Such inactive cells are usually compressed out of reservoir simulation solution arrays prior to the memory and time-intensive flow solution stage, and enable reservoirs with irregular boundaries to be represented within extended simulation grids. Some cells may be inactive, representing volumes of the reservoir with zero porosity. Not all the elements in the grid need represent active solution variables in the simulation. For a 3D system, regular grids yield seven-point schemes, in which the flow equations for a cell involve solution values for just the cell and its six neighbors. The resulting grid is cylindrical and is important for the special case of near-well studies dominated by radial inflow. Such a two-point transmissibility assumes a permeability tensor with primary axes aligned along the grid axes.Īlthough regular grids are normally defined in normal Cartesian coordinates, it is also possible to use an ( r, Ф, z) radial system. d the dimension of the cell in that direction.A is the area of the cell orthogonal to the direction of flow.K is the cell permeability in that direction.cell b is the neighbor to cell a in some direction.In this case, it is possible to obtain the transmissibility value as a harmonic average: Cells in such a grid may be simply identified using their ( i, j, k) index values.Įach of the grid elements will be assigned a single permeability or porosity value. Regular cartesian gridsĪ simple 3D grid is the regular Cartesian grid ( Fig. It is sometimes possible to cast a finite-element Galerkin discretization into the upstreamed transmissibility-based form. Other options for discretization are available, such as Galerkin finite elements and mixed finite-element. When solution values from other cells are required, the flow takes a multipoint form. The flow expression then takes a simple form: When the flows between two cells a and b can be expressed as a function of the solution values in just those two cells, so that the summation over cells includes just x = b, the flow expression takes a two-point form. The constant coefficients of mobility and potential difference products, T ax, are commonly termed the transmissibilities. ΔФ pax is the potential difference of phase p between cell a and cell x, which includes pressure, gravity and capillary pressure contributions: This is often set to an upstream value of the mobility, depending upon the sign of the potential difference. K rp is the relative permeability of phase p.x cp is the concentration of component c in phase p.M cpax is the mobility of component c in phase p for the contribution to the flow between a and x, given by x cp. The linear pressure dependence of flows given by Darcy’s law leads to an expression of the type: In general, the flows F cpab may involve the solution values of a number of cells, the number of cells involved defining the stencil of the numerical scheme. F cpab is the flow rate of component c in phase p from cell a to its neighbor b.Q ca is the injection or production rate of component c because of wells.m ca is the density of conserved component c in cell a.The mass conservation equations for a timestep from T to T + Δ T then become: This yields a discretization scheme which is conservative (each outflow from one cell is an inflow to another) and for which the fluid in place may be obtained straightforwardly. Rock properties such as porosity are assumed constant over the cell or controlled volume. This yields a set of equations in which the mass conservation conditions for the fluid in the simulation cell volumes are related to the flows through the interfaces between those cell volumes. An alternative approach is to use an integral finite-difference or finite-volume method in which the fluid-flow equations are integrated over a set of cell volumes. ![]() In a classical finite-difference scheme, the point values of pressures and saturations are used as solution variables, and the differential operators that appear in the fluid-flow equations may be expanded as difference expressions of these point values to some order. Accuracy with which the geological description of the reservoir is matched. ![]() Two related issues are involved in choosing a grid for reservoir simulation: ![]() Even after this process, the geology to be represented is rarely homogeneous at the scale of the simulation grid. It is usually possible to identify many more layers in the geological model than it is practical to include in reservoir flow simulation, so some upscaling of rock properties will normally be carried out. The reservoir may contain faults, at which the strata are displaced. The basic structure of an oil reservoir is a set of geological horizons representing bedding planes.
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