# Analytical Solution for 2D Inter-Well Porous Flow in a Rectangular Reservoir
## 4 Concluding Remarks  
An analytical solution for inter-well porous flow in a rectangular reservoir is proposed in this paper. The applications of the solution to nine cases of enhanced geothermal systems are demonstrated as examples, and the convergence properties of the analytical solution are discussed accordingly. It should be noted that the nine ideal cases investigated in Section 3 are chosen just as a demonstration. In fact, the analytical solution proposed in this paper has sufficient flexibility to deal with different aspect ratios of the rectangular reservoir and different distributions of isolated sources/sinks, as long as the condition of overall mass conservation for steady porous flow, Equation (6), is satisfied. In principle, the analytical solution proposed in this paper can be readily applied to other scientific and engineering fields, as long as the involved problem is governed by 2D Laplace equation in a rectangular domain, with uniform Dirichlet or homogeneous Neumann boundary conditions, and subject to isolated point sources/sinks.  
This work may be furthered in the following directions. First, the method of mirror images may be employed to derive analytical solutions for the 2D Laplace equation in other polygon regions. In principle, the procedure for deriving such solutions is essentially the same as that for a rectangular domain, but the mirror image distributions will follow more complicated patterns for polygons with non-orthogonal boundaries. Second, the methodology adopted in this work may be extended  
Figure 6: Convergence processes of inter-well pressure difference for the nine cases, with Euler transform being invoked for Cases 1, 2, 3, 5, and 6.  
straightforward to derive analytical solutions for problems governed by 3D Laplace equation. Finally, the methodology of superposition of analytical functions may be applied to solve problems with regularly arranged sources/sinks in unbounded domains. For example, Case 9 of this paper represents an infinite well array with staggered injection and production wells arranged in orthogonal straight lines. If we replace the locations of the mirror images in Equation (15) by those of true sources/sinks distributed in unbound domains, the requirement of orthogonal distribution of the sources/sinks can be eliminated as a consequence of that the symmetric boundary conditions no longer need to be satisfied. Then, modified forms of Equation (15) might be applied to solve similar problems, such as the lattice sums problems [22; 23] and the Helmholtz equation problems with vanishing index of refraction [24; 25].  
We thank the anonymous reviewers for insightful and constructive reviews. This work was supported by the National Natural Science Foundation of China (Grant Numbers 41374090 and 41674086) and the Hundred Talents Program of Chinese Academy of Sciences.  
[PERSON] and [PERSON] jointly conceived and designed the research; [PERSON] performed mathematical derivation and calculation under the supervision of [PERSON]; [PERSON] wrote the paper.  
The authors declare no conflict of interest. The funding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results.