Cell to node projections: An assessment of error

Abstract

SUMMARYReservoir simulators typically use cell-centered finite volume schemes and do not model directly the coupling of the flow processes with the geomechanics. Coupling of geomechanics with fluid flow can be important in many cases, but introducing fully coupled geomechanical effects in those simulators is not a trivial issue, because the geomechanics is better done by using the Galerkin vertex-centered finite element methods by which the solid displacements are computed at the vertices of the cells. This creates difficulties in interfacing cell variables with nodal variables. Uncoupled or loosely coupled models are used by many researchers/practitioners by which a reservoir model is coupled to a geomechanical model by staggering in-time flow and deformation via a sophisticated interface that repeatedly calls first flow and then mechanics. The method therefore requires projection of the reservoir cell variables onto the nodes of the geomechanics Galerkin finite element mesh.In this note, we attempt to quantify the errors associated with cell to node projection operations. For that purpose, we use a simple model of the pressure equation for a heterogeneous medium in one dimension. We are able to derive the exact analytical solution for this problem for both nodal and cell pressures. This allows us to compute the errors due to projection analytically, function of meshing refinement and permeability field variations. We compute upper and lower bounds for the errors, and analyze their magnitude for a variety of cases. We conclude that, in general, cell to node projection operations lead to substantial errors.

Coupled multi-phase thermo-poromechanical effects. Case study: CO2 injection at In Salah, Algeria

“…For the saturation equation, which is a transport equation, a more sophisticated approach is needed in order to avoid the typical instabilities that usually occur when standard Galerkin is used. The finite volume method with one-point upwinding is known to avoid such instabilities, and a vertex-centered implementation avoids a cumbersome and inaccurate remapping of values from cell centers to nodes (for an error analysis of cell-to-node projections the reader is referred to Goumiri and Prévost, 2010). The saturation equation is solved in a different stagger than the other equations since its coupling with pressure, geomechanics and energy is weak.…”

“…This can be particularly significant when sharp contrasts of material properties occur, leading to high pressure and strain gradients. This aspect has been investigated by Goumiri and Prévost (2010).…”

One-Way versus Two-Way Coupling in Reservoir-Geomechanical Models

“…Difficult issues related to interfacing a cell-centered Reservoir simulator with a vertex centered Geomechanical simulator are discussed in (Goumiri and Prevost, 2011) but are not addressed hereafter. At the core of the one-way coupling solution procedure is the sequential integration of first the pressure equation, then the stress equation.…”

Two‐way coupling in reservoir–geomechanical models: vertex‐centered Galerkin geomechanical model cell‐centered and vertex‐centered finite volume reservoir models

“…Difficult issues related to interfacing a cell‐centered reservoir simulator with a vertex‐centered geomechanical simulator are discussed in but are not addressed hereafter. For completeness, we quickly review the ‘cheap’ one‐way coupling solution schemes.…”