New Analytical and Numerical Solutions for the Radial Convection-Dispersion Problem
- D.H.E. Tang (Arco Oil & Gas Co.) | D.W. Peaceman (Consultant)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Engineering
- Publication Date
- August 1987
- Document Type
- Journal Paper
- 343 - 359
- 1987. Society of Petroleum Engineers
- 2.5.2 Fracturing Materials (Fluids, Proppant), 5.3.2 Multiphase Flow, 5.6.5 Tracers, 5.5 Reservoir Simulation, 4.3.4 Scale, 5.2 Reservoir Fluid Dynamics
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New analytical and numerical solutions are presented for the radial convection-dispersion differential equation with a velocity-dependent dispersion coefficient. The finite difference numerical scheme is shown to have negligible numerical dispersion for this problem. Very close agreement is obtained between the analytical and numerical solutions. The effect of the type of boundary condition used at the wellbore is discussed.
Numerical simulations are routinely performed by engineers to predict the movement of reservoir fluids and to screen effective injection schemes. Unfortunately, capabilities of existing simulators are severely limited by numerical dispersion problems, particularly simulators associated with tertiary EOR processes. In the EOR process, a slug of valuable hydrocarbon or surfactant is injected. Because of numerical dispersion, the simulator will often predict significant smearing of the injected slug in the reservoir. Therefore, the use of miscible displacement simulators to design EOR processes may lead to serious uncertainties.
Obviously, the analytical solution of a given problem can provide a valuable check on the accuracy of a numerical solution that simulates the solute transport in porous media, such as in a miscible displacement problem. One prototype is the radial dispersion problem: analyzing convective-dispersive transport of a slug in a steady, radial flow from an injection well that fully penetrates a homogeneous reservoir of uniform thickness and infinite areal extent. Aside from the obvious importance in the study of solute transport from injection wells, the radial dispersion problem also is probably the simplest case for which the dispersion coefficient is a function of a spatially varying velocity field.
During the past two decades, numerous investigators have presented theoretical models and experimental measurements. (Work before 1965 has been summarized by Hoopes and Harleman.1) Most of the analytical solutions available to date have dealt with problems with a constant dispersion coefficient. Tang and Babu2 have presented an analytical solution for the radial dispersion problem with first-type boundary conditions at the injection well. In the area of numerical methods, Chase3 has presented a finite-element numerical solution for the physical problem discussed in this paper. However, Chase concluded that, even with a finite-element method, significant numerical dispersion could still exist.
The purpose of this paper is first to present analytical solution for the radial convective-dispersive equation with slug injection. Then, the analytical solution is used to check against the solution by a numerical approach. The finite-difference numerical method adopted is a conventional time-implicit, center-differenced method. We show, however, that the numerical method used actually results in a negligible amount of numerical dispersion for the problem of interest.
When one fluid is miscibly displacing another in a porous medium and when the displacement is stable so that viscous fingers do not form, the following convection-dispersion equation describes the overall transport and mixing of incompressible fluids flowing through a porous medium.
For the linear system with constant diffusivity, D, several analytical solutions can be found in the literature. In general, they incorporate various combinations of the error function, differing according to the boundary condition imposed. The most common boundary conditions at the wellbore are
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