Steady-state two-phase flow in porous media: statistics and transport properties

Inertia onset in disordered porous media flow

We investigate the onset of the inertial regime in the fluid flow at the pore level in three-dimensional, disordered, highly porous media. We analyze the flow structure in a wide range of Reynolds numbers starting from 0.01 up to 100. We focus on qualitative and quantitative changes that appear with increasing Reynolds number. To do that, we investigate the weakening of the channeling effect, defined as the existence of preferred flow paths in a system. We compute tortuosity, spatial kinetic energy localization, and the pore-space volume fraction containing negative streamwise velocity to assess accompanying changes quantitatively. Our results of tortuosity and participation number derivatives show that the onset of inertia is apparent for Reynolds number Re∼0.1, an order of magnitude lower than indicated by analyzing relations of friction factor with the Reynolds number. Moreover, we show that the vortex structures appear at Reynolds number two orders of magnitude higher than the onset of inertia.

A Statistical Analysis of Fluid Interface Fluctuations: Exploring the Role of Viscosity Ratio

Understanding multiphase flow through porous media is integral to geologic carbon storage or hydrogen storage. The current modelling framework assumes each fluid present in the subsurface flows in its own continuously connected pathway. The restriction in flow caused by the presence of another fluid is modelled using relative permeability functions. However, dynamic fluid interfaces have been observed in experimental data, and these are not accounted for in relative permeability functions. In this work, we explore the occurrence of fluid fluctuations in the context of sizes, locations, and frequencies by altering the viscosity ratio for two-phase flow. We see that the fluctuations alter the connectivity of the fluid phases, which, in turn, influences the relative permeability of the fluid phases present.

Development of stochastically reconstructed 3D porous media micromodels using additive manufacturing: numerical and experimental validation

We propose an integrated methodology for the design and fabrication of 3D micromodels that are suitable for the pore-scale study of transport processes in macroporous materials. The micromodels, that bear the pore-scale characteristics of sandstone, such as porosity, mean pore size, etc, are designed following a stochastic reconstruction algorithm that allows for fine-tuning the porosity and the correlation length of the spatial distribution of the solid material. We then construct a series of 3D micromodels at very fine resolution (i.e. 16 μ m) using a state-of-the-art 3D printing infrastructure, specifically a ProJet MJP3600 3D printer, that utilizes the Material Jetting technology. Within the technical constraints of the 3D printer resolution, the fabricated micromodels represent scaled-up replicas of natural sandstones, that are suitable for the study of the scaling between the permeability, the porosity and the mean pore size. The REV- and pore-scale characteristics of the resulting physical micromodels are recovered using a combination of X-ray micro-CT and microfluidic studies. The experimental results are then compared with single-phase flow simulations at pore-scale and geostatistic models in order to determine the effects of the design parameters on the intrinsic permeability and the spatial correlation of the velocity profile. Our numerical and experimental measurements reveal an excellent match between the properties of the designed and fabricated 3D domains, thus demonstrating the robustness of the proposed methodology for the construction of 3D micromodels with fine-tuned and well-controlled pore-scale characteristics. Furthermore, a pore-scale numerical study over a wider range of 3D digital domain realizations reveals a very good match of the measured permeabilities with the predictions of the Kozeny-Carman formulation based on a single control parameter, k 0 , that is found to have a practically constant value for porosities ϕ ≥ 0.2 . This, in turn, enables us to customize the sample size to meet REV constraints, including enlarging pore morphology while considering the Reynolds number. It is also found that at lower porosities there is a significant increase in the fraction of the non-percolating pores, thus leading to different k 0 , as the porosity approaches a numerically determined critical porosity value, ϕ c , where the domain is no longer percolating.

Steady-State Two-Phase Flow of Compressible and Incompressible Fluids in a Capillary Tube of Varying Radius

We study immiscible two-phase flow of a compressible and an incompressible fluid inside a capillary tube of varying radius under steady-state conditions. The incompressible fluid is Newtonian and the compressible fluid is an inviscid ideal gas. The surface tension associated with the interfaces between the two fluids introduces capillary forces that vary along the tube due to the variation in the tube radius. The interplay between effects due to the capillary forces and the compressibility results in a set of properties that are different from incompressible two-phase flow. As the fluids move towards the outlet, the bubbles of the compressible fluid grows in volume due to the decrease in pressure. The volumetric growth of the compressible bubbles makes the volumetric flow rate at the outlet higher than at the inlet. The growth is not only a function of the pressure drop across the tube, but also of the ambient pressure. Furthermore, the capillary forces create an effective threshold below which there is no flow. Above the threshold, the system shows a weak nonlinearity between the flow rates and the effective pressure drop, where the nonlinearity also depends on the absolute pressures across the tube.

The Co-Moving Velocity in Immiscible Two-Phase Flow in Porous Media

We present a continuum (i.e., an effective) description of immiscible two-phase flow in porous media characterized by two fields, the pressure and the saturation. Gradients in these two fields are the driving forces that move the immiscible fluids around. The fluids are characterized by two seepage velocity fields, one for each fluid. Following Hansen et al. (Transport in Porous Media, 125, 565 (2018)), we construct a two-way transformation between the velocity couple consisting of the seepage velocity of each fluid, to a velocity couple consisting of the average seepage velocity of both fluids and a new velocity parameter, the co-moving velocity. The co-moving velocity is related but not equal to velocity difference between the two immiscible fluids. The two-way mapping, the mass conservation equation and the constitutive equations for the average seepage velocity and the co-moving velocity form a closed set of equations that determine the flow. There is growing experimental, computational and theoretical evidence that constitutive equation for the average seepage velocity has the form of a power law in the pressure gradient over a wide range of capillary numbers. Through the transformation between the two velocity couples, this constitutive equation may be taken directly into account in the equations describing the flow of each fluid. This is, e.g., not possible using relative permeability theory. By reverse engineering relative permeability data from the literature, we construct the constitutive equation for the co-moving velocity. We also calculate the co-moving constitutive equation using a dynamic pore network model over a wide range of parameters, from where the flow is viscosity dominated to where the capillary and viscous forces compete. Both the relative permeability data from the literature and the dynamic pore network model give the same very simple functional form for the constitutive equation over the whole range of parameters.

Pore-Scale Investigation of Dynamic Immiscible Displacement in Layered Media using Synchrotron X-ray Microtomography

Understanding the dynamics of immiscible fluid in a porous media is critical in many chemical and environmental engineering processes. However, the geological heterogeneity effect on multiphase flow behavior remains unclear. Here, the dynamics of immiscible fluid displacement and entrapment were experimentally demonstrated at pore-level using time-lapse synchrotron X-ray microtomography. A drainage-imbibition experiment was designed using an unconsolidated layered sand pack that comprised coarse sand and fine sand zones. There were significant differences between the two zones, with regard to the temporal variations in fluid saturation and morphological evolution of nonwetting fluid (oil) during imbibition. Highly connected oil clusters in the coarse zone broke up into many small fragments, whereas the cluster in the fine zone remained connected while spanning multiple pores. To further understand the impacts of pore size and connectivity on multiphase fluid dynamics, a new approach that tracks the temporal variation of immiscible fluid in individual pores was conducted. The surface area at the oil-water interface increased during imbibition, which is expected to facilitate mass transfer and surface interactions. Understanding immiscible fluid displacement in layered porous media at the pore-level could lead to more effective environmental remediation.

Inversion of Solvent Migration in Charged Membranes

We theoretically demonstrate the possibility of inversion of solvent migration in charged membranes, opposing osmosis. Inversion of solvent migration is ascribed to the finite volume of ions in the solution permeating the membrane, a quantity that has been neglected in the literature so far. We propose a model of the electrochemistry in the proximity of an electrode, illustrating the range of the molar volume of ions that can yield the inversion of solvent migration. This study poses the basis for novel applications in microfluidics, nanofluidics, and electrochemistry, along with new inquiries in biology.

Rheology of Immiscible Two-phase Flow in Mixed Wet Porous Media: Dynamic Pore Network Model and Capillary Fiber Bundle Model Results

Immiscible two-phase flow in porous media with mixed wet conditions was examined using a capillary fiber bundle model, which is analytically solvable, and a dynamic pore network model. The mixed wettability was implemented in the models by allowing each tube or link to have a different wetting angle chosen randomly from a given distribution. Both models showed that mixed wettability can have significant influence on the rheology in terms of the dependence of the global volumetric flow rate on the global pressure drop. In the capillary fiber bundle model, for small pressure drops when only a small fraction of the tubes were open, it was found that the volumetric flow rate depended on the excess pressure drop as a power law with an exponent equal to 3/2 or 2 depending on the minimum pressure drop necessary for flow. When all the tubes were open due to a high pressure drop, the volumetric flow rate depended linearly on the pressure drop, independent of the wettability. In the transition region in between where most of the tubes opened, the volumetric flow depended more sensitively on the wetting angle distribution function and was in general not a simple power law. The dynamic pore network model results also showed a linear dependence of the flow rate on the pressure drop when the pressure drop is large. However, out of this limit the dynamic pore network model demonstrated a more complicated behavior that depended on the mixed wettability condition and the saturation. In particular, the exponent relating volumetric flow rate to the excess pressure drop could take on values anywhere between 1.0 and 1.8. The values of the exponent were highest for saturations approaching 0.5, also, the exponent generally increased when the difference in wettability of the two fluids were larger and when this difference was present for a larger fraction of the porous network.

Dynamic fluid configurations in steady-state two-phase flow in Bentheimer sandstone

Fast synchrotron tomography is used to study the impact of capillary number, Ca, on fluid configurations in steady-state two-phase flow in porous media. Brine and n-decane were co-injected at fixed fractional flow, f_{w}=0.5, in a cylindrical Bentheimer sandstone sample for a range of capillary numbers 2.1×10^{-7}≤Ca≤4.2×10^{-5}, while monitoring the pressure differential. As we have demonstrated in Gao et al. [Phys. Rev. Fluids 5, 013801 (2020)2469-990X10.1103/PhysRevFluids.5.013801], dependent on Ca, different flow regimes have been identified: at low Ca only fixed flow pathways exist, while after a certain threshold dynamic effects are observed resulting in intermittent fluctuations in fluid distribution which alter fluid connectivity. Additionally, the flow paths, for each capillary number, were imaged multiple times to quantify the less frequent changes in fluid occupancy, happening over timescales longer than the duration of our scans (40 s). In this paper we demonstrate how dynamic connectivity results from the interaction between oil ganglia populations. At low Ca connected pathways of ganglia are fixed with time-independent small, medium, and large ganglia populations. However, with an increase in Ca we see fluctuations in the size and numbers of the larger ganglia. With the onset of intermittency, fluctuations occur mainly in pores and throats of intermediate size. When Ca is further increased, we see rapid changes in occupancy in pores of all size. By combining observations on pressure fluctuations and flow regimes at various capillary numbers, we summarize a phase diagram over a range of capillary numbers for the wetting and nonwetting phases, Ca_{w} and Ca_{nw}, respectively, to quantify the degree of intermittent flow. These different regimes are controlled by a competition between viscous forces on the flowing fluids and the capillary forces acting in the complex pore space. Furthermore, we plot the phase diagrams of the transition from Darcy flow to intermittent flow over a range of Reynolds and Weber numbers for the wetting and nonwetting phases to evaluate the balance among capillary, viscous, and inertial forces, incorporating data from the literature. We demonstrate that pore geometry has a significant control on flow regime.

Pore-scale effects during the transition from capillary- to viscosity-dominated flow dynamics within microfluidic porous-like domains

We perform a numerical and experimental study of immiscible two-phase flows within predominantly 2D transparent PDMS microfluidic domains with disordered pillar-like obstacles, that effectively serve as artificial porous structures. Using a high sensitivity pressure sensor at the flow inlet, we capture experimentally the pressure dynamics under fixed flow rate conditions as the fluid–fluid interface advances within the porous domain, while also monitoring the corresponding phase distribution patterns using optical microscopy. Our experimental study covers 4 orders of magnitude with respect to the injection flow rate and highlights the characteristics of immiscible displacement processes during the transition from the capillarity-controlled interface displacement regime at lower flow rates, where the pores are invaded sequentially in the form of Haines jumps, to the viscosity-dominated regime, where multiple pores are invaded simultaneously. In the capillary regime, we recover a clear correlation between the recorded inlet pressure and the pore-throat diameter invaded by the interface that follows the Young–Laplace equation, while during the transition to the viscous regime such a correlation is no longer evident due to multiple pore-throats being invaded simultaneously (but also due to significant viscous pressure drop along the inlet and outlet channels, that effectively mask capillary effects). The performed experimental study serves for the validation of a robust Level-Set model capable of explicitly tracking interfacial dynamics at sub-pore scale resolutions under identical flow conditions. The numerical model is validated against both well-established theoretical flow models, that account for the effects of viscous and capillary forces on interfacial dynamics, and the experimental results obtained using the developed microfluidic setup over a wide range of capillary numbers. Our results show that the proposed numerical model recovers very well the experimentally observed flow dynamics in terms of phase distribution patterns and inlet pressures, but also the effects of viscous flow on the apparent (i.e. dynamic) contact angles in the vicinity of the pore walls. For the first time in the literature, this work clearly shows that the proposed numerical approach has an undoubtable strong potential to simulate multiphase flow in porous domains over a wide range of Capillary numbers.

Characterizing Dissipation in Fluid-Fluid Displacement Using Constant-Rate Spontaneous Imbibition

When one fluid displaces another in a confined environment, some energy is dissipated in the fluid bulk and the rest is dissipated near the contact line. Here we study the relative strengths of these two sources of dissipation with a novel experimental setup: constant-rate spontaneous imbibition experiments, achieved by introducing a viscous oil slug in front of the invading fluid inside a capillary tube. We show that a large fraction of dissipation can take place near the contact line, and rationalize the observations by means of a theoretical analysis of the dynamic contact angles of the front and back menisci of the oil slug. Our results bear important implications for macroscopic descriptions of multiphase flows in microfluidic systems and porous media.

Suitability of 2D modelling to evaluate flow properties in 3D porous media

The employment of 2D models to investigate the properties of 3D flows in porous media is ubiquitous in the literature. The limitations of such approaches are often overlooked. Here, we assess to which extent 2D flows in porous media are suitable representations of 3D flows. To this purpose, we compare representative elementary volume (REV) scales obtained by 2D and 3D numerical simulations of flow in porous media. The stationarity of several quantities, namely porosity, permeability, mean and variance of velocity, is evaluated in terms of both classical and innovative statistics. The variance of velocity, strictly connected to the hydrodynamic dispersion, is included in the analysis in order to extend conclusions to transport phenomena. Pore scale flow is simulated by means of a Lattice Boltzmann model. The results from pore scale simulations point out that the 2D approach often leads to inconsistent results, due to the profound difference between 2D and 3D flows through porous media. We employ the error in the evaluation of REV as a quantitative measure for the reliability of a 2D approach. Moreover, we show that the acceptance threshold for a 2D representation to be valid strongly depends on which flow/transport quantity is sought.

Collective behaviors of Drosophila-derived retinal progenitors in controlled microenvironments

Collective behaviors of retinal progenitor cells (RPCs) are critical to the development of neural networks needed for vision. Signaling cues and pathways governing retinal cell fate, migration, and functional organization are remarkably conserved across species, and have been well-studied using Drosophila melanogaster. However, the collective migration of heterogeneous groups of RPCs in response to dynamic signaling fields of development remains incompletely understood. This is in large part because the genetic advances of seminal invertebrate models have been poorly complemented by in vitro cell study of its visual development. Tunable microfluidic assays able to replicate the miniature cellular microenvironments of the developing visual system provide newfound opportunities to probe and expand our knowledge of collective chemotactic responses essential to visual development. Our project used a controlled, microfluidic assay to produce dynamic signaling fields of Fibroblast Growth Factor (FGF) that stimulated the chemotactic migration of primary RPCs extracted from Drosophila. Results illustrated collective RPC chemotaxis dependent on average size of clustered cells, in contrast to the non-directional movement of individually-motile RPCs. Quantitative study of these diverse collective responses will advance our understanding of retina developmental processes, and aid study/treatment of inherited eye disease. Lastly, our unique coupling of defined invertebrate models with tunable microfluidic assays provides advantages for future quantitative and mechanistic study of varied RPC migratory responses.

Intermittent fluid connectivity during two-phase flow in a heterogeneous carbonate rock

Subsurface fluid flow is ubiquitous in nature, and understanding the interaction of multiple fluids as they flow within a porous medium is central to many geological, environmental, and industrial processes. It is assumed that the flow pathways of each phase are invariant when modeling subsurface flow using Darcy's law extended to multiphase flow, a condition that is assumed to be valid during steady-state flow. However, it has been observed that intermittent flow pathways exist at steady state even at the low capillary numbers typically encountered in the subsurface. Little is known about the pore structure controls or the impact of intermittency on continuum scale flow properties. Here we investigate the impact of intermittent pathways on the connectivity of the fluids for a carbonate rock. Using laboratory-based micro computed tomography imaging we observe that intermittent pathway flow occurs in intermediate-sized pores due to the competition between both flowing fluids. This competition moves to smaller pores when the flow rate of the nonwetting phase increases. Intermittency occurs in poorly connected pores or in regions where the nonwetting phase itself is poorly connected. Intermittent pathways lead to the interrupted transport of the fluids; this means they are important in determining continuum scale flow properties, such as relative permeability. The impact of intermittency on flow properties is significant because it occurs at key locations, whereby the nonwetting phase is otherwise disconnected.

Mechanisms controlling fluid breakup and reconnection during two-phase flow in porous media

The use of Darcy's law to describe steady-state multiphase flow in porous media has been justified by the assumption that the fluids flow in continuously connected pathways. However, a range of complex interface dynamics have been observed during macroscopically steady-state flow, including intermittent pathway flow where flow pathways periodically disconnect and reconnect. The physical mechanisms controlling this behavior have remained unclear, leading to uncertainty concerning the occurrence of the different flow regimes. We observe that the fraction of intermittent flow pathways is dependent on the capillary number and viscosity ratio. We propose a phase diagram within this parameter space to quantify the degree of intermittent flow.

Pressure evolution and deformation of confined granular media during pneumatic fracturing

By means of digital image correlation, we experimentally characterize the deformation of a dry granular medium confined inside a Hele-Shaw cell due to air injection at a constant overpressure high enough to deform it (from 50 to 250 kPa). Air injection at these overpressures leads to the formation of so-called pneumatic fractures, i.e., channels empty of beads, and we discuss the typical deformations of the medium surrounding these structures. In addition we simulate the diffusion of the fluid overpressure into the medium, comparing it with the Laplacian solution over time and relating pressure gradients with corresponding granular displacements. In the compacting medium we show that the diffusing pressure field becomes similar to the Laplace solution on the order of a characteristic time given by the properties of the pore fluid, the granular medium, and the system size. However, before the diffusing pressure approaches the Laplace solution on the system scale, we find that it resembles the Laplacian field near the channels, with the highest pressure gradients on the most advanced channel tips and a screened pressure gradient behind them. We show that the granular displacements more or less always move in the direction against the local pressure gradients, and when comparing granular velocities with pressure gradients in the zone ahead of channels, we observe a Bingham type of rheology for the granular paste (the mix of air and beads), with an effective viscosity μ_{B} and displacement thresholds ∇[over ⃗]P_{c} evolving during mobilization and compaction of the medium. Such a rheology, with disorder in the displacement thresholds, could be responsible for placing the pattern growth at moderate injection pressures in a universality class like the dielectric breakdown model with η=2, where fractal dimensions are found between 1.5 and 1.6 for the patterns.

X-ray Microtomography of Intermittency in Multiphase Flow at Steady State Using a Differential Imaging Method

We imaged the steady state flow of brine and decane in Bentheimer sandstone. We devised an experimental method based on differential imaging to examine how flow rate impacts impact the pore-scale distribution of fluids during coinjection. This allows us to elucidate flow regimes (connected, or breakup of the nonwetting phase pathways) for a range of fractional flows at two capillary numbers, Ca, namely 3.0 × 10-7 and 7.5 × 10-6. At the lower Ca, for a fixed fractional flow, the two phases appear to flow in connected unchanging subnetworks of the pore space, consistent with conventional theory. At the higher Ca, we observed that a significant fraction of the pore space contained sometimes oil and sometimes brine during the 1 h scan: this intermittent occupancy, which was interpreted as regions of the pore space that contained both fluid phases for some time, is necessary to explain the flow and dynamic connectivity of the oil phase; pathways of always oil-filled portions of the void space did not span the core. This phase was segmented from the differential image between the 30 wt % KI brine image and the scans taken at each fractional flow. Using the grey scale histogram distribution of the raw images, the oil proportion in the intermittent phase was calculated. The pressure drops at each fractional flow at low and high flow rates were measured by high-precision differential pressure sensors. The relative permeabilities and fractional flow obtained by our experiment at the mm-scale compare well with data from the literature on cm-scale samples.

Pneumatic fractures in confined granular media

We perform experiments where air is injected at a constant overpressure P_{in}, ranging from 5 to 250 kPa, into a dry granular medium confined within a horizontal linear Hele-Shaw cell. The setup allows us to explore compacted configurations by preventing decompaction at the outer boundary, i.e., the cell outlet has a semipermeable filter such that beads are stopped while air can pass. We study the emerging patterns and dynamic growth of channels in the granular media due to fluid flow, by analyzing images captured with a high speed camera (1000 images/s). We identify four qualitatively different flow regimes, depending on the imposed overpressure, ranging from no channel formation for P_{in} below 10 kPa, to large thick channels formed by erosion and fingers merging for high P_{in} around 200 kPa. The flow regimes where channels form are characterized by typical finger thickness, final depth into the medium, and growth dynamics. The shape of the finger tips during growth is studied by looking at the finger width w as function of distance d from the tip. The tip profile is found to follow w(d)∝d^{β}, where β=0.68 is a typical value for all experiments, also over time. This indicates a singularity in the curvature d^{2}d/dw^{2}∼κ∼d^{1-2β}, but not of the slope dw/dd∼d^{β-1}, i.e., more rounded tips rather than pointy cusps, as they would be for the case β>1. For increasing P_{in}, the channels generally grow faster and deeper into the medium. We show that the channel length along the flow direction has a linear growth with time initially, followed by a power-law decay of growth velocity with time as the channel approaches its final length. A closer look reveals that the initial growth velocity v_{0} is found to scale with injection pressure as v_{0}∝P_{in}^{3/2}, while at a critical time t_{c} there is a cross-over to the behavior v(t)∝t^{-α}, where α is close to 2.5 for all experiments. Finally, we explore the fractal dimension of the fully developed patterns. For example, for patterns resulting from intermediate P_{in} around 100-150 kPa, we find that the box-counting dimensions lie within the range D_{B}∈[1.53,1.62], similar to viscous fingering fractals in porous media.

Effective Rheology of Two-Phase Flow in Three-Dimensional Porous Media: Experiment and Simulation

We present an experimental and numerical study of immiscible two-phase flow of Newtonian fluids in three-dimensional (3D) porous media to find the relationship between the volumetric flow rate (Q) and the total pressure difference ([Formula: see text]) in the steady state. We show that in the regime where capillary forces compete with the viscous forces, the distribution of capillary barriers at the interfaces effectively creates a yield threshold ([Formula: see text]), making the fluids reminiscent of a Bingham viscoplastic fluid in the porous medium. In this regime, Q depends quadratically on an excess pressure drop ([Formula: see text]). While increasing the flow rate, there is a transition, beyond which the overall flow is Newtonian and the relationship is linear. In our experiments, we build a model porous medium using a column of glass beads transporting two fluids, deionized water and air. For the numerical study, reconstructed 3D pore networks from real core samples are considered and the transport of wetting and non-wetting fluids through the network is modeled by tracking the fluid interfaces with time. We find agreement between our numerical and experimental results. Our results match with the mean-field results reported earlier.

Generalized three-dimensional lattice Boltzmann color-gradient method for immiscible two-phase pore-scale imbibition and drainage in porous media

This article presents a three-dimensional numerical framework for the simulation of fluid-fluid immiscible compounds in complex geometries, based on the multiple-relaxation-time lattice Boltzmann method to model the fluid dynamics and the color-gradient approach to model multicomponent flow interaction. New lattice weights for the lattices D3Q15, D3Q19, and D3Q27 that improve the Galilean invariance of the color-gradient model as well as for modeling the interfacial tension are derived and provided in the Appendix. The presented method proposes in particular an approach to model the interaction between the fluid compound and the solid, and to maintain a precise contact angle between the two-component interface and the wall. Contrarily to previous approaches proposed in the literature, this method yields accurate solutions even in complex geometries and does not suffer from numerical artifacts like nonphysical mass transfer along the solid wall, which is crucial for modeling imbibition-type problems. The article also proposes an approach to model inflow and outflow boundaries with the color-gradient method by generalizing the regularized boundary conditions. The numerical framework is first validated for three-dimensional (3D) stationary state (Jurin's law) and time-dependent (Washburn's law and capillary waves) problems. Then, the usefulness of the method for practical problems of pore-scale flow imbibition and drainage in porous media is demonstrated. Through the simulation of nonwetting displacement in two-dimensional random porous media networks, we show that the model properly reproduces three main invasion regimes (stable displacement, capillary fingering, and viscous fingering) as well as the saturating zone transition between these regimes. Finally, the ability to simulate immiscible two-component flow imbibition and drainage is validated, with excellent results, by numerical simulations in a Berea sandstone, a frequently used benchmark case used in this field, using a complex geometry that originates from a 3D scan of a porous sandstone. The methods presented in this article were implemented in the open-source PALABOS library, a general C++ matrix-based library well adapted for massive fluid flow parallel computation.

Influence of phase connectivity on the relationship among capillary pressure, fluid saturation, and interfacial area in two-fluid-phase porous medium systems

Multiphase flows in porous medium systems are typically modeled at the macroscale by applying the principles of continuum mechanics to develop models that describe the behavior of averaged quantities, such as fluid pressure and saturation. These models require closure relations to produce solvable forms. One of these required closure relations is an expression relating the capillary pressure to fluid saturation and, in some cases, other topological invariants such as interfacial area and the Euler characteristic (or average Gaussian curvature). The forms that are used in traditional models, which typically consider only the relationship between capillary pressure and saturation, are hysteretic. An unresolved question is whether the inclusion of additional morphological and topological measures can lead to a nonhysteretic closure relation. Relying on the lattice Boltzmann (LB) method, we develop an approach to investigate equilibrium states for a two-fluid-phase porous medium system, which includes disconnected nonwetting phase features. A set of simulations are performed within a random close pack of 1964 spheres to produce a total of 42 908 distinct equilibrium configurations. This information is evaluated using generalized additive models to quantitatively assess the degree to which functional relationships can explain the behavior of the equilibrium data. The variance of various model estimates is computed, and we conclude that, except for the limiting behavior close to a single fluid regime, capillary pressure can be expressed as a deterministic and nonhysteretic function of fluid saturation, interfacial area between the fluid phases, and the Euler characteristic. To our knowledge, this work is unique in the methods employed, the size of the data set, the resolution in space and time, the true equilibrium nature of the data, the parametrizations investigated, and the broad set of functions examined. The conclusion of essentially nonhysteretic behavior provides support for an evolving class of two-fluid-phase flow in porous medium systems models.

Viscoelastic polymer flows and elastic turbulence in three-dimensional porous structures

Viscoelastic polymer solutions flowing through reservoir rocks have been found to improve oil displacement efficiency when the aqueous-phase shear-rate exceeds a critical value. A possible mechanism for this enhanced recovery is elastic turbulence that causes breakup and mobilization of trapped oil ganglia. Here, we apply nuclear magnetic resonance (NMR) pulsed field gradient (PFG) diffusion measurements in a novel way to detect increased motion of disconnected oil ganglia. The data are acquired directly from a three-dimensional (3D) opaque porous structure (sandstone) when viscoelastic fluctuations are expected to be present in the continuous phase. The measured increase in motion of trapped ganglia provides unequivocal evidence of fluctuations in the flowing phase in a fully complex 3D system. This work provides direct evidence of elastic turbulence in a realistic reservoir rock - a measurement that cannot be readily achieved by conventional laboratory methods. We support the NMR data with optical microscopy studies of fluctuating ganglia in simple two-dimensional (2D) microfluidic networks, with consistent apparent rheological behaviour of the aqueous phase, to provide conclusive evidence of elastic turbulence in the 3D structure and hence validate the proposed flow-fluctuation mechanism for enhanced oil recovery.

Flow of concentrated viscoelastic polymer solutions in porous media: effect of M(W) and concentration on elastic turbulence onset in various geometries

Viscoelastic polymer solutions exhibit a variety of flow instabilities and in particular, in mixed shear and extensional flow, elastic turbulence. Coincident with the transition to turbulence is additional dissipation that, in porous flow, may be characterised as an increased apparent viscosity. We report elastic turbulence and apparent thickening in the flow of polymer solutions both in rock samples and in microfluidic analogues and we correlate the onset of thickening and turbulence with rheological measurements. Contrary to expectations, the characteristic relaxation time associated with the transition to turbulence is found to be independent of polymer concentration over the range studied (10c* ≲c≲ 100c*). Furthermore, this characteristic time scales with the square of molecular weight. Thus the characteristic time associated with the transition to turbulence is not the linear-viscoelastic timescale usually measured but rather scales as a dilute Rouse time despite being an entangled system.

History effects on nonwetting fluid residuals during desaturation flow through disordered porous media

We investigate experimentally the sweeping of a nonwetting fluid by a wetting one in a quasi-two-dimensional porous medium consisting of random obstacles. We focus primarily on the resulting phase distributions and the residual nonwetting phase saturation as a function of the normalized wetting fluid flow rate-the capillary number Ca-at steady state. The wetting liquid is then flowing in the medium partially saturated by immobile nonwetting liquid blobs. The decrease of the nonwetting saturation is an irreversible process that depends strongly on flow history and more specifically on the highest value of Ca reached in the past. At lower Ca values, when capillary forces are dominant, the residual steady state saturation depends significantly on the initial phase configuration. However, at higher Ca, the saturation becomes independent of the history and thus follows a master curve that converges to an asymptotic residual value. Blob sizes range over four orders of magnitude in our experimental domain, following a probability distribution function P that scales with the blob size s as P(s)∝s(-2) for blob sizes larger than the typical pore size. It also exhibits a maximum size cutoff s(max), that decreases as s(max)∝Ca(-1). To determine the flow properties, we have measured the pressure drop (B) versus the flow rate (Ca). In the ranges of low and high Ca values, the relationship between Ca and B is found to be linear, following Darcy's law (B∝Ca). In the intermediate regime, the progressive mobilization of blobs leads to a nonlinear dependence B∝Ca(0.65), due to an increase of the available flow paths.

Non-Darcy behavior of two-phase channel flow

We study the macroscopic behavior of two-phase flow in porous media from a phase-field model. A dissipation law is first derived from the phase-field model by homogenization. For simple channel geometry in pore scale, the scaling relation of the averaged dissipation rate with the velocity of the two-phase flow can be explicitly obtained from the model which then gives the force-velocity relation. It is shown that, for the homogeneous channel surface, Dacry's law is still valid with a significantly modified permeability including the contribution from the contact line slip. For the chemically patterned surfaces, the dissipation rate has a non-Darcy linear scaling with the velocity, which is related to a depinning force for the patterned surface. Our result offers a theoretical understanding on the prior observation of non-Darcy behavior for the multiphase flow in either simulations or experiments.

Inertial forces affect fluid front displacement dynamics in a pore-throat network model

The seemingly regular and continuous motion of fluid displacement fronts in porous media at the macroscopic scale is propelled by numerous (largely invisible) pore-scale abrupt interfacial jumps and pressure bursts. Fluid fronts in porous media are characterized by sharp phase discontinuities and by rapid pore-scale dynamics that underlie their motion; both attributes challenge standard continuum theories of these flow processes. Moreover, details of pore-scale dynamics affect front morphology and subsequent phase entrapment behind a front and thereby shape key macroscopic transport properties of the unsaturated zone. The study presents a pore-throat network model that focuses on quantifying interfacial dynamics and interactions along fluid displacement fronts. The porous medium is represented by a lattice of connected pore throats capable of detaining menisci and giving rise to fluid-fluid interfacial jumps (the study focuses on flow rate controlled drainage). For each meniscus along the displacement front we formulate a local inertial, capillary, viscous, and hydrostatic force balance that is then solved simultaneously for the entire front. The model enables systematic evaluation of the role of inertia and boundary conditions. Results show that while displacement patterns are affected by inertial forces mainly by invasion of throats with higher capillary resistance, phase entrapment (residual saturation) is largely unaffected by inertia, limiting inertial effects on hydrological properties behind a front. Interfacial jump velocities are often an order of magnitude larger than mean front velocity, are strongly dependent on geometrical throat dimensions, and become less predictable (more scattered) when inertia is considered. Model simulations of the distributions of capillary pressure fluctuations and waiting times between invasion events follow an exponential distribution and are in good agreement with experimental results. The modeling approach provides insights into the rich pore-scale dynamics of displacement fronts; these insights not only improve the basic understanding of these ubiquitous processes, but could shed light on solute dispersion and colloids mobilization at fronts and the mechanical consequences of passing fronts.

History independence of steady state in simultaneous two-phase flow through two-dimensional porous media

It is well known that the transient behavior during drainage or imbibition in multiphase flow in porous media strongly depends on the history and initial condition of the system. However, when the steady-state regime is reached and both drainage and imbibition take place at the pore level, the influence of the evolution history and initial preparation is an open question. Here, we present an extensive experimental and numerical work investigating the history dependence of simultaneous steady-state two-phase flow through porous media. Our experimental system consists of a Hele-Shaw cell filled with glass beads which we model numerically by a network of disordered pores transporting two immiscible fluids. From measurements of global pressure evolution, histograms of saturation, and cluster-size distributions, we find that when both phases are flowing through the porous medium, the steady state does not depend on the initial preparation of the system or on the way it has been reached.

Interface dynamics of immiscible two-phase lattice-gas cellular automata: a model with random dynamic scatterers and quenched disorder in two dimensions

We use a lattice gas cellular automata model in the presence of random dynamic scattering sites and quenched disorder in the two-phase immiscible model with the aim of producing an interface dynamics similar to that observed in Hele-Shaw cells. The dynamics of the interface is studied as one fluid displaces the other in a clean lattice and in a lattice with quenched disorder. For the clean system, if the fluid with a lower viscosity displaces the other, we show that the model exhibits the Saffman-Taylor instability phenomenon, whose features are in very good agreement with those observed in real (viscous) fluids. In the system with quenched disorder, we obtain estimates for the growth and roughening exponents of the interface width in two cases: viscosity-matched fluids and the case of unstable interface. The first case is shown to be in the same universality class of the random deposition model with surface relaxation. Moreover, while the early-time dynamics of the interface behaves similarly, viscous fingers develop in the second case with the subsequent production of bubbles in the context of a complex dynamics. We also identify the Hurst exponent of the subdiffusive fractional Brownian motion associated with the interface, from which we derive its fractal dimension and the universality classes related to a percolation process.

Effective rheology of bubbles moving in a capillary tube

We calculate the average volumetric flux versus pressure drop of bubbles moving in a single capillary tube with varying diameter, finding a square-root relation from mapping the flow equations onto that of a driven overdamped pendulum. The calculation is based on a derivation of the equation of motion of a bubble train, considering the capillary forces and the entropy production associated with the viscous flow. We also calculate the configurational probability of the positions of the bubbles.

Macroscopic phase-field model of partial wetting: bubbles in a capillary tube

Drops and bubbles are nonspreading, local, compactly supported features. They are also equilibrium configurations in partial wetting phenomena. Yet, current macroscopic theories of capillary-dominated flow are unable to describe these systems. We propose a framework to model multiphase flow in porous media with nonspreading equilibrium configurations. We illustrate our approach with a one-dimensional model of two-phase flow in a capillary tube. Our model allows for the presence of compactons: nonspreading steady-state solutions in the absence of external forces. We show that local rate dependency is not needed to explain globally rate-dependent displacement patterns, and we interpret dynamic wetting transitions as the route from equilibrium, capillary-dominated configurations, towards viscous-dominated flow. Mathematically, these transitions are possible due to nonclassical shock solutions and the role of bistability and higher-order terms in our model.

Strong influence of geometrical heterogeneity on drainage in porous media

We present an experimental study of drainage in two-dimensional porous media exhibiting bimodal pore size distributions. The role of the pore size heterogeneity is investigated by measuring separately the desaturation curves of the two pore populations. The displaced wetting fluid remains trapped in small pores at low capillary numbers and is swept only above a critical capillary number proportional to the permeability of the big pores network. Based on this observation, we derive a simple criterion for phase trapping based on the balance of viscous to capillary forces. Numerical implementation of this theory in a pore network model quantitatively fits our experimental results. This combination of approaches demonstrates quantitatively the influence of geometrical heterogeneities on drainage in porous media.

Patterns and flow in frictional fluid dynamics

Pattern-forming processes in simple fluids and suspensions have been studied extensively, and the basic displacement structures, similar to viscous fingers and fractals in capillary dominated flows, have been identified. However, the fundamental displacement morphologies in frictional fluids and granular mixtures have not been mapped out. Here we consider Coulomb friction and compressibility in the fluid dynamics, and discover surprising responses including highly intermittent flow and a transition to quasi-continuodynamics. Moreover, by varying the injection rate over several orders of magnitude, we characterize new dynamic modes ranging from stick-slip bubbles at low rate to destabilized viscous fingers at high rate. We classify the fluid dynamics into frictional and viscous regimes, and present a unified description of emerging morphologies in granular mixtures in the form of extended phase diagrams.

Pattern-forming processes in simple fluids and suspensions are well understood, but displacement morphologies in frictional fluids and granular mixtures have not been studied extensively. Sandnes et al. consider the effects of Coulomb friction and compressibility on the fluid dynamics of granular mixtures.

Flow coupling during three-phase gravity drainage

We measure the three-phase oil relative permeability k(ro) by conducting unsteady-state drainage experiments in a 0.8 m water-wet sand pack. We find that when starting from capillary-trapped oil, k(ro) shows a strong dependence on both the flow of water and the water saturation and a weak dependence on oil saturation, contrary to most models. The observed flow coupling between water and oil is stronger in three-phase flow than two-phase flow, and cannot be observed in steady-state measurements. The results suggest that the oil is transported through moving gas-oil-water interfaces (form drag) or momentum transport across stationary interfaces (friction drag). We present a simple model of friction drag which compares favorably to the experimental data.

Drainage in two-dimensional porous media: from capillary fingering to viscous flow

This paper reports some experimental results on two-phase flows in model two-dimensional porous media. Standard microfluidic techniques are used to fabricate networks of straight microchannels and to control the throat size distribution. We analyze both the invasion mechanism of the medium by a nonwetting fluid and the drainage after the percolation for capillary numbers lying between 10(-7) and 10(-2). We propose a crude model allowing a description of the observed capillary fingering that captures its scaling properties. This model is supported by numerical simulations based on a pore-network model. Numerical simulations and experiments agree quantitatively.

Oscillation-induced displacement patterns in a two-dimensional porous medium: a lattice Boltzmann study

We present a numerical study of the statistical behavior of a two-phase flow in a two-dimensional porous medium subjected to an oscillatory acceleration transverse to the overall direction of flow. A viscous nonwetting fluid is injected into a porous medium filled with a more viscous wetting fluid. During the whole process sinusoidal oscillations of constant amplitude and frequency accelerates the porous medium sideways, perpendicular to the overall direction of flow. The invasion process displays a transient behavior where the saturation of the defending fluid decreases, before it enters a state of irreducible wetting fluid saturation, where there is no net transport of defending fluid toward the outlet of the system. In this state the distribution of sizes of the remaining clusters are observed to obey a power law with an exponential cutoff. The cutoff cluster size is found to be determined by the flow and oscillatory stimulation parameters. This cutoff size is also shown to be directly related to the extracted amount of defending fluid. Specifically, the results show that the oscillatory acceleration of the system leads to potentially a large increase in extracted wetting fluid.

Steady-state, simultaneous two-phase flow in porous media: an experimental study

We report on experimental studies of steady-state two-phase flow in a quasi-two-dimensional porous medium. The wetting and the nonwetting phases are injected simultaneously from alternating inlet points into a Hele-Shaw cell containing one layer of randomly distributed glass beads, initially saturated with wetting fluid. The high viscous wetting phase and the low viscous nonwetting phase give a low viscosity ratio M=10(-4). Transient behavior of this system is observed in time and space. However, we find that at a certain distance behind the initial front a "local" steady-state develops, sharing the same properties as the later "global" steady state. In this state the nonwetting phase is fragmented into clusters, whose size distribution is shown to obey a scaling law, and the cutoff cluster size is found to be inversely proportional to the capillary number. The steady state is dominated by bubble dynamics, and we measure a power-law relationship between the pressure gradient and the capillary number. In fact, we demonstrate that there is a characteristic length scale in the system, depending on the capillary number through the pressure gradient that controls the steady-state dynamics.

Flux-dependent percolation transition in immiscible two-phase flows in porous media

Using numerical simulations, we study immiscible two-phase flow in a pore network reconstructed from Berea sandstone under flow conditions that are statistically invariant under translation. Under such conditions, the flow is a state function which is not dependent on initial conditions. We find a second-order phase transition resembling the phase inversion transition found in emulsions. The flow regimes under consideration are those of low surface tension-hence high capillary numbers Ca-where viscous forces dominate. Nevertheless, capillary forces are imminent, we observe a critical stage in saturation where the transition takes place. We determine polydispersity critical exponent tau=2.27+/-0.08 and find that the critical saturation depends on how fast the fluids flow.