Symmetric Cohesive Element Formulation for Fully Coupled 3D Hydraulic Fracture Modeling
Authors: Kirane K. S. , Xia L. , Garzon J. , Sanz P. F. 
Published: September 2023

ABSTRACT
A novel formulation for a symmetric cohesive element is presented here, for hydraulic fracture modeling problems that involve one fluid driven crack with a plane of symmetry coinciding with the fracture plane. The formulation enables analyses by modeling only half the domain for such problems. The modeling framework employed is the recently developed 3D finite element model in Abaqus which fully couples the rock deformation with the pore fluid Darcy flow. The symmetric cohesive element is an enhanced version of the previously developed 3D pore pressure cohesive element, uniquely capable of simulating Darcy flow in the undamaged state and a smooth transition to a tangential Poiseuille flow when damaged completely. Flow into the formation/fracture is modeled via specialized 1D pipe elements that can account for the wellbore fluid flow and frictional losses. Fluid leak-off into the formation is also modeled by means of a filter cake leak-off coefficient. The symmetric cohesive element is formulated by imposing appropriate constraint conditions on the displacement and pore pressure degrees of freedom, that enforce symmetric crack face behavior on either side of the fracture plane, i.e., the plane of symmetry. In addition, appropriate scaling factors are applied to various model parameters viz. the rock fracture energy, the pipe diameter, pipe friction factor and last but not the least, the viscosity of the fracturing fluid. The proposed approach with the aforementioned constraints and scaling factors is demonstrated to correctly simulate just half the domain of a symmetric hydraulic fracture, and yield results virtually identical to the full model, at half or less computational cost. Both lab and field scale examples are presented to demonstrate the effectiveness of the proposed approach.

INTRODUCTION
Symmetry has long been exploited in mechanics to simplify problem solving and to reduce domain and model sizes during numerical analysis e.g., using finite element method (FEM). For instance, it is common to employ an axisymmetric solution to engineering problems with a radial symmetry. In many problems involving fractures, the plane of the fracture is also a plane of symmetry, e.g., in three-point bending of a simply supported notched beam, or during delamination of a double cantilever beam (DCB). In these analyses, it suffices to model just one half of the structure and one face of the fracture, thus reducing the computational cost by half. However, while doing so, there are some additional considerations required to ensure that the solution is the same as that from the full domain. E.g., when the fracture is modeled using cohesive elements, additional constraint conditions on the displacements must be enforced, as discussed in Brocks et. al (2013) and Abaqus manual (2016), so that the displacements (openings) of top and bottom surfaces of cohesive elements are equal, to guarantee a symmetric fracture opening. Due to these constraints, the specified cohesive strength (which can be expressed as a function of the Young's modulus and fracture energy) needs to be reduced to half, to obtain the same results as the full model.