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We present a review of pore-scale simulations of immiscible fluid transport with focus on two of the most popular approaches: lattice Boltzmann modeling for direct simulations on digital models of the pore space and simulations on network models extracted from the pore space. This review focuses on covering basic theory and implementation strategies and gives the readers input and motivation to start their own pore-scale simulations and relate them to realistic porous media. We present a review of recent and relevant applications and how a digital workflow that combines advanced pore-scale imaging and simulations can give very useful input to different fields of science and industry, including reservoir characterization. Given the large span in methods and applications, this review does not aim to cover all methods or applications. However, it covers popular methods and describes to some extent their applicability to different types of transport problems.

Abstract X-ray computerized tomography (CT) images as digital representations of whole cores can provide valuable information on the composition and internal structure of cores extracted from wells. Incorporation of millimeter-scale core CT data into lithology classification workflows can result in high-resolution lithology description. In this study, we use 2D core CT scan image slices to train a convolutional neural network (CNN) whose purpose is to automatically predict the lithology of a well on the Norwegian continental shelf. The images are preprocessed prior to training, i.e., undesired artefacts are automatically flagged and removed from further analysis. The training data include expert-derived lithofacies classes obtained by manual core description. The trained classifier is used to predict lithofacies on a set of test images that are unseen by the classifier. The prediction results reveal that distinct classes are predicted with high recall (up to 92%). However, there are misclassification rates associated with similarities in gray-scale values and transport properties. To postprocess the acquired results, we identified and merged similar lithofacies classes through ad hoc analysis considering the degree of confusion from the prediction confusion matrix and aided by porosity–permeability cross-plot relationships. Based on this analysis, the lithofacies classes are merged into four rock classes. Another CNN classifier trained on the resulting rock classes generalize well, with higher pixel-wise precision when detecting thin layers and bed boundaries compared to the manual core description. Thus, the classifier provides additional and complementing information to the already existing rock type description. Article Highlights A workflow for automatic lithofacies classification using whole core 2D image slices and CNN is introduced. The proposed classifier shows lithology-dependent accuracies. The prediction confusion matrix is exploited as a tool to identify lithofacies classes with similar transport properties and to automatically generate lithofacies hierarchies.

In this work, we calculate contact angles in X-ray tomography images of two-phase flow in order to investigate the wettability. Triangulated surfaces, generated using the images, are smoothed to calculate the contact angles. As expected, the angles have a spread rather than being a constant value. We attempt to shed light on sources of the spread by addressing the overlooked mesh corrections prior to smoothing, poorly resolved image features, cluster-based analysis, and local variations of contact angles. We verify the smoothing algorithm by analytical examples with known contact angle and curvature. According to the analytical cases, point-wise and average contact angles, average mean curvature and surface area converge to the analytical values with increased voxel grid resolution. Analytical examples show that these parameters can reliably be calculated for fluid–fluid surfaces composed of roughly 3000 vertices or more equivalent to 1000 pixel2. In an experimental image, by looking into individual interfaces and clusters, we show that contact angles are underestimated for wetting fluid clusters where the fluid–fluid surface is resolved with less than roughly 500 vertices. However, for the fluid–fluid surfaces with at least a few thousand vertices, the mean and standard deviation of angles converge to similar values. Further investigation of local variations of angles along three-phase lines for large clusters revealed that a source of angle variations is anomalies in the solid surface. However, in the places least influenced by such noise, we observed that angles tend to be larger when the line is convex and smaller when the line is concave. We believe this pattern may indicate the significance of line energy in the free energy of the two-phase flow systems. Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Com-mons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

A widely used approach for upscaling relative permeability is based on a steady-state assumption. For small time intervals and at small scales, the flooding process can be approximated as being in a steady state. However, at large scales with large time steps, water flooding of a reservoir is an unsteady process. In this article, we first investigate the balance of viscous, capillary and gravity forces on the fine scale during the water flooding of a reservoir at different flow velocities. We introduce a semi-analytical method to find the low-rate limit solution, while the high-rate limit solution is found by running a simulation without gravity and capillary pressure. These limit solutions allow us to understand when rate-dependent simulations approach a point where some forces become negligible. We perform a series of numerical simulations on the fine scale to construct solution transitions between the established outer limits. Simulations are run both on homogeneous models, on different layered models and on a more complex two-dimensional model. The rate-dependent simulations show smooth transitions between the low- and high-rate limits, and these transitions are in general non-trivial. In all our example cases, one of the limit solutions gives a lower bound for the rate dependent results, while they do not in general provide an upper bound. Based on the rate-dependence of the force balance, we evaluate when different steady-state upscaling procedures are applicable for an unsteady flooding process. We observe that the capillary-limit upscaling, which also takes gravity into account, reproduces the low-rate limit fine-scale simulations. Such capillary-limit upscaling is also able to reproduce the transition to capillary equilibrium normal to the flow direction. As already known, the viscous-limit upscaling is only applicable when we have close to constant fractional flow within each coarse grid block.

In this article, we present fundamental transport property relations incorporating direct descriptors of the pore structure. The pore structure descriptors are defined from streamline decomposition of the numerical solutions of the transport equations. These descriptors have been introduced earlier, while the calculations are extended to voxel-based microstructures in this article. The pore structure descriptors for the respective transport equations are used in turn to obtain rigorous cross-property relations for porous media. We derive such cross-property relations exemplarily for computed tomography (CT) data and digital rock models of Fontainebleau sandstone, and CT data of two reservoir sandstone facies. Pore structure parameterizations of these porous media are given for electrical conductance and fluid permeability in the microstructure, yielding correlations for the transport property-dependent descriptors of effective porosity, tortuosity and constriction. These relations are shown to be well-correlated functions over the range of sample porosities for the Fontainebleau sandstone. Differences between the outcrop Fontainebleau sandstone and the reservoir samples are observed mainly in the derived hydraulic length descriptor. A quantitative treatment of the obtained cross-property functions is provided that could be applied for porous medium characterization. It is suggested that such cross-property investigation honoring the detailed microstructure will lead to more fundamental relations between porous medium properties, which could be exploited for example in rock typing or wire-line log interpretation.

Permeability and formation factor are important properties of a porous medium that only depend on pore space geometry, and it has been proposed that these transport properties may be predicted in terms of a set of geometric measures known as Minkowski functionals. The well-known Kozeny–Carman and Archie equations depend on porosity and surface area, which are closely related to two of these measures. The possibility of generalizations including the remaining Minkowski functionals is investigated in this paper. To this end, two-dimensional computer-generated pore spaces covering a wide range of Minkowski functional value combinations are generated. In general, due to Hadwiger’s theorem, any correlation based on any additive measurements cannot be expected to have more predictive power than those based on the Minkowski functionals. We conclude that the permeability and formation factor are not uniquely determined by the Minkowski functionals. Good correlations in terms of appropriately evaluated Minkowski functionals, where microporosity and surface roughness are ignored, can, however, be found. For a large class of random systems, these correlations predict permeability and formation factor with an accuracy of 40% and 20%, respectively. © The Author(s) 2019 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Steady-state upscaling of relative permeability is studied for a range of reservoir models. Both rate-dependent upscaling and upscaling in the capillary and viscous limits are considered. In particular, we study fluvial depositional systems, which represent a large and important class of reservoirs. Numerical examples show that steady-state upscaling is rate dependent, in accordance with previous work. In this respect we introduce a scale-dependent capillary number to estimate the balance between viscous and capillary forces. The difference between the limit solutions can be large, and we show that the intermediate flow rates can span several orders of magnitude. This substantiate the need for rate-dependent steady-state upscaling in a range of flow scenarios. We demonstrate that steady-state upscaling converges from the capillary to the viscous limit solution as the flow rate increases, and we identify a simple synthetic model where the convergence fails to be monotone. Two different sets of boundary conditions were tested, but had only minor effects on the presented reservoir models. Finally, we demonstrate the applicability of steady-state upscaling by performing dynamic flow simulation at the reservoir scale, both on fine-scaled and on upscaled models. The considered model is viscous dominated for realistic flow rates, and the simulation results indicate that viscous limit upscaling is appropriate., Comment: 25 pages, 18 figures, 4 tables

In this article we investigate the permeability of a porous medium as given in Darcy's law. The permeability is described by an effective hydraulic pore radius in the porous medium, the fluctuation in local hydraulic pore radii, the length of streamlines, and the fractional volume conducting flow. The effective hydraulic pore radius is related to a characteristic hydraulic length, the fluctuation in local hydraulic radii is related to a constriction factor, the length of streamlines is characterized by a tortuosity, and the fractional volume conducting flow from inlet to outlet is described by an effective porosity. The characteristic length, the constriction factor, the tortuosity and the effective porosity are thus intrinsic descriptors of the pore structure relative to direction. We show that the combined effect of our pore structure description fully describes the permeability of a porous medium. The theory is applied to idealized porous media, where it reproduces Darcy's law for fluid flow derived from the Hagen-Poiseuille equation. We also apply this theory to full network models of Fontainebleau sandstone, where we show how the pore structure and permeability correlate with porosity for such natural porous media. This work establishes how the permeability can be related to porosity, in the sense of Kozeny-Carman, through fundamental and well-defined pore structure parameters: characteristic length, constriction, and tortuosity., Comment: 20 pages, 8 figures, 1 table