Modeling of Off-Center Wells in Reservoir Simulation
- Ho-Jeen Su (Chevron Petroleum Technology Co.)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Engineering
- Publication Date
- February 1995
- Document Type
- Journal Paper
- 47 - 51
- 1995. Society of Petroleum Engineers
- 1 in the last 30 days
- 166 since 2007
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In the conventional formulation of reservoir simulation, wells are assumedto be at cell centers. Thus, the fluid flow pattern around an off-center wellis incorrectly represented. This is especially true for large field-widemodels, which are gridded by very coarse blocks resulting in numerousoff-center wells.
This paper describes a new and rigorous method to model single or multipleoff-center wells in a cell block. Fluids can be directly drawn from the fourclosest neighboring cells into an off-center producer. The well location isexactly honored. Using the new method, the calculated wellblock pressure can beinterpreted at cell center rather than at a distance from cell center in theconventional approach.
The new method is fully implicitly formulated, and although it adds morenon-zero elements to the Jacobian matrix, its dimension remains unchanged.Therefore, the implementation of this method is not difficult and the increasein computing costs is minimal.
This work has been validated by fine-grid simulation results. Severalsingle- and multiple-phase test cases are presented. It is shown that the newmethod can improve the accuracy of numerical results for large field-widemodels.
In field-wide simulation models, grid blocks are usually very coarse andmost of the wells are not located at centers of the grid blocks. Unfortunately,in the conventional simulation formulation, wells are assumed to be at the cellcenters. Instead of flowing towards the off-center wells, as it should be,fluids flow towards the cell centers in the simulation calculation.
Williamson and Chappelear realized the problem and presented a localanalytical solution for the well source term. Based on the off-center welllocation, geometric weighting factors can be derived to quantify thecontribution from neighboring cells to the source term. According to the wellpressure, the neighboring cell pressures, and the weighting factors, the wellrate can be calculated. However, as suggested in their paper, once the rate isdetermined, the simulator still assumes that fluids are produced only from thecell containing the well. Thus, the location of the source within the wellblock has no influence on the finite difference scheme results. In other words,they honored the exact off-center well location when calculating the well rate,and did not take the exact well location into account when solving the finitedifference equations. The cell pressure is still calculated under the same oldassumption that the wellbore fluids are produced from the cell center.Therefore, the fluid flow pattern around an off-center well is incorrectlyrepresented. They did not present any numerical test results in their work.
Peaceman derived a correct formula for evaluating the equivalent wellblockradius for a grid block containing an off-center well. He demonstrated that awell pressure value, very close to the analytical solution, can be obtainedusing the formula in a single-phase test case. Abou-Kassem and Aziz derived ageneral analytical expression for the equivalent well-block radius applicableto off-center wells, and their solution matched well with the numerical resultspresented by Kuniansky and Hillestad. Since the conventional well equation wasstill used in the finite difference scheme, the flow pattern was notrepresented correctly, as explained in the last paragraph.
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