Variable discretization method for flow simulation on complex geological models

摘要

A variable discretization method for general multiphase flow simulation in a producing hydrocarbon reservoir. For subsurface regions for which a regular or Voronoi computational mesh is suitable, a finite difference/finite volume method (“FDM”) is used to discretize numerical solution of the differential equations governing fluid flow (101). For subsurface regions with more complex geometries, a finite element method (“FEM”) is used. The invention combines FDM and FEM in a single computational framework (102). Mathematical coupling at interfaces between different discretization regions is accomplished by decomposing individual phase velocity into an averaged component and a correction term. The averaged velocity component may be determined from pressure and averaged capillary pressure and other properties based on the discretization method employed, while the velocity correction term may be computed using a multipoint flux approximation type method, which may be reduced to two-point flux approximation for simple grid and permeability fields.

主权项

1. A method for simulating multi-phase flow in a producing hydrocarbon reservoir, comprising:
generating a model of the reservoir made up of discrete cells; using a computer to numerically solve differential equations for pressure and for fluid saturation, by phase, for the reservoir, wherein:
a finite element discretization method (“FEM”) is used for the numerical solution in at least one region of the model, and a finite difference or finite volume discretization method (collectively, “FDM”) is used for remaining regions of the model, wherein at least one of the remaining regions is a neighboring region to the at least one region of the model;wherein the flow rate computation is made in at least one FEM region using a mixed finite-element method (“MFEM”) generalized to handle multiphase flow, accounting for phase equilibrium, compressibility, capillary and gravitational effects, in which an MFEM mass conservation equation for single-phase flow is replaced with a volume balance equation for multiphase flow and said phase velocity correction term, δuv, accounts for capillary and gravity effects, and is computed using a using a modified multi-point flux approximation (“MPFA”) method wherein:
an interaction region is set up around each vertex point in the FEM's grid, each such interaction region containing portions of all FEM grid cells that include the vertex point, each such portion being called a sub-interaction region;a linear pressure function, δpv, is found for each sub-interaction region such that the following conditions are satisfied:(i) εpv=pc,v−pc,a at cell centers, where pc,v is capillary pressure of phase v, and pc,a is average value of capillary pressure;(ii) δpv is continuous at centers of cell interfaces; and(iii) for each sub-face between adjoining sub-interaction regions, δuv is computed from δuv=−K˜[∇δpv+(ρv−ρa)g{right arrow over (z)}] where ρv is fluid density for phase v, ρa is average density of the fluids, g is acceleration due to gravity constant, and {right arrow over (z)} is a unit vector in z-direction, which expression for δuv, when evaluated in each sub-interaction region, is constrained to be continuous across each sub-face in its normal direction; andphase flow rate computation is made consistent in all regions and across interfaces between regions by decomposing each individual phase velocity into an averaged component and a correction term; and downloading, storing in memory or other data storage, or displaying at least one of hydrocarbon saturations, flow rates, and masses, for at least one phase, at one or more locations in the model.