An Improved Streamline-Generating Technique That Uses the Boundary (Integral) Element Method

Abstract

Summary. The boundary element method (BEM) has been used to generate streamlines for homogeneous and sectionally homogeneous reservoirs having irregular boundaries. This technique is superior to earlier methods that use "image wells" to simulate the reservoir boundary, especially for cases where the reservoir boundary is irregularly shaped. It has wider applicability to different kinds of reservoirs and eliminates the need for trial-and-error solutions. The errors caused by discretization and numerical approximation arise on and adjacent to the boundaries only, therefore the pressures and pressure gradients on which the streamlines are derived can be calculated with very high accuracy in the interior of the reservoir. Thus, the streamlines generated with this method are expected to be more realistic and representative of the actual physical system. Introduction Stream-line models are widely used in the petroleum industry to delineate the theoretical paths to be taken by injected fluids before expensive secondary and tertiary recovery processes are initiated. These paths can help to determine the area that can be swept by the injected fluids, and by experimenting with various injection arrangements, an optimum pattern and injection strategy can be developed. Sometimes, the streamlines are converted to so-called stream-tubes, which can be incorporated into predictive models to estimate the expected recovery performance for the proposed project. Considering its widespread use, the current "method of images" used to simulate the boundary effects when generating the stream-lines has some limitations that make it inadequate for reservoirs with irregularly shaped boundaries and for nonhomogeneous reservoirs. The image-well method is best suited to reservoirs whose boundaries can be considered as straight lines, such as square or rectangular reservoirs. But even in these cases, the image-well technique results in an infinite array of image wells and thus poses the problem of knowing when to terminate the array. The image-well problem of knowing when to terminate the array. The image-well technique cannot be used in nonhomogeneous reservoirs except for the simplest cases with one or two linear boundaries. These limitations have been recognized for some time, and several attempts have been made to rectify them. Chan presented an improved image-well method for reservoirs with straight boundaries that involves a rearrangement of the image pattern depending on the boundary configurations. However, the number of image wells required still prohibits its use in all but simple geometric shapes. Leblanc used the image-well technique to bound the streamlines in a homogeneous reservoir of arbitrarily shaped boundary by using a trial-and-error procedure to determine the image-well rates required to confine the streamlines adequately. Lin extended the method to include reservoirs that have some restricted flow, as in the case of partial water encroachment. AH the above improvements are only partially satisfactory because they all use the method of images, which has the limitations mentioned above. The problem of irregular boundaries was not satisfactorily resolved either, because the improvements of Leblanc and Lin give rise to a trial-and-error solution to determine which combination of image-well location and rate best confines the streamlines within the domain of the porous medium. This procedure may produce streamlines that may not necessarily represent the pattern of the streamlines in a real system. A different approach is used that is based on the BEM or the boundary integral element method (BIEM). In this method, the governing differential equation is transformed into an equivalent integral equation. Discretizing the integral equation to solve it nu-merically results in the discretization of the boundary of the problem only (without discretizing the domain). This breaking of the boundary into discrete segments is what enables irregular boundaries to be modeled. This method of streamline modeling removes most of the limitations encountered with the image-well method. It is not only more accurate but also has more general applicability. For example, the shape of the boundary does not have to be defined by straight lines, but can be curved and completely irregular; the boundary conditions can be of any type; and the method can be applied to a nonhomogeneous reservoir as long as the reservoir can be reduced to a few regions having homogeneous properties. This paper explains the method and shows examples of it's applications to streamline modeling in homogeneous and sectional homogeneous reservoirs. Problem Formulation Problem Formulation Consider a two-dimensional (2D) homogeneous porous medium having n, sources and sinks located arbitrarily within its domain, omega, (see Fig. 1). Its bounding surface, S, is assumed to be of irregular shape. The surface, S, can be made up of any combination of three surfaces denoted by S, S, and S, respectively. Over S, the pressures are prescribed. Over S, the pressure gradients normal to the boundary are prescribed. Over S, a mixed boundary condition can be prescribed. Neglecting gravitational effects, the equation describing the steady flow of a slightly compressible fluid in a homogeneous, 2D porous medium containing n, sources and sinks is given as (1) subject to p=p(X, Y) specified on S and delta p/delta n=p (X, Y) specified on S, wherek = permeability, p = pressure, mu = fluid viscosity, h = thickness of pay section, q = rate per unit area of jth well (positive for injectorsand negative for producers), delta = Dirac delta function, X, Y = coordinate axes, X, Y = coordinates of source/sink locations for j = 1,2....n SPERE P. 1061