A finite-volume discretization for deformation of fractured media

Abstract

Simulating the deformation of fractured media requires the coupling of different models for the deformation of fractures and the formation surrounding them. We consider a cell-centered finite-volume approach, termed the multi-point stress approximation (MPSA) method, which is developed in order to discretize coupled flow and mechanical deformation in the subsurface. Within the MPSA framework, we consider fractures as co-dimension one inclusions in the domain, with the fracture surfaces represented as line pairs in 2D (face pairs in 3D) that displace relative to each other. Fracture deformation is coupled to that of the surrounding domain through internal boundary conditions. This approach is natural within the finite-volume framework, where tractions are defined on surfaces of the grid. The MPSA method is capable of modeling deformation, considering open and closed fractures with complex and nonlinear relationships governing the displacements and tractions at the fracture surfaces. We validate our proposed approach using both problems, for which analytical solutions are available, and more complex benchmark problems, including comparison with a finite-element discretization.

Appendix

The analytical solutions of the induced displacements (u_x, u_y) and stresses (σ_xx, σ_yy, σ_xy) at any point (x,y) for an infinite 2D homogeneous and isotropic elastic nonporous medium containing a finite small thin fracture with constant normal- and shear-displacement discontinuities are given by Crouch and Starfield [12] as

$$\begin{array}{@{}rcl@{}} u_{x} &=&{\Delta} u_{\boldsymbol{n}_{+}}^{{\Gamma}} \left( {2\left( {1-\nu} \right)\frac{\partial f}{\partial y}-y\frac{\partial^{2}f}{\partial x^{2}}} \right)\\ &&+{\Delta} u_{\boldsymbol{\tau} _{+}}^{{\Gamma}} \left( {-\left( {1-2\nu} \right)\frac{\partial f}{\partial x}-y\frac{\partial^{2}f}{\partial x\partial y}} \right){,} \\ u_{y} &=&{\Delta} u_{\boldsymbol{n}_{+}} ^{{\Gamma}} \left( {\left( {1-2\nu} \right)\frac{\partial f}{\partial x}-y\frac{\partial^{2}f}{\partial x\partial y}} \right)\\ &&+{\Delta} u_{\boldsymbol{\tau}_{+}}^{{\Gamma}} \left( {2\left( {1-\nu} \right)\frac{\partial f}{\partial y}-y\frac{\partial^{2}f}{\partial y^{2}}} \right){,} \end{array} $$

(1)

and

$$\begin{array}{@{}rcl@{}} \sigma_{xx} &\,=\,&2\mu {\Delta} u_{\boldsymbol{n}_{+}}^{{\Gamma}} \left( {2\frac{\partial^{2}f}{\partial x\partial y}+y\frac{\partial^{3}f}{\partial x\partial y^{2}}} \right)\\ &&+ 2\mu {\Delta} u_{\boldsymbol{\tau}_{+}}^{{\Gamma}} \left( {\frac{\partial^{2}f}{\partial y^{2}}+y\frac{\partial^{3}f}{\partial y^{3}}} \right){,} \\ \sigma_{yy} &\,=\,&2\mu {\Delta} u_{\boldsymbol{n}_{+}}^{{\Gamma}} \left( {-y\frac{\partial^{3}f}{\partial x\partial y^{2}}} \right)\,+\,2\mu {\Delta} u_{\boldsymbol{\tau}_{+}}^{{\Gamma}} \left( {\frac{\partial^{2}f}{\partial y^{2}}\,-\,y\frac{\partial^{3}f}{\partial y^{3}}} \right){,} \\ \sigma_{xy} &\,=\,&2\mu {\Delta} u_{\boldsymbol{n}_{+}}^{{\Gamma}} \left( {\frac{\partial^{2}f}{\partial y^{2}}\,+\,y\frac{\partial^{3}f}{\partial y^{3}}} \right)\,+\,2\mu {\Delta} u_{\boldsymbol{\tau} _{+}}^{{\Gamma}} \left( {\!-y\frac{\partial^{3}f}{\partial x\partial y^{2}}} \right),\\ \end{array} $$

(2)

where ${{\Delta } u}_{\boldsymbol {n}_{+}}^{{{\Gamma }} }$ and ${{\Delta } u}_{\boldsymbol {\tau }_{+}}^{{{\Gamma }} }$ are the displacement discontinuities in the normal and shear directions, respectively; μ is the shear modulus; v is Poisson’s ratio; and f is the function of the position (x,y) of the field point relative to the center of the fracture. Denoting the half radius of the fracture as a, f is given as

$$\begin{array}{@{}rcl@{}} f\left( x\mathrm{,}y \right)&\,=\,&-\frac{1}{4\pi \left( 1\,-\,v \right)}\left( y\left( \tan^{-1}\frac{y}{x\,-\,a}-\tan^{-1}\frac{y}{x\,+\,a} \right) \right)\\ &&-\left( x-a \right)\ln \sqrt {\left( x\,-\,a \right)^{2}+y^{2}}\\ &&+\left( x+a \right) \ln \sqrt {\left( x+a \right)^{2}+y^{2}} \mathrm{.} \end{array} $$

(3)