The motivation for reservoir simulation is to optimise the production of oil for different scenarios of enhanced recovery, such as water flooding, or polymer flooding. To give accurate predictions, one is faced with the problem of tracking the motion of the resident and injected fluid (i.e., oil and water). Assuming incompressible flow, this motion is typically modelled by a coupled system of nonlinear equations in total pressure and fluid saturation; an elliptic equation for the pressure and a viscous conservation law for the saturation.

In many situations, the fluid motion from injection to production wells is dominated by convection. In such cases it is reasonable to neglect capillary pressure. The saturation equation then reduces to a nonlinear first order hyperbolic conservation law, which may develop discontinuous solutions even for smooth initial data. These discontinuities correspond to sharp interfaces between fluids, and are of great importance in reservoir simulation. In recent years, many methods have been devised to resolve these sharp fluid interfaces numerically.

A group of Norwegian researchers at NTNU, University of Bergen, and University of Oslo have developed efficient solution methods based upon operator splitting. Two members of our staff, Lie and Natvig, have their university background in this group and have continued working with operator splitting methods at SINTEF. By operator splitting methods we mean methods that split the time evolution into partial steps to separate the effects of convection and diffusion. There are two main motivations for these methods:

1. It is easy to combine modern methods developed for tracking discontinuous solutions with efficient methods for solving implicit discretisations of the parabolic diffusive step, thus giving efficient and powerful methods designed for solving sharp gradients.
2. By separating out the convection in a separate (explicit) step, most of the nonlinearity in the equation is removed and an iterative solution methods for implicit discretisations of the diffusive terms will converge more rapidly.

By using an efficient front tracking method to discretise the convective step, we have developed a very efficient and unconditionally stable method that resolves (almost) all balances of convective and diffusive forces.

References

1. H. Holden, K. H. Karlsen, K.-A. Lie, and N. H. Risebro. Splitting Methods for Partial Differential Equations with Rough Solutions: Analysis and Matlab Programs. EMS Series of Lectures in Mathematics, Vol. 11, European Mathematical Society Publishing House, 2010.
2. H. Holden, K. Hvistendahl Karlsen, and K.-A. Lie. Operator splitting methods for degenerate convection-diffusion equations II: numerical examples with emphasis on reservoir simulation. Comput. Geo., 4(4):287-322, 2000.
3. K. Hvistendahl Karlsen, K.-A. Lie, J. R. Natvig, H. F. Nordhaug, and H. K. Dahle. Operator splitting methods for systems of convection-diffusion equations: nonlinear error mechanisms and correction strategies. J. Comput. Phys., Vol. 173, Issue 2, pp. 636-663, 2001