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Steady-state upscaling techniques for multi-phase transport properties have been proposed as an alternative to dynamic methods because they are faster and simpler to implement: there is no need for fine grid simulation and the upscaled transport properties are not case dependant. However, although their speed and ease of implementation is attractive, it is still not clear under what circumstances the assumption of steady state is valid and whether the upscaled transport properties properly capture flow. We have run numerical experiments on some simple geological systems, to investigate the validity of the capillary equilibrium limit (CEL) method, in which it is assumed that capillary forces dominate the distribution of fluids. The CEL method is likely to be valid at small length scales and low flow rates. We begin by demonstrating that existing criteria are not sufficient to accurately specify the conditions under which the method is valid. Upscaling using steady-state methods outside this range can yield significant errors in predicted recovery. However, we are able to match upscaled and fine grid simulation results under a different range of conditions. We have derived a new set of criteria to specify the conditions under which the CEL method is valid, which are applicable to a variety of different geological heterogeneities including those in which capillary trapping occurs. We find that the CEL method can be applied over length-scales of up to tens of meters for thinly laminated systems (i.e. high aspect ratios) and with low to near typical flow rates. Our results demonstrate that the CEL method can be used in the first step of a two-step upscaling process, to generate effective multi-phase transport properties quickly and cheaply which capture the impact of small-scale heterogeneities at the scale of a geological model grid-block. |
