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We measure the pressure difference during two‐phase flow across a sandstone sample for a range of injection rates and fractional flows of water, the wetting phase, during an imbibition experiment. We quantify the onset of a transition from a linear relationship between flow rate and pressure gradient to a nonlinear power‐law dependence. We show that the transition from linear (Darcy) to nonlinear flow and the exponent in the power‐law is a function of fractional flow. We use energy balance to accurately predict the onset of intermittency for a range of fractional flows, fluid viscosities, and different rock types. Plain Language Summary: Multiphase flow in porous media is described by an empirical extension of Darcy's law where flow rate is proportional to pressure gradient. Darcy's law is used to quantify and design a wide variety of processes including carbon dioxide storage in the subsurface, and air and moisture flows through materials as diverse as soils and surgical masks. However, as flow rate increases, this linear Darcy law breaks down with a transition to a nonlinear intermittent flow regime. Through an extensive set of experimental measurements, we quantify the relationship between flow rate and pressure gradient. A theory based on energy balance accurately predicts the transition to intermittency. Our work quantifies multiphase flow over a wide range of conditions. Key Points: Flow rate and pressure gradient during steady‐state multiphase flow are measured experimentallyThe transition to a non‐Darcy flow regime is quantified and predicted theoreticallyThe theory makes accurate predictions for several rock types, fractional flows, and mobility ratios [ABSTRACT FROM AUTHOR]

We use synchrotron X-ray micro-tomography to investigate the displacement dynamics during three-phase—oil, water and gas—flow in a hydrophobic porous medium. We observe a distinct gas invasion pattern, where gas progresses through the pore space in the form of disconnected clusters mediated by double and multiple displacement events. Gas advances in a process we name three-phase Haines jumps, during which gas re-arranges its configuration in the pore space, retracting from some regions to enable the rapid filling of multiple pores. The gas retraction leads to a permanent disconnection of gas ganglia, which do not reconnect as gas injection proceeds. We observe, in situ, the direct displacement of oil and water by gas as well as gas–oil–water double displacement. The use of local in situ measurements and an energy balance approach to determine fluid–fluid contact angles alongside the quantification of capillary pressures and pore occupancy indicate that the wettability order is oil–gas–water from most to least wetting. Furthermore, quantifying the evolution of Minkowski functionals implied well-connected oil and water, while the gas connectivity decreased as gas was broken up into discrete clusters during injection. This work can be used to design CO2 storage, improved oil recovery and microfluidic devices. [ABSTRACT FROM AUTHOR]

We identify a distinct two-phase flow invasion pattern in a mixed-wet porous medium. Time-resolved high-resolution synchrotron X-ray imaging is used to study the invasion of water through a small rock sample filled with oil, characterized by a wide non-uniform distribution of local contact angles both above and below 90°. The water advances in a connected front, but throats are not invaded in decreasing order of size, as predicted by invasion percolation theory for uniformly hydrophobic systems. Instead, we observe pinning of the three-phase contact between the fluids and the solid, manifested as contact angle hysteresis, which prevents snap-off and interface retraction. In the absence of viscous dissipation, we use an energy balance to find an effective, thermodynamic, contact angle for displacement and show that this angle increases during the displacement. Displacement occurs when the local contact angles overcome the advancing contact angles at a pinned interface: it is wettability which controls the filling sequence. The product of the principal interfacial curvatures, the Gaussian curvature, is negative, implying well-connected phases which is consistent with pinning at the contact line while providing a topological explanation for the high displacement efficiencies in mixed-wet media. [ABSTRACT FROM AUTHOR]

At the pore scale, slow invasion of a wetting fluid in porous materials is often modeled with quasi‐static approximations which only consider capillary forces in the form of simple pore‐filling rules. The appropriateness of this approximation, often applied in pore network models, is contested in the literature, reflecting the difficulty of predicting imbibition relative permeability with these models. However, validation by sole comparison to continuum‐scale experiments is prone to induce model overfitting. It has therefore remained unclear whether difficulties generalizing the model performance are caused by errors in the predicted filling sequence or by subsequent calculations. Here, we address this by examining whether such a model can predict the pore‐scale fluid distributions underlying the behavior at the continuum scale. To this end, we compare the fluid arrangement evolution measured in fast synchrotron micro‐CT experiments on two rock types to quasi‐static simulations which implement capillary‐dominated pore filling and snap‐off, including a sophisticated model for cooperative pore filling. The results indicate that such pore network models can, in principle, predict fluid distributions accurately enough to estimate upscaled flow properties of strongly wetted rocks at low capillary numbers. Key Points: A new workflow to compare pore network simulations of two‐phase flow to time‐resolved micro‐CT data is presentedThe evolution of fluid distributions were validated with respect to pore size, connectivity, and flow pathsQuasi‐static pore network models put the fluids in approximately the right pores to predict flow properties [ABSTRACT FROM AUTHOR]

We propose a spatial Markov model to simulate transport in three‐dimensional complex porous media flows. Our methodology is inspired by the concept of training images from geostatistics. Instead of using a training image we use highly resolved training trajectories obtained by high‐resolution particle tracking, from which we sample increments in our random walk model. To reflect higher‐order processes, subsequent increments are correlated. The approach can be split into three steps. First, we subdivide (cut) the training trajectories to form an archive of trajectory segments. Next, we recursively sample segments, where subsequent samples are chosen conditioned to the previous one to ensure continuity and smoothness of velocity (conditional copy). Finally, we merge (paste) consecutive segments together to generate simulated trajectories of arbitrary length. This training trajectory approach aims to overcome three common shortcomings of spatial Markov models: (1) We simulate finite‐Péclet transport in three dimensions without commonly made simplifications (e.g., dimensionality reduction, and neglecting diffusion). (2) We do not parameterize dependence via a high‐dimensional transition matrix. (3) We simulate transport at the resolution of the (highly resolved) training trajectories, which can be important for processes such as mixing and reaction. To validate our methodology, we apply it to simulate transport within a three‐dimensional sandstone sample and compare predictions of a broad range of benchmark metrics against measurements from direct numerical simulations. We demonstrate that the training trajectories approach accurately represents three‐dimensional particle motion, suggesting that this method can capture the governing dependence structure and simulate transport processes in full complexity. Key Points: We propose a data‐driven spatial Markov model to effectively simulate transport in 3‐D without common simplificationsWe capture process dependence by a systematic rearrangement of highly resolved training trajectoriesWe simulate transport at the resolution of these highly resolved training trajectories [ABSTRACT FROM AUTHOR]

We have investigated the effect of wettability of carbonate rocks on the morphologies of remaining oil after sequential oil and brine injection in a capillary-dominated flow regime at elevated pressure. The wettability of Ketton limestone was altered in situ using an oil phase doped with fatty acid which produced mixed-wet conditions (the contact angle where oil contacted the solid surface, measured directly from the images, θ=180°, while brine-filled regions remained water-wet), whereas the untreated rock (without doped oil) was weakly water-wet (θ=47 ± 9°). Using X-ray micro-tomography, we show that the brine displaces oil in larger pores during brine injection in the mixed-wet system, leaving oil layers in the pore corners or sandwiched between two brine interfaces. These oil layers, with an average thickness of 47 ± 17 µm, may provide a conductive flow path for slow oil drainage. In contrast, the oil fragments into isolated oil clusters/ganglia during brine injection under water-wet conditions. Although the remaining oil saturation in a water-wet system is about a factor of two larger than that obtained in the mixed-wet rock, the measured brine-oil interfacial area of the disconnected ganglia is a factor of three smaller than that of oil layers. [ABSTRACT FROM AUTHOR]

We study capillary trapping in porous media using direct pore-scale simulation of two-phase flow on micro-CT images of a Berea sandstone and a sandpack. The trapped non-wetting phase saturations are predicted by solving the full Navier–Stokes equations using a volume-of-fluid based finite-volume framework to simulate primary drainage followed by water injection. Using these simulations, we analyse the effects of initial non-wetting-phase saturation, capillary number and flow direction on the residual saturation. The predictions from our numerical method are in agreement with published experimental measurements of capillary trapping curves. This shows that our direct simulation method can be used to elucidate the effect of pore structure and flow pattern of capillary trapping and provides a platform to study the physics of multiphase flow at the pore scale. [ABSTRACT FROM AUTHOR]

A physically based description is provided for the transverse dispersion coefficient in porous media as a function of Péclet number, Pe. We represent the porous medium as lattices of bonds with square cross section whose radius distribution is the same as computed for Berea sandstone and describe flow (Stokes equation) and diffusion (random walk method) at the pore scale (∼μm) to compute the transverse dispersion coefficient at a larger scale (∼cm to ∼m). We show that the transverse dispersion coefficient D T ∼ Pe for all Pe 1. A comprehensive comparative study of transverse dispersion with experiment indicates that the model can successfully predict the trends for the asymptotic macroscopic dispersion coefficient over a broad range of Péclet numbers, 0 < Pe < 105. We discuss the relation between transverse and longitudinal dispersion coefficient and show that unless one studies solute transport in the advection dominated regime, it is not appropriate to take D T to be 1 order of magnitude less than D L. [ABSTRACT FROM AUTHOR]

We provide a physically based explanation for the complex macroscopic behavior of dispersion in porous media as a function of Peclet number, Pe, using a pore-scale network model that accurately predicts the experimental dependence of the longitudinal dispersion coefficient, D L, on Pe. The asymptotic dispersion coefficient is only reached after the solute has traveled through a large number of pores at high Pe. This implies that preasymptotic dispersion is the norm, even in experiments in statistically homogeneous media. Interpreting transport as a continuous time random walk, we show that (1) the power law dispersion regime is controlled by the variation in average velocity between throats (the distribution of local Pe), giving D L ∼ Peδ with δ = 3 − β ≈ 1.2, where β is an exponent characterizing the distribution of transit times between pores, (2) the crossover to a linear regime D L ∼ Pe for Pe > Pe crit ≈ 400 is due to a transition from a diffusion-controlled late time cutoff to transport governed by advective movement, and (3) the transverse dispersion coefficient D T ∼ Pe for all Pe 1. [ABSTRACT FROM AUTHOR]

We study macroscopic (centimeter scale) dispersion using pore-scale network simulation. A Lagrangian-based transport model incorporating flow and diffusion is applied in a diamond lattice of throats with square cross section whose radius distribution is the same as computed for Berea sandstone. We use physically consistent rules using a combination of stream-tube routing and ideal mixing to transport particles across pore junctions. The influence of both heterogeneity and high Peclet numbers results in asymptotic behavior only being seen after movement through many throats. A comprehensive comparative study of longitudinal dispersion with experiments in consolidated and unconsolidated media indicates that the model can quantitatively predict the asymptotic macroscopic dispersion coefficient over a broad range of Peclet numbers, 0 < Pe < 105. In the low Peclet number region, molecular diffusion is more restricted for consolidated media as compared with unconsolidated media. The first effects of advection on dispersion are observed at Pe ∼ 0.1. In the advection-dominated regions the longitudinal dispersion coefficient follows a weak nonlinear dependence on Peclet number ( D L ∼ Pe1.19) followed by a linear dependence D L ∼ Pe for Pe > 400. [ABSTRACT FROM AUTHOR]

The development of high-resolution in situ imaging has allowed contact angles to be measured directly inside porous materials. We evaluate the use of concepts in integral geometry to determine contact angle. Specifically, we test the hypothesis that it is possible to determine an average contact angle from measurements of the Gaussian curvature of the fluid/fluid meniscus using the Gauss-Bonnet theorem. We show that it is not possible to unambiguously determine an average contact angle from the Gauss-Bonnet theorem. We instead present an approximate relationship: 2 π n (1 - cos θ) = 4 π - ∫ κ G 12 dS 12 , where n is the number of closed loops of the three-phase contact line where phases 1 and 2 contact the surface, θ is the average contact angle, while κ G 12 is the Gaussian curvature of the fluid meniscus which is integrated over its surface S 12 . We then use the results of pore-scale lattice Boltzmann simulations to assess the accuracy of this approach to determine a representative contact angle for two-phase flow in porous media. We show that in simple cases with a flat solid surface, the approximate expression works well. When applied to simulations on pore space images, the equation provides a robust estimate of contact angle, accurate to within 3 ° , when averaged over many fluid clusters, although individual values can have significant errors because of the approximations used in the calculation. [ABSTRACT FROM AUTHOR]

Based on energy balance during two-phase displacement in porous media, it has recently been shown that a thermodynamically consistent contact angle can be determined from micro-tomography images. However, the impact of viscous dissipation on the energy balance has not been fully understood. Furthermore, it is of great importance to determine the spatial distribution of wettability. We use direct numerical simulation to validate the determination of the thermodynamic contact angle both in an entire domain and on a pore-by-pore basis. Two-phase direct numerical simulations are performed on complex 3D porous media with three wettability states: uniformly water-wet, uniformly oil-wet, and non-uniform mixed-wet. Using the simulated fluid configurations, the thermodynamic contact angle is computed, then compared with the input contact angles. The impact of viscous dissipation on the energy balance is quantified; it is insignificant for water flooding in water-wet and mixed-wet media, resulting in an accurate estimation of a representative contact angle for the entire domain even if viscous effects are ignored. An increasing trend in the computed thermodynamic contact angle during water injection is shown to be a manifestation of the displacement sequence. Furthermore, the spatial distribution of wettability can be represented by the thermodynamic contact angle computed on a pore-by-pore basis. [ABSTRACT FROM AUTHOR]

The change of wettability toward more water-wet by the injection of low salinity water can improve oil recovery from porous rocks, which is known as low salinity water flooding. To simulate this process at the pore-scale, we propose that the alteration in surface wettability mediated by thin water films which are below the resolution of simulation grid blocks has to be considered, as observed in experiments. This is modeled by a wettability alteration model based on rate-limited adsorption of ions onto the rock surface. The wettability alteration model is developed and incorporated into a lattice Boltzmann simulator which solves both the Navier-Stokes equation for oil/water two-phase flow and the advection-diffusion equation for ion transport. The model is validated against two experiments in the literature, then applied to 3D micro-CT images of a rock. Our model correctly simulated the experimental observations caused by the slow wettability alteration driven by the development of water films. In the simulations on the 3D rock pore structure, a distinct difference in the mixing of high and low salinity water is observed between secondary and tertiary low salinity flooding, resulting in different oil recoveries. [ABSTRACT FROM AUTHOR]

Pore-scale forces have a significant effect on the macroscopic behaviour of multiphase flow through porous media. This paper studies the effect of these forces using a new volume-of-fluid based finite volume method developed for simulating two-phase flow directly on micro-CT images of porous media. An analytical analysis of the relationship between the pore-scale forces and the Darcy-scale pressure drops is presented. We use this analysis to propose unambiguous definitions of Darcy-scale viscous pressure drops as the rate of energy dissipation per unit flow rate of each phase, and then use them to obtain the relative permeability curves. We show that this definition is consistent with conventional laboratory/field measurements by comparing our predictions with experimental relative permeability. We present single and two-phase flow simulations for primary oil injection followed by water injection on a sandpack and a Berea sandstone. The two-phase flow simulations are presented at different capillary numbers which cover the transition from capillary fingering at low capillary numbers to a more viscous fingering displacement pattern at higher capillary numbers, and the effect of capillary number on the relative permeability curves is investigated. Overall, this paper presents a new finite volume-based methodology for the detailed analysis of two-phase flow directly on micro-CT images of porous media and upscaling of the results to the Darcy scale. [ABSTRACT FROM AUTHOR]

Abstract: We present a stable numerical scheme for modelling multiphase flow in porous media, where the characteristic size of the flow domain is of the order of microns to millimetres. The numerical method is developed for efficient modelling of multiphase flow in porous media with complex interface motion and irregular solid boundaries. The Navier–Stokes equations are discretised using a finite volume approach, while the volume-of-fluid method is used to capture the location of interfaces. Capillary forces are computed using a semi-sharp surface force model, in which the transition area for capillary pressure is effectively limited to one grid block. This new formulation along with two new filtering methods, developed for correcting capillary forces, permits simulations at very low capillary numbers and avoids non-physical velocities. Capillary forces are implemented using a semi-implicit formulation, which allows larger time step sizes at low capillary numbers. We verify the accuracy and stability of the numerical method on several test cases, which indicate the potential of the method to predict multiphase flow processes. [Copyright &y& Elsevier]

Conservation of energy is used to derive a thermodynamically-consistent contact angle, θ t , when fluid phase 1 displaces phase 2 in a porous medium. Assuming no change in Helmholtz free energy between two local equilibrium states we find that Δ a 1 s cos θ t = κ ϕ Δ S 1 + Δ a 12 , where a is the interfacial area per unit volume, ϕ is the porosity, S is the saturation and κ the curvature of the fluid-fluid interface. The subscript s denotes the solid, and we consider changes, Δ , in saturation and area. With the advent of high-resolution time-resolved three-dimensional X-ray imaging, all the terms in this expression can be measured directly. We analyse imaging datasets for displacement of oil by water in a water-wet and a mixed-wet sandstone. For the water-wet sample, the curvature is positive and oil bulges into the brine with almost spherical interfaces. In the mixed-wet case, larger interfacial areas are found, as the oil resides in layers. The mean curvature is close to zero, but the interface tends to bulge into brine in one direction, while brine bulges into oil in the other. We compare θ t with the values measured geometrically in situ on the pore-scale images, θ g . The thermodynamic angle θ t provides a robust and consistent characterization of wettability. For the water-wet case the calculated value of θ t gives an accurate prediction of multiphase flow properties using pore-scale modelling. [ABSTRACT FROM AUTHOR]

Highlights • We obtained micro scale three-dimensional images of three-phase flow in situ at high temperature and pressure in a water-wet carbonate rock. • Pore occupancy is quantified: brine, oil and gas occupy the centre of respectively small, intermediate and large pores. • Double drainage and double imbibition are visualized in three-dimensions. • Spreading oil layers and water wetting layers are observed and their thickness allows for effective flow. • Wettability and the formation of oil layers favour trapping of gas over oil when the three phases are simultaneously present in the pore space. Abstract X-ray micro-tomography is used to image the pore-scale configurations of fluid in a rock saturated with three phases - brine, oil and gas - mimicking a subsurface reservoir, at high pressure and temperature. We determine pore occupancy during a displacement sequence that involves waterflooding, gas injection and water re-injection. In the water-wet sample considered, brine occupied the smallest pores, gas the biggest, while oil occupied pores of intermediate size and is displaced by both water and gas. Double displacement events have been observed, where gas displaces oil that displaces water or vice versa. The thickness of water and oil layers have been quantified, as have the contact angles between gas and oil, and oil and water. These results are used to explain the nature of trapping in three-phase flow, specifically how oil preferentially traps gas in the presence of water. [ABSTRACT FROM AUTHOR]