Macroscopic Two-phase Flow in Porous Media Assuming the Diffuse-interface Model at Pore Level

Modeling multiphase flow in porous structures remains a challenge due to the complexity of handling multiple interfaces. This paper presents a one-domain pore-resolved simulation approach for immiscible two-phase flows in porous media, using a monolithic fluid–solid coupling framework to implicitly consider the existence of solid objects, with the fluid–fluid interfaces captured through solving an algebraic volume of fluid equation. Fluid interfacial tension is considered by integrating a continuum surface force, and the wall wettability condition is imposed by modifying the contact angle of the fluid interface at the embedded solid surface. The resulting equations are simple and stable, as there are no empirical models or parameters involved for the interface representation. This approach has been validated through performing a series of test-case simulations, including capillary-dominated flow, capillary rise with gravity, Taylor film formation, and finally two-phase flow in a heterogeneous porous structure. The numerical approach is demonstrated to be well suited for investigating pore-scale two-phase flows in realistic porous media.

Purpose – This paper aims to present a computationally efficient finite element model for the simulation of isothermal immiscible two-phase flow in a rigid porous media with a particular application to CO2 sequestration in underground formations. Focus is placed on developing a numerical procedure, which is effectively mesh-independent and suitable to problems at regional scales. Design/methodology/approach – The averaging theory is utilized to describe the governing equations of the involved unsaturated multiphase flow. The level-set (LS) method and the extended finite element method (XFEM) are utilized to simulate flow of the CO2 plume. The LS is employed to trace the plume front. A streamline upwind Petrov-Galerkin method is adopted to stabilize possible occurrence of spurious oscillations due to advection. The XFEM is utilized to model the high gradient in the saturation field front, where the LS function is used for enhancing the weighting and the shape functions. Findings – The capability of the proposed model and its features are evaluated by numerical examples, demonstrating its accuracy, stability and convergence, as well as its advantages over standard and upwind techniques. The study showed that a good combination between a mathematical model and a numerical model enables the simulation of complicated processes occurring in complicated and large geometry using minimal computational efforts. Originality/value – A new computational model for two-phase flow in porous media is introduced with basic requirements for accuracy, stability, and convergence, which are met using relatively coarse meshes.

We study three different time integration methods for a dynamic pore network model for immiscible two-phase flow in porous media. Considered are two explicit methods, the forward Euler and midpoint methods, and a new semi-implicit method developed herein. The explicit methods are known to suffer from numerical instabilities at low capillary numbers. A new time-step criterion is suggested in order to stabilize them. Numerical experiments, including a Haines jump case, are performed and these demonstrate that stabilization is achieved. Further, the results from the Haines jump case are consistent with experimental observations. A performance analysis reveals that the semi-implicit method is able to perform stable simulations with much less computational effort than the explicit methods at low capillary numbers. The relative benefit of using the semi-implicit method increases with decreasing capillary number $\mathrm{Ca}$, and at $\mathrm{Ca} \sim 10^{-8}$ the computational time needed is reduced by three orders of magnitude. This increased efficiency enables simulations in the low-capillary number regime that are unfeasible with explicit methods and the range of capillary numbers for which the pore network model is a tractable modeling alternative is thus greatly extended by the semi-implicit method.

Use of preconditioned bi-conjugate gradient method in modeling two-phase fluid flow in porous media

The Mixed Multiscale Finite Element method (MMsFE) is a promising alternative to traditional upscaling techniques in order to accelerate the simulation of flows in large heterogeneous porous media. Indeed, in this method, the calculation of the basis functions which encompass the fine-scale variations of the permeability field, can be performed in parallel and the size of the global linear system is reduced. However, we show in this work that a two-level MPI strategy should be used to adapt the calculation resources at these two steps of the algorithm and thus obtain a better scalability of the method. This strategy has been implemented for the resolution of the pressure equation which arises in two-phase flow models. Results of simulations performed on complex reservoir models show the benefits of this approach.

The Diffuse-Interface model for two-phase flow in porous media is applied to the invesigation of relative permeabilities in unsteady-state flows. It is shown that relative permeabilities have a transient regime where they can be significantly larger than 1. It is also shown that, when normalized by their endpoint values, relative permeabilities can be calculated from formulas involving thermodynamical and capillary pressure functions. The end-point relative permeabilities are linearly related to the fluid velocity, but the coefficients seem to be flow dependent.

We compute the fluid flow time-correlation functions of incompressible, immiscible two-phase flow in porous media using a 2D network model. Given a properly chosen representative elementary volume, the flow rate distributions are Gaussian and the integrals of time correlation functions of the flows are found to converge to a finite value. The integrated cross-correlations become symmetric, obeying Onsager's reciprocal relations. These findings support the proposal of a non-equilibrium thermodynamic description for two-phase flow in porous media.

This study aims to bridge length scales in immiscible multiphase flow simulation by connecting two published governing equations at the pore-scale and continuum-scale through a novel validation framework. We employ Niessner and Hassnaizadeh's [“A model for two-phase flow in porous media including fluid-fluid interfacial area,” Water Resour. Res. 44(8), W08439 (2008)] continuum-scale model for multiphase flow in porous media, combined with the geometric equation of state of McClure et al. [“Modeling geometric state for fluids in porous media: Evolution of the Euler characteristic,” Transp. Porous Med. 133(2), 229–250 (2020)]. Pore-scale fluid configurations simulated with the lattice-Boltzmann method are used to validate the continuum-scale results. We propose a mapping from the continuum-scale to pore-scale utilizing a generalized additive model to predict non-wetting phase Euler characteristics during imbibition, effectively bridging the continuum-to-pore length scale gap. Continuum-scale simulated measures of specific interfacial area, saturation, and capillary pressure are directly compared to up-scaled pore-scale simulation results. This research develops a numerical framework capable of capturing multiscale flow equations establishing a connection between pore-scale and continuum-scale simulations.

Monte Carlo simulation of two-phase flow in porous media: Invasion with two invaders and two defenders

One of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect. It is well known that classical methods involving sharp interfaces fail to describe this type of phenomena. Following some previous work in this area, we suggest a physically motivated regularization of the Euler equations which allows topological transitions to occur smoothly. In this model, the sharp interface is replaced by a narrow transition layer across which the fluids may mix. The model describes a flow of a binary mixture, and the internal structure of the interface is determined by both diffusion and motion. An advantage of our regularization is that it automatically yields a continuous description of surface tension, which can play an important role in topological transitions. An additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the Euler (or Navier–Stokes) and the Cahn–Hilliard equations. The model takes into account weak non–locality (dispersion) associated with an internal length scale and localized dissipation due to mixing. The non–locality introduces a dimensional surface energy; dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes. In particular, we study a non–trivial limit when both components are incompressible, the pressure is kinematic but the velocity field is non–solenoidal (quasi–incompressibility). To demonstrate the effects of quasi–incompressibility, we analyse the linear stage of spinodal decomposition in one dimension. We show that when the densities of the fluids are not perfectly matched, the evolution of the concentration field causes fluid motion even if the fluids are inviscid. In the limit of infinitely thin and well–separated interfacial layers, an appropriately scaled quasi–incompressible Euler–Cahn–Hilliard system converges to the classical sharp interface model. In order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient, we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity.

An understanding of the pore-level interactions that affect multi-phase flow in porous media is important in many subsurface engineering applications, including enhanced oil recovery, remediation of dense non-aqueous liquid contaminated sites, and geologic CO2 sequestration. Standard models of two-phase flow in porous media have been shown to have several shortcomings, which might partially be overcome using a recently developed model based on thermodynamic principles that includes interfacial area as an additional parameter. A few static experimental studies have been previously performed, which allowed the determination of static parameters of the model, but no information exists concerning the interfacial area dynamic parameters. A new experimental porous flow cell that was constructed using stereolithography for two-phase gas-liquid flow studies was used in conjunction with an in-house analysis code to provide information on dynamic evolution of both fluid phases and gas-liquid interfaces. In this paper, we give a brief introduction to the new generalized model of two-phase flow model and describe how the stereolithography flow cell experimental setup was used to obtain the dynamic parameters for the interfacial area numerical model. In particular, the methods used to determine the interfacial area permeability and production terms are shown.

A 6M digital twin for modeling and simulation in subsurface reservoirs

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.

In this paper, the lattice-Boltzmann method is developed to investigate the behavior of isothermal two-phase fluid flow in porous media. The method is based on the Shan-Chen multiphase model of nonideal fluids that allow coexistence of two phases of a single substance. We reproduce some different idealized situations (phase separation, surface tension, contact angle, pipe flow, and fluid droplet motion, et al) in which the results are already known from theory or laboratory measurements and show the validity of the implementation for the physical two-phase flow in porous media. Application of the method to fluid intrusion in porous media is discussed and shows the effect of wettability on the fluid flow. The capability of reproducing critical flooding phenomena under strong wettability conditions is also proved.

Models of instabilities in porous media usually assume that the capillary pressure (the difference of pressure between the nonwetting and the wetting phase) depends on the radii of the macroscopic curvature of the two-phase front. However, this definition is not taken into account for modeling stable immiscible displacements in porous media whenever the heterogeneity of the porous medium may lead to high macroscopic curvature of the front. Before trying to solve flow equations in porous media under unstable conditions, a more accurate and complete set of equations for immiscible two-phase flow in porous media is required. Space averaging of microscopic equations valid at the pore level is used to define variables and equations that link these variables at the macroscopic scale. The thermodynamics of irreversible processes completes the set of equations. If some coupling between the flow of both phases is introduced, the relative permeability equation is proved to be valid, even with moving interfaces. Capillary pressure appears to be twofold; i. e. a static capillary pressure taking into account 1) the amount of interface (Gibbs-Duhem like equation) and 2) the area swept by the three-phase lines (Laplace like equation), as well as a dynamic capillary pressure related to fluid inertia. This is the first time that capillary pressure in porous media can be proved to be composed of three terms, each having an evident physical meaning. Under the assumptions of the present paper, the capillary pressure does not depend on macroscopic curvature, therefore.