The solutions of Sec. 2.1 are summarized in Table using the variables 𝐴 𝑐 , 𝐶 1 , 𝐶 2 , and 𝑥 0 or 𝑟 0 from Equation . The final column shows the value of 𝐶 1 ln 𝐶 2 , which is proportional to the impedance, with an assumed value of 𝐿/𝐷 = 1000. Notably, the differences in impedance are small amongst all analytical solutions (the range for 𝐶 1 ln 𝐶 2 is 1.8 to 2.2), which indicates that the ID does not have a significantly different impedance than the TDF. It is also notable that the TDF hybrid solution has a value between the TDF radial solution and the TDF cartesian solution. We used the TDF hybrid solution moving forward because it honors the radial flow near the well and the linear flow near 𝑥 = 𝐿/2, as discussed in Sec. 2.1.4. We use the "ID, potential" solution for the ID cases because it includes all the swept area. We compare the impedance as a function of time for the numerical and two analytical solutions in Fig. . The analytical solutions represent the most-accurate TDF and ID solutions (i.e., the "ID, streamlines" and "TDF, hybrid" entries from Table ). Fluid properties are pressure-and temperaturedependent, which is accounted for internally in TOUGH2. For the analytical solution, the temperature from Equation and the hydrostatic pressure are inputs to the CoolProp python library, which calculates the fluid properties. In all three solutions, the impedance increases with time due to the increase in water's kinematic viscosity as the reservoir cools, although the shape of the impedance curves differ because of to the way the viscosity is calculated. In the analytical solution, the kinematic viscosity varies both spatially and temporally as the reservoir cools. Since the region near the injection well starts cooling almost immediately, the kinematic viscosity near the injection well also increases at early times. The impedance increases steadily throughout time as the cooled region grows, and the impedance doubles by the tenth year of operation. In contrast, the analytical solution uses the average reservoir temperature and pressure to calculate the average kinematic viscosity of the reservoir, which does not account for spatial variations in fluid properties. It takes approximately ten years for the average temperature to change by 10% (see Sec 3.4), and therefore the average kinematic viscosity and the average impedance only increase by approximately 10% after ten years. However, as the reservoir continues to cool, the impedance continues to increase, doubling after approximately 200 years.