The numerical solution gives a lower impedance than either of the analytical solutions, which are similar to each other, differing by only 13%. At early and late times when the fluid properties are in agreement, the analytical solutions calculate reservoir impedance roughly twice the numerical solution. found that reservoir impedance increases with the number of nodes. Thus, the numerical solution in Fig. , with its relatively few nodes, likely provides a lower bound for the reservoir impedance. It is possible that the early-and late-time numerical solution would converge to the analytical solution with finer and finer discretization.