Counter-current 1D spontaneous imbibition in scaled form is investigated using Boltzmann transform before and after water meets the closed boundary (early and late time). At early time the system is self-similar and only depends on position divided by square root of time. At late time it also depends on the interaction with the no-flow boundary and hence a second variable, which is set as the square root of time. Diffusion coefficients shifted to high saturations result in early time spatial saturation profiles with shorter front distance, higher average saturation within the imbibition profile and larger imbibed amount. Strongly water-wet systems have zero oil mobility at the inlet, while mixed-wet systems have finite non-zero mobility. The imbibition rate is proportional to inlet diffusion coefficient, inlet saturation gradient (regarding position divided by square root of time) and inverse square root of time. Accordingly, the saturation gradient is infinite and finite for strongly water-wet and mixed-wet systems. At early time, the profile does not change, thus recovery is proportional to square root of time. When the front meets the no-flow boundary (critical time), the saturation profile deviates from the early time profile first at the no-flow boundary, then towards the inlet. When the inlet gradient changes, imbibition rate declines faster than inverse square root of time. The interaction at the inlet and not the closed boundary, thus determines when recovery stops being proportional to square root of time and explains why such proportionality after critical time is reported. The findings were confirmed by matching experimental data.

Mercury injection capillary pressure analysis is a methodology for determining different petrophysical properties, including bulk density, porosity, and pore throat distribution. In this work, distinct parameters derived from mercury injection capillary pressure tests was considered for the prediction of permeability by coupling machine learning and theoretical approaches in a dataset composed of 246 tight sandstone samples. After quality checking the dataset, the feature selection was carried out by correlation analysis of different theoretical permeability models and statistical parameters with the measured permeability. Finally, porosity, median capillary pressure, Winland model, and mean pore-throat radius (corresponding to the saturation range 0.4-0.8) were chosen as the input features of the machine learning model. As the machine learning approach, a support vector machine (SVM) model with a radial basis function kernel was proposed. Furthermore, the model and its metaparameters were trained with a particle swarm optimization (PSO) algorithm to avoid over-fitting or under-fitting. In contradiction to the theoretical models, the implemented SVM-PSO model could acceptably predict the experimentally measured permeability values with an R ${}^2$ rate of over 0.88 for training and testing datasets. The introduced approach could reduce the mean relative errors from about 10 to values less than 0.45. The improvements were more significant for low permeability samples. This successful implementation shows the potential of coupled usage of theoretical and machine learning methodologies for improved prediction of permeability of tight sandstone rocks.

In tight shales, gas is stored in both free and adsorbed form. A one-dimensional model is derived for shale gas production by pressure depletion where the adsorbed layer thickness is of similar magnitude as the pore radius and can affect flow performance. The adsorbed layer thickness is a function of pressure. Different pore geometries are assumed varying continuously between spherical pores to more fracture shaped pores. The shale is assumed compressible and its porosity and pore radius reduce with pressure depletion. The effective pore radius (pore radius minus adsorption layer thickness) controls intrinsic and apparent permeability. The impact of the adsorption layer, compressibility and geometry are investigated. A given adsorbed layer thickness fills more of the pores when they are more spherical and concentrates more of the volumetric flow to the effective pore boundaries giving lower permeability for a given effective radius. Increasing the adsorbed thickness increases the adsorbed fraction initial gas in place. A high volume fraction adsorbed gas reduces apparent permeability and delays production. Pressure depletion causes both pore radius and adsorbed layer to be reduced. The change in adsorbed layer thickness is low at high pressure and greater at low pressure, while pore radius changes more linearly and more with higher compressibility. The free gas saturation and slip increases with pressure depletion for low compressible cases, but if matrix compression dominates there can be a net reduced permeability. Recovery was linear with the square root of time for all cases. Adsorbed gas is less effectively produced by pressure depletion than free gas and higher adsorbed content by more spherical pores or higher layer thickness reduces end recovery. Higher compressibility reduces permeability and delays recovery but increases end recovery.

The shape, size, and connectivity of porous structures control the overall storage capacity and flow in oil and gas reservoirs. The mercury intrusion capillary pressure (MICP) technique is widely utilized to measure capillary pressure and calculate pore size distribution of core samples in the geo-energy industry. Combining the MICP capillary pressure data with parameters from other experimental methods (such as scanning electron microscopy, and nuclear magnetic resonance) or theoretical approaches (such as fractal theory) can more accurately describe the pore structure of reservoirs. In this paper, the latest advances on the application of primary drainage MICP curves from reservoir porous structures are reviewed in three main aspects: The measurement and calculation of MICP capillary pressure, estimation of pore size distributions making use of fractal characteristics, and determination of permeability. Experimental measurements and numerical simulation methods of MICP capillary pressure with its influencing factors are also discussed. MICP capillary pressure combined with other methods are argued to be one of the main directions for future research on reservoir pore structures.