# Analytical Solution for 2D Inter-Well Porous Flow in a Rectangular Reservoir
## 1 Introduction  
The Laplace equation is a very important equation commonly occurring in many branches of physics as well as in other fields of science and engineering, notably in studies of electromagnetic phenomena, hydrodynamics, heat flow, and gravitation [1; 2]. In this paper, we focus on analytically solving inter-well flows through porous media in a rectangular reservoir. Such problems are governed by the Laplace equation and frequently encountered in the production of groundwater, oil, and geothermal energy [3; 4]. Although geological formations are inevitably involved in 3D flows through heterogeneous [5] and anisotropic [6] reservoirs, we consider the inter-well flow in this paper as a 2D homogenous and isotropic porous flow, with the permeability of the reservoir being treated as a constant scalar, instead of a more general form of tensor with spatial variation, in order to derive an analytical solution. The significance of such an analytical solution lies in that it can clearly reveal the fundamental functional dependence of inter-well flow on its controlling factors (e.g., driving pressure, well distribution, as well as reservoir geometry and properties) and provide first order approximation to many practical engineering problems [4]. Furthermore, an analytical solution has its unique advantages over a numerical solution. First, an analytical solution does not require discretizing the computational domain, and the solution is thus free of discretization error. Second, the computational cost for evaluating an analytical solution ismuch lower than obtaining a numerical solution. For inter-well flows, as the size of wells are normally much smaller than typical inter-well distances, very fine meshes are required for numerically capturing the sharp variations of physical quantities near the well regions, which results in huge computations in numerical modeling. Third, an analytical solution provides an ideal tool for quantifying the accuracy of numerical modeling. It is a common practice in numerical modeling studies to verify the accuracy of the employed numerical methods and computational meshes by numerically solving some benchmark problems and comparing the obtained numerical solutions directly with available analytical solutions. With the analytical solution proposed in this paper, inter-well porous flow problems can be established as benchmark problems for verifying numerical simulations of porous flows, including 3D heterogeneous and anisotropic porous flows in industrial applications. Numerical modeling provides a powerful approach for simulating complicated engineering problems, but the accuracy of numerical modeling results is difficult to quantify, especially for 3D modeling results (partly due to the fact that 3D simulation normally involves huge computations that almost exhausted the computer resources and thus cannot afford full examination for the modeling accuracy by carrying on standard mesh and time step independence tests). The Richardson extrapolation method [7] is frequently invoked to assist accuracy assessment, but the required condition of systematically refining meshes often cannot be satisfied and may lead to underestimating the modeling errors. In such circumstances, comparing with a relevant 2D benchmark analytical solution will be a useful and practical test for estimating the 3D modeling error. For example, the error associated with a 3D modeling of a depth-dependent, heterogeneous, and anisotropic problem can be indirectly estimated from the error associated with a 3D modeling of a depth-independent, homogenous and isotropic but otherwise similar problem solved with comparable mesh and time step, while the error associated with the 3D modeling of the depth-independent, homogenous and isotropic problem can be well quantified through a comparison with an available 2D analytical solution.