Self-Similar Solutions for the Foam Drainage Equation

We provide travelling wave solutions of the equation for foam drainage in porous media, taking into account an additional symmetry requirement. The method of solution used is reminescent of the approach developed to treat the Rapoport–Leas equation for two-phase flow. Numerical solutions are also presented and compared to the analytical ones.     

Coupling of Foam Drainage and Viscous Fingering in Porous Media Revealed by X-ray Computed Tomography

We investigate the development foam in granular porous media and the subsequent flow of the surfactant solution, where the fluid fraction variations are visualized and quantified using X-ray computed tomography. It is found that foam flows in a front like manner leading to a residual liquid fraction of 0.18±0.01, far from the inlet surface of the porous sample. A desaturation backward wave is also observed during foam development. We provided direct evidence that the flow of surfactant solution in porous media containing foam gives rise to superposition of a drainage wave and a characteristic viscous fingering pattern. In the wave the liquid fraction ranges from the above residual value to nearly 0.25±0.01. The liquid fraction associated with the viscous fingering decays as a function of distance but the inlet value increases up to 0.06±0.01. Certain ideas about the physics of foam flow in porous media are revised in the light of our findings.     

Accurate and Efficient Implementation of Pore-Morphology-based Drainage Modeling in Two-dimensional Porous Media

We have developed an efficient and accurate numerical implementation for pore-morphological modeling of drainage in two-dimensional, totally wetting porous media. The new numerical method uses level sets to describe the fluid distribution and polygons that can be defined with subgrid scale accuracy for the pore boundaries, while a previously developed approach represents the phases by pixels arranged on a square lattice. We analyze and compare the previous and new method. For both approaches, the simulated fluid saturations are first-order accurate. For the level-set approach, the simulated interfacial lengths converge to the real values, while the pixel approach yields biased results. The level-set method is orders of magnitudes faster than the pixel method.     

Foam Mobility in Heterogeneous Porous Media

It is shown experimentally that in situ generation of foam is an effective method for achieving gas mobility control and diverting injected fluid to low permeability strata within heterogeneous porous media. The experimental system is composed of a 0.395 porosity, 5.35 µm² synthetic sandstone and a 0.244 porosity, 0.686 µm² natural sandstone. The cores are arranged in parallel and communicate through common injection and production conditions. Nitrogen is the gas phase and alpha-olefin sulfonate (AOS 1416) in brine is the foamer. Three types of experiments were conducted. First, gas alone was injected into the system after presaturation with the foamer solution. Second, gas and foamer solution were coinjected at an overall gas fraction of 90% into cores presaturated with surfactant. Each core accepted a portion of the injected gas and liquid according to the mobility within the core. Lastly, gas and foamer solution were coinjected into the individual, isolated porous media in order to establish baseline behavior. The results are striking. It is possible to achieve total diversion of gas injection to the low permeability medium in some cases. The results also confirm previous predictions that foamed gas can be more mobile in lower permeability porous media.     

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.     

Combined Free and Forced Convection in Vertical Channels of Porous Media

In this paper we investigate the combined free and forced convection of a fully developed Newtonian fluid within a vertical channel composed of porous media when viscous dissipation effects are taken into consideration. The flow is analysed in the region of a first critical Rayleigh number in order to interpret the multiple-valued solutions and discuss their validity. The governing fourth-order, ordinary differential equation, which contains the Darcy and the viscous dissipation terms, is solved analytically using perturbation techniques and numerically using D02HBF NAG Library. A detailed investigation of the governing O.D.E. is performed on both clear fluid and porous medium for various values of the viscous dissipation parameter, , when the wall temperature decreases linearly with height, and the pressure gradient is both above and below its hydrostatic value. Although mathematically the results in all cases show that there are two solution branches, producing four possible solutions, the study of the velocity and buoyancy profiles together with the Darcy effect indicate that only one of the two solutions at any value of the Rayleigh number appears to be physically acceptable. It is shown that the effect of the Darcy number decreases as the critical Rayleigh numbers increase.     

Asymptotic Analysis of Solute Transport with Linear Nonequilibrium Sorption in Porous Media

We analysed the asymptotic behaviour of breakthrough curves (BTCs) obtained after a single pulse injection in a 1D flow domain. Five different types of solute transport with nonequilibrium sorption were considered. The properties of the porous medium were assumed to be spatially constant. For long times, the concentration at a fixed position in time was found to decay like exp(–t) where depends on both the transport parameters and the parameters describing the nonequilibrium process. The results from the asymptotic analysis were compared with 1D numerical transport calculations. For all cases examined a good agreement between numerical calculations and the asymptotic analysis was found. The results from the asymptotic analysis provide an alternative way to determine transport and sorption related parameters from BTCs. The derived relationships between and the model parameters are however only valid for large times. This requires that the very low concentrations need to be measured and not only the bulk mass, too. By either increasing or decreasing the velocity during BTC experiments additional asymptotic equations are obtained which can be used to determine the value of the model parameters. The results from the asymptotic analysis can also be used in standard inverse modelling techniques to either obtain good initial guesses or to reduce the parameter space. The fact that linear nonequilibrium processes decay like exp(–t) can be used to qualitatively evaluate observed BTCs. The asymptotic analysis can also be easily extended to a larger class of transport problems (e.g. transport of solutes with microbial decay) provided that the overall transport problem remains linear in the concentration.     

Surfactant Concentration and End Effects on Foam Flow in Porous Media

Foaming injected gas is a useful and promising technique for achieving mobility control in porous media. Typically, such foams are aqueous. In the presence of foam, gas and liquid flow behavior is determined by bubble size or foam texture. The thin-liquid films that separate foam into bubbles must be relatively stable for a foam to be finely textured and thereby be effective as a displacing or blocking agent. Film stability is a strong function of surfactant concentration and type. This work studies foam flow behavior at a variety of surfactant concentrations using experiments and a numerical model. Thus, the foam behavior examined spans from strong to weak.Specifically, a suite of foam displacements over a range of surfactant concentrations in a roughly 7m², one-dimensional sandpack are monitored using X-ray computed tomography (CT). Sequential pressure taps are employed to measure flow resistance. Nitrogen is the gas and an alpha olefin sulfonate (AOS 1416) in brine is the foamer. Surfactant concentrations studied vary from 0.005 to 1wt%. Because foam mobility depends strongly upon its texture, a bubble population balance model is both useful and necessary to describe the experimental results thoroughly and self consistently. Excellent agreement is found between experiment and theory.     

Stochastic Finite Element Analysis for Multiphase Flow in Heterogeneous Porous Media

This study is concerned with developing a two-dimensional multiphase model that simulates the movement of NAPL in heterogeneous aquifers. Heterogeneity is dealt with in a probabilistic sense by modeling the intrinsic permeability of the porous medium as a stochastic process. The deterministic finite element method is used to spatially discretize the multiphase flow equations. The intrinsic permeability is represented in the model via its Karhunen–Loeve expansion. This is a computationally expedient representation of stochastic processes by means of a discrete set of random variables. Further, the nodal unknowns, water phase saturations and water phase pressures, are represented by their stochastic spectral expansions. This representation involves an orthogonal basis in the space of random variables. The basis consists of orthogonal polynomial chaoses of consecutive orders. The relative permeabilities of water and oil phases, and the capillary pressure are expanded in the same manner, as well. For these variables, the set of deterministic coefficients multiplying the basis in their expansions is evaluated based on constitutive relationships expressing the relative permeabilities and the capillary pressure as functions of the water phase saturations. The implementation of the various expansions into the multiphase flow equations results in the formulation of discretized stochastic differential equations that can be solved for the deterministic coefficients appearing in the expansions representing the unknowns. This method allows the computation of the probability distribution functions of the unknowns for any point in the spatial domain of the problem at any instant in time. The spectral formulation of the stochastic finite element method used herein has received wide acceptance as a comprehensive framework for problems involving random media. This paper provides the application of this formalism to the problem of two-phase flow in a random porous medium.     

Transport of Three-Phase Hyper-Saline Brines in Porous Media: Examples

This paper presents a number of applications of a new code which can simulate the transport of high temperature three-phase (gas, liquid, solid) hyper-saline fluids in a porous medium. The examples presented demonstrate that multiple phase changes occur as the fluid state evolves across the H₂O–NaCl phase diagram. Multi-phase flows occur in a variety of situations, including a horizontal domain with fluid withdrawal, a vertical counter-flowing salt-pipe, and a horizontal domain with saturation shocks and expansion waves. The code is also used to simulate heat, water and salt flows in a large scale model (10s of km) of the Taupo Volcanic Zone, New Zealand.     

Mixed Convection in a Vertical Porous Channel

A numerical study of mixed convection in a vertical channel filled with a porous medium including the effect of inertial forces is studied by taking into account the effect of viscous and Darcy dissipations. The flow is modeled using the Brinkman–Forchheimer-extended Darcy equations. The two boundaries are considered as isothermal–isothermal, isoflux–isothermal and isothermal–isoflux for the left and right walls of the channel and kept either at equal or at different temperatures. The governing equations are solved numerically by finite difference method with Southwell–Over–Relaxation technique for extended Darcy model and analytically using perturbation series method for Darcian model. The velocity and temperature fields are obtained for various porous parameter, inertia effect, product of Brinkman number and Grashof number and the ratio of Grashof number and Reynolds number for equal and different wall temperatures. Nusselt number at the walls is also determined for three types of thermal boundary conditions. The viscous dissipation enhances the flow reversal in the case of downward flow while it counters the flow in the case of upward flow. The Darcy and inertial drag terms suppress the flow. It is found that analytical and numerical solutions agree very well for the Darcian model. An erratum to this article is available at .     

Effect of Wetting on Gravity Drainage in Porous Media

A series of benchmark experiments on the effect of the wetting state on the flow properties in porous media were performed, allowing us to relate the wetting properties at the pore scale to the macroscale hydrodynamics. Drainage of n-alkanes (oils) displaced by air in a model porous medium consisting of water-wet sand was studied using gamma-ray densitometry and weight measurements. The enormous advantage of our system is that we know and control the wetting properties perfectly: we can tune the wetting properties by changing the salinity of the water. This allows us to perform porous medium flow experiments for the different wetting states without changing the transport properties (viscosity and density) of the oil. Drainage is found to be more efficient, and consequently oil recovery more important for partial wetting.     

Comments on On an Exact Analytical Solution of the Boussinesq Equation,Transport in Porous Media 39, 339–345, 2000

Transport in Porous Media -     

Phenomenological Meniscus Model for Two-Phase Flows in Porous Media

A new macroscale model of a two-phase flow in porous media is suggested. It takes into consideration a typical configuration of phase distribution within pores in the form of a repetitive field of mobile menisci. These phase interfaces give rise to the appearance of a new term in the momentum balance equation, which describes a vectorial field of capillary forces. To derive the model, a phenomenological approach is developed, based on introducing a special continuum called the Meniscus-continuum. Its properties, such as a unique flow velocity, an averaged viscosity, a compensation mechanism and a duplication mechanism, are derived from a microscale analysis. The closure relations to the phenomenological model are obtained from a theoretical model of stochastic meniscus stream and from numerical simulations based on network models of porous media. The obtained transport equation remains hyperbolic even if the capillary forces are dominated, in contrast to the classic model which is parabolic. For the case of one space dimension, the analytical solutions are obtained, which manifest non-classical effects as double displacement fronts or counter-current fronts.     

Foam Generation Hysteresis in Porous Media: Experiments and New Insights

Foam application in subsurface processes including environmental remediation, geological carbon-sequestration, and gas-injection enhanced oil recovery (EOR) has the potential to enhance contamination remediation, secure \(\hbox {CO}_{2}\) storage, and improve oil recovery, respectively. Nanoparticles are a promising alternative to surfactants in creating foam in harsh environments. We conducted \(\hbox {CO}_{2}\)-in-brine foam generation experiments in Boise sandstones with surface-treated silica nanoparticle in high-salinity conditions. All the experiments were conducted at the fixed \(\hbox {CO}_{2}\) volume fraction and fixed flow rate which changed in steps. The steady-state foam apparent viscosity was measured as a function of injection velocity. The foam flowing through the cores showed higher apparent viscosity as the flow rate increased from low to medium and high velocities. At very high velocities, once foam bubbles were finely textured, the foam apparent viscosity was governed by foam rheology rather than foam creation. A noticeable hysteresis occurred when the flow velocity was initially increased and then decreased, implying multiple (coarse and strong) foam states at the same superficial velocity. A normalized generation function was combined with CMG-STARS foam model to cover full spectrum of foam behavior in the experiments. The new model successfully captures foam generation and hysteresis trends in presented experiments in this study and data from the literature. The results indicate once foam is generated in porous media, it is possible to maintain strong foam at low injection rates. This makes foam more feasible in field applications where foam generation is limited by high injection rates that may only exist near the injection well.     

Modeling Nanoparticle Transport in Porous Media in the Presence of a Foam

Transport in Porous Media - Nano-remediation is a promising in situ remediation technology. It consists in injecting reactive nanoparticles (NPs) into the subsurface for the displacement or the...     

Effect of Wetting on the Dynamics of Drainage in Porous Media

We examine the consequences of the wettability properties on the dynamics of gravity drainage in porous media. The relation between the wetting properties at the pore scale and the macroscale hydrodynamics is studied. Model porous media consisting of hydrophilic and hydrophobic glass beads or sand with well defined wetting properties, are prepared for this study. Gravity drainage experiments with air displacing water (two-phase flow), are performed for different Bond numbers, and using different techniques such as gamma-ray densitometry, magnetic resonance imaging (MRI) and weight measurements. The dynamics of drainage is found to be different for hydrophilic and hydrophobic porous media in the transition zone (funicular regime). Moreover, for hydrophilic (water-wet) porous media, MRI experiments reveal the importance of drainage through the continuous water film, which leads to an increase of the residual quantity of water in the transition zone with time.     

Onset of Mobilization and the Fraction of Trapped Foam in Porous Media

Usually, foam in a porous medium flows through a small and spatially varying fraction of available pores, while the bulk of it remains trapped. The trapped foam is under a pressure gradient corresponding to the pressure gradient imposed by the flowing foam and continuous wetting liquid. The imposed pressure gradient and coalescence of the stationary foam lamellae periodically open flow channels in the trapped foam region. Foam lamellae in each of these channels flow briefly, but channels are eventually plugged by smaller bubbles entering into the trapped region. The result is a cycling of flow channels that open and close throughout the trapped foam, leading to intermittent pulsing of foam flow in that region.The dynamic behavior of foam trapped in porous media is modeled here with a pore network simulator. We predict the magnitude of the pressure drop leading to the onset of flow of foam lamellae in the region containing trapped foam. This mobilization pressure drop depends only on the number of lamellae in the flow path and on the geometry of the pores that make up this path.The principles learned in this study allow us to predict the fraction of foam that is trapped in a porous medium under given flow conditions. We present here the first analytic expression for the trapped foam fraction as a function of the pressure gradient, and of the mean and standard deviation of the pore size distribution. This expression provides a missing piece for the continuum foam flow models based on the moments of the volume-averaged population balance of foam bubbles.