An Accurate Numerical Technique for Solution of Convection-Diffusion Equations Without Numerical Dispersion
- P.D. Fleming (Gencorp) | J. Mansoori (Phillips Petroleum Corp.)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Engineering
- Publication Date
- August 1987
- Document Type
- Journal Paper
- 373 - 386
- 1987. Society of Petroleum Engineers
- 5.5 Reservoir Simulation, 2.5.2 Fracturing Materials (Fluids, Proppant), 5.4.6 Thermal Methods, 5.4.9 Miscible Methods, 4.1.2 Separation and Treating, 4.1.5 Processing Equipment, 5.2.1 Phase Behavior and PVT Measurements, 5.4.7 Chemical Flooding Methods (e.g., Polymer, Solvent, Nitrogen, Immiscible CO2, Surfactant, Vapex)
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Summary. A technique is presented that optimizes the accuracy of numerical solutions to convection-diffusion equations while eliminating the detrimental effects of numerical dispersion. Such equations arise in the simulation of EOR processes where front smearing as a result of numerical dispersion can distort the sharp displacement fronts that are such an important part of the processes. Such front smearing can also lead to false predictions processes. Such front smearing can also lead to false predictions of breakthrough times. More accurate, and hence more reliable, simulation of oil recovery processes should facilitate the assessment of the economics of EOR processes. Our technique uses a parameterized, third order-correct algorithm and Fourier analysis. parameterized, third order-correct algorithm and Fourier analysis. The Fourier analysis allows for analytical solutions of both the differential equation and the algorithm. This enables the formulation of a concise method for requiring that the numerical solution conforms as closely as possible to the correct solution. The method is illustrated by performing the optimization on several examples with different degrees of convection and dispersion. The solutions have no numerical dispersion and little truncation error. Our method can serve as a starting point for generalization to the solution of more complicated equations that occur in multicomponent, multiphase reservoir simulators. Suggestions for such generalizations are presented.
Virtually all reservoir simulators obtain solutions to fluid flow equations, usually nonlinear partial-differential equations, by making discrete approximations to derivatives. Whether finite-difference or finite-element methods are used, these approximations always introduce truncation errors that often can distort the accuracy and stability of the solution. The truncation error is often referred to as numerical dispersion because, to lowest order, it can be represented as a second spatial derivative term added to any true dispersion term in the problem.
Distortion of the numerical solution is most significant in the simulation of EOR processes where sharp displacement, concentration, and/or temperature fronts are an important part of the efficiency of the processes, and artificial smearing as a result of numerical dispersion can render the simulation meaningless. For processes involving CO2, surfactant, polymer, or steam, it is also necessary to perform calculations of phase behavior, relative permeabilities, viscosities, interfacial tension, heat and mass transfer, and even chemical reactions at each mesh point and timestep. Therefore, resolving the sharp fronts merely by resorting to very fine meshes is impractical. Hence it is essential to be able to derive solution algorithms that have minimal truncation error for a given mesh size. A simple equation that exhibits many of the features of reservoir simulator equations is the one-dimensional (1D) linear convection-diffusion equation
where v is the convection rate (velocity) and D is the diffusion coefficient. This equation can be viewed as a good approximation to oil recovery by miscible displacement, immiscible displacement, or thermal methods. Therefore, most discussions of the effects of numerical dispersion have used Eq. 1 as a prototype, although some have preferred to work with specific simulator equations.
Lantz was the first to quantify numerical dispersion in ID equations like Eq. 1. His formulation was used by Peaceman to characterize numerical solutions to such equations and proved useful in interpreting the results of a simple chemical-flood model, where the predicted oil recovery was strongly dependent on gridblock size. Fanchi generalized Lantz's formulation to three dimensions. Larson, Jensen and Finlayson, and Laprea-Bigott and Morse used different forms of front-tracking schemes to resolve sharp fronts. Jensen and Finlayson actually used a moving coordinate system and finite-element methods in tracking fronts; the resultant moving boundary conditions were the price paid for the effective front tracking. price paid for the effective front tracking. SPERE
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