The Convergence of Upstream Collocation in the Buckley-Leverett Problem(includes associated papers 14810 and 14970 )
- Myron B. Allen (U. of Wyoming) | George F. Pinder (Princeton U.)
- Document ID
- Society of Petroleum Engineers
- Society of Petroleum Engineers Journal
- Publication Date
- June 1985
- Document Type
- Journal Paper
- 363 - 370
- 1985. Society of Petroleum Engineers
- 4.1.2 Separation and Treating, 5.3.2 Multiphase Flow, 5.1.5 Geologic Modeling, 1.2.3 Rock properties, 5.5 Reservoir Simulation, 4.3.4 Scale
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Upstream collocation is a fast and accurate scheme for simulating multiphase flows in oil reservoirs. In contrast to standard orthogonal collocation, upstream collocation yields numerical solutions to the Buckley-Leverett problem that converge to correct solutions physically. The failure of standard orthogonal collocation is not surprising, since the Buckley-Leverett problem as commonly stated is posed incompletely. The equal-area rule of Buckley and Leverett and the Welge tangent construction both specify additional constraints needed to close the problem properly. An error analysis of upstream collocation shows that this method forces convergence through an artificial dissipative term analogous to the "vanishing viscosity" used in shock fitting. This constraint is mathematically equivalent to the more familiar constructions and should prove beneficial in stimulating EOR schemes based on frontal displacement.
The Buckley-Leverett saturation equation is of fundamental importance in the mechanics of oil recovery, yet solving the equation poses difficulties when saturation shocks are present. Analytic or graphic methods must negotiate triple-valued saturations, while naively applied numerical solutions may yield incorrect solutions. Orthogonal collocation is a noteworthy example: it conserves mass but, like centered difference schemes, misplaces shocks. All these problems reflect the fact that the usual statements of Cauchy problems for the Buckley-Leverett saturation equation are incomplete. To guarantee uniqueness of discontinuous solutions requires, in addition to initial data, the specification of a shock condition that is mathematically equivalent to several physically reasonable constraints. The Buckley-Leverett equal-area rule and Welge's tangent construction both implement this extra condition. A recently developed numerical method called upstream collocation overcomes the convergence failures of orthogonal collocation, generating solutions with steep gradients at the correct shock location. The intent of this paper is to demonstrate that upstream collocation enforces the correct shock condition through an error term that mimics dissipation but vanishes on refinement of the spatial grid. This error term parallels the lowest-order error terms in upstream-weighted finite differences and achieves the same effects as the artificial capillary pressures used in several earlier finite-element formulations. The analysis leading to the form of the error rests on a correspondence between collocation and Galerkin schemes and follows a line of reasoning originally developed for the linear, parabolic convection-diffusion equation. The gist of the argument is that upstream collocation corresponds to an erroneous approximation of the integrals arising in Galerkin's method. Calculation of the quadrature error leads to an expression for the artificial dissipation induced by upstream collocation. To clarify why such an error term is appropriate, we precede the error analysis with a brief review of the physical setting, solutions, and mathematics of the Buckley-Leverett problem.
The Buckley-Leverett saturation equation describes the simultaneous flow of two immiscible, incompressible fluids through a homogeneous porous medium. The equation is important in oil production because fluid injection and displacement are common to essentially all EOR schemes. In simple applications the displaced fluid is oil, and the displacing fluid may be either water or gas. The equation arises from a material balance on water and the two-phase extension of Darcy's law, as Buckley and Leverett describe in their original paper. For a one-dimensional reservoir with uniform rock properties, combining this set of governing equations yields
where S is water saturation, q is the effective total flow rate [L/T], and f(s) signifies the fraction of flowing fluid that is water. We assume a consistent set of units. In a horizontal, vertically uniform reservoir, the fractional flow function off is related to saturation-dependent rock and fluid properties as follows:
where (S) and (S) are the oil and water mobilities, respectively, and(S) is the capillary pressure.
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