Injection-induced fracture reactivation during hydraulic fracturing processes in shale gas development as well as coal bed methane (CBM) and other unconventional oil and gas recovery is widely investigated because of potential permeability enhancement impacts. Less attention is paid to induced fracture reactivation during oil and gas production and its impacts on reservoir permeability, despite its relatively common occurrence. During production, a reservoir tends to shrink as effective stresses increase, and the deviatoric effective stresses also increase. These changes in the principal effective stresses may cause Coulomb fracture slip in existing natural fractures, depending on their strength, orientation, and initial stress conditions. In this work, an extended finite element model with contact constraints is used to investigate different fracture slip scenarios induced by general reservoir pressure depletion. The numerical experiments assess the effect of Young’s modulus, the crack orientation, and the frictional coefficient of the crack surface on the distribution of stress and displacement after some reservoir depletion. Results show that the crack orientation significantly affects the state of stress and displacement, particularly in the vicinity of the crack. Slip can only occur in permitted directions, as determined by the magnitudes of the principal stresses and the frictional coefficient. Lastly, a larger frictional coefficient (i.e., a rougher natural fracture surface) makes the crack less prone to shear slip.

Reservoir permeability enhancement through injection-based hydraulic fracturing is widely studied for improved oil production and shale gas development as well as coal bed methane (CBM) and other unconventional oil and gas recovery [

The contact problem, an issue central to solid mechanics, refers to the stress, strain and damage phenomena that take place when two solid surfaces interact [

A challenge to the analysis of contact-slip problems using conventional finite element methods lies in the need to adequately remesh the domain during crack nucleation and propagation, including the use of extremely fine meshes and issues of mesh-dependency of crack propagation. To help overcome these challenges, the extended finite element method (XFEM) is used to model cracks with arbitrary geometric shapes within the framework of FEM approaches based on the Partition of Unity method; this enriches the standard FEM approximations by using additional discontinuous interpolations near the propagation crack tips [

XFEM is widely applied to solve crack problems in reservoir engineering, including branched and intersecting faults [

In this article, the XFEM and frictional contact model are employed to simulate the behavior of rock discontinuities in producing reservoirs. In the numerical experiments, we analyze the effect of Young’s modulus, the crack orientation, and the frictional coefficient of the crack surface on the distribution of stress and displacement after some reservoir depletion.

The constitutive model for contact friction must be stipulated before numerical modeling of the contact problem because it provides the physical description of the contact mechanism between the two bodies. Assuming two bodies, a slave and a master body denoted by

where

where

If the two parts of the surface are in contact, there will be traction along the contact surfaces. The contact surfaces can be defined by the standard Kuhn–Tucker relations [

In this study, the penalty method is selected to cope with the contact constraints, and the penalty parameters _{N}_{T}

where

A stick-slip criterion is needed to determine whether the contact surface is in a state of adhesion or slip, and the most widely used is Coulomb’s friction law [

where _{f}

The “stick” condition applies if the magnitude of the right side is greater than that of the left side; in contrast, “slip” arises if the magnitude of the right side is less than the left side. If slip along the surface takes place, the tangential tractions need to be recalculated and returned to the magnitude of the right side to achieve a static, non-slipping state.

The general equations of contact problems using XFEM are needed, and the situation is sketched in

where _{c}

where ^{enr} denote the set of all nodal points and the set of enriched nodal points in the domain, respectively.

The Heaviside function is given by:

where

in which

Based on the relationship between the strain vector and the approximate displacement, the corresponding strain vector

where

Various modeling techniques are employed to introduce the contact constraints stipulated in Section 2.2 into the weak form of the FEM equation for the contact problem, such as the penalty method [

Rearranging the integral equilibrium equation, and it can be written as

According to the

where

The Newton–Raphson iterative procedure [

where

In which

whereas for the plane strain case,

The return mapping algorithm is central to the numerical solution of plasticity problems, taking much of the computational time [

where

If the equation is not satisfied, then both the tangential contact force and the tangential penalty parameter are updated using the following correction

The residual force vector can be used to evaluate solution convergence by

In this section, the frictional contact codes developed here are verified using an elastic plate with an oblique fracture [^{3}, and the fracture has a parameter

During oil and gas production, the pore pressure drawdown changes the stress state in the reservoir and surrounding strata, which may cause fault reactivation, slip of natural fractures and bedding planes, and alteration of reservoir permeability by fracture dilation, especially in the case of fracture-flow dominated reservoirs. We focus on the issue of a single fracture slip arising from a stipulated depletion. Assume a depleted reservoir with the vertical maximum principal stress ^{3} and the tangential stiffness ^{3}. The fracture has a friction coefficient of

To investigate the effect of Young’s modulus on the distribution of displacements and stresses around the fracture, simulations are performed on the same XFEM mesh and physical parameters but with three different Young’s moduli, 5 GPa, 20 GPa, and 80 GPa respectively.

The distribution of stresses around the fracture can be seen in

The displacement field is plotted in

The horizontal and vertical displacements for different cases where the fracture has different frictional coefficients are shown in

Using the penalty method, this paper introduces a frictional contact algorithm to XFEM, and a Coulomb friction criterion is employed to govern tangential contact slip. In the numerical experiments, the effect of Young’s modulus, crack orientation, and Coulomb frictional coefficient of the crack surface on the distribution of stress and displacement after depletion of a reservoir are analyzed. The results indicate that depletion-induced shear failure is much more likely to occur in a reservoir of lower Young’s modulus. Also, the angle of the crack (fault, bedding plane, joint) significantly affects the state of stress and displacement, particularly in the vicinity of the crack. Slip only can occur in permitted directions, as determined by the magnitudes of the principal stresses and the frictional coefficient. Finally, a larger frictional coefficient makes the crack less prone to shear failure, and this parameter can be related to the roughness of the slip surface.

This research received no external funding.