Two-phase Flow In Porous Media Research Articles

Numerical method for hydraulic fracture propagation in a two-phase poroelastoplasticity model

Abstract The present work introduces a numerical method for mathematical modelling of the hydraulic fracture propagation process within a naturally fractured reservoir. We consider a poroelastoplasticity model with rock failure described by an advanced constitutive material model representing the tensile, shear, and compressional failure of the formation. We propose a discretization method for the elastoplastic model coupled to the equations for two-phase fluid flow in porous media. The discretization is based on a collocated cell-centered finite volume method that uses an advanced approximation of the traction vector for mechanics and the conventional single-point upstream-weighted two-point approximation of the Darcy flux. The Biot term coupling of the mechanical and flow models is approximated assuming piecewise constant pore pressure, leading to an inf-sup stable method. To solve the plasticity problem, we use the cutting plane algorithm. The plastic strain tensor derivatives are obtained in the solution process, thus bypassing the necessity of the consistent tangent elastoplastic stiffness tensor. We study the grid convergence for the discrete solution of poroelasticity equations on a set of problems with analytical solutions and demonstrate the application of proposed methods for poroplasticity to tensile fractures near a borehole and hydraulic fracturing experiments.

Nonequilibrium Phase Behavior of Condensate Gas Flow in Porous Media.

The description and quantification of condensate gas flow in porous media are often inadequate when considering nonequilibrium phase behavior. This study presents a systematic approach that integrates experimental research with advanced mathematical techniques. A pressure depletion test conducted on a long core demonstrates that a higher condensate recovery can be achieved with faster pressure drop rates. A novel mathematical nonequilibrium compositional model, which is based on kinetic interphase mass transfer between gas and oil phases, was developed. This model accurately captures dynamic changes in the interface area between phases and corrects nonequilibrium thermodynamic equations. It can seamlessly transition between equilibrium and nonequilibrium phase behavior during two-phase flow in porous media. The model exhibits deviations of less than 5% from experimental results for condensate recovery and gas-oil ratio predictions when comparing pressure drop rates of 2.5 and 5 MPa/h. Its predictions closely match experimental data, showing a superior accuracy over the standard equilibrium model. The characteristic phase behavior parameters, such as condensate saturation and pseudocomponent composition, are highly sensitive to pressure drop rate, highlighting the nonequilibrium model's ability to address condensate hysteresis. This study provides a solid theoretical foundation for reservoir simulations of the pressure depletion process in condensate gas reservoirs and other scenarios involving nonequilibrium subsurface fluid flow and recovery.

Two-phase flow in porous media under the influence of external electric and magnetic field: Homogenization approach

Multiphase flow in porous media under the influence of electric and magnetic fields is of significant interest in applications, such as enhanced oil recovery, microfluidics, and electrochemical systems. This study aims to develop a novel theoretical and numerical framework to model immiscible two-phase flow through a rigid, homogeneous porous medium exposed to external electromagnetic fields. The fluids are assumed to be Newtonian and incompressible, and the porous matrix is considered periodic. A diffuse interface approach is employed using the Navier–Stokes–Cahn–Hilliard system, and the influence of external fields is incorporated through Lorentz force terms. Asymptotic two-scale homogenization is applied to upscale the coupled system, leading to a macroscopic model governed by an electromagneto (EM)-permeability tensor. The model equations are solved using the finite element method. The results reveal that the capillary number (Ca), Hartmann number (Ha), and electric field intensity (S) significantly influence the permeability tensor. Off-diagonal components, indicating anisotropy, emerge under the combined action of electric and magnetic fields. This anisotropic control enables field-responsive manipulation of flow characteristics. The proposed model offers insights into the design of field-tunable porous systems and provides a foundation for future exploration in three-dimensional and reactive transport settings.

Geometric structure of parameter space in immiscible two-phase flow in porous media

In a recent paper, a continuum theory of immiscible and incompressible two-phase flow in porous media based on generalized thermodynamic principles was formulated (Transport in Porous Media, 125, 565 (2018)). In this theory, two immiscible and incompressible fluids flowing in a porous medium are treated as a single effective fluid, substituting the two interacting subsystems for a single system with an effective viscosity and pressure gradient. In assuming Euler homogeneity of the total volumetric flow rate and comparing the resulting first-order partial differential equation to the total volumetric flow rate in the porous medium, one can introduce a novel velocity that relates the two pairs of velocities. This velocity, the co-moving velocity, describes the mutual co-carrying of fluids due to immiscibility effects and interactions between the fluid clusters and the porous medium itself. The theory is based upon general principles of classical thermodynamics and allows for many relations and analogies to draw upon in analyzing two-phase flow systems in this framework. The goal of this work is to provide additional connections between geometric concepts and the variables appearing in the thermodynamics-like theory of two-phase flow. In this endeavor, we will encounter two interpretations of the velocities of the fluids: as tangent vectors (derivations) acting on functions or as coordinates on an affine line. The two views are closely related, with the former viewpoint being more useful in relation to the underlying geometrical structure of equilibrium thermodynamics and the latter being more useful in concrete computations and finding examples of constitutive relations. We apply these relatively straightforward geometric contexts to interpret the relations between velocities and, from this, obtain a general form for the co-moving velocity.

Interface instability of two-phase flow in a three-dimensional porous medium

Interface instability of two-phase flow in a three-dimensional porous medium

A class of high-order physics-preserving schemes for thermodynamically consistent model of incompressible and immiscible two-phase flow in porous media

A class of high-order physics-preserving schemes for thermodynamically consistent model of incompressible and immiscible two-phase flow in porous media

A comprehensive study of two-phase flow in deformable porous media

This study explored the initial-boundary value problem of an immiscible, incompressible two-phase flow within a deformable porous medium system, modeled in an N-dimensional (N≥3) Euclidean space. The primary objective was to devise a novel model that effectively captured previously overlooked physical phenomena and established the global existence of a weak solution to the system. After proving the existence of a classical solution to the approximate problem using Schauder's fixed point theorem under a series of reasonable assumptions, we obtained the convergence to a weak solution of the original problem in two different cases by applying the Fréchet–Kolmogorov theorem. Subsequently, we verified the applicability of this model to the commonly used van Genuchten model in the dynamic simulation of oil storage, providing theoretical support for engineering applications.

THE HYDRODYNAMİCS OF GAS-LİQUİD MİXTURE FLOW IN POROUS MEDIA

Equations and models that describe the hydrodynamics of gas-liquid two-phase flows in porous media are becoming increasingly crucial for predicting their primary behaviors within porous structures. These models are essential in understanding the flow dynamics in various applications, such as enhanced oil recovery, soil treatment, and reactor design. The focus of this research was to examine the effects of capillary forces, viscous forces, inertial forces, and flow configurations on the hydrodynamic features of a gas-liquid two-phase flow within a glass micromodel. By conducting experiments, results were gathered and then compared with predictions made by three different established models. The two models, Fundamental Forces Balance and Fluid-Fluid Interface models, did not accurately capture the experimental behavior, despite the fact that the Fundamental Forces Balance model incorporates particular flow pattern characteristics. These models were not able to fully describe that real-word behavior of gas-liquid two-phase flow, indicating that improvements are necessary. On the other hand, semi-empirical models, such as the Relative Permeability model, provide a better representation of the physical flow characteristics. One key advantage of semi-empirical models is that they can be adjusted to account for additional effects, such as varying flow configurations and interfacial interactions, which are not initially included in the basic theoretical models. Traditionally, relative permeabilities have been almost exclusively linked to saturation conditions in porous media. However, this research concluded that liquid relative permeability is not solely dependent on saturation levels. Instead, is also depends on flow patterns and the Capillary number. These findings highlight the importance of incorporating multiple factors when developing more accurate and reliable models for gas-liquid two-phase flow in porous media. Keywords: porous media; hydrodynamic models; flow patterns; gas-liquid two-phase flow; flow patterns; relative permeabilities.

Microscopic simulations of oil–water two-phase flow in high-porosity and low-permeability carbonate porous media

The pore types in carbonate reservoirs are highly diverse, and a detailed characterization of the flow behavior of oil–water two-phase flow at the pore scale within different pore storage types holds significant importance. In this work, digital core reconstruction based on computed tomography scanning technology has quantitatively characterized the micro-pore structures of these rocks, and typical core samples representing diverse pore storage types have been selected for microscopic visualization simulation studies. Utilizing the volume of fluid method, we conducted visual simulations of oil–water two-phase flow in porous media. Comparisons were made under varying conditions of wettability, displacement pressure, and viscosity ratio regarding breakthrough time, residual oil distribution, and changes in residual oil saturation, revealing the dynamic flow characteristics of oil–water phases within different pore types during water flooding. The results demonstrate that complex pore-throat structures (large pores, small throats) significantly reduce the displacement efficiency during microscopic water flooding. Specifically, moldic pores exhibit high permeability, leading to oil phase retention; biological chamber pores (intraparticle pores) are characterized by the most pronounced high-porosity and low-permeability features, with numerous blind-end voids and poor connectivity, resulting in limited displacement effectiveness, whereas intergranular dissolution pores show good connectivity, achieving more efficient oil recovery. The mobility of the water phase among the three pore types follows the order: intergranular dissolution pores > moldic pores > biological chamber pores. Furthermore, improvements in wettability, increased displacement pressure, and reduced oil–water viscosity ratio serve to optimize the flow process at the microscopic level, thereby enhancing overall oil recovery efficiency.

Pore-scale simulation of the effects of the interfacial viscoelasticity on oil displacement by low-salinity water in porous media

Enhanced oil recovery through low-salinity waterflooding hinges not only on rock–fluid interactions but also on the intricate interaction between the fluids. Adsorption of polar macromolecules of the oil phase onto the oil-brine interface leads to non-isotropic interfacial tension and a complex rheological reaction to deformations. This phenomenon becomes more complicated by the pore structure during two-phase flow in porous media. This paper represents a pioneering effort to model the influence of interfacial viscoelasticity on oil-brine displacement at the pore-scale using computational fluid dynamics simulations. Following model validation with experiments, we delve into the distinct salinity-dependent effects of shear and dilatational properties of the interface in different geometries with different pore-to-throat aspect ratios and at a capillary-dominated flow regime. The findings indicate that shear stresses arising from interfacial viscoelasticity influence the relative movement of the wetting and non-wetting phases within the pores, resulting in the reduction of trapped oil by improving its mobility. It is also found that the dilatational effects stretch the oil phase passing through the throat. This effect provides additional time for the oil to be released, culminating in a reduction in the trapped oil volume by as much as 5%–7%. This is while the interfacial viscous forces tend to hamper oil flow and increase its snap-off. The viscoelastic forces become more pronounced as the aspect ratio (pore-to-throat size) exceeds 2–3. While conventional waterflooding would result in high remaining oil saturation, this phenomenon arising from lowering water salinity proves to be particularly efficient for oil recovery.

Derivation of A Representative Elementary Volume (REV) for Upscaled Two-Phase Flow in Porous Media

Relative permeability plays an important role in the upscaling of multiphase flow in porous media from the pore scale to the Darcy scale. The entire concept of relative permeability is contingent on the existence of a representative elementary volume (REV). As we move to smaller samples to measure relative permeability, such as with digital core analysis, the concept of a classical REV has become increasingly unlikely when using the conventional approach to defining a representative volume. The “‘conventional”’ understanding of an REV is that a large enough volume must be considered such that spatial variability averages out. In digital rock methods, such as pore-scale simulations based on micro-computed tomography (CT) images, the domain size is typically 2 to 4 mm. This is approximately the length scale of a single-phase flow REV using the classic REV approach. However, the single-phase perspective does not consider the complex dynamics and fluctuations often observed in multiphase flow systems, even at centimeter-scale experiments and/or simulations. A fundamental question is, therefore, whether the domain size commonly used in digital rock simulations can provide a consistent energy budget such that the concept of relative permeability exists. Based on first principles, relative permeability accounts for the rate of energy dissipated in a stationary process. If the dynamics are fluctuating, the energy dissipated can vary but will average out over a long enough timescale. The key to determining the validity of the relative permeability is the timescale of the measurement, not the spatial scale. The conventional REV theory assumes that spatial, temporal, and ensemble averages are equivalent in an ergodic system, but it does not provide a way to test this assumption. We provide a formal way to identify the timescale where the relative permeability accurately captures energy dissipation as a way to validate relative permeability measurements and quantitatively assess their accuracy. This result will be tested for a practical SCAL test, determining how long a flow experiment needs to be run to accurately characterize the rate of energy dissipation by the flow. The outcome will be a best practice guide for the determination of relative permeability from core-scale experiments and/or digital core simulations that ensure the energy budget is fully accounted for in the relative permeability coefficient.

Thermodynamics-like Formalism for Immiscible and Incompressible Two-Phase Flow in Porous Media

It is possible to formulate an immiscible and incompressible two-phase flow in porous media in a mathematical framework resembling thermodynamics based on the Jaynes generalization of statistical mechanics. We review this approach and discuss the meaning of the emergent variables that appear, agiture, flow derivative, and flow pressure, which are conjugate to the configurational entropy, the saturation, and the porosity, respectively. We conjecture that the agiture, the temperature-like variable, is directly related to the pressure gradient. This has as a consequence that the configurational entropy, a measure of how the fluids are distributed within the porous media and the accompanying velocity field, and the differential mobility of the fluids are related. We also develop elements of another version of the thermodynamics-like formalism where fractional flow rather than saturation is the control variable, since this is typically the natural control variable in experiments.

The Characteristics of Two-phase Flow in Porous Media over Wide Range of Capillary, Viscous, and Inertial Force

The Characteristics of Two-phase Flow in Porous Media over Wide Range of Capillary, Viscous, and Inertial Force

Study of Gas–Liquid Two-Phase Flow Characteristics at the Pore Scale Based on the VOF Model

To study the effects of liquid properties and interface parameters on gas–liquid two-phase flow in porous media. The volume flow model of gas–liquid two-phase flow in porous media was established, and the interface of the two-phase flow was reconstructed by tracing the phase fraction. The microscopic imbibition flow model was established, and the accuracy of the model was verified by comparing the simulation results with the classical capillary imbibition model. The flow characteristics in the fracturing process and backflow process were analyzed. The influence of flow parameters and interface parameters on gas flow was studied using the single-factor variable method. The results show that more than 90% of the flowing channels are invaded by fracturing fluid, and only about 50% of the fluid is displaced in the flowback process. Changes in flow velocity and wetting angle significantly affect Newtonian flow behavior, while variations in surface tension have a pronounced effect on non-Newtonian fluid flow. The relative position of gas breakthrough in porous media is an inherent property of porous media, which does not change with fluid properties and flow parameters.

Two-phase flow in heterogeneous porous media based on Brinkman and Darcy models

Two-phase flow in heterogeneous porous media based on Brinkman and Darcy models

Multimodality Imaging of Fluid Saturation and Chemical Transport for Two-Phase Surfactant/Polymer Floods in Porous Rocks

Multicomponent, two-phase flow in porous media is a problem of practical relevance that remains difficult to study experimentally. Advanced methodologies are needed that enable the monitoring of both the saturation of each fluid phase within the pore space and the concentration of the chemical species within the fluids. We present an approach based on multimodality imaging and apply it to the case study of surfactant/polymer flooding in a sandstone for enhanced oil recovery. X-ray computed tomography and positron emission tomography (PET) are applied for the asynchronous acquisition of dynamic profiles of saturations (aqueous and oleic) and of the solute concentration within the surfactant/polymer slug, respectively. This complementary dataset enables precise investigation of the evolution of both the oil bank and the induced mixing at its rear arising from the surfactant/polymer flooding process. The dilution index, intensity of segregation and the spreading length are used to quantify the degree of mixing within the surfactant/polymer slug as a function of time from the spatial structure of the solute concentration field. Relative to the single-phase flow scenario, a threefold increase in dispersivity is observed. We demonstrate that mixing is systematically overestimated if only the PET dataset is used—highlighting the importance of implementing multimodality imaging. We also show that the advection–dispersion equation model, parameterised using the dispersivity derived from the experiments, provides reasonable estimates for the rate of both mixing and spreading.

Mathematical Modeling of Dynamic Adsorption of Amphiphilic Catalyst During Two-phase Flow in Porous Media

Mathematical Modeling of Dynamic Adsorption of Amphiphilic Catalyst During Two-phase Flow in Porous Media

Convergence of a CVFE finite volume scheme for nonisothermal immiscible incompressible two-phase flow in porous media

Convergence of a CVFE finite volume scheme for nonisothermal immiscible incompressible two-phase flow in porous media

A combined mixed finite element method and discontinuous Galerkin method for hybrid-dimensional fracture models of two-phase flow

A combined mixed finite element method and discontinuous Galerkin method for hybrid-dimensional fracture models of two-phase flow

Interplay of pore geometry and wettability in immiscible displacement dynamics and entry capillary pressure

Recent studies highlight the significant role of pore geometry and wettability in determining fluid–fluid interface dynamics in two-phase flow in porous media. However, current entry capillary pressure equations, rooted in the Young–Laplace equation, consider only cross-sectional details and apply wettability data measured on flat surfaces to complex three-dimensional (3D) pore structures, overlooking the coupled effect of contact angle and pore morphologies along the flow direction. This study employs the volume-of-fluid method to investigate the following: (a) How do combined effects of pore geometry and wettability control capillary pressure change, displacement efficiency, and residual saturations? (b) Can continuous two-phase flow be achieved at the pore scale? Through direct numerical simulations in constricted idealized-geometry capillary tubes and real pore structures, we vary the contact angle to characterize its impact on fluid–fluid interface morphology, entry capillary pressure (pce), and displacement efficiency. Our results show that during the drainage, pce temporarily decreases/turns negative under intermediate wettability conditions due to forced curvature rearrangement/reversal in the converging section. Local orientation angles along the flow direction are important in controlling the interface morphology and pce evolution. Moreover, intermediate contact angles enhance displacement efficiency due to curvature reversal, while insufficient corner flow during imbibition causes pore snap-off of the receding fluid, leading to higher residual saturation. The results challenge conventional methods in predicting entry capillary pressure, highlighting the need for incorporating 3D geometry in predictive models. Eventually, the insights underscore the importance of considering corner flow in controlling displacement efficiency within constricted geometries in pore network modelling studies.