Two-phase Flow In Porous Media Research Articles

Immiscible Two-Phase Flow in Porous Media: Effective Rheology in the Continuum Limit

AbstractWe consider steady-state immiscible and incompressible two-phase flow in porous media. It is becoming increasingly clear that there is a flow regime where the volumetric flow rate depends on the pressure gradient as a power law with an exponent larger than one. This occurs when the capillary forces and viscous forces compete. At higher flow rates, where the viscous forces dominate, the volumetric flow rate depends linearly on the pressure gradient. This means that there is a crossover pressure gradient that separates these two flow regimes. At small enough pressure gradient, the capillary forces dominate. If one or both of the immiscible fluids percolate, the volumetric flow rate will then depend linearly on the pressure gradient as the interfaces will not move. If none of the fluids percolate, there will be a minimum pressure gradient threshold to mobilize the interfaces and thereby get the fluids moving. We now imagine a core sample of a given size. The question we pose is what happens to the crossover pressure gradient that separates the power-law regime from the high-flow rate linear regime and the threshold pressure gradient that blocks the flow at low pressure gradients when the size of the core sample is increased. Based on analytical calculations using the capillary bundle model and on numerical simulations using a dynamical pore-network model, we find that the crossover pressure gradient and the threshold pressure gradient decrease with two distinct power laws in the size. This means that the power-law regime disappears in the continuum limit where the pores are infinitely small compared to the sample size.

Two-phase Porous Media Flow Model Based on the Incompressible Navier-Stokes Equation.

Two-phase porous media flow is important in many applications from drug delivery to groundwater diffusion and oil recovery and is of particular interest to biomedical diagnostic test developers using cellulose and nitrocellulose membranes with limited fluid sample volumes. This work presents a new two-phase porous media flow model based on the incompressible Navier-Stokes equation. The model aims to address the limitations of existing methods by incorporating a partial saturation distribution in porous media to account for limited fluid volumes. The basic parameters of the model are the pore size distribution and the contact angle. To validate the model, we solved five analytical solutions and compared them to corresponding experimental data. The experimentally measured penetration length data agreed with the model predictions, demonstrating model accuracy. Our findings suggest that this new two-phase porous media flow model can provide a valuable tool for researchers developing fluidic assays in paper and other porous media.

Disorder-induced non-linear growth of fingers in immiscible two-phase flow in porous media

Immiscible two-phase flow in porous media produces different types of patterns depending on the capillary number Ca and viscosity ratio M. At high Ca, viscous instability of the fluid–fluid interface occurs when the displaced fluid is the more viscous, and leads to viscous fingering, which is believed to exhibit the same growth behavior as the viscously-unstable fingers observed in Hele–Shaw cells by Saffman and Taylor [“The penetration of a fluid into a porous medium or Hele–Shaw cell containing a more viscous liquid,” Proc. R. Soc. London 245, 312 (1958)], or as diffusion-limited aggregates (DLA). In such Laplacian growth processes, the interface velocity depends linearly on the local gradient of the physical field that drives the growth process (for two-phase flow, the pressure field). However, a non-linear power-law dependence between the flow rate and the global pressure drop, reminiscent of what has also been observed for steady-state two-phase flow in porous media, was evidenced experimentally for the growth of viscously-unstable drainage fingers in two-dimensional porous media, 20 years ago. Here, we revisit this flow regime using dynamic pore-network modeling and explore the non-linearity in the growth properties. We characterize the previously unstudied dependencies of the statistical finger width and non-linear growth law's exponent on Ca, and discuss quantitatively, based on theoretical arguments, how disorder in the capillary barriers controls the growth process' non-linearity, and why the flow regime crosses over to Laplacian growth at sufficiently high Ca. In addition, the statistical properties of the fingering patterns are compared to those of Saffman–Taylor fingers, DLA growth patterns, and the results from the aforementioned previous experimental study.

Effect of surfactant adsorption on the efficiency of drainage displacement at different viscosity ratios: A lattice Boltzmann study

Surfactant flooding is an effective method for enhanced oil recovery, accompanied by the adsorption of surfactant molecules during their transport in porous media. This paper studies the effects of surfactant adsorption on immiscible two-phase flows in porous media and focuses on two primary objectives. The first is to identify the influence of the water/oil viscosity ratio on the maximum adsorbed amount of surfactant. The second scope involves establishing a relationship between the viscosity ratio and the effect of surfactant adsorption on the displacement efficiency during drainage displacement. The novelty lies in the identification of new patterns characterizing the adsorption effect on the displacement efficiency depending on the viscosity ratio. The main results are based on statistical data collected from numerical simulations performed on various synthetic 2D micromodels. The lattice Boltzmann method was used as a mathematical model. It has been established that the adsorbed amount is influenced by the viscosity ratio as follows: the higher the oil viscosity, the less the adsorbed amount. However, the effect is not strong. An increase in oil viscosity leads to a decrease in the adsorption effect on displacement efficiency. This finding is determined by two factors: the maximum adsorbed amount and the adsorbed amount at the moment of surfactant breakthrough. It has been found that if surfactant adsorption is neglected, the viscosity ratio does not affect the increment in oil recovery compared to the injection of pure water. However, if adsorption is taken into account, an increase in oil viscosity enhances the effect of surfactant flooding on displacement efficiency. The dependence of the viscosity ratio on the adsorption effect on oil recovery obeys linear law. This trend was confirmed in a digital 3D model of natural sandstone obtained using X-ray computed tomography.

Bridging the gap: Connecting pore-scale and continuum-scale simulations for immiscible multiphase flow in porous media

Bridging the gap: Connecting pore-scale and continuum-scale simulations for immiscible multiphase flow in porous media

Machine Learning-Based Interfacial Tension Equations for (H2 + CO2)-Water/Brine Systems over a Wide Range of Temperature and Pressure.

Large-scale underground hydrogen storage (UHS) plays a vital role in energy transition. H2-brine interfacial tension (IFT) is a crucial parameter in structural trapping in underground geological locations and gas-water two-phase flow in subsurface porous media. On the other hand, cushion gas, such as CO2, is often co-injected with H2 to retain reservoir pressure. Therefore, it is imperative to accurately predict the (H2 + CO2)-water/brine IFT under UHS conditions. While there have been a number of experimental measurements on H2-water/brine and (H2 + CO2)-water/brine IFT, an accurate and efficient (H2 + CO2)-water/brine IFT model under UHS conditions is still lacking. In this work, we use molecular dynamics (MD) simulations to generate an extensive (H2 + CO2)-water/brine IFT databank (840 data points) over a wide range of temperature (from 298 to 373 K), pressure (from 50 to 400 bar), gas composition, and brine salinity (up to 3.15 mol/kg) for typical UHS conditions, which is used to develop an accurate and efficient machine learning (ML)-based IFT equation. Our ML-based IFT equation is validated by comparing to available experimental data and other IFT equations for various systems (H2-brine/water, CO2-brine/water, and (H2 + CO2)-brine/water), rendering generally good performance (with R2 = 0.902 against 601 experimental data points). The developed ML-based IFT equation can be readily applied and implemented in reservoir simulations and other UHS applications.

A benchmark study on reactive two-phase flow in porous media: Part II - results and discussion

A benchmark study on reactive two-phase flow in porous media: Part II - results and discussion

A benchmark study on reactive two-phase flow in porous media: Part I - model description

A benchmark study on reactive two-phase flow in porous media: Part I - model description

Drainage dynamics: From capillary to viscous fingering under different wettability

The displacement of two-phase fluids within porous media is a critical process in various engineering and industrial fields. However, the inherent instability of the fluid interface often leads to an irregular morphology of the displacement front, which can impact the efficiency of displacement and limit its application in engineering activities. In this study, the color gradient model based on the lattice Boltzmann method was employed to simulate displacement, with the aim of investigating the dynamic evolution of the displacement front under the combined effects of wettability and injection velocity, specifically during the capillary to viscous fingering transition. The results demonstrate that the impact of wettability on the displacement front morphology increases as the injection velocity decreases. The displacement process with the maximum fingering distribution range and compact morphology is found at the midpoint of the capillary-viscous fingering crossover zone. Furthermore, an in-depth analysis of the phase field evolution and pressure difference changes between the inlet and outlet unveiled the pore-filling mechanisms associated with the unique variation characteristics of the displacement front and the fluid topological structure. These comprehensive pore-scale findings offer theoretical insight supporting the engineering application of the two-phase flow in porous media from multiple perspectives.

Fractal theory and dynamic contact angle-based imbibition model for two-phase flow in porous media

During the development of tight oil reservoirs, there are significant occurrences of spontaneous imbibition. Understanding the spontaneous imbibition behavior at the core scale of tight sandstone holds significant importance in improving the recovery rate. This study presents a novel mathematical model for characterizing the spontaneous imbibition phenomenon in tight porous media, drawing upon the fractal theory and the dynamic contact angle in capillary bundles. The proposed model has been verified by the results of core imbibition experiments in the literature. Furthermore, we conducted spontaneous imbibition simulation studies using core structures of different pore types extracted from real tight reservoirs to validate the applicability of the new mathematical model. Comparative analysis shows that the derived mathematical approach fits well with the simulation results, but the heterogeneity of the pore space can lead to certain errors between the model and the simulation results. The influencing factors analysis suggests that the higher the porosity, the higher the final recovery rate, whereas an increase in pore fractal dimension has little effect on the final recovery rate.

A New Fully Implicit Two-Phase Pore-Network Model by Utilizing Regularization Strategies

In this paper, we address the expensive computational cost resulting from limited time-step sizes during numerical simulations of two-phase flow in porous media using dynamic pore-network models. To overcome this issue, we propose a numerical method for dynamic pore-network models using a fully implicit approach. The proposed method introduces a regularization strategy considering the historical fluid configuration at the pore throat, which smooths the discontinuities in local conductivity caused by invasion and snap-off events. The results demonstrate the superiority of the proposed method in terms of accuracy, efficiency and consistency in comparison with other numerical schemes. With similar computational cost, determined by time-step sizes and number of Newton iterations, the developed method in this work yields more accurate results compared to similar schemes presented in the literature. Additionally, our results highlight the enhanced robustness of the our scheme, as it exhibits reduced sensitivity to variations in time-step sizes.

Flow Rate Dependency of Steady-State Two-Phase Flows in Pore Networks: Universal, Relative Permeability Scaling Function and System-Characteristic Invariants

The phenomenology of steady-state two-phase flow in porous media is conventionally recorded by the relative permeability diagrams in terms of saturation. Yet, theoretical, numerical and laboratory studies of flow in artificial pore network models and natural porous media have revealed a significant dependency on the flow rates—especially when the flow regime is capillary to capillary/viscous and part of the disconnected non-wetting phase remains mobile. These studies suggest that relative permeability models should incorporate the functional dependence on flow intensities. In the present work, a systematic dependence of the pressure gradient and the relative permeabilities on flow rate intensity is revealed. It is based on extensive simulations of steady-state, fully developed, two-phase flows within a typical 3D model pore network, implementing the DeProF mechanistic–stochastic model algorithm. Simulations were performed across flow conditions spanning 5 orders of magnitude, both in the capillary number, Ca, and the flow rate ratio, r, and for different favorable /unfavorable viscosity ratio fluid systems. The systematic, flow rate dependency of the relative permeabilities can be described analytically by a universal scaling function along the entire domain of the independent variables of the process, Ca and r. This universal scaling comprises a kernel function of the capillary number, Ca, that describes the asymmetric effects of capillarity across the entire flow regime—from capillarity-dominated to mixed capillarity/viscosity- to viscosity-dominated flows. It is shown that the kernel function, as well as the locus of the cross-over relative permeability values, are single-variable functions of the capillary number; they are both identified as viscosity ratio invariants of the system. Both invariants can be correlated with the structure of the pore network, through a function of Ca. Consequently, the correlation is associated with the wettability characteristics of the system. Among the potential applications, the proposed, universal, flow rate dependency scaling laws are the improvement of core analysis and dynamic rock-typing protocols, as well as integration into field-scale simulators or associated machine learning interventions for improved specificity/accuracy.

Generalized finite difference method-based numerical modeling of oil–water two-phase flow in anisotropic porous media

This paper applies generalized finite difference method (GFDM) to a compressible two-phase flow in anisotropic porous media with the aim of further extending the wider application of this class of meshless methods. We develop an implicit Euler scheme in time and a GFDM discretization in space based on two treatments of the anisotropic permeability tensor in continuous function expression and discrete distribution. The effectiveness and generality of GFDM for two-phase flow problems in anisotropic porous media are verified by three examples with rectangular, irregular, and complex boundaries. Also, the computational performance of the method is verified according to the error calculation with L2 absolute error functions in different node collocation schemes. In addition, the sensitivity analysis of the radius of the influence domain to the transient pressure equation (parabolic equation) and the saturation equation (hyperbolic equation) is considered. It generally holds that the larger the radius of the influence domain, the lower the calculation accuracy in the case of Cartesian collocation. This may be a preliminary rule for the radius choice of the influence domain for GFDM. In sum, this work provides an efficient and accurate meshless solver to handle two-phase flow problems in anisotropic porous media under the GFDM framework, which reveals the great application potential of GFDM in reservoir numerical simulation.

Flow-Dependent Relative Permeability Scaling for Steady-State Two-Phase Flow in Porous Media: Laboratory Validation on a Microfluidic Network

Conventionally, the relative permeabilities of two immiscible fluid phases flowing in porous media are considered and expressed as functions of saturation. Yet, this has been put into challenge by theoretical, numerical, and laboratory studies of flow in artificial pore network models and real porous media. These works have revealed a significant dependency of the relative permeabilities on the flow rates, especially when the flow regime is capillary to capillary-viscous dominated, and part of the disconnected nonwetting phase remains mobile. These studies suggest that relative permeability models should include the functional dependence on flow intensities. However, revealing the explicit form of such dependence remains a persistent problem. Just recently, a general form of dependence was inferred based on extensive simulations with the DeProF model for steady-state two-phase flows in pore networks. The simulations revealed a systematic dependence of the relative permeabilities on the local flow rate intensities. This dependence can be described analytically by a universal scaling functional form of the actual independent variables of the process, namely, the capillary number, Ca, and the flow rate ratio, r. The proposed scaling incorporated a kernel function, the intrinsic dynamic capillary pressure (IDCP) function, describing the transition between capillarity- and viscosity-dominated flow phenomena. In a parallel laboratory study, SCAL measurements provided a preliminary proof-of-concept on the applicability of the model. In the laboratory study presented here, we examine the applicability of the scaling model by taking extensive, ex-core measurements of relative permeabilities for steady-state co-injections of two immiscible fluids within an artificial microfluidic pore network, across different flow regimes in Ca and r. From these measurements, we calculate the values of the mobility ratio, and we compare these to the corresponding values of the flow rate ratio. We also extract the IDCP curve, the locus of critical flow conditions, whereby the process is more efficient in terms of energy utilization – accounted by the nonwetting phase flow rate per unit of total power provided to the process, as well as the locus of flow conditions of equal relative permeabilities. We show that the degree of consistency between flow rate ratio and mobility ratio values, the IDCP curve, the locus of critical flow conditions, and the locus of equal relative permeabilities, as well as some associated invariant characteristic values, can be used for assessing the extent of end effects and for characterizing the flow as capillary- or viscous-dominated. The proposed scaling introduces new opportunities for enhancing SCAL protocols and their associated applications. These include the characterization of systems and flow conditions, dynamic rock typing, evaluation of capillary end effects, as well as the advancement of more efficient field-scale simulators. Additionally, it paves the way in designing more energy-efficient EOR interventions.

Fully coupled hydro-mechanical modeling of two-phase flow in deformable fractured porous media with discontinuous and continuous Galerkin method

Discrete fracture and matrix modeling of coupled flow and geomechanics is instrumental for understanding flow in fractured media for various geoengineering applications such as enhanced geothermal energy systems and groundwater remediation. We employ the governing equations for two-phase flow in deformable fractured porous media with a stress-dependent porosity and permeability model of matrix, and variable fracture aperture and the corresponding permeability. A finite element framework is presented, in which fractures are regarded as low-dimensional objects, to discretize the coupled two-phase flow and geomechanics in fractured porous media. A hybrid method, combining discontinuous Galerkin (DG) and continuous Galerkin (CG) finite element methods (FEM), is utilized to solve for the two-phase flow, while the solid deformation is approximated using a discontinuous Galerkin FEM approach. Several benchmark cases are utilized to examine the accuracy of the proposed hybrid DG-CG FEM method. Further validation is performed using more complex, realistic fracture configurations. As the mechanical characteristics of the fracture and the surrounding matrix differ, the simulation results demonstrate that displacement and stress are discontinuous on both sides of the fracture. While the case with low permeability fractures exhibits pressure jump across the fractures, the pressure changes are reasonably smooth for the conduit fractures.

A study of non-equilibrium wave groups in two-phase flow in high-contrast porous media with relative permeability hysteresis

Non-equilibrium models are applicable in various physical situations, including phase transitions, hysteresis, and chemical reactions, among others. To model the dynamics of such phenomena, partial differential equations are employed with source terms, as seen in Eq. (1.1). This work focuses on situations where we connect states in equilibrium while allowing for non-equilibrium times. In our model, we model non-equilibrium through relaxation. The source term, in this case, acts as an attractor, and the model’s dynamics allow only a small-time far from equilibrium. We use this theory to model a two-equation system in a two-phase fluid in porous media, where different permeabilities of the flow lead to hysteresis phenomena. We show several examples and study the Riemann solution to gain a better understanding of the solution. We investigate the interplay between nonlinear wave groups of water–oil saturation with different permeabilities in the non-wetting phase and the high-contrast heterogeneity of the porous medium in two-space dimensions. This phenomenon is described by a coupled set of time-dependent partial differential equations of hyperbolic–parabolic–elliptic mixed type. We present and discuss a new projection method with relaxation to obtain 1D-analytical Riemann solutions in an idealized homogenized medium, and we use this solution to validate our numerical method. The primary components of obtaining these solutions are shocks, rarefactions, and bifurcations. We obtain 2D-numerical solutions using an operator splitting approach based on physics, highlighting the discretization methods used. We provide robust analytical–numerical examples to verify the theory and illustrate the approach’s capabilities. In particular, we discuss the 1D-2D behavior of wave groups in situations driven by water–oil infiltration problems of permeable reservoir rocks.

Fragmentation and coalescence dynamics of non-wetting blobs during immiscible two-phase flows in porous media

Fragmentation and coalescence dynamics of non-wetting blobs during immiscible two-phase flows in porous media

Investigation of porosity variation on water retention behaviour of unsaturated granular media by using pore scale Micro-CT and lattice Boltzmann method

Most soils in nature are in unsaturated states. The key to correctly understanding its hydraulic characteristics lies in mastering the correlation between its microstructure and macro water retention characteristics. Based on an independently developed miniaturized in-situ unsaturated soil–water characteristic instrument, three-phase topological morphology and micrographic information (pore throat, liquid cluster, interface area) of unsaturated granular media were explored, supplemented by CT scanning under a high resolution (5.6 μm/pixel). In addition, three-dimensional reconstruction of CT images was performed for simulating two-phase flow in porous granular media using the lattice Boltzmann method. The experimental results indicate that the water retention capacity, air-entry value, and residual saturation of unsaturated granular specimens are negatively correlated with the porosity of specimens. As the saturation decreases, the specific air–water interfacial area first increases and then decreases, with the peak value appearing around 20% of saturation. Based on LBM simulation, the water retention capacity of the specimen is negatively correlated with the contact angle, while the liquid–solid characteristic interface area and relative permeability are positively correlated. The experimental and simulation results demonstrate good consistency with each other.

Macroscopic model for generalised Newtonian inertial two-phase flow in porous media

A closed macroscopic model for quasi-steady, inertial, incompressible, two-phase generalised Newtonian flow in rigid and homogeneous porous media is formally derived. The model consists of macroscopic equations for mass and momentum balance as well as an expression for the macroscopic pressure difference between the two fluid phases. The model is obtained by upscaling the pore-scale equations, employing a methodology based on volume averaging, the adjoint method and Green's formulation, only assuming the existence of a representative elementary volume and the separation of scales between the microscale and the macroscale. The average mass equations coincide with those for Newtonian flow. The macroscopic momentum balance equation in each phase expresses the seepage velocity in terms of a dominant and a coupling Darcy-like term, a contribution from interfacial tension effects and another one from interfacial inertia. Finally, the expression of the macroscopic pressure difference is obtained in terms of the macroscopic pressure gradient and body force in each phase, and interfacial terms that account for capillary effects and inertia, if present when the interface is not stationary. All terms involved in the macroscale equations are predicted from the solution of adjoint closure problems in periodic representative domains. Numerical predictions from the upscaled models are compared with direct numerical simulations for two-dimensional configurations, considering flow of a Newtonian non-wetting fluid and a Carreau wetting fluid. Excellent agreement between the two approaches confirms the pertinence of the macroscopic models derived here.

A Dimension-Reduced Line Element Method for 3D Transient Free Surface Flow in Porous Media

In order to reduce the numerical difficulty of the 3D transient free surface flow problems in porous media, a line element method is proposed by dimension reduction. Different from the classical continuum-based methods, homogeneous permeable pores in the control volume are conceptualized by a 3D orthogonal network of tubes. To obtain the same hydraulic solution with the continuum model, the equivalent formulas of flow velocity, continuity equation and transient free surface boundary are derivable from the principle of flow balance. In the solution space of transient free surface flow, the 3D problem is transformed into 1D condition, and then a finite element algorithm is simply deduced. The greatest advantage of the line element method is line integration instead of volume/surface integration, which has dramatically decreased the integration difficulty across the jump free surface. Through the analysis of transient free surface flow in the unconfined aquifer, trapezoidal dam, sand flume and wells, the transient free surface locations predicted from the proposed line element method generally agree well with the analytical, experimental and other numerical data in the available literatures, the numerical efficiency can also be well guaranteed. Furthermore, the hydraulic anisotropy has significant effect on the evolution of free surface locations and the shape of depression cones in spatial. The line element method can be expanded to model the 3D unsaturated seepage flow, two-phase flow and thermos problems in porous media because of the similarity between the similarity of Darcy’s law, Buckingham Law and Fourier’s law.