# Analytical Solution for 2D Inter-Well Porous Flow in a Rectangular Reservoir
## 2 Analytical Solution
The analytical results shown in Figure 4 can be used to quantify modeling errors associated with numerical simulation results. For example, the relative difference for Case 9 between the analytical solution shown in this figure and a finite element solution based on 1.2 million degrees of freedom [19] is less than \(10^{-9}\). \[u=-\frac{k}{\mu}\frac{\partial p}{\partial x}=0,\text{ at }x=\pm a, \tag{3}\]0 Predicted pore pressure contours (blue lines) and streamlines (red lines) for the nine well layout cases of enhanced geothermal systems shown in \[u=-\frac{k}{\mu}\frac{\partial p}{\partial x}=0,\text{ at }x=\pm a, \tag{3}\]1  
The convergence properties of the inter-well pressure difference predicted by the series solution along with increasing \(M\) and \(N\) values are quantitatively investigated for the nine well layout cases. It turns out that the truncated series solutions for Cases 4, 7, 8, and 9 converge rapidly and monotonically, while those for Cases 1, 2, 3, 5, and 6 are associated with alternating series and converge slowly. This result suggests that the convergence behavior of the analytical solution is closely related to the symmetric nature of the well layouts. As is clear from Figure 2, the wells for the four cases with rapid and monotonic convergence are all distributed symmetrically about both the \(x\) and \(y\) axes, while the wells for the other five cases are all distributed asymmetrically about the \(x\) axis. By invoking Euler transform [20,21]  
\[\sum_{n=1}^{N}\left(-1\right)^{n-1}a_{n}=\sum_{n=0}^{N}\frac{1}{2^{n+1}}\sum_{ m=0}^{n}\left(-1\right)^{m}\left( \begin{array}{c}n\\ m\end{array} \right)a_{m+1}, \tag{23}\]  
to the alternating series solutions for Cases 1, 2, 3, 5, and 6, the convergence processes for all the five cases are accelerated dramatically, as evident in Figure 5, indicating that the adoption of Euler transform significantly accelerates the convergence of alternating series cases associated with asymmetric well distribution. However, the five cases exhibit two different trends: the convergent rates for Cases 2, 3, and 6 constantly increase with \(M\) and \(N\) values, as shown in Figure 5a, while those for Cases 1 and 5 only increase with \(M\) and \(N\) values for \(M=N<20\), beyond which the convergent rates are nearly constant, as shown in Figure 5b. Such trends can also be attributed to the symmetric properties of the well layouts, as the wells for Case 2 are strictly anti-symmetric about the \(x=-y\) diagonal and those for Cases 3 and 6 are strictly anti-symmetric about the \(x\) axis, while the wells for Cases 1 and 5 are only roughly anti-symmetric about the \(x=-y\) diagonal and the \(x\) axis, respectively (Figure 2). Figure 4: Predicted variations of pore pressure relative to the pressure on the production well wall versus the nondimensional \(X\) (distance from the injection well center scaled by the corresponding inter-well distance) for the nine cases. The convergence processes of inter-well pressure difference for the nine cases are compared in Figure 6. Although Cases 4, 7, 8, and 9 all converge with nearly constant rates, the convergent rates of Cases 7, 8, and 9 are much higher than Case 4. This difference can be well explained by the additional symmetric property of the well layouts associated with Cases 7, 8, and 9: the wells for these three  
Figure 5: Convergence processes of inter-well pressure difference predicted by the monotonic series invoking Euler transform, compared with those predicted by the original alternating series for (**a**) Cases 2, 3, and 6 and (**b**) Cases 1 and 5.  
cases are also symmetric about the \(x=y\) and \(x=-y\) diagonals. Moreover, it is interesting to observe that the convergent curves of Cases 1 and 5 for \(M=N>20\) approach to a nearly constant slope that is comparable to that of Case 4. These results are clear evidence for the dependence of the convergent rate of the analytical solution upon the symmetric level of the well distribution inside the reservoir.