A stabilized extended finite element framework for hydraulic fracturing simulations

Abstract

We present a stabilized extended finite element formulation to simulate the hydraulic fracturing process in an elasto-plastic medium. The fracture propagation process is governed by a cohesive fracture model, where a trilinear traction-separation law is used to describe normal contact, cohesion and strength softening on the fracture face. Fluid flow inside the fracture channel is governed by the lubrication equation, and the flow rate is related to the fluid pressure gradient by the 'cubic' law. Fluid leak off happens only in the normal direction and is assumed to be governed by the Carter's leak-off model. We propose a 'local' U-P (displacement-pressure) formulation to discretize the fluid-solid coupled system, where volume shape functions are used to interpolate the fluid pressure field on the fracture face. The 'local' U-P approach is compatible with the extended finite element framework, and a separate mesh is not required to describe the fluid flow. The coupled system of equations is solved iteratively by the standard Newton-Raphson method. We identify instability issues associated with the fluid flow inside the fracture channel, and use the polynomial pressure projection method to reduce the pressure oscillations resulting from the instability. Numerical examples demonstrate that the proposed framework is effective in modeling 3D hydraulic fracture propagation.where the fluid pressure is uniformly equal to zero. Therefore, the traction on dry fracture face S d is purely governed by the traction-separation (cohesive) lawwhere t c represents the cohesive traction. We note that until now, we have not assumed any particular constitutive law for the bulk material. In other words, the previous equations are valid for both linear and nonlinear bulk material laws.

Numerical simulation of particle plugging in hydraulic fracture by element partition method

“…This method is extremely convenient for fracture simulations and has been applied to hydraulic fracture simulations. [7][8][9] Despite the strength of the extended finite element method, additional degrees of freedom resulting from interpolation enrichment may increase computation burden. Although the boundary element method performs well in reducing the degree of freedom, it does not consider nonlinear mechanical properties.…”

A 2D explicit numerical scheme–based pore pressure cohesive zone model for simulating hydraulic fracture propagation in naturally fractured formation

“…10 In addition, some advanced models have been implemented in various numerical frameworks such as finite element method (FEM), boundary element method (BEM), and discrete element method (DEM). In the framework of FEM, adaptive remeshing technique, [17][18][19][20] cohesive element, [21][22][23][24][25] and the partition of unity methods (eg, XFEM and GFEM) [26][27][28][29][30][31][32][33][34] are the most widely used tools for simulating hydraulic fracturing. Displacement discontinuity method (DDM), a special boundary element method which was first proposed by Crouch et al, 35 is particularly suitable for hydraulic fracture propagation modeling due to its relatively computational efficiency as well as simplicity of meshing and remeshing.…”

Fully coupled hydraulic fracture simulation using the improved element partition method

“…Recently, the XFEM has been extensively used to simulate HF. [2][3][4][5][6][7][8][9] Liu et al 6,7 proposed a stabilized XFEM framework for simulating the fully coupled HF process. In their method, the cohesive fracture model is used to characterize the fracture propagation.…”

“…Consider a cracked triangular element in its local mesh, as shown in Figure 3A. The real node set of this cracked element is [1,2,3], and the virtual node set is [4,5,6,7]. The cracked element is divided into three subelements (Figure 3 B), namely, subjoint element with nodes [4,5,6,7], subtriangular element [7,6,3], and subquadrilateral element [1,2,5,4].…”