Pressure-Transient Analysis of Arbitrarily Shaped Reservoirs With the Boundary-Element Method

Abstract

Summary Two boundary-element-method (BEM) formulations are proposed for the solutionof pressure-transient problems in proposed for the solution ofpressure-transient problems in homogeneous, anisotropic reservoirs. Pressuresolutions in arbitrary reservoir shapes with multiple sources and/or sinks anda variety of constant and/or time-dependent boundary conditions can begenerated. This technique is superior to numerical methods because it preservesthe analytical nature of the solution and because numerical dispersion andgrid-orientation effects are nonexistent. Procedures for the convolution and Laplace-domain solution procedures are compared, and problems illustrating thevarious aspects of the BEM are solved. Introduction A considerable amount of time and effort is spent analyzing and interpretingpressure data collected from wells to determine reservoir parameters so thatreservoir performance can be predicted. Pressure data are matched to an assumedmodel whose behavior Pressure data are matched to an assumed model whosebehavior is known by analytical or numerical means. Once a reasonable match isobtained, the parameters are evaluated against the solution of the assumedmodel. The basic requirement is that reasonably accurate solutions can begenerated by analytical or numerical means for the particular reservoir inquestion. Analytical methods have several deficiencies. One of the major shortcomingsis that only a few simple geometries lend themselves to the limited number ofexisting solution procedures. Other simple boundary shapes can be generated bysuperposing image wells of a particular type. These methods have been studiedextensively. Earlougher et al. and Earlougher discuss the solution andinterpretation of pressure behavior in bounded rectangular systems. Larsengives a general algorithm for the location of images to generate solutions in afew regular boundary shapes. Extension of this methodology, however, does notnecessarily carry over to other regular and irregular shapes. The image method of generating solutions to various types of boundaryconditions may become too tedious for calculations at long times when theimages fill the entire space and consequently require considerable computingeffort. Also, mixed-boundary-value problems (i.e., problems in which differenttypes of boundary problems (i.e., problems in which different types of boundaryconditions are specified on one section of the boundary) cannot be treated bythese techniques. An example of a mixed-boundary-value problem is the symmetryelement of a constant-pressure, vertical problem is the symmetry element of aconstant-pressure, vertical fracture in a closed square. In such numerical methods as finite differences and finite elements, thegoverning equations are solved approximately in the solution domain. Finite-difference methods, which are widely used in the oil industry, haveproblems with numerical dispersion, grid-orientation effects, and conformanceto the boundaries. In recent years, the boundary integral equation method, alsoknown as BEM, has been considered as a viable alternative to finite-differenceand finite-element methods for the solution of linear differential equationsover irregular boundaries. The advantage of the technique stems from thereduction in the dimensionality of the problem by one; i.e., a 3D problem isreduced to a 2D problem and so on (this advantage is not always realized ifdistributed sources or nonhomogeneous initial conditions are present in thedifferential equation). Also, because it is a present in the differentialequation). Also, because it is a surface or boundary procedure (as a result ofthe reduction in dimensionality), the BEM conforms well to the boundaries. In the BEM, the governing linear differential operator is solved exactlywithin an arbitrarily shaped problem domain. Approximations are made only onthe boundaries. The exact solution of the governing operator in terms of theapproximation at the boundaries provides an analytical flavor to the method andretains the accuracy of the solutions. Thus, the grid-orientation effects andnumerical dispersion suffered by the finite-difference and finite-elementmethods are eliminated. Also, because it is a smoothing procedure, theresulting integrals, even if they cannot be performed analytically, incurminimal error when accurate Gaussian-type, panel-integration schemes areused. The early development of this technique took place within the realm ofpotential theory. The horizons of the method expanded tremendously during the1970's in such engineering disciplines as elastodynamics, heat transfer, fluidmechanics, and groundwater hydrology. Since then, many studies have beenconducted to solve various physical problems. In petroleum engineeringapplications, Numbere and Tiab and Masukawa and Horne used the BEM as astreamline-generating technique for steady-state, water-injection problems forfavorable mobility ratios. Different methods of solving time-dependent problemsgoverned by the diffusivity equation have also been considered, with eachmethod having some useful aspects and drawbacks. Before discussing the specifics of any particular formulation, we outlinethe basic development of a BEM procedure. The governing linear differentialoperator is cast into an integral equation form by the use of the fundamentalsolution of an adjoint operator, which is also known as the free-space Green'sfunction or the derivative function. With Green's identity, the integrals arecast in terms of boundary and domain integrals. The domain integrals representthe inhomogeneity in the governing equation in terms of initial condition orsource terms. The boundary of the problem is then discretized in elements, onwhich interpolation functions are defined. The integral equation is used firstto evaluate the unknown boundary data and then used as quadratures to evaluatesolutions at any interior point. Two formulations, the convolution BEM and the Laplace-space BEM arepresented next. The differences between the two methods and between them andother published formulations are highlighted, and the verification of bothformulations with simple problems that have analytical solutions are shown. Afew practical pressure-transient problems are solved to show the efficacy ofthe pressure-transient problems are solved to show the efficacy of thetechniques. Problem Definition Problem Definition Fig. 1 shows a typical reservoirgeometry. The domain of the problem is defined by, and the correspondingbounding surface is problem is defined by, and the corresponding boundingsurface is represented by on sections of which different types of boundaryconditions could be applied. The nondimensionalized equation for 2D flow of aslightly compressible, single-phase fluid in a homogeneous, anisotropic, finitereservoir in the Cartesian coordinate system, with the coordinate axes alignedwith the principal permeability directions, is permeability directions, is ..........................(1) SPEFE P. 53