Reservoir simulators estimate fluid flow from grid block to grid block until the flow reaches a well somewhere in the simulation model. Inter grid block flow is approximated using Darcy's law, with one noticeable exception: Fluid flow from well blocks to wells within those grid blocks introduces some additional difficulties. Peaceman (1978) noted that "in numerical reservoir simulation, the pressure calculated for a well block is the same as the flowing pressure at an equivalent radius, ro. For a square grid, ro = 0.2 ∆x." The problem is that the equivalent radius should demonstrably be much greater than that. In fact:where ∆x is the length of the sides of the square grid block. The video above shows that Peaceman's correction does not solve the problem noted by Peaceman, it only glosses over it. And in doing so, it introduces the distinct error stated above. Failing to solve the problem correctly may cause significant errors in the calculated fluid saturations and viscosities in the well block.Consequently, ExcSim does not use the Peaceman well correction at all. Instead, ExcSim attacks the cause of the problem and solves it by means of transmissibility corrections. See the video for details. For square grid blocks the equivalent well block radius is given by Fig2, and the transmissibility multiplier is: If the grid blocks are rectangular of size (A x B), then the transmissibility multiplier is a function of the ratio between the grid block sides, C=B/A. The formula below gives the transmissibility multiplier for the colored edges of the well block. The transmissibility multiplier for the other edges is found by inverting C, C’=1/C. The equivalent well block radius for rectangular well blocks is given by Fig. 6 below
6 Comments
Alan King
9/1/2017 17:23:20
What about in 3d?
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Michael Hovdan
9/18/2017 08:45:28
In 3D the same kind of logic applies. Obviously, the mathematical formulas in 3D become much more complicated, and the expressions much longer (extending over several pages).
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6/3/2018 13:21:00
Thank you for a very interesting presentation. This would make a very good SPE paper. The question on modifications to transmissibilities (Tx, Ty and Tz) in 3D simulation needs an answer. Typically, the largest pressure and saturation changes occur in the near wellbore region, particularly in high drawdown cases. Is the correctness of avg pressure in the well block that important, when phase change affecting saturations occurs at or near the entry to the well?
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Michael Hovdan
9/1/2018 22:36:55
David
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11/16/2018 02:03:47
To prove your corrections correct, you need to show that the solution to some problem converges to the correct unchanging fine-grid solution as the grid is refined while keeping the well in the exact same position, as in the refined cases determining the fine-grid solution for spe3 using Peaceman's eqns for WI at https://www.coatsengineering.com/spe3.htm. I believe that those cases show that Peaceman's formulae are correct, since the solution is not a function of wellblock size when numerical dispersion is sufficiently controlled.
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Michael Hovdan
12/3/2018 10:55:56
Brian
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