Abstract:
We propose and investigate formulation and numerical simulation for largely deformable anisotropic reservoirs in this study. We employ the total Lagrangian method (TL) for coupled flow and geomechanics, which does not need to update the coordinate system each time step. The use of the deformation gradient as well as the first and second Piola stresses reflects the change of reservoir configuration, being mathematically equivalent to the updated Lagrangian method. To accurately model full-tensor permeability derived from the Piola transformation in permeability, we use the multi-point flux approximation (MPFA). The total Lagrangian method with MPFA can provide high accurate and rigorous modeling. We thus consider the total Lagrangian method with MPFA as the reference method in this study. Then, we compare it with two other methods: Total Lagrangian method with the two-point flux approximation (TPFA) and infinitesimal transformation assumption with TPFA. From numerical simulation, we find differences between the reference method and the other two methods. Displacement based on the assumption of the infinitesimal transformation is different from that of the total Lagrangian method. Also, we find that volumetric strain and pressure of TL-MPFA are different from those of TL-TPFA. As the anisotropy ratio of permeability increases, the errors between MPFA and TPFA increases.
Introduction
Small deformation (i.e., infinitesimal transformation) is typically assumed in reservoir geomechanics [1, 2, 3, 4]. This assumption is usually valid in reservoir engineering problems associated with rock, which induce small deformation. However, the assumption might be invalid in largely deformable reservoirs, such as oceanic gas hydrate deposits and fractured/crashed salt domes [5, 6]. Anisotropic reservoirs are profoundly sensitive to substantial changes in reservoir configuration, having full-tensor permeability and elastic moduli during deformation. This causes non-orthogonal grids in flow. However, the modeling of largely deformable anisotropic reservoirs has little been investigated.
In this study, we employ the total Lagrangian method for coupled flow and geomechanics. The coordinate system remained fixed both for flow and geomechanics. Instead, the deformation gradient reflects the change of reservoir configuration, which yields mathematical equivalence to the updated Lagrangian method [7, 8, 9]. The total Lagrangian method also induces full-tensor permeability from the Piola transformation, even if the initial permeability tensor is diagonal [11]. To accurately model full-tensor permeability, we use the multi-point flux approximation (MPFA) [12]. Then, the total Lagrangian method with MPFA can provide high accurate and rigorous modeling, honoring the objective stress rates (i.e., Lie derivatives). We thus consider the total Lagrangian method with MPFA as the reference model in this study.