Poroelastic Solutions of Hydraulic Fracturing Problems Using Fast Multipole Boundary Element Method
Hydraulic fracturing (HF) of horizontal wells is an essential part of any unconventional field development activity. Hence, understanding the behavior of this process in horizontal wells is extremely important. A comprehensive study on the behavior of hydraulic fractures is highly dependent on the mathematical formulation of the problem, assumptions that are made to make the problem tractable, and numerical method that is utilized to solve the corresponding equations. Although the host medium of hydraulic fractures is porous and saturated, often this problem is solved using elasticity equations and adding a leak-off, or other parameters in an un-coupled fashion to the model. This approach ignores the fully-coupled essence of the problem and often gives inaccurate results. On the other hand, adding the necessary details makes the model inefficient when degree of freedom (DOF) increases. Another complexity that exists in the hydraulic fracturing problem is movement of the boundary, which makes it difficult to re-mesh the domain that is required by most of the domain-based numerical methods after each propagation event. The objective of this study is to develop a comprehensive yet efficient model to investigate the behavior of hydraulic fractures in poroelastic media. The theory of poroelasticity together with the displacement discontinuity method, and an efficient fast multipole algorithm are utilized to develop an efficient hydraulic fracture model.
Displacement Discontinuity Method (DDM) is a numerical method that is used in this study to solve the field equations of poroelasticity. This method belongs to a class of boundary element methods (BEM), which are suitable when the ratio of volume to surface is high. DDM is a special formulation of BEM for a medium containing discontinuities (e.g., fractures). It is based on considering the fracture as a line in 2D (or a surface in 3D) along which one defines quantities that take into account the discontinuity in displacements from one side of the fracture to the other. Despite the great advantages of this method over the domain methods, it suffers from dense coefficients matrices. These dense matrices are multiplied by a vector in several places, and make the method computationally inefficient in circumstances that time and/or space are discretized with so many elements. This may happen in hydraulic fracturing problems where multiple fractures propagating from one or more wells. It would also occur in three-dimensional problems where typically the number of elements exceeds a few thousands.
Additionally, including poroelasticity makes this method even more computationally intensive because of the necessity to build a time-marching procedure. The Fast Multipole Method (FMM) is a computational method to efficiently compute matrix-vector products with a controllable error. Unlike the conventional BEM, in FMM the interaction between far-field sources and influenced points are calculated by initially concentrating a cluster of far-field sources into a separate point, then the effect of these concentrated forces on each influenced point is calculated. A technique known as black-box fast multipole method (bbFMM) is used in a part of this study to improve the computational efficiency of the conventional fully-coupled poroelastic displacement discontinuity model while keeping its accuracy. This method uses the conventional hierarchical tree structure in FMM to separate the influencing (source) and influenced (target) points. It also uses the Chebyshev polynomials for kernel expansion instead of the analytical expansion used in a conventional FMM formulation.
In this study, the fully-coupled formulation of displacement discontinuity method with a fast multipole algorithm are used to develop an efficient poroelastic hydraulic fracture simulator. The simulator is used to study several problems such as refracturing of a horizontal well and fracturing of an infill well in a partially depleted reservoir. It is also used to examine the computational effectiveness and the accuracy of the new approach against the conventional displacement discontinuity approach in calculating stresses, displacements and pore pressure. A comparison between efficiencies and precisions of the developed fast multipole model and the conventional displacement discontinuity model is presented. The main contribution of this research can be categorized in two parts, namely modifications of the DDM algorithm and applications of the model. In the first part, a new algorithm is presented for the fully-coupled fast multipole displacement discontinuity method. Also, a new procedure is suggested for analyzing problems such as refracturing horizontal wells that requires coupling between pore pressure and stresses. The new approach, unlike the conventional approach, can handle pore pressure and stress changes, and refracture propagation in one simulator run. Moreover, solutions that are obtained by the fast multipole algorithm is compared with the conventional fully-poroelastic displacement discontinuity method. Also, the applicability of this algorithm in the fully-coupled displacement discontinuity method is established. On the application side, several applications such as refracturing a single well, infill drilling, and problems involving N-fractures are studied using the suggested algorithm and approaches.