The problem of multiphase flow through underground geological fractured formations arises in various areas of geosciences such as geology, hydrology, petroleum, soil, and environmental engineering. Considering the fact that flow in fractured formations is fracture dominated, relative permeabilities through fractures must be well understood. The comprehensive literature review of this paper shows that few experimental, numerical, and analytical studies have been performed to describe two- and multiphase flow in fractures. Contrary to reservoir rocks, there is no standard experimental procedure for measurement of relative permeabilities through fractures. Furthermore, existing techniques are generally inconsistent, time consuming, expensive, complicated, and may fail due to technical difficulties encountered while performing the experiments. Thus, developing an analytical model for accurate prediction of relative permeabilities through fractures is of vital importance for treating one of significant uncertainties associated with characterization and simulation of fractured reservoirs.

Historically, the straight-line relative permeability model (X-curve) has been widely used in numerical simulation of fractured reservoirs. However, experimental measurements of gas-liquid relative permeabilities through fractures have revealed high degrees of phase interference which cannot be well represented by the X-model. In this work, idealized flow configurations of stratified and segregated flow are assumed in order to make the analytical derivation tractable without the need of using numerical techniques. Then, basic fluid flow equations are combined to develop simple analytical models for gas-liquid flow in each of the patterns. Combining the models with Darcy's law leads to closed-form analytical expressions for explicit calculation of gas and liquid relative permeabilities through fractures. The relationships simply state that the relative permeabilities depend on several parameters one of which being the saturation, whereas they were considered to be only functions of the saturation in the classic X-model. Afterwards, the formulas are further simplified by adopting a realistic assumption. Eventually, the validity of the derived relative permeability expressions is checked by comparing their predictions with available experimental data. The results show that the ability of the segregated model to accurately predict experimental data is considerably poorer than that of the stratified model. Therefore, the generalized relative permeability expressions proposed for the stratified pattern must be used to describe the dynamics of gas-liquid flow in fractures. It addition to robustness, other advantages of the newly presented formulas such as simplicity, cost effectiveness, and being explicit and analytic make their implementation in numerical simulators straightforward.

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