Interface control volume finite element method for modelling multi-phase fluid flow in highly heterogeneous and fractured reservoirs

Abstract

We present a new control volume finite element method that improves the modelling of multi-phase fluid flow in highly heterogeneous and fractured reservoirs, called the Interface Control Volume Finite Element (ICVFE) method. The method drastically decreases the smearing effects in other CVFE methods, while being mass conservative and numerically consistent. The pressure is computed at the interfaces of elements, and the control volumes are constructed around them, instead of at the elements' vertices. This assures that a control volume straddles, at most, two elements, which decreases the fluid smearing between neighbouring elements when large variations in their material properties are present. Lowest order Raviart-Thomas vectorial basis functions are used for the pressure calculation and first-order Courant basis functions are used to compute fluxes. The method is a combination of Mixed Hybrid Finite Element (MHFE) and CVFE methods. Its accuracy and convergence are tested using three dimensional tetrahedron elements to represent heterogeneous reservoirs. Our new approach is shown to be more accurate than current CVFE methods.

Velocity dependent up-winding scheme for node control volume finite element method for fluid flow in porous media

“…(8) for the pressure Eq. (6) produce unphysical fluid saturation profiles, as discussed by Abushaikha 43 . In this paper, we introduce an equation to allocate the upstream direction of the fluid flow over each element.…”

“…In the NCFVE method, a multi-phase flow problem is solved in two steps. First, the primary variable, pressure, is calculated using the finite element method; in this paper we use the well-established Galerkin method 12,37,39 Then, the advection of fluid between the node control volumes is calculated using the finite volume method. In this paper, we do not detail the discretization procedures of the governing equations and the construction of the secondary control volume mesh, as they are thoroughly discussed by Abushaikha et al 37 .…”

“…This mismatch in the mobility calculation between the pressure and transport equations along with the misalignment of the corresponding meshes produce inaccurate fluid saturation profiles. Especially, in heterogeneous media since the material properties are imposed on the elements and the transport values are computed on the control volumes, as discussed by Abushaikha et al 37 . Higher order schemes for the mobility calculation between the control volumes have also been applied 38,39 .…”

Improving the robustness of the control volume finite element method with application to multiphase porous media flow

“… This result is important because the close‐to‐linear convergence obtained in Gomes et al is very uncommon in the literature. Usually the observed convergence is 0.5 (see, for example, Schmid et al, Abushaikha et al, and Hoteit and Firoozabadi ). Discretising the pressure using CVs does not affect the order of convergence of the method.…”

“…Although widely used, complex geometries are often poorly represented in this approach because of the limitations of the 2‐point flux approximation. Alternative methods use unstructured meshes to discretise space and control volume finite element methods (CVFEMs) or variants thereof to solve the governing equations (see the references herein ). An unstructured mesh better captures the often complex, multiscale, and time‐dependent solution fields such as pressure, saturation, or composition, especially when used in conjunction with dynamic adaptive mesh optimisation .…”