We implement a novel up-winding scheme for the mobility calculation using the computed velocities in an adaptive finite element (FE) unstructured-mesh reservoir simulator.

In the finite-element finite-volume (FEFV) numerical method, the pressure and transport equations are decoupled. The pressure is calculated using finite elements, and the saturation is calculated using finite volumes. Each element is shared between several control volumes -- three for triangles (2D) and four for tetrahedral (3D). Consequently, the saturations used in calculating the mobilities hence updating pressure - are unclear. Some researchers use the average value between the elemental control volumes, or the integration points of the finite elements. For two-dimensional radial flow, this does not produce accurate saturations profiles when compared to the Buckley-Leverett reference solution.

In this paper, we present a new formulation to calculate the FE mobility. We use the velocity vector, which is piece-wise constant in first order elements, to find the upstream saturation—where the tail of velocity vector intersects an element. This novel approach produces more accurate saturation profiles than previous methods even with higher order methods.

Then, we present some benchmark simulation results where we model vertical spontaneous imbibition driven by capillarity and gravity disequilibrium between a fracture network at the bottom of the simulation domain and the matrix. The results compare favourably with semi-analytical treatments of this problem and experimental measurements.

The method presented better models multi-phase displacements in complex reservoirs using FEFV. It can, also, be easily implemented in current FEFV based simulators.


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