Spontaneous imbibition (SI) is a key process in many petrophysical applications, ranging from the mass transfer in fractured reservoirs during a waterflood to wettability characterization of rock samples, or steam migration in geothermal reservoirs. Scaling groups are an essential tool for upscaling laboratory data and modeling and describing SI. A general form has been debated for over 90 years, and several dozen specific groups have been proposed. Here, we give the first general scaling group for arbitrary wettability state, viscosity ratios, rock type, initial water content, and boundary conditions. The result is obtained by extending recent findings for water-wet systems but otherwise arbitrary properties (Schmid and Geiger, 2012) to the mixed-wet case. The group is based on the only known exact, general solution to Darcy's equation with capillarity, and we show that this solution can be viewed as the capillary analogue to the Buckley-Leverett solution for viscous dominated flow. Our group serves as a 'master equation' that contains many of the previously obtained groups as special cases, and its generality can be used to give the first predictive theory for the validity range of specific groups. Based on the universal group, we show that SI is best characterized by the cumulative inflow of the wetting phase and not by the movement of the wetting front, as has been conjectured. Furthermore, our results give strong evidence that Darcy's equation is suitable for describing SI, contrary to what has been hypothesized. The general correlation can be fitted by an exponential model for mass transfer that closely correlates 45 published water-oil, and water-air SI experiments obtained for widely different petrophysical properties.

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Multi-phase flow in carbonate reservoirs, which hold about half of the world's remaining oil reserves, is strongly influenced by fractures present in the geological formations. Fractures are often the main flow conduits, leaving most of the oil behind in the low permeability rock matrix, and cause early water breakthrough. An accurate characterization of fracture flow and fluid exchange between fracture and matrix is needed to forecast oil recovery and optimise production in fracture-dominated and fracture-assisted reservoirs. Dual-porosity simulations are traditionally used to model naturally fractured reservoirs. However, the classical dual-porosity models miss some key physics of fracture - matrix fluid exchange: (1) They tend to simplify mass transfer due to spontaneous imbibition since they typically cannot include arbitrary petro-physical properties due to e.g. wettabilities, viscosity ratios, initial water content, etc. (2) They cannot account for the fact that matrix blocks within a single simulation grid cell have various sizes and permeabilities, giving rise to different transfer rates, which should be captured by a distribution of transfer functions in each grid cell. In this paper we present a novel multi-rate dual-porosity model. It is based on an unstructured finite element - finite volume technique, which solves the governing equations for two-phase flow fully implicitly. This allows us to represent complex large-scale geological structures (e.g., non-orthogonal faults and fracture corridors) accurately while small-scale diffuse fractures are modeled with an improved dual-porosity approach. Until recently, calculating the mass transfer between fractures and matrix blocks due to spontaneous imbibition for arbitrary petro-physical and fluid properties exactly was not possible because a general and exact transfer rate for arbitrary petro-physical and fluid properties was not known. In our new dual-porosity model we compute fracture-matrix transfer using the only known analytical and general solution of the Darcy equation including capillarity. This provides us with a generalised transfer function for arbitrary wettability, viscosity ratios, rock types, initial water content and boundary conditions. Each reservoir simulation grid block can contain multiple of these generalised transfer functions to model different matrix permeabilities and/or matrix block sizes present at the sub-grid scale. Using a series of proof-of-concept simulations, we show that the difference in oil recovery using a standard single-rate dual-porosity model and our new multi-rate dual-porosity model cannot be neglected. This demonstrates that our proposed model with the generalised transfer function predicts oil recovery more accurately compared to a classical dual-porosity model.@en

Multi-phase multi-component flow processes are fundamental to engineering applications in hydrocarbon and geothermal reservoirs but also to many classical geological processes. This review will highlight recent developments in the mathematical modelling and numerical simulations of the underlying physical processes from the pore- to the reservoir scale. Many modern approaches now rely on integrating numerical and analytical methods and incorporate results across the different length scales. This provides new insights into the fundamental properties of multi-phase multi-component flow and helps to mitigate some of the inherent difficulties in quantifying them in subsurface reservoirs.

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We derive a set of analytical solutions for the transport of adsorbing solutes in an immiscible, incompressible two-phase system. This work extends recent results for the analytical description for the movement of inert tracers due to capillary and viscous forces and dispersion to the case of adsorbing solutes. We thereby obtain the first known analytical expression for the description of the effect of adsorption, dispersion, capillary forces and viscous forces on solute movement in two-phase flow. For the purely advective transport, we combine a known exact solution for the description of flow with the method of characteristics for the advective transport equations to obtain solutions that describe both co and spontaneous counter-current imbibition and advective transport in one dimension. We show that for both cases, the solute front can be located graphically by a modified Welge tangent. For the dispersion, we derive approximate analytical solutions by the method of singular perturbation expansion. The solutions reveal that the amount of spreading depends on the flow regime and that adsorption diminishes the spreading behavior of the solute. We give some illustrative examples and compare the analytical solutions with numerical results.

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