almost all flotation systems, including mineral and coal flotation.
Since this is a quiescent model, the results predicted do not match industrial flotation systems.
This model did provide a key basis for the current model by the use of the extended DLVO
theory and its relationship to the energies of the system.
Model
Due to the turbulent environment experienced inside a flotation cell, particle-bubble
collisions are modeled far differently than in a quiescent environment. For laminar flows, the
collision frequency may be modeled using the volume that the bubble travels through and the
percent solids of the slurry (Mao and Yoon 1997). A collision efficiency would then be applied
that would account for streamline effects. On the other hand, particles and bubbles in turbulent
flows generally do not follow the path of the fluid and are modeled based upon their deviation
from the fluid path. The Stokes number, a ratio of the particle relaxation (response) time to the
smallest fluid relaxation time (Kolmogorov timescale), represents how well a particle follows the
fluid flow and is a way to measure this deviation. If a particle’s relaxation time is equal to or
less than the Kolmogorov timescale then the lag time between the fluid movement and the
particle movement will be zero and the particle will follow the fluid completely. Any particles
with relaxation times above the Kolmogorov timescale will have some departure from the fluid
path.
1
τ ⎛ν⎞ 2
St = i =τ [2]
⎜ ⎟
τ i ⎝ε⎠
η
Particle and bubble relaxation times are given by various authors throughout the literature
(Govan 1989; Ceylan, Altunbas et al. 2001; Bourloutski and Sommerfeld 2002).
There are two mechanisms involved in turbulent collisions. The shear mechanism
accounts for relative motions of particles (fluid, solid, or gas) in a shear field. These collisions
always occur in a turbulent field, even among fluid particles. Collisions between particles with
Stokes numbers less than 1 occur by shear only. The second mechanism, the accelerative
mechanism, accounts for inertial effects due to large and/or heavy particles. Collisions due to
the accelerative mechanism occur above a Stokes number of 1 where there is some lag between
the particle and fluid.
Saffman and Turner (1956) proposed a collision model based upon the shear mechanism,
where the Stokes number is zero, while Abrahamson (1975) proposed a model based entirely on
the accelerative mechanism, where the Stokes number is infinity. There are collision models that
combine the shear and accelerative mechanisms (Williams and Crane 1983; Kruis and Kusters
1997), but nothing to date accounts for bubble-particle collisions in a liquid environment. This is
complicated due to the fact that bubbles are lighter than the surrounding fluid, while particles are
heavier. Since Abrahamson’s model more closely predicts real world collisions when Stokes
numbers are greater than 1 (Sundaram and Collins 1997) combined with the fact that most
particles in flotation are within this range, leads to the current use of Abrahamson’s model.
Abrahamson’s model
27 |
Virginia Tech | ( )
Z =232π12N N d2 U2 +U2 [3]
12 1 2 12 1 2
incorporates the number densities, N, of both the particles (1) and bubbles (2), the average
i
diameter of the collision, d (= r +r ), and the turbulent root-mean-squared velocities of the
12 1 2
particles and bubbles, U2 . The particle turbulent root-mean-squared velocity is given by Liepe
i
and Moeckel (1976) in Equation [4].
2
ε49d79 ⎛ρ−ρ ⎞ 3
U2 =0.4 1 ⎜ 1 3 ⎟ [4]
1 ν13 ρ
⎝ ⎠
3
The bubble turbulent mean-squared velocity is given by Lee et al. (1987) in Equation [5].
U2 =C (εd )23 [5]
2 0 2
Rate Constant
Knowing the collision frequency and the corresponding probability of collection, the
change in particle concentration within the entire flotation cell can be calculated as the number
of collisions between particles and bubbles that occur (Z ) that lead to attachment (P ), once
12 A
attached do not detach (1-P ), and are able to rise within the froth (R ).
D F
dN
1 =−Z P (1−P )R [6]
dt 12 A D F
Comparing equation [6] with a first order rate equation [7], that most researchers believe model
flotation (Kelsall 1961; Arbiter and Harris 1962; Mao and Yoon 1997),
dN
1 =−kN [7]
dt 1
results in an equation for the rate constant that is dependent on hydrodynamics of the flotation
cell as well as surface forces of the particles and bubbles.
Z P (1−P )R
k = 12 A D F [8]
N
1
Reducing the collision frequency to its individual components
Z =βN N [9]
12 1 2
produces the final rate constant equation.
k =βN P (1−P )R [10]
2 A D F
Particle Collection
The probability of collection is dependent on the attachment and detachment processes
and is a combination of their probabilities. These are both influenced by surface properties of
the particles and bubbles as well as hydrodynamics of the system. Surface energies are modeled
based upon the Extended DLVO theory. This incorporates the electrostatic, V , van der Waals
E
(dispersion), V , and hydrophobic, V , surface forces (Rabinovich and Churaev 1979; Shaw
D H
1992; Mao and Yoon 1997). V is a function of K and K , which can be obtained from
H 131 232
experimental results (Rabinovich and Yoon 1994; Yoon and Ravishankar 1994; Yoon and Mao
1996; Yoon and Ravishankar 1996; Yoon, Flinn et al. 1997; Vivek 1998; Pazhianur 1999; Yoon
28 |
Virginia Tech | 80
60
40
20
0
-20
-40
-60
-80
0 50 100 150 200
Separation Distance, H (nm)
and Aksoy 1999). The surface forces are additive and combine to form the total energy of
interaction, V , as shown in Figure 1.
T
For attachment there exists a maximum repulsive (positive) energy, E , that must be
1
overcome. This maximum energy occurs at the critical rupture thickness, H . For separation
c
distances less than H , the free energy continuously drops. Therefore, nothing will prevent the
c
particle and bubble from coming together once H is overcome.
c
The probability of attachment is dependent on the energy barrier that must be overcome
and the kinetic energy of attachment, E .
k-A
⎛ E ⎞
P =exp⎜− 1 ⎟ [11]
A E
⎝ ⎠
k−A
The probability of detachment is dependent on the kinetic energy of detachment, E ,
k-D
and the work of adhesion, W , that must be overcome for detachment to occur (Figure 1).
A
⎛ W ⎞
P =exp⎜− A ⎟ [12]
D E
⎝ ⎠
k−D
The work of adhesion is the energy needed to return the free energy of interaction to a
zero value. It is assumed, that when detachment occurs, the equilibrium edge of the bubble will
be past the critical rupture thickness. This is due to the small critical rupture thickness
(approximately 100 nm) as compared to the deformation that a bubble goes through (tens to
hundreds of microns) (Schimann 2004). Since the bubble edge will already be past the critical
rupture thickness, E does not need to be overcome and therefore is not used in detachment. The
1
work of adhesion is the energy required to take apart a bubble-particle aggregate into a separate
bubble and particle. This energy is obtained by surface tensions (gas-solid, gas-liquid, solid-
liquid) and their respective areas. A well known model used by Mao (1997)
29
)J(
7101x
V
V
E
V
T
E
1
V
D
H
C
W
V A
H
Figure 1. Surface energy vs. distance of separation between two
particles (i.e. particle-bubble). |
Virginia Tech | W
=γπr2(1−cosθ)2
[13]
A lv 1
assumes that the bubble surface is completely flat. Since the bubble and particle sizes are within
two orders of magnitude of each other, a more accurate way to calculate W would be to assume
A
a spherical bubble attached to a spherical particle. With simple geometry, this can be easily
worked out and the current model uses this approach.
It is known that the contact angle for spherical particles is smaller than flat plate
measurements (Preuss and Butt 1998). There can be up to a 10 degree contact angle reduction
for colloidal sized particles. The contact angle used in the work of adhesion equation is usually
obtained by measurements upon flat plates. Since it is assumed that this reduction is a function
of particle size and that particles in flotation are much larger than colloidal sized particles, a
constant 5 degree reduction is included in this model.
Energies
The main source of energy within a flotation cell comes from the motion of the impeller.
The impeller motion produces the turbulent environment within the flotation cell. The impeller
itself creates the largest eddies within the cell, roughly equal to the impeller size. The energy
input into the cell (through the impeller) is transferred from the largest turbulent scale
(corresponding to the impeller size) to the smallest turbulent scale (Kolmogorov microscale). A
certain range of these eddy sizes will have an effect on the particles and give them their turbulent
kinetic energies.
The wave number (κ=2π/d) equivalent to the largest eddy size (impeller) gives a kinetic
energy equal to the tip-velocity of the impeller squared.
U2 =( R ω)2 [14]
T−large Imp
The kinetic energy then cascades, at a slope of -8/3 on a log-log scale, to the Kolmogorov
microscale. In a pure liquid system the slope is theoretically -5/3, but with the introduction of
bubbles, the slope is reduced to -8/3 (Wang, Lee et al. 1990). Particles are also found to reduce
the theoretically predicted single-phase slope (Buurman 1990). In these dispersed phase
systems, the energy is dissipated much more quickly, due to the fact that a portion of the energy
is transferred to the particles and bubbles (Gore and Crowe 1991). Because of this, at each wave
number there is less energy with particles and bubbles than without. It is assumed that with a
combination of all three phases the slope will follow the -8/3 two-phase prediction. The energy
spectrum is shown in Figure 2.
Eddies corresponding to the particle/bubble size through the Kolmogorov microscale will
allow the particle/bubble to deviate from the fluid flow. The fluid within this range will have a
different relaxation time than the particles and bubbles, as opposed to large eddies, where
bubbles and particles follow their movement. This out of phase motion allows the particles and
bubbles to move independent of each other and will produce collisions. It is assumed that the
average amount of this energy (U2 ) over the associated wave numbers (κ to κ) directly
T-A P/B K
corresponds to the particle and bubble attachment energy.
1
E = (m +m )U2 [15]
k−A 2 1 2 T−A
Large eddies, on the other hand, provide the energy for detachment. For detachment,
bubbles and particles are already combined and, therefore, do not need a corresponding
relaxation time as they do in the attachment process. Since all aggregates are subjected to large
eddies, and these eddies contain the largest energies within the system, they provide the greatest
30 |
Virginia Tech | Wave number (κ)
energy for detachment. Detachment follows from the centrifugal motion of these eddies, in
which bubbles travel in towards the center of vortices and particles travel outward (Chahine
1995; Crowe and Trout 1995).
1
E = (m +m )U2 [16]
k−D 2 1 2 T−D
The turbulent energy corresponding to the largest eddy, U2 , is equal to the turbulent
T−large
detachment energy, U2 .
T−D
Froth Recovery
Froth behavior can be very complex. The coalescence of bubbles and the subsequent loss
of surface area, and the drainage of liquid between bubbles, to name a few, can affect the
recovery of particles attached to bubble surfaces and particles entrained within the froth liquid.
Froth recovery, R, is the percentage of particles which enter the froth, subsequently pass through
f
the froth and are collected. All particles not recovered from the froth are returned to the slurry or
never truly enter the froth phase. A simple approach to modeling this is to consider only the
particles attached to the bubble surface. The only factor affecting the bubble surface would then
be the coalescence of bubbles and loss of surface area. Once bubbles coalesce, a portion of their
carrying capacity, for that volume of air, is lost. Once that carrying capacity is lost, it is assumed
that those particles that were attached will drain back into the slurry.
This loss of surface area must be calculated with respect to the volume flow rate of air
through the cell, which is assumed to be uniform throughout the slurry and froth section. A way
to account for the loss of surface area is to calculate the surface area rates at both the initial (S )
0
and final (S) stages of the froth. The fraction of surface area that is still useful for particle
f
recovery would then be the ratio of these two surface area rates (Equation [17]).
31
)2U(
ygrene
citenik
tnelubruT
κ κ κ κ
I P/B avg K
2U
2U
egral-T
A-T
Figure 2. Turbulent kinetic energy spectrum |
Virginia Tech | S ⎛3V ⎞ ⎛3V ⎞ r
R = f =⎜ g ⎟ ⎜ g ⎟= 2−0 [17]
F S ⎜r ⎟ r r
0 ⎝ 2−f ⎠ ⎝ 2−0 ⎠ 2−f
V is the volumetric air flow rate divided by the cross-sectional area of the flotation cell, and is
g
usually referred to as the superficial air flow rate. The froth recovery then becomes a ratio of the
initial to final bubble sizes. This recovery assumes that the same percentage of particles adhere
to the bubble surface at the initial and final stages of the froth. This recovery then becomes the
maximum recovery allowed for a given froth.
Three-phase froths are highly complex. Aside from liquid and gas effects; particle size,
shape, smoothness, hydrophobicity, contact angle, and concentration can affect froth behavior
(Harris 1982; Knapp 1990; Johansson and Pugh 1992; Aveyard, Binks et al. 1999). Froth
recovery is also thought to be a function of these variables, with particle size having a large
effect. An empirical model proposed by Gorain et al. (1998) is thought to give the best results
for froth recovery, to date (Equation [18]).
( )
R =exp −ατ [18]
F f
This model formulates the froth recovery as a function of the froth retention time, τ, and a
f
parameter, α, that incorporates both physical and chemical properties of the froth (Mathe, Harris
et al. 1998). Froth retention time is usually defined as the ratio of the froth height to superficial
air flow rate, V . α is an empirical parameter that must be determined by experiments, for each
g
system. α usually ranges, in industrial flotation cells, between 0.1 and 0.5 (Gorain, Harris et al.
1998).
Given the fact that there is a maximum recovery that can not be overcome, a modification
of Gorain’s model is proposed. All recoveries calculated using Equation [18] must be scaled
using the maximum froth recovery. This is shown in Equation [19].
( )
R = R exp −ατ [19]
F max f
Assuming that the maximum froth recovery is given by Equation [17], and substituting this into
Equation [19], gives the following formula for the froth recovery within a flotation cell.
r
( )
R = 2−0 exp −ατ [20]
F r f
2−f
It is also known that particles within a flotation froth have varying retention times, due to,
in large part, particle size (Mathe, Harris et al. 1998). Average retention time within a froth is
given by the following formula (Equation [21]) where h is the froth height (Gorain, Harris et al.
f
1998).
h
τ = f [21]
avg−f V
g
Superficial airflow rate is constant throughout the froth, as long as there is no diversion of the
froth (secondary product output such as lower froth diversion) and steady state has been reached.
Knowing that small particles within a liquid environment follow the flow, small particles are
thought to have a froth retention time equal to the average retention time within the froth. It is
also known that larger particles have a more difficult time traveling upward in the froth
(Bikerman 1973). It is assumed that particles take longer to travel through the froth, due to their
density and/or size. Knowing this, a froth retention time model is proposed
32 |
Virginia Tech | h
τ ( d ) = f P [22]
f p V fr
g
that is a function of particle size, d , and takes this into account by the addition of a froth particle
p
effect, P .
fr
The particle effect on the retention time is given the functional form
⎛ d ⎞
P =exp⎜B p ⎟ [23]
fr ⎜ d ⎟
⎝ p−n ⎠
where B is a constant and d is the particle diameter whose bubble-particle aggregate mass is
p-n
equal to the fluid mass as shown in Equation [24].
1
⎛ρ −ρ ⎞ 3
d =⎜ 3 2 ⎟ d [24]
p−n ρ−ρ 2
⎝ ⎠
1 3
d takes into account the density of the particle as well as the size of the attached bubble.
p-n
A graphical representation of P and its effect on froth recovery can be seen in Figure 3.
fr
The neutrally buoyant particle, d , affecting Equation [23] is used to take into account the
p-n
buoyancy of the bubble with an attached particle. Smaller or less dense particles allow the
bubble to travel upward within the froth more quickly. The smaller a particle is compared to the
neutrally buoyant particle, the closer the particle effect is to 1. Therefore, the closer the particle
is to following the fluid flow the closer the particle retention time is to the average retention time
within the froth. The larger or more dense the particle is compared to the neutrally buoyant
particle, the greater the particle effect becomes. Large particles will take longer to travel through
the froth, if at all. Particle effect, using this functional form, must always be greater than 1 and
therefore, particle froth retention time must always be equal to or greater than average froth
retention time. B is thought to be a function of frother type and cell-dynamics and is found
empirically for each system. The effect of B can be seen in Figure 3.
100
10
1
0 200 400 600 800
d (µm)
p
33
)P(
tceffE
elcitraP
htorF
rf
33.5%
33.0%
32.5%
32.0%
31.5%
31.0%
30.5%
30.0%
29.5%
29.0%
28.5%
yrevoceR
htorF
B = 1
B = 3
d = 1 mm
p
h f = 0.05 m B = 5
V = 1 m/s
g
α = 0.03
r = 0.5 mm 2-0
r = 1.5 mm
2-f
B = 3
B = 1
Figure 3. Particle size effect on froth particle effect, P , (―) and froth
fr
recovery (- -) |
Virginia Tech | 0.2
0.175
0.15
0.125
0.1
0.075
0.05
0.025
0
10 100 1000
D (µm)
p
Results
The effect of the individual components on the theoretical rate constant can be
determined using Equation [10]. The importance of each individual process on the overall
flotation process can be shown. These processes include the collision frequency, which is a
function of the collision kernel (Figure 4), the froth recovery (Figure 5) and the probability of
attachment and detachment (Figure 6).
Figure 4 through Figure 6 reveal a great deal of information about the entire flotation
process. The overall magnitude of the rate constant is determined by the collision kernel, as well
as the number density of the bubbles as shown in Figure 4. As the particle size increases, the
collision kernel exponentially increases. This would result in greater and greater rate constants
as particle size increases. This is not seen in flotation due to the combined effect of froth
recovery and detachment. As shown in Figure 5, as particle size increases, the froth recovery
decreases. Depending on the constant B being used (Equation [23]), the effect of particle size
can be increased or reduced. Therefore, froth is a limiting factor to large particle flotation.
Detachment also plays a roll in lowering the rate constant as shown in Figure 6. As particle size
increases, detachment increases, until eventually no particle can stay attached to a bubble. The
drop in probability of detachment corresponding to the maximum rate constant particle size is
due to the combined effect of work of adhesion and turbulent detachment energy. Both increase
as particle size increases. Because the relationship between the work of adhesion and turbulent
detachment energy to particle size is not linear, initially work of adhesion increases much slower
than turbulent detachment energy, until a certain particle size is reached. After this particle size
is reached turbulent detachment energy increases more rapidly than work of adhesion which
34
)1-nim(
k
1.5E-06
1.3E-06
1.0E-06
7.5E-07
5.0E-07
2.5E-07
0.0E+00
lenreK
noisilloC
Rate Constant
Collision Kernel
Figure 4. Effect of the collision kernel (―) on the rate constant
(- -). Same conditions shown in Figure 9, contact angle 60
degrees. |
Virginia Tech | 0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
10 100 1000
d (µm)
p
angle, and specific energy input.
Figure 7 and Figure 8 show the effect of physical variables on the rate constant. The
outcome from varying the bubble diameter, along with the particle diameter, can be found in
Figure 7. As the bubble diameter decreases, the rate constant increases. In addition, when the
bubble diameter is decreased, the particle diameter, where the maximum rate constant occurs, is
decreased. This affirms the phenomenon found in column flotation, where smaller bubbles are
more beneficial to small particle flotation. The first phenomenon (increased small particle rate
constant) is mainly due to the fact that bigger bubbles have lower specific surface area (surface
area per unit volume of air flow). This results in a lower amount of sites for potential particles to
adhere to. The smaller the bubble, the greater the specific surface area, and therefore the greater
potential for particles to become attached. For the same volume of air, smaller bubbles have the
potential to float more particles.
The second phenomenon (shift where maximum rate constant occurs) is due, mainly, to
kinetic energies, as well as surface forces. As a bubble becomes smaller, its kinetic energy
decreases. The energy difference between smaller and larger bubbles is more greatly felt, in
interactions, by smaller particles, because of the small particles’ already low kinetic energies.
Surface forces are also affected by the smaller bubbles but more so in the energy barrier than in
the work of adhesion. The energy barrier will decrease, like kinetic energy, when the bubble size
is reduced. Since both main variables (energy barrier and attachment kinetic energy) are reduced
for the probability of attachment, it is not thought that the attachment process is affecting this
movement. Probability of detachment, on the other hand, has only one variable greatly affected
by bubble size. It is thought that the detachment process of smaller bubbles will lower the
maximum rate constant particle size.
36
)1-nim(
k
d b= 500 µm θ = 45°
γlv = 60.0 mN/m
εsp = 2 kW/m3
ζp = -20 mV
ζb = -30 mV
κ-1 = 96.0 nm
ρp = 2.475 g/cm3
d b= 750 µm
d b= 1000 µm
Figure 7. Effect of bubble size on the flotation rate constant.
Smaller bubbles are more beneficial to small particle flotation while
large particles may benefit more from bigger bubbles. This is due to
a combination of specific surface area, kinetic energies, and surface
forces. |
Virginia Tech | 0.06
0.05
0.04
0.03
0.02
0.01
0
10 100 1000
d (µm)
p
It should also be noted that as bubble size decreases, large particles become more
difficult to float. This is due to the froth recovery factor, R . The neutral density particle size,
F
d , decreases with decreasing bubble size, which makes recovery of large particles more
p-n
difficult. The smaller the bubble size becomes in the froth, the greater the difficulty large
particles have in traveling upward within the froth.
The outcome from varying the energy input, in the form of impeller speed, along with the
particle diameter, is shown in Figure 8. Although not much difference is seen in the particle size
where the maximum rate constant occurs, there is a meaningful difference in the magnitude of
the rate constant. As the specific energy input increases, the rate constant decreases. At first,
this seems counter-intuitive, and the opposite effect is seen in industrial machines. It was
thought that as the energy input into the system increases the rate constant would also increase.
This is true up to a point after which the potential for detachment increases substantially more
than the potential for attachment. Lower energy input values, which reverse the effect on the rate
constant, are not shown, since most industrial flotation machines are run with much higher
impeller speeds. Actual industrial flotation machines see an opposite effect because of the
production of bubbles. As the energy input into a system is increased, the bubble size is
decreased. The reduced bubble size is what causes the increase in the rate constant. There is no
relationship between bubble size and energy input in the present flotation model. The bubble
size is constant throughout this simulation. When the bubble production model given by
Johansen et al. (1997) is input into the current model, the opposite effect is shown. Higher
energy inputs produce higher rate constants because of smaller bubbles. Johansen’s model is not
used to calculate flotation rate constants due it’s applicability to flotation systems. It was derived
for bubble formation in a liquid metal environment. It was input here only to state that there can
37
)1-nim(
k
d = 1 mm
b
θ = 45°
1100 rpm γlv = 60.0 mN/m
ζp = -20 mV
ζb = -30 mV
κ-1 = 96.0 nm
ρp = 2.475 g/cm3
1500 rpm
1900 rpm
Figure 8. Effect of energy input, in the form of impeller rpm, on
the flotation rate constant. Higher energy inputs decrease rate
constant due to higher probabilities of detachment from higher
kinetic energies of detachment. Bubble size changes due to
increased impeller speeds are not taken into account. |
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