Abstract
In petroleum reservoirs, it is common to assume that the superficial velocity of a fluid phase equals a mobility coefficient (permeability over viscosity) multiplying the opposite of the pressure gradient. Phase mobility is typically a function of phase saturations, compositions, temperature, and to a lesser extent pressure, but in specific circumstances it may also depend on the velocity itself. For example, in polymer flooding the aqueous phase viscosity is velocity-dependent due to non-Newtonian effects.
The numerical treatment of velocity-dependent mobilities in reservoir simulators based on the finite-volumes method is still an unsettled topic. For a fully implicit discretization, the main difficulty lies in the fact that velocities are not necessarily aligned with the grid; therefore the velocity-dependent mobility governing the flux across a cell face is not merely a function of the normal pressure drop. In addition, the implicit relationship between velocity and pressure gradient needs to be inverted for each flux, at each nonlinear iteration. To get around the above difficulties, several commercial or academic simulators implement a semi-implicit scheme where the pressure gradient driving the flow is evaluated implicitly, while the velocity used in mobility calculations is evaluated explicitly based on the previously converged time step. However, a semi-implicit formulation may be subject to stability restrictions.
In this work, we first review the derivation of a linear stability criterion for the two-level semi-implicit discretization of simplified monophasic, non-Newtonian or non-Darcy flow equations. Based on this criterion, we propose an adaptive-implicit strategy where for each individual flux the velocity argument in the mobility function is evaluated either explicitly or implicitly. We discuss the numerical accuracy of this scheme and its benefits in terms of computational cost. Finally, using a specifically designed MATLAB code, we validate our adaptive-implicit strategy on representative 1D and 2D nonlinear test cases.
