A New Representation of Wells in Numerical Reservoir Simulation
- Yu Ding (Inst. Français du Petrole) | Gerard Renard (Inst. Français du Petrole)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Engineering
- Publication Date
- May 1994
- Document Type
- Journal Paper
- 140 - 144
- 1994. Society of Petroleum Engineers
- 4.3.4 Scale, 5.1.5 Geologic Modeling, 5.5 Reservoir Simulation
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Numerical PI's are used to relate wellblock and wellbore pressures and the flow rate of a well in reservoir simulations by finite differences. This approach is based on an "equivalent wellblock radius," req,o. When nonuniform grids are used, req,o may create an error in wellbore pressure or oil rate. This paper presents a new well representation. The analytical solution for near-well pressure is included by modifying the transmissibilities between gridblocks so that flow around a well is described fully. The new method is applicable to nonuniform grids and nonisolated wells.
Because finite-difference methods are used in reservoir simulation, well models are used to relate wellbore to wellblock pressure.1-9 These models assume a radial flow around the well so that a logarithmic relation is satisfied between the wellblock pressure, po, and the wellbore pressure, pw, by
where req,o=equivalent wellblock radius. Peaceman6 first demonstrated numerically that the req,o should be equal to 0.2?x for square gridblocks and derived an analytical expression for req,o. In a later study,2 he showed that this analytical expression could not be used in cases of grids with nonsquare blocks. However, he provided other formulas to calculate req,o in uniform grids in isotropic and anisotropic systems. Babu et al.3 gave a general analytical expression for req,o in uniform grids when the drainage area of the well is a rectangle. As pointed out by both Peaceman1,2 and Babu et al.,3 the formulas they provided are irrelevant to simulations on nonuniform grids.
Wellblock pressure calculated by finite-difference methods depends mainly on grid configuration6,9 and numerical scheme7 Therefore, in nonuniform grids, a general expression cannot be found for req,o that relates po to pw with certainty.
In this paper, we propose a new approach based on the finite-volume method to compute pw. It does not involve the req,o concept but does require that the near-well pressure distribution be known. The new approach implies a modification in the calculation of transmissibilities between the wellblock and its neighboring blocks. The approach is general. It can be applied to nonuniform grids and can be viewed as a generalization of Peaceman's1 analytical method for req,o.
Following Peaceman's approach, the new well representation is shown for a 2D single-phase flow problem for which the pressure distribution is known. Expressions of the new block transmissibilities are derived for isolated and nonisolated wells in nonuniform grids. Both isotropic and anisotropic cases are treated. The numerical results are more accurate with the new approach than with the numerical PI method, especially in cases of nonuniform grids and nonisolated wells.
New Isolated Well Representation
To illustrate the new approach, we consider the steady-state flow of a single-phase fluid toward an isolated well in a homogeneous porous media. The flow is governed by
A standard five-point finite-difference scheme is generally used for its discretization:
In particular, for the wellblock (Fig. 1), we have
where, po=wellblock pressure.
Assuming the isolated well is at (xo,yo), the pressure at a distance r from the wellbore is given by
As pressure varies rapidly (logarithmically) near the well, its gradient cannot be approximated by linear relation. Because of the singularity created by the well, the calculated po is very different from pw. To improve pressure-gradient calculations, a finite-volume method10 is considered.
Formulation With Finite-Volume Method
The finite-volume method applied to the flow equation for the wellblock yields
We must find a good approximation for flow terms
When the near-well pressure distribution is known, as in an isolated well case, good approximations can be obtained.
This section shows the derivation of q?1. Extension to the three other directions can be obtained easily.
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