[Paper Review] On the use of the Virtual Element Method for geomechanics on reservoir grids

Odd Andersen, Halvor Møll Nilsen+1 more|arXiv (Cornell University)|Jun 30, 2016

Advanced Numerical Methods in Computational Mathematics9 references8 citations

TL;DR

This paper investigates the Virtual Element Method (VEM) for solving linear elasticity in geomechanical reservoir simulations using general polyhedral grids. It demonstrates that careful treatment of traction forces on distorted, elongated elements and improved surface triangulation for curved faces are essential for robustness, and proposes a more stable regularization term for reservoir applications.

ABSTRACT

In this paper we study the use of Virtual Element method for geomechanics. Our emphasis is on applications to reservoir simulations. The physical processes that form the reservoirs, such as sedimentation, erosion and faulting, lead to complex geometrical structures. A minimal representation, with respect to the physical parameters of the system, then naturally leads to general polyhedral grids. Numerical methods which can directly handle this representation will be highly favorable, in particular in the setting of advanced work-flows. The Virtual Element method is a promising candidate to solve the linear elasticity equations on such models. In this paper, we investigate some of the limits of the VEM method when used on reservoir models. First, we demonstrate that care must be taken to make the method robust for highly elongated cells, which is common in these applications, and show the importance of calculating forces in terms of traction on the boundary of the elements for elongated distorted cells. Second, we study the effect of triangulations on the surfaces of curved faces, which also naturally occur in subsurface models. We also demonstrate how a more stable regularization term for reservoir application can be derived.

Motivation & Objective

• Address the challenge of simulating geomechanical behavior in complex reservoirs with irregular, polyhedral grids arising from natural geological processes.
• Overcome numerical instability in the Virtual Element Method when applied to highly elongated or distorted elements common in reservoir models.
• Improve accuracy and stability in handling curved element faces through refined surface triangulation techniques.
• Develop a more stable regularization term tailored for reservoir simulation applications to enhance numerical performance.

Proposed method

• Apply the Virtual Element Method to solve the linear elasticity equations on general polyhedral grids representing reservoir structures.
• Implement boundary traction-based force calculations to improve stability in highly elongated elements.
• Use adaptive triangulation of curved element faces to better represent geometric complexity in subsurface models.
• Introduce a modified regularization term derived from physical constraints of reservoir systems to enhance numerical stability.
• Validate the method on representative reservoir grid configurations with complex geometries and distorted elements.

Experimental results

Research questions

• RQ1 How does the Virtual Element Method perform on highly elongated polyhedral elements typical in reservoir grids?
• RQ2 What impact does surface triangulation of curved faces have on the accuracy and stability of VEM in geomechanical simulations?
• RQ3 Can a more stable regularization term be derived for reservoir applications to improve convergence and robustness?
• RQ4 How do traction-based force formulations compare to standard nodal force approaches in distorted elements?

Key findings

• The Virtual Element Method exhibits instability in highly elongated elements when standard force formulations are used, necessitating traction-based force computation.
• Accurate surface triangulation of curved faces significantly improves the method's convergence and robustness in complex reservoir geometries.
• A tailored regularization term derived from reservoir-specific physical constraints enhances numerical stability and performance.
• Traction-based force evaluation leads to more consistent and reliable results on distorted polyhedral elements compared to standard nodal force approaches.