Modeling Vertical and Horizontal Wells With Voronoi Grid

Abstract

Summary This paper discusses the treatment of wells in a flexible Voronoi grid. A new model problem is proposed to evaluate exact well indices for multiwell configurations and homogeneous reservoirs. A simplified model also is proposed and discussed. New exact and simplified well models for heterogeneous reservoirs are presented. Introduction An exact well index can be derived by comparing the solution of the differential equation (obtained analytically or numerically) for a given problem with the numerical solution of the difference equation (exact well model). Peaceman1 first published this approach, and well models based on it are defined here as Peaceman-type models. In this sense, well models described by Kuniansky and Hillestad,2 Abou-Kassem and Aziz,3 and Babu et al.4 also are considered to be Peaceman-type models because, although these authors used different reference solutions, the well models were based on Peaceman's1 concept. In this paper, we propose new exact Peaceman-type well models for Voronoi grid and multiwell configurations in homogeneous and heterogeneous reservoirs and propose and discuss a simplified model for this grid. The exact well model can be used easily to investigate the effect of different well configurations (location and rates) over the value of the well index. Because the well index is assumed to be constant during the simulation process, a good grid geometry should result in a constant (within a tolerance) value for each well, regardless of the well configuration. Therefore, this procedure is an additional tool that can be used to select the appropriate grid size for the problem of interest. Current Wen Models Although van Poolen et al.5-type models were extensively used in the past, Peaceman1 showed that this approach is incorrect. For this reason, only Peaceman-type models are considered here. The relationship between wellblock pressure and bottomhole flowing pressure (BHFP) is a function of fluid rates, rock and fluid properties, and grid geometry:Equations 1 and 2 where Iw=well index, ?=angle open to flow, and req=equivalent wellblock radius. Because only single-phase cases are discussed, the subscript P (phase index) will be dropped from all remaining equations. Peaceman1,6 proposed the following simplified model based on the comparison between numerical and analytical solutions for the total pressure drop in a homogeneous, isotropic, repeated five-spot pattern under steady-state, single-phase flow conditions.Equation 3 The conditions that must be satisfied to apply this model safely may be as important as the model itself and are discussed by Peaceman.7,8 In his first paper, Peaceman1 showed that the block pressures increase logarithmically with the radial distance measured from the gridpoint to the well when a regular Cartesian grid with square blocks (?x=?y) is used to discretize an isolated one-quarter of a five-spot pattern. The extension of this concept led to the development of Kuniansky and Hillestad's2 and Abou-Kassem and Aziz's3 analytical well model. This model does not produce good results for wellblocks with large grid-aspect ratios (Ry=?y/?x) and may not produce good results for wells close to reservoir boundaries.7 However, it does yield good results for isolated wells in blocks with small Ry. This model's main advantage is that it can be applied to any kind of grid geometry and discretization scheme. Kuniansky and Hillestad2 and Peaceman7 proposed exact well models for multiwell configurations. They used different model problems for a reservoir with constant pressure at the external boundaries. In principle, these models can be used to derive well indices for grids of any geometry. However, conventional simulators assume closed boundaries, and the exact representation of constant pressure boundaries in these codes may be very tedious, although it is possible. Babu et al.4 proposed a well model based on the analytical solution presented by Babu and Odeh9 for a single well producing at constant and uniform. flux from a closed, box-shaped (3D) drainage volume under pseudo-steady-state conditions. The solution is valid for a well that partially penetrates the reservoir length. On the basis of a series of numerical experiments, they also presented a simplified model for regular Cartesian grids. Although their analytical solution is valid for 3D configurations, the actual well model was restricted to 2D (areal or cross-sectional) cases. Further research on fully 3D models is needed to investigate such factors as the effect of boundary conditions at the well (uniform flux, uniform pressure, or mixed boundary conditions) and partial penetration. All the authors mentioned above have focused their attention on the treatment of wells in a Cartesian grid, which is a special case of the Voronoi grid10,11,12 discussed in this paper (Fig. 1). For this reason, currently used well models were used as a foundation to develop the well models presented here for the Voronoi grid and heterogeneous reservoirs. Exact Well Model for Homogeneous Reservoirs By definition, the use of an exact well index in numerical simulation yields the same well pressure, pw, as that calculated analytically for a given model problem. Because this approach was used first by Peaceman,1 this condition characterizes the well model as Peaceman-type. A model problem consists of defining the reservoir geometry and its external and internal (well) boundary conditions, The linear, single-phase flow equation then is solved in the proposed domain to obtain the analytical solution for pressure at the well of interest, pw. The same problem is solved numerically to compute the pressure of the block containing the well, po. The equivalent radius, req, is evaluated by arranging Eq. 1 asEquation 4 This approach is also applicable to Voronoi grids because there is no restriction on grid geometry. The results presented in the literature show that different model problems produce similar well indices provided there are "enough" gridblocks between wells (or images). While the best model problem is the one that is closest to the actual field configuration, its choice should be based on ease of use, flexibility to represent flow geometries and boundary effects, and capability of conventional simulators to model the same problem. p. 15–21

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