Integral Equations for the Problem of a 3D Crack in an Infinite, Fluid Filled, Poroelastic Solid

Abstract

Summary Two integral equations are derived for the 3D problem of a pressurized plane crack in an infinite, fluid-saturated, poroelastic solid. These equations relate normal tractions and fluid pressures on the crack faces to crack openings and fluid injection rate per unit fracture area. An important application of these equations is the prediction of hydraulic fractures induced during waterflooding of reservoirs to enhance gas and oil recovery. Introduction Hydraulic fracturing is commonly used to enhance oil and gas production from low-permeability reservoirs. A fracturing fluid injected through perforations in the well casing is used to cause a large vertical fracture that is understood to be oriented so that the normal to the fracture surface has the direction of the minimum in-situ compressive stress. One fundamental problem in hydraulic fracturing is the prediction of the size and shape of the fracture produced by a given schedule of fluid injection. Initially, it was thought that the fractures did not extend vertically into the relatively impermeable bounding layers. Hence, predictions of fracture geometry were based on assumed fracture heights equal to the height of the hydrocarbon-bearing layer. However, field experience and fracture mechanics analysis indicate that in many applications the fractures are not contained by the bounding layers. Clifton and Abou-Sayed1,2 and Clifton3,4 presented a computational method for modeling the 3D development of hydraulic fractures caused by fluid injection at the wellbore. The rock formation is modeled as an infinite, homogeneous, isotropic, elastic solid with in-situ stresses that vary with depth. The 3D elasticity problem is formulated as an integral equation5 that relates pressure on the crack faces to crack openings by use of the fundamental solution for the stress field resulting from an infinitesimal segment of a dislocation line. This integral equation has been solved numerically by an approach similar to finite elements where the unknown crack opening is represented in terms of local trial functions, which vary linearly over triangular elements. Results of fully 3D simulations of hydraulic fracturing for field cases have been presented.4,6-8 To extend this approach to cases where the rock formation is modeled as a fluid-filled, poroelastic solid, we present here a pair of integral equations that relate normal traction and pore pressure on the crack surfaces to the crack opening and the fluid mass supply rate per unit fracture area. These equations are derived by starting with Cleary's9,10 reciprocal theorem for a fluid-saturated porous medium and converting the resulting Burgers integral expressions for displacements and pore pressure into equations for the stress and pore-pressure fields caused by infinitesimal dislocation line segments. The pair of integral equations, which include convolution integrals, are intended for applications where leakoff is large, such as in waterflood-induced hydraulic fractures and other EOR processes. Initial results based on these equations have been obtained recently11 for the case of hydraulic fractures induced during waterflooding. Basic Equations Consider Cleary's9,10 formulation of the classic Biot theory for a homogeneous, isotropic, fluid-saturated, poroelastic solid. The pore fluid is assumed to diffuse freely (any trapped fluid is regarded as contributing to the net constitutive behavior of the solid component), and the deformation is assumed to be isothermal and quasistatic. The deformation of the solid is described by a suitable average vector displacement ui; average strains are defined by Equation 1 where commas denote partial differentiation. The mass of free fluid per unit volume is denoted by m, which has an associated apparent volume fraction, Vf, defined by m=?Vf, where? is the fluid density. Changes in the strains, eij and the free-fluid mass, ?m, can be related elastically to total stresses, sij and pore-fluid pressure, p, by: Equations 2 and 3 where the usual summation convention holds for the repeated indices.12 The four elastic constants are the shear modulus, G; the "drained" and "undrained" Poisson ratios, µ and µu, respectively; and the induced pore-pressure parameter, B, defined as the quotient Equation 4 where the changes in the pore pressure, p, and the total hydrostatic pressure -?kk/3 are measured in an undrained test.