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We develop an efficient computational model for simulating fluid invasion patterns emerging in variable aperture fractures. This two-dimensional model takes into account the effect of capillary force on the fluid-fluid interfaces and viscous pressure drop in both fluid phases. The pressure distribution is solved at each time step based on mass balance and local cubic law, considering an imposed pressure jump condition at the fluid-fluid interface. This pressure jump corresponds to the Laplace pressure which includes both terms related to the out-of-plane (aperture-spanning) curvature and to the in-plane curvature. Simulating a configuration that emulates viscous fingering in two-dimensional random porous media confirms that the model accounts properly for the role of viscous forces. Furthermore, direct comparison with previously obtained experimental results shows that the model reproduces the observed drainage patterns in a rough fracture reasonably well. The evolutions of tip location, the inlet pressures, and the invading phase fractal dimensions are analyzed to characterize the transition from capillary fingering to viscous fingering regimes. A radial injection scenario of immiscible invasion is also studied with varying modified capillary number and viscosity ratio, showing displacement patterns ranging from capillary fingering to viscous fingering to stable displacement. Such simulations using two contact angles show that the invading phase becomes more compact when the wetting condition changes from strong to weak drainage, as already observed in 2-D porous media. The model can be used to bridge the gap in spatial scales of two-phase flow between pore-scale modeling approaches and the continuum Darcy-scale models.

Plain Language Summary The flow of two or more fluids in fractured media is an important process involved in many industrial and environmental applications in the subsurface with examples spanning from contaminant transport to petroleum recovery to geological storage of carbon dioxide. Understanding and controlling two-phase flow in fractures is critical from both the scientific and technological points of view. In this study, we focus on the displacement of one fluid by another immiscible one in a rough fracture, a fundamental process underlying two-phase flow in fractured media. We develop an efficient computational model for simulating fluid invasion patterns emerging in rough fractures. This model takes into account the effect of capillary force on the fluid-fluid interfaces and viscous pressure drop in both fluid phases. Direct comparison with experimental results shows that the model output matches the observed patterns reasonably well. In addition, generic simulations demonstrate the ability of the model to produce flow patterns that fall into regimes whose dependence on the viscosity ratio and capillary number is similar to that of the classical phase diagram. The model can be used to bridge the gap in spatial scales of two-phase flow between pore-scale modeling approaches and the continuum Darcy-scale models.