Well Index in Reservoir Simulation for Slanted and Slightly Curved Wells in 3D Grids
- I. Aavatsmark (Norsk Hydro Research Center) | R.A. Klausen (Norsk Hydro Research Center and U. of Oslo)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- March 2003
- Document Type
- Journal Paper
- 41 - 48
- 2003. Society of Petroleum Engineers
- 4.1.5 Processing Equipment, 4.1.2 Separation and Treating, 2.2.2 Perforating, 5.5 Reservoir Simulation
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A new, accurate method for computing the well index (WI) for a well that is oriented in an arbitrary direction in a 3D grid is presented. The method is simple and seminumerical and takes well direction and neighboring gridblocks into account. It is assumed that the medium is homogeneous. The method can be applied for any grid or discretization. However, only Cartesian grids with a seven-point stencil are used here. Properties of the seminumerical WI are investigated for both synthetic and field examples. The investigated properties are convergence rate and dependency on well direction, well location, grid geometry, and anisotropy. The new method shows the importance of taking the neighboring gridblocks into account.
The subject of this paper is the problem of finding a good well index for a slanted and slightly curved well in a reservoir simulation model. In this context, a slanted well is a well that is not aligned with the axes of the grid. Other names used on the same topic in the literature are connection transmissibility factor, used in the Eclipse manual;1 well transmissibility; or numerical productivity index. The well index WIi for single-phase flow is defined as
for the connection in gridblock number i. Here, qi represents the volumetric flow rate into the part of the well perforated in block i, mui=the viscosity in block i, Pi=the discrete pressure in the gridblock, and Pwf=the pressure at the well boundary. The first derivation of a WI was given by Peaceman for 2D grids in 1978,2 extended to include anisotropy in 1983.3 Until recently, this has been the dominant method for finding WIs used in reservoir simulation. This 2D formula is still frequently used in different weighting formulations in 3D. Peaceman's solution handles anisotropic homogeneous media with rectangular uniform grids, and introduces the numerical factor 0.14. The derivation of this factor is founded on the assumption of uniform grids. The well must be aligned with one of the axes of the grid, which again must be equal to the principal axes of the permeability. The problem of finding a proper well index in 3D has been addressed by many authors in recent years. Different weightings of Peaceman's 2D solution or of its factors are given in Refs. 4 through 6. Other authors start out with a mathematical derivation and define a WI built on the numerical factor 0.14 (see Ref. 7). Assuming an orthogonal, box-shaped reservoir, Babu and Odeh et al.8,9 use separation of variables and Green's functions to compute a WI. A Fourier transform is used to include the boundary condition, leading to a slowly converging solution, particularly for slanted wells; see Ref. 10 for discussion of the convergence. The ideas are utilized further by Refs. 11 through 13, among others. One of the problems with finite difference discretization of the pressure is the strong nonlinearity in the pressure near the well. Hales14 omits this problem and the need for a WI by subtracting the well singularity in the finite difference pressure solution. Ding15 also suggests a change in the reservoir-model discretization in order to find a better well model for estimation of Pwf or qi. He modifies the near well transmissibilities, taking an analytical model into account. In Ref. 16, slender body theory is used to find an analytical pressure drop. This is then used directly in the seven-point discretization molecule in 3D, with adjustment for anisotropy. The accuracy is discussed for 2D.
The method presented here is based on a different approach. This method applies a seminumerical technique, which will converge to the correct WI under the restrictions made. As in Peaceman's 2D solution, the medium has to be homogeneous, but anisotropy is allowed. Because we do not compute an analytic expression for the well index, we allow nonuniform grids. We do not discuss inhomogeneous media, and in a real case, one has to assume that the medium has the permeability of the block for which the well index is to be calculated. This is exactly as when Peaceman's 2D solution or its 3D weighting extensions4-6 are applied in present simulators.
This paper is organized as follows. In the next section, the seminumerical WI is presented, including a derivation of the analytical solution for slanted wells in 3D. In the section after that, properties of the WI are discussed, including a discussion of convergence properties. Next, some numerical examples are demonstrated. In the last section, a conclusion is given.
Seminumerical Well Index
The problem of finding the WI is one of finding a relation between the well pressure, the well rate and the discrete pressure value from a numerical simulation for the entire gridblock. A typical well radius is about 10 cm, while one numerical pressure value for a gridblock typically represents a volume of 100×100×1 m3. The well represents a sink (or source) for the pressure distribution. This means that in the wellblock there will be a huge variation in the real pressure. An illustration is given in Fig. 1. Some gridblocks away from the well, the variation in the pressure is small, and the numerical pressure for the gridblock is close to any value of the real pressure in the gridblock.
The idea is first to solve the pressure equation analytically with a linear, infinite well in an infinite reservoir, and then to solve the same problem numerically. The difference between these two solutions is zero at infinity and gives Pi-Pwf in the actual wellblock. Far away from the well, the analytical solution is almost equal to the numerical solution. At this distance, the boundary of the numerical problem is defined. We solve the pressure equation discretely as a Dirichlet problem, with the boundary condition derived from the analytical solution. This represents a presimulation, resulting in the well index for each wellblock.
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