Abstract
The one-dimensional (1D) material balance equations for multiphase multicomponent transport in porous media can be cast into forms, analogous to characteristic equations, that express explicitly the velocities at which fixed values of concentration are propagated. Use of these concentration-velocity equations to control the frequency with which component fluxes from finite-difference gridblocks ate updated leads to greatly reduced numerical dispersion, as demonstrated in miscible flooding, waterflooding, surfactant flooding, and other example problems.
Introduction
Accurate numerical simulation of enhanced oil-recovery processes, such as CO2, surfactant, thermal, or caustic flooding can involve calculations of phase behavior, interfacial tension, relative permeabilities, viscosities, heat and mass transfer, and even chemical reactions, thereby requiring considerable computational effort for each meshpoint or gridblock at each timestep. It is therefore impractical to resolve the steep concentration or thermal gradients often present in these processes by resorting to ultrafine meshes. Because the mathematical description of such processes is often unavoidably complex, it is important that the numerical technique be simple and ruggedly insensitive to the details of the process description, if one is to avoid becoming ensnarled in cumbersome and tedious programming and debugging.Although the finite-difference method's simplicity is its great advantage, its accuracy is seriously deficient, at least when one is using the simplest and most obvious discretizations. Central-difference discretization leads to artificial oscillations and overshoot, and upstream differencing leads to artificial smearing of sharp fronts-i.e., numerical dispersion or truncation error. Upstream difference solutions in two or three dimensions often show a significant dependence on grid orientation. Suggested improvements in the finite-difference technique, such as "transfer of overshoot," "truncation error analysis," or "two-point upstream weighting," still have significant numerical dispersion, grid orientation or oscillation errors.The method of characteristics, or point tracking, incurs no numerical dispersion or overshoot errors, but for general multicomponent, multidimensional problems, computer programs based on these techniques can become labyrinthine in their complexity.The finite-element, or variational, methods hold the potential of significantly reducing overshoot and/or numerical dispersion below that produced by finite difference, but implementation is considerably more complicated and time-consuming.The method of random choice, a technique developed for solving sets of multidimensional hyperbolic equations that appear in gas dynamics, recently has been employed in reservoir simulation. This method is somewhat akin to point tracking, propagating discontinuous fronts without smearing or overshoot errors.A new numerical technique is presented here that has the form and simplicity of finite difference, but utilizes variably timed flux updating (VTU) to gain a considerable improvement in accuracy. The technique is potentially applicable to general multicomponent, multidimensional problems. In this and a companion paper (see Pages 409-419), however, the technique is restricted to problems governed by the following equations.
SPEJ
P. 399^
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