Analytical solution and mechanisms of fluid production from hydraulically fractured wells with finite fracture conductivity

Abstract

A comprehensive study of the physical mechanisms of fluid production from a well intersected by a narrow elliptically shaped vertical fracture with finite fracture conductivity is carried out using a newly obtained analytical solution. The flow pattern, flux density distribution along the fracture surface, and fluid production rate are analyzed systematically with respect to finite fracture conductivity. The simplicity of the new analytical solution reveals many physical insights not attainable from existing analytical or numerical solutions. It is shown that the nearly singular pressure gradient developed at the fracture tip induces the reservoir flow to converge to and focus at the tip region, promoting flux density along the entire fracture surface and enhancing the production rate. It is established that the flow in the reservoir is a superposition of two basic flows, a confocal elliptical flow responsible for the fluid production, and a redistributive nonproducing flow induced by a finite fracture conductivity that draws fluids out of the fracture from the part near the tip and redeposits them back to the part of the fracture close to the wellbore. An explicit analytical formula for the fluid production rate is also derived that exhibits a simple dependency on the dimensionless fracture conductivity.

Fluid extraction from porous media by a slender permeable prolate-spheroid

“…Recently, we have performed an analysis of the Darcy flow pattern in the reservoir and the flux density distribution along the fracture-reservoir surface, using a newly obtained analytical solution for finite fracture conductivities [2,3]. It is found that when the well pressure is lowered, the induced-flow in the reservoir develops a converging pattern that focuses the fluid to the fracture tip, creating a nearly singular flux density around the fracture tip.…”

Extended finite element modeling of production from a reservoir embedded with an arbitrary fracture network

“…According to some simple exact solutions, 24,25 in fractured porous media, pore fluid flow towards a fracture develops a pressure-gradient singularity at the fracture tip. Because the fracture possesses sharp ends, the potential flows around a thin plate develop a velocity singularity at the ends of the plate.…”

“…25 The used formulations in the exact solution can be found in Appendix. 25 The used formulations in the exact solution can be found in Appendix.…”

“…6,7 Compared with the "double-porosity" model, the discrete fracture model based on FEM or FVM can discretize a geometrically complex reservoir with an optimal use of mesh points and exactly represent the flow in the 2 media (matrix and fractures). [22][23][24][25] Thus, the fractures in the reservoir should be regarded as weak discontinuities which possess kink shape along the fracture interfaces and pressure-gradient singularity at the fracture tip. 8,9 Another alternative is the use of the eXtended Finite Element Method (XFEM), which was initially developed to simulate crack growth in an elastic medium 10,11 and then applied to study thermoelasticity, 12 wave propagation in composites, 13 and dislocations, 14,15 to mention just a few.…”

Productivity-Index Optimization for Hydraulically Fractured Vertical Wells in a Circular Reservoir: A Comparative Study With Analytical Solutions

“…Mathematically, the solution to the boundary-value problem for a domain slightly distorted from a circular area can be rigorously pursued by the method of domain perturbation in which the solution is expanded in a Tayler-series expansion in terms of the small geometrical distortion (Joseph 1973). In this spirit, the solutions obtained by Prats and coworkers and Chen and coworkers with shape approximation are actually the leading-order solutions in these Taylor-series expansions.…”