An Application of Green's Function Technique for Computing Well Inflow without Radial Flow Assumption

Abstract

Well modeling plays an important role in numerical reservoir simulation. The main difficulty in well modeling is the difference in scale between the wellbore radius and the well block grid dimension used in the simulation. Peaceman’s formula is widely used in reservoir simulation in order to match the cell pressure to the local solution of the diffusivity equation describing the flow near the well. However, it was developed under the assumption of radial flow. The objective of this study is to calculate a semi-analytical expression for the well productivity index without making any assumption about radial flow, and to subsequently use it in numerical reservoir simulation. Radial flow may not occur due to boundary conditions at the top (bottom) of reservoir or well trajectory. The well inflow equation can be solved through the Green’s function method (GFM), which may take into account various boundary conditions and different well trajectories. The GFM solution of the diffusivity equation is presented as a series over the eigenvalues of the Laplace differential operator, but the series converges conditionally and its direct summation is time-consuming. This makes the GFM solution impractical for well modeling. Reference [1] presented the method of fast summation of such a series, which was successfully applied for analyzing pressure build up curves. In this paper we apply the same techniques for calculating the well productivity index for horizontal, deviated and partially penetrating wells . It is shown that for reservoirs with a gas cap (or underline water), using Peaceman’s model for the well productivity index leads to a significant discrepancy between the numerical and presented semi-analytical solution. Although local grid refinement around the well leads to a reduction in the discrepancy, it introduces its own set of numerical problems. It is demonstrated that using new expression for the well index models the well inflow with high accuracy even on a coarse grid. [1] E.S. Makarova, D.V.Posvyanskii, V.S.Posvyanskii, A.B. Starostin ECMOR XI P26 2008