On the Barenblatt Model for Non-Equilibrium Two Phase Flow in Porous Media

We prove global existence and uniqueness of solutions to a quasilinear Goursat problem, which was proposed by G. I. Barenblatt to describe non-equilibrium two phase fluid flow in permeable porous media. When the equilibrium relaxation time tends to zero, the solution is shown to converge to the entropy solution of the corresponding initial-boundary value problem for the classical Buckley-Leverett equation.     

A New Model for Immiscible Displacement in Porous Media

Immiscible displacement is regarded as the superposition of forward flows of both water and oil, due to injection of water into the medium, and of additional forward flow of water coupled with reverse flow of oil, caused by the existence of capillary pressure gradients. The model has been evaluated numerically for the prediction of the evolution of saturation profiles in waterfloods covering a wide range of water injection rates. In agreement with experimentation, saturation profiles ranging from a completely flat shape to piston-shape, depending on the injection rate, have been obtained. Also in agreement with experimentation, numerical evaluation of the model for the case of a closed system with an initial step-function saturation profile has predicted a gradual spreading of the piston front into S-shaped profiles with an increasing variance. The final profile corresponds to uniform saturation everywhere in the medium.     

A Monte Carlo Algorithm for Immiscible Two-Phase Flow in Porous Media

We present a Markov Chain Monte Carlo algorithm based on the Metropolis algorithm for simulation of the flow of two immiscible fluids in a porous medium under macroscopic steady-state conditions using a dynamical pore network model that tracks the motion of the fluid interfaces. The Monte Carlo algorithm is based on the configuration probability, where a configuration is defined by the positions of all fluid interfaces. We show that the configuration probability is proportional to the inverse of the flow rate. Using a two-dimensional network, advancing the interfaces using time integration, the computational time scales as the linear system size to the fourth power, whereas the Monte Carlo computational time scales as the linear size to the second power. We discuss the strengths and the weaknesses of the algorithm.     

On Capillary Slowdown of Viscous Fingering in Immiscible Displacement in Porous Media

Hydrodynamic instability in immiscible porous media flows in the presence of capillarity is investigated here. The analysis and arguments presented here show that the slowdown of instabilities due to capillarity is usually very rapid which makes the flow almost, but not entirely, stable. The profiles of the stable and unstable waves in the far-field are characterized using a novel but very simple approach.     

A Spatially Non-Local Model for Flow in Porous Media

A general mathematical model of steady-state transport driven by spatially non-local driving potential differences is proposed. The porous medium is considered to be a network of short-, medium-, and long-range interstitial channels with impermeable walls and at a continuum of length scales, and the flow rate in each channel is assumed to be linear with respect to the pressure difference between its ends. The flow rate in the model is thus a functional of the non-local driving pressure differences. As special cases, the model reduces to familiar forms of transport equations that are commonly used. An important situation arises when the phenomenon is almost, but not quite, locally dependent. The one-dimensional form of the model discussed here can be extended to multiple dimensions, temporal non-locality, and to heat, mass, and momentum transfer.     

A Subgrid-Scale Model for Turbulent Flow in Porous Media

Transport in Porous Media - Given the analogy between the filtered equations of large eddy simulation and volume-averaged Navier–Stokes equations in porous media, a subgrid-scale model is...     

A Stochastic Model for Particulate Suspension Flow in Porous Media

A population balance model for a particulate suspension transport with size exclusion capture of particles by porous rock is derived. The model accounts for particle flux reduction and pore space accessibility due to restriction for large particles to move through smaller pores – a particle is captured by a smaller pore and passes through a larger pore. Analytical solutions are obtained for a uniform pore size medium, and also for a medium with small pore size variation. For both cases, the equations for averaged concentrations significantly differ from the classical deep bed filtration model.     

A Dynamic Network Model for Two-Phase Flow in Porous Media

We present a dynamic model of immiscible two-phase flow in a network representation of a porous medium. The model is based on the governing equations describing two-phase flow in porous media, and can handle both drainage, imbibition, and steady-state displacement. Dynamic wetting layers in corners of the pore space are incorporated, with focus on modeling resistivity measurements on saturated rocks at different capillary numbers. The flow simulations are performed on a realistic network of a sandpack which is perfectly water-wet. Our numerical results show saturation profiles for imbibition in agreement with experiments. For free spontaneous imbibition we find that the imbibition rate follows the Washburn relation, i.e., the water saturation increases proportionally to the square root of time. We also reproduce rate effects in the resistivity index for drainage and imbibition.     

A Mathematical Model for Hysteretic Two-Phase Flow in Porous Media

We develop a mathematical model for hysteretic two-phase flow (of oil and water) in waterwet porous media. To account for relative permeability hysteresis, an irreversible trapping-coalescence process is described. According to this process, oil ganglia are created (during imbibition) and released (during drainage) at different rates, leading to history-dependent saturations of trapped and connected oil. As a result, the relative permeability to oil, modelled as a unique function of the connected oil saturation, is subject to saturation history. A saturation history is reflected by history parameters, that is by both the saturation state (of connected and trapped oil) at the most recent flow reversal and the most recent water saturation at which the flow was a primary drainage. Disregarding capillary diffusion, the flow is described by a hyperbolic equation with the connected oil saturation as unknown. This equation contains functional relationships which depend on the flow mode (drainage or imbibition) and the history parameters. The solution consists of continuous waves (expansion waves and constant states), shock waves (possibly connecting different modes) and stationary discontinuities (connecting different saturation histories). The entropy condition for travelling waves is generalized to include admissible shock waves which coincide with flow reversals. It turns out that saturation history generally has a strong influence on both the type and the speed of the waves from which the solution is constructed.     

A Theoretical Model of Hysteresis and Dynamic Effects in the Capillary Relation for Two-phase Flow in Porous Media

It is well known that the relationship between capillary pressure and saturation, in two-phase flow problems demonstrates memory effects and, in particular, hysteresis. Explicit representation of full hysteresis with a myriad of scanning curves in models of multiphase flow has been a difficult problem. A second complication relates to the fact that P ^c–S relationships, determined under static conditions, are not necessarily valid in dynamics. There exist P ^c–S relationships which take into account dynamic effects. But the combination of hysteretic and dynamic effects in the capillary relationship has not been considered yet. In this paper, we have developed new models of capillary hysteresis which also include dynamic effects. In doing so, thermodynamic considerations are employed to ensure the admissibility of the new relationships. The simplest model is constructed around main imbibition and drainage curves and assumes that all scanning curves are vertical lines. The dynamic effect is taken into account by introducing a damping coefficient in P ^c–S equation. A second-order model of hysteresis with inclined scanning curves is also developed. The simplest version of proposed models is applied to two-phase incompressible flow and an example problem is solved.     

Modeling and Numerical Simulations of Immiscible Compressible Two-Phase Flow in Porous Media by the Concept of Global Pressure

A new formulation is presented for the modeling of immiscible compressible two-phase flow in porous media taking into account gravity, capillary effects, and heterogeneity. The formulation is intended for the numerical simulation of multidimensional flows and is fully equivalent to the original equations, contrary to the one introduced in Chavent and Jaffré (Mathematical Models and Finite Elements for Reservoir Simulation, 1986). The main feature of this formulation is the introduction of a global pressure. The resulting equations are written in a fractional flow formulation and lead to a coupled system which consists of a nonlinear parabolic (the global pressure equation) and a nonlinear diffusion–convection one (the saturation equation) which can be efficiently solved numerically. A finite volume method is used to solve the global pressure equation and the saturation equation for the water and gas phase in the context of gas migration through engineered and geological barriers for a deep repository for radioactive waste. Numerical results for the one-dimensional problem are presented. The accuracy of the fully equivalent fractional flow model is demonstrated through comparison with the simplified model already developed in Chavent and Jaffré (Mathematical Models and Finite Elements for Reservoir Simulation, 1986).     

Relations Between Seepage Velocities in Immiscible,Incompressible Two-Phase Flow in Porous Media

Based on thermodynamic considerations, we derive a set of equations relating the seepage velocities of the fluid components in immiscible and incompressible two-phase flow in porous media. They necessitate the introduction of a new velocity function, the co-moving velocity. This velocity function is a characteristic of the porous medium. Together with a constitutive relation between the velocities and the driving forces, such as the pressure gradient, these equations form a closed set. We solve four versions of the capillary tube model analytically using this theory. We test the theory numerically on a network model.     

Immiscible Displacement in Cross-Bedded Heterogeneous Porous Media

Direct insight into the mechanisms of flow and displacements within small-scale (cm) systems having permeability heterogeneities that are not parallel to the flow direction (cross-bedding and fault zones) have been carried out. In our experiments, we have used visual models with unconsolidated glass bead packs having carefully controlled permeability contrasts to observe the processes with coloured fluids and streamlines. The displacements were followed visually and by video recording for later analysis. The experiments show the significance that heterogeneities have on residual saturations and recovery, as well as the displacement patterns themselves. During a waterflood, high permeability regions can be by-passed due to capillary pressure differences, giving rise to high residual oil saturations in these regions. This study demonstrates the importance of incorporating reservoir heterogeneity into core displacement analysis, but of course the nature of the heterogeneity has to be known. In general, the effects created by the heterogeneities and their unknown boundaries hamper interpretation of flood experiments in heterogeneous real sandstone cores. Our experiments, therefore, offer clear visual information to provide a firmer understanding of the displacement processes during immiscible displacement, to present benchmark data for input to numerical simulators, and to validate the simulator through a comparison with our experimental results for these difficult flow problems.     

A Model for Flow and Deformation in Unsaturated Swelling Porous Media

A thermomechanical theory for multiphase transport in unsaturated swelling porous media is developed on the basis of Hybrid Mixture Theory (saturated systems can also be modeled as a special case of this general theory). The aim is to comprehensively and non-empirically describe the effect of viscoelastic deformation on fluid transport (and vice versa) for swelling porous materials. Three phases are considered in the system: the swelling solid matrix s, liquid l, and air a. The Coleman–Noll procedure is used to obtain the restrictions on the form of the constitutive equations. The form of Darcy’s law for the fluid phase, which takes into account both Fickian and non-Fickian transport, is slightly different from the forms obtained by other researchers though all the terms have been included. When the fluid phases interact with the swelling solid porous matrix, deformation occurs. Viscoelastic large deformation of the solid matrix is investigated. A simple form of differential-integral equation is obtained for the fluid transport under isothermal conditions, which can be coupled with the deformation of the solid matrix to solve for transport in an unsaturated system. The modeling theory thus developed, which involves two-way coupling of the viscoelastic solid deformation and fluid transport, can be applied to study the processing of biopolymers, for example, soaking of foodstuffs and stress-crack predictions. Moreover, extension and modification of this modeling theory can be applied to study a vast variety of problems, such as drying of gels, consolidation of clays, drug delivery, and absorption of liquids in diapers.     

Large Scale Mixing for Immiscible Displacement in Heterogeneous Porous Media

We derive a large scale mixing parameter for a displacement process of one fluid by another immiscible one in a two-dimensional heterogeneous porous medium. The mixing of the displacing fluid saturation due to the heterogeneities of the permeabilities is captured by a dispersive flux term in the large scale homogeneous flow equation. By making use of the stochastic approach we develop a definition of the dispersion coefficient and apply a Eulerian perturbation theory to determine explicit results to second order in the fluctuations of the total velocity. We apply this method to a uniform flow configuration as well as to a radial one. The dispersion coefficient is found to depend on the mean total velocity and can therefore be time varying. The results are compared to numerical multi-realization calculations. We found that the use of single phase flow stochastics cannot capture all phenomena observed in the numerical simulations.     

A New Stochastic Bubble Population Model for Foam Flow in Porous Media

Foam has been widely used as a mobility control agent for Improved and Enhanced Oil Recovery IOR/EOR, gas blocking, and acid diversion during matrix stimulation. The prediction of foam performance relies on macroscopic modeling. Traditionally, foam modeling approaches include fractional flow theories and population balance models. However, fractional foam models assume implicitly that foam is incompressible and do not account directly for the evolution of bubble population. The population balance models, instead, rely on the idea that foam mobility depends on bubble density and are more comprehensive. Yet, population balance models did not gain full acceptance thus far, because of their perceived complexity, with parameters that are hard to obtain experimentally. This article presents an improved foam model based on a simpler but realistic foam rheology and stochastic bubble generation ideas. Physical ideas in agreement with pictures emerging from recent foam studies using X-ray computed tomography form the basis for the new model.     

Influence of Relative Permeability on the Stability Characteristics of Immiscible Flow in Porous Media

Relative permeability functions for immiscible displacements in porous media show a wide range of profiles. Although, this behavior is well known, its impact on the stability of the displacement process is unexplored. Our analysis clearly demonstrates for the first time that the viscous instability characteristics of two-phase flows are governed not only by their end point values, but are strongly dependent on the actual profile of relative permeability functions. Linear stability analysis predicts the capacity of the flow to develop large scale fingers which can result in substantial bypassing of the resident fluid. It is observed that relative permeability functions attributed to drainage processes yield a more unstable displacement as compared to functions related to imbibition processes. Moreover, instability is observed to increase for those relative permeability functions which result from increased wettability of the wetting fluid. High accuracy numerical simulations show agreement with these predictions and demonstrate how large amplitude viscous fingers result in significant bypassing for certain relative permeability functions. In the nonlinear regime, the finger amplitude grows at a rate ∝ t^1/2 initially, drops to t^1/4 at a later time and finally grows ∝ t. The basic mechanisms of finger interaction, however, are not substantially influenced by relative permeability functions.     

A Criterion for Non-Darcy Flow in Porous Media

Non-Darcy behavior is important for describing fluid flow in porous media in situations where high velocity occurs. A criterion to identify the beginning of non-Darcy flow is needed. Two types of criteria, the Reynolds number and the Forchheimer number, have been used in the past for identifying the beginning of non-Darcy flow. Because each of these criteria has different versions of definitions, consistent results cannot be achieved. Based on a review of previous work, the Forchheimer number is revised and recommended here as a criterion for identifying non-Darcy flow in porous media. Physically, this revised Forchheimer number has the advantage of clear meaning and wide applicability. It equals the ratio of pressure drop caused by liquid–solid interactions to that by viscous resistance. It is directly related to the non-Darcy effect. Forchheimer numbers are experimentally determined for nitrogen flow in Dakota sandstone, Indiana limestone and Berea sandstone at flowrates varying four orders of magnitude. These results indicate that superficial velocity in the rocks increases non-linearly with the Forchheimer number. The critical Forchheimer number for non-Darcy flow is expressed in terms of the critical non-Darcy effect. Considering a 10% non-Darcy effect, the critical Forchheimer number would be 0.11.     

A Multi-Scale Approach to Model Two-Phase Flow in Heterogeneous Porous Media

The immiscible displacement of a wetting fluid by a non-wetting one in heterogeneous porous media is modeled using a multi-scale network-type analysis: (1) The pressure-controlled immiscible displacement of water by oil in pore-and-throat networks (1st length scale ~ 1?mm) is simulated as a capillary-driven process. (2) The pressure-controlled immiscible displacement in uncorrelated cubic lattices (2nd length scale ~ 1?cm) is simulated as a site percolation process governed by capillary and gravity forces. At this scale, each node represents a network of the previous scale. (3) The rate-controlled immiscible displacement of water by oil in cubic networks (3rd length scale ~ 10?cm), where each node represents a lattice of the previous scale, is simulated by accounting for capillary, gravity, and viscous forces. The multi-scale approach along with the information concerning the pore structure properties of the porous medium can be employed to determine the transient responses of the pressure drop and axial distribution of water saturation, and estimate the effective (up-scaled) relative permeability functions. The method is demonstrated with application to data of highly heterogeneous soils.