Analytical Solutions for Productivity Evaluation of Multifractured Wells in Multilayered and Bounded Reservoirs

Abstract

Abstract We propose a semi-analytical model to represent the transient and pseudo-steady state behavior of hydraulic fractures in multilayered reservoirs where cross-flow is permitted between layers. Fractures can be horizontal or vertical. We only consider vertical fractures which are parallel to x or y axes. Fractures are fully or partially penetrating in the formation and can extend across several layers. We consider single or multiple fractures where influence between each other is modeled. The reservoir is bounded by plane surface perpendicular to the main axes, and we can impose on these limits either no-flow, constant pressure or mixed boundary conditions. The solutions are obtained from plane (zero thickness) source solutions developed in the Laplace-Fourier domain. Basquet et al.1 already used this method for the determination of point source solutions, the reasoning is still based on the « quadripole » method to account for layer cross-flow and on the Fourier transform to describe boundary effects. Vertical line source solutions are developed and are then integrated over the horizontal fracture extension to obtain the plane source solution under the assumption of uniform flux solution. The infinite conductivity solution is obtained by splitting the fractures into basic slabs with uniform flux and solving a linear system. The solution is validated against existing analytical solutions (using concept of instantaneous sources and Green's functions) modeling vertical or horizontal fractures in single layer reservoir. To check that the cross-flow between layers is well represented, we compared our vertical line source transient solution versus well known analytical solutions in multilayered reservoirs.