Representing Wells in Numerical Reservoir Simulation: Part 1 – Theory

Abstract

A reservoir simulation system uses an analytical model to represent flow within a grid block as it enters or leaves a well. This model is called a well model. We give a description here of the theoretical background of a well model, including how the sandface pressure and saturation boundary conditions can be calculated and how the well boundary itself can be replaced (approximately) by a source function. This paper and the following companion paper, SPE 9770, present a unified viewpoint of material, some of which may be already familiar to simulator developers. Introduction Our concern in this paper is the theory of representation of wells and the well boundary condition in a reservoir simulator.It frequently has been noted that, except in the case of a central well in a problem involving cylindrical coordinates, it is impractical to represent a well with an internal boundary. The ratio of well radius to desired grid-block length can be of order 0.001 or less. In such cases, an alternative procedure has evolved in which the well is represented by a source. The relationship between the source strength, the wellbore flow, and the flow in the surrounding grid blocks composes an essential part of the well model. Even when the grid around a well is sufficiently fine to represent the well as an internal boundary, other features such as partial perforation, partial penetration, or skin may be important to the local flow but extend over a "small" interval in relation to the appropriate grid-block dimension. Here also, a suitable source representation is advantageous. We shall develop the source representation of a well for a variety of circumstances.The well boundary condition generally involves the sandface pressure and flow rate. However, these quantities also must be consistent with the requirements of wellbore flow - i.e., reservoir and wellbore flows are coupled, and a wellbore flow model is required. We describe a means of treating a wide variety of wellbore flows without creating a numerically cumbersome simulator. We hope that this paper may provide a basis for further work and discussion of this essential topic. Review of Literature The source representation of a well can be described as a local, approximate, steady, singular solution of the flow equations. The idea of separating a singularity of this type for special treatment is an old idea in applied mathematics. In series solutions to certain elliptic and parabolic equations, it was found that the convergence of the series could be improved considerably by first extracting the singular part. In these cases the singular solution extended through the entire domain. The analogous approach using numerical methods in place of the series solution is also well known. The use of singular solutions in a purely local role in numerical solutions was introduced before the general use of digital computers. Woods' use of a local logarithmic expression in a solution of Poisson's equation by relaxation methods corresponds closely to the source representation of a well recently proposed by Peaceman. SPEJ P. 323＾

Analysis of Some Difference Schemes Used in Numerical Simulation of Production/Injection Wells

“…These results are a modification of approximations from [2] for an interior grid block with a well located at the center of a block. The results obtained in [7] are similar to those of [2], except that in [7] no restriction was placed on the position of the well within a grid block. This approach was summarized in [11]- [15] In [4], [8], and [9], the differential problem (1) - (3) is approximated with an 'adjustment' method in two different ways which yield varying accuracies.…”

“…It can be shown using results from [21] that difference scheme (9) from [4], [8], [9] is the same as from [2], [5]- [7], and [10]. Consequently, the latter do not provide an asymptotical approximation (in a mathematical sense) either.…”

“…Examination of the numerical solution for Laplace's equation using the repeated five-spot pattern/stencil showed that the numerical solution for pressure plotted versus radius exhibits a linear trend. The results from [2], [5]- [7], [10] achieve a better description of well block pressure by modifying the approximation in terms 'equivalent well radius'. The authors of [4] approached the problem using the five-point pattern for a well located at a node, and introduced a correction factor for each used approximation.…”

Upscaling and Simulation of Waterflooding in Heterogeneous Reservoirs Using Wavelet Transformations: Application to the SPE-10 Model

“…Separating a singularity of this type for special treatment is an old idea in applied mathematics (Collatz 1960;Carslaw and Jaeger 1963;Fox 1971). We use the ideas described by Crichlow (1977) and Williamson and Chappelear (1981).…”

Analytical Well Models for Reservoir Simulation

“…2,3 While the numerical approach to obtaining r 0 may be necessary in some situations, in many cases (1/2< a < 2) the radial flow assumption used to derive Eq. 6 may be generalized.…”