An Improved Method for Simulating Reservoir Pressures Through the Incorporation of Analytical Well Functions

Abstract

The simulation of reservoir performance is particularly difficult, compared with the simulation of similar processes, because of the extreme behavior of reservoir pressures and saturations. In particular, the reservoir pressures are very non-linear and exhibit near-singularities at the wells as a result of injection and production. Finite difference methods, which are used almost universally for reservoir simulation, are troublesome with such highly non-linear solutions. Accuracy is reduced and the solutions become more time consuming. This paper describes an attempt to eliminate these nonlinearities from the finite difference solutions by incorporating analytical solutions in the pressure solution. The reservoir pressures are the sum of analytical functions and traditional finite difference solutions. The analytical functions are based on well test equations which describe the pressures around the wells in an infinite, homogeneous system. Although heterogeneities and reservoir geometries make these analytical functions grossly inaccurate for predicting actual reservoir pressures, they do exhibit nonlinearities much like the actual solutions. As a result, the finite difference components become much more linear, improving both the accuracy and the speed of the solution. Several other advantages of this new technology are also demonstrated:–Empirical well equations, which relate cell pressures to well pressures, are unnecessary.–The technique is applicable to Cartesian grids as well as other grid types.–Wells need not be centered in the cells for best results.–Accurate simulations of early-time, pressure transients make well test simulations practical. Introduction Despite the similarity of reservoir simulation with other engineering problems such as laminar fluid flow, conductive heat transfer, and convective/diffusive mass transfer, reservoir simulation is uniquely difficult. These difficulties must be due, at least in part, to the large non-linearities inherent in the solutions. For example, the reservoir pressures develop near-singularities at the wells as illustrated in Figure 1. Unlike the other engineering problems, reservoir simulation results depend on flow through very small areas, the well bores. The wells act much as line sources resulting in the very sharp pressure gradients near them. The accuracy of finite difference solutions deteriorates when the solutions are highly non-linear, as can be easily demonstrated through the Taylor's series derivation of the finite difference approximations to the partial differential terms. The solution speed is also affected. This work investigates the elimination of the near-singularities in the finite difference equations, through the combination of analytical functions. As illustrated in Figure 1, an analytical function which accurately represents the pressure gradients around the well bores is added to the finite difference solution. The analytical solution may not accurately represent the actual reservoir pressure, particularly at large distances from the wells, but it does reduce the non-linear behavior of the finite difference component of the solution. Aftab has tried a similar approach to eliminate both the pressure singularities and the saturation discontinuities. Well Equations This new treatment of reservoir pressures eliminates the need for well equations. In the past, well equations have been used to relate well bore pressures to the well cell pressures. This approach follows Peaceman's classical work of 1978.

A singularity removal method for coupled 1D–3D flow models

“…It follows that the p constructed in (13) indeed solves (12) in a suitably weak sense. By a similar argument as the one given in [20, Section 3.2], one finds that F given by (24) belongs to L 2− (Ω) for arbitrarily small > 0. It follows that there exists v ∈ H 2− (Ω) solving (23a)-(23b).…”

“…where E : H 2 (Λ) → H 2 (Ω) is the same extension operator as before,G w is given by (22) with a = a w and b = b w , Ψ w is some smooth cut-off function with respect to line segment Λ w , and v solves…”

“…Many authors have contributed to the extension to generalized well placements, we mention here the work of King et. al in [24], Aavatsmark in [4], and of special relevance to our work, that of Wolfsteiner et al in [39] and Babu et al in [7]. In this work, we take a different approach, in which the singularities are explicitly removed from the governing equations.…”

“…It has previously been studied in the context of reservoir models by e.g. Hales, who used it to improve well modelling for 2D reservoir models [22]. A splitting of the type (6) was introduced by Ding in [16] for the point source problem, where it was used to formulate grid refinement strategies.…”

A new and consistent well model for one-phase flow in anisotropic porous media using a distributed source model

“…Discussion of other discretization schemes can be found in [7], again with the restriction of two-dimensional grids. Enhancements include, among others, three-dimensional slanted wells [8,9], Green's functions for the computation of well indices [10,11,12], or the singularity subtraction method to obtain smooth solutions in the near-well region [13,14].…”