Analytical solutions from integral transforms for transient fluid flow in naturally fractured porous media with and without boundary flux

TL;DR

This paper develops analytical solutions for transient fluid flow in fractured porous media using integral transforms, addressing inhomogeneous boundary conditions and providing insights into flow characteristics and pressure behavior.

Contribution

It introduces a method combining finite Hankel and Laplace transforms to solve inhomogeneous boundary problems in fractured reservoirs, extending previous homogeneous solutions.

Findings

Closed-form long-time solutions for specific boundary conditions.

Inverse transform expressions are incomplete for Neumann-Neumann conditions.

Flow behavior in reservoirs with influx recharge resembles double-porosity reservoirs.

Abstract

A kind of problems of radially symmetric transient fluid flow in a medium with a geometry similar to a hollow-disk can be addressed using the finite Hankel transform. However, the inverse Hankel transform [G. Cinelli, Int. J. Engng. Sci., 3, 539 (1965)] works well only for homogeneous boundary conditions. We use the finite Hankel transform, together with the Laplace transform, to solve partial dfferential equations with inhomogeneous boundary conditions. With this aim, we propose a method to obtain an analytical solution of the problem of fluid flow in a finite naturally fractured reservoir with inner and outer boundaries having constant and time-dependent conditions, respectively. We assume that the reservoir has a producing well, with either constant terminal pressure or constant terminal rate, while the outer boundary has either an influx recharge or constant pressure. Using these case studies, we show that a part of the inverse transformation given by Cinelli can be expressed as closed formulas for long time solutions, at the same time that make it possible to capture the inhomogeneous condition and speed up the convergence of solutions. For Neumann-Neumann conditions, we show that the inverse expression of Cinelli is incomplete. In addition, an analysis of the flow characteristic curves in a reservoir with influx recharge is presented. The drawdown pressure curves are used to elucidate the statement that the pressure drop of a single-porosity reservoir with influx recharge resembles the flow behavior in a double-porosity closed reservoir, establishing a criteria to distinguish between both.