The modelling of capillary pressure is a classic multi-scale phenomenon, associated with the fluid surface tension at the scale of the pore, while producing macroscopic effects at the scale of the reservoir. Typically, both the measurement and prediction of capillary effects are challenging enough at the pore scale, but once we look on the scale of the reservoir any model will inevitably be a relatively simple correlated model rather than one derived from first principles.
The modelling at reservoir scale needs to honour known measurements, for example the water saturation at equilibrium, and to cover any effect that could have macroscopic implications. The change in capillary pressure is hysteretic in nature, in part, because processes active on geological time-scales may not be relevant over the time-scale of production. This hysteretic nature is typically captured in reservoir scale simulation via simple correlated models that typically honour input drainage and imbibition curves. An example is the widely used Killough model (ref 2).
Unfortunately use of these hysteresis models in complex full field simulations can lead to increased non-linearity in the problem and hence to slow and unreliable simulation runs. In such cases, engineers will often be tempted to turn off the hysteresis model, using fixed capillary pressure curves and risking biasing their results and potentially seeing non-physical behaviour.
This paper addresses some of the shortcoming of existing hysteresis models and proposes an alternative formulation, where the gradient of the intermediate scanning curves matches the gradient of the bounding drainage curve. This feature prevents a discontinuity in the derivative of capillary pressure that occurs at exactly the conditions in a transition zone where multiple phases are mobile. Taking out this gradient discontinuity has led to improved convergence in a fully implicit black-oil simulator.