New solution to pressure transient equation in a two-layer reservoir with crossflow

A two-layer reservoir with crossflow was erroneously taken as a dual-porosity-dual-permeability naturally fractured reservoir in the literature, and the solutions obtained from dual-porosity-dual-permeability model cannot account for the characteristics of two-layer reservoir. This paper presents a new mathematical model and an analytical solution to the pressure transient equation of a uniform-flux fully penetrating vertical well in a two-layer reservoir with crossflow. The pressure of a vertical well producing a two-layer reservoir with a constant rate is examined. The pressure response is observed in three stages: first, it is identical to the commingled system, because the crossflow is so weak at the production beginning that we can ignore it. Second stage is transition stage and the third stage, which is same as the homogeneous system. What's more important, we propose a parameter to describe the crossflow in a two-layer reservoir quantitatively. It is verified that the crossflow coefficient is not constant, but a function of the time and distance away from the wellbore. These solutions are obtained by using Laplace transform, double Fourier transform and Green's functions method with numerical approximations. These solutions give results accurate enough for practical applications and allow to estimate the effects of formation properties on pressure behavior both at large distance from the wells and at the wells. When all the corresponding parameters are identical in each layer, the proposed solution for a two-layer reservoir reduces to the solution for a single layer reservoir. The solution procedure proposed in this paper is a fast tool to evaluate a vertical well performance in a two-layer reservoir.