Interior point methods for an algebraic system involving complementarity equations for geomechanical fractures

TL;DR

This paper explores the use of interior point methods to solve complex algebraic systems arising from boundary integral equations in geomechanical fracture modeling, demonstrating improved robustness and efficiency over traditional algorithms.

Contribution

It introduces an adaptation of interior point methods for solving non-optimization algebraic systems in fracture modeling, ensuring convergence and robustness even with complex geometries.

Findings

Interior point methods improve solution robustness for fracture systems.

Numerical results show high accuracy and efficiency.

Methods outperform empirical algorithms in convergence guarantees.

Abstract

Many applications like subseismic fault modeling, fractured reservoir modeling and interpretation/validation of fault connectivity involve the solution to an elliptic boundary value problem in a background medium perturbed by the presence of cracks that take the form of one or many pieces of surface (with boundary). When the background medium can be considered as homogeneous, boundary integral equations appear as a method of choice for the numerical solution to fractures problems. With such an approach, the problem is reformulated as a fully non-local equation posed at the surface of cracks. Discretization of boundary integral resulting in the so-called Boundary Element Method (BEM) leads to densely populated matrices due to the full non-locality of the operators under consideration. After the discretization process, geologists are faced with a system of equations that turns out difficult to solve numerically. Many empirical algorithms have been proposed by geologists to solve this system of equations. Unfortunately, none of them is guaranteed to converge in theory (in particular when faults (fractures) intersect each other forming a geometrically highly irregular structure). In practice, none of them appears to be either robust or efficient. We investigate another approach, referred to as interior point methods, for which convergence can be ensured (even if faults are too close). Interior point methods have proved their efficiency in a wide variety of domains, most notably for linear programming. Here, even though we do not have any optimization problem, we can adapt ideas from interior point methods for the numerical resolution of the system considered. The numerical results obtained demonstrate computational efficiency and accuracy, highlighting the robustness and effectiveness of the implemented methods.