Nonequilibrium effects in models of three-phase flow in porous media

In this paper we extend to three-phase flow the nonequilibrium formalism proposed by Barenblatt and co-workers for two-phase porous media flow. The underlying idea is to include nonequilibrium effects by introducing a pair of effective water and gas saturations, which are linked to the actual saturations by a local evolution equation. We illustrate and analyze how nonequilibrium effects lead to qualitative and quantitative differences in the solution of the three-phase flow equations.     

A stochastic model for air injection into saturated porous media

Air injection into porous media is investigated by laboratory experiments and numerical modelling. Typical applications of air injection into a granular bed are aerated bio-filters and air sparging of aquifers. The first stage of the dynamic process consists of air injection into a fixed or a quasi-fixed water-saturated granular bed. Later stages could include stages of movable beds as well, but are not further investigated here. A series of laboratory experiments were conducted in a two-dimensional box of the size 60 cm × 38 cm × 0.55 cm consisting of glass walls and using glass beads of diameter 0.4–0.6 mm as granular material. The development of the air flow pattern was optically observed and registered using a digital video camera. The resulting transient air flow pattern can be characterized as channelled flow in a fixed porous medium with dynamic tree-like evolution behaviour. Attempts are undertaken to model the air injection process. Multiphase pore-scale modelling is currently disregarded since it is restricted to very small scales. Invasion percolation models taking into account gravity effects are usually restricted to slow processes. On the other hand a continuum-type two-phase flow modelling approach is not able to simulate the observed air flow pattern. Instead a stochastic continuum-type approach is discussed here, which incorporates pore-scale features on a subscale, relevant for the immiscible processes involved. Consequently, the physical process can be modelled in a stochastic manner only, where the single experiment represents one of many possible realizations. However, the present procedure retains realistic water and air saturation patterns and therefore produces similar finger lengths and widths as observed in the experiments. Monte Carlo type modelling leads to ensemble mean water saturation and the related variance.     

Multi-physics modeling of flow and transport in porous media using a downscaling approach

Multi-phase flow and transport processes generally occur on different spatial and temporal scales. Very often also, within a physical system, they vary in space meaning that different kinds of processes might take place in different parts of the system. In order to account for the variety of processes and to take their scale-dependence into account, the development of multi-scale multi-physics techniques can be envisaged.     

Modeling non-equilibrium mass transport in biologically reactive porous media

We develop a one-equation non-equilibrium model to describe the Darcy-scale transport of a solute undergoing biodegradation in porous media. Most of the mathematical models that describe the macroscale transport in such systems have been developed intuitively on the basis of simple conceptual schemes. There are two problems with such a heuristic analysis. First, it is unclear how much information these models are able to capture; that is, it is not clear what the model's domain of validity is. Second, there is no obvious connection between the macroscale effective parameters and the microscopic processes and parameters. As an alternative, a number of upscaling techniques have been developed to derive the appropriate macroscale equations that are used to describe mass transport and reactions in multiphase media. These approaches have been adapted to the problem of biodegradation in porous media with biofilms, but most of the work has focused on systems that are restricted to small concentration gradients at the microscale. This assumption, referred to as the local mass equilibrium approximation, generally has constraints that are overly restrictive. In this article, we devise a model that does not require the assumption of local mass equilibrium to be valid. In this approach, one instead requires only that, at sufficiently long times, anomalous behaviors of the third and higher spatial moments can be neglected; this, in turn, implies that the macroscopic model is well represented by a convection–dispersion–reaction type equation. This strategy is very much in the spirit of the developments for Taylor dispersion presented by Aris (1956). On the basis of our numerical results, we carefully describe the domain of validity of the model and show that the time-asymptotic constraint may be adhered to even for systems that are not at local mass equilibrium.     

Fluid dispersion effects on density-driven thermohaline flow and transport in porous media

This study introduces the dispersive fluid flux of total fluid mass to the density-driven flow equation to improve thermohaline modeling of salt and heat transports in porous media. The dispersive fluid flux in the flow equation is derived to account for an additional fluid flux driven by the density gradient and mechanical dispersion. The coupled flow, salt transport and heat transport governing equations are numerically solved by a fully implicit finite difference method to investigate solution changes due to the dispersive fluid flux. The numerical solutions are verified by the Henry problem and the thermal Elder problem under a moderate density effect and by the brine Elder problem under a strong density effect. It is found that increment of the maximum ratio of the dispersive fluid flux to the advective fluid flux results in increasing dispersivity for the Henry problem and the brine Elder problem. The effects of the dispersive fluid flux on salt and heat transports under high density differences and high dispersivities are more noticeable than under low density differences and low dispersivities. Values of quantitative indicators such as the Nusselt number, mass flux, salt mass stored and maximum penetration depth in the brine Elder problem show noticeable changes by the dispersive fluid flux. In the thermohaline Elder problem, the dispersive fluid flux shows a considerable effect on the shape and the number of developed fingers and makes either an upwelling or a downwelling flow in the center of the domain. In conclusion, for the general case that involves strong density-driven flow and transport modeling in porous media, the dispersive fluid flux should be considered in the flow equation.     

Lattice-Boltzmann simulations of the capillary pressure–saturation–interfacial area relationship for porous media

Hysteresis in the relationship between capillary pressure (P_c), wetting phase saturation (S_w) and nonwetting–wetting interfacial area per volume (a_nw) is investigated using multiphase lattice-Boltzmann simulations of drainage and imbibition in a glass bead porous system. In order to validate the simulations, the P_c–S_w and a_nw–S_w main hysteresis loops were compared to experimental data reported by Culligan et al. [Culligan KA, Wildenschild D, Christensen BS, Gray WG, Rivers ML, Tompson AB. Interfacial area measurements for unsaturated flow through porous media. Water Resour Res 2004;40:W12413]. In general, the comparison shows that the simulations are reliable and capture the important physical processes in the experimental system. P_c–S_w curves, a_nw–S_w curves and phase distributions (within the pores) show good agreement during drainage, but less satisfactory agreement during imbibition. Drainage and imbibition scanning curves were simulated in order to construct P_c–S_w–a_nw surfaces. The root mean squared error (RMSE) and mean absolute error (MAE) between drainage and imbibition surfaces was 0.10 mm⁻¹ and 0.03 mm⁻¹, respectively. This small difference indicates that hysteresis is virtually nonexistent in the P_c–S_w–a_nw relationship for the multiphase system studied here. Additionally, a surface was fit to the main loop (excluding scanning curves) of the drainage and imbibition P_c–S_w–a_nw data and compared to the surface fit to all of the data. The differences between these two surfaces were small (RMSE = 0.05 mm⁻¹ and MAE = 0.01 mm⁻¹) indicating that the P_c–S_w–a_nw surface is adequately represented without the need for the scanning curve data, which greatly reduces the amount of data required to construct the non-hysteretic P_c–S_w–a_nw surface for this data.     

基于时空守恒元和解元(CE/SE)方法的孔隙介质多相流动计算

将时空守恒元/解元(CE/SE)方法推广到二维孔隙介质多相流问题的数值计算中,采用人工压缩法耦合速度和压力,同时结合杂交粒子水平集方法捕捉物质界面.提出一套完整的二维欧拉型孔隙介质非稳态多相不可压缩黏性流动计算方案.通过对溃坝和液滴在重力作用下的运动和变形问题的数值模拟,验证了方法的精度和有效性.在此基础上,提出了一个新的孔隙介质两相流物理模型——双层流体顶盖驱动方腔流.     

Comment on the effect of anisotropy on the onset of convection in a porous medium

The effect of anisotropy on the onset of convection in a saturated porous medium is discussed. In particular, the case of time-dependent density-driven convection is examined. The applicability of the value of an equivalent Rayleigh number as the criterion for the onset of convection is discussed.     

Seismic wave attenuation in fluid-saturated porous media

Contrary to the traditional view, seismic attenuation in Biot's theory of fluid-saturated porous media is due to viscous damping of local (not global) pore-fluid motion. Since substantial inhomogeneities in fluid permeability of porous geological materials are to be expected, the regions of highest local permeability contribute most to the wave energy dissipation while those of lowest permeability dominate the fluid flow rate if they are uniformly distributed. This dichotomy can explain some of the observed discrepancies between computed and measured attenuation of compressional and shear waves in porous earth. One unfortunate consequence of this result is the fact that measured seismic wave attenuation in fluid-filled geological materials cannot be used directly as a diagnostic of the global fluid-flow permeability.     

周期性层状含孔隙、裂隙介质模型纵波衰减特征

地震波在含孔隙、裂隙斑块饱和介质传播过程中会诱发多个尺度孔隙流体流动而产生衰减和速度频散.在含有宏观尺度“Biot流”和介观尺度“局域流”衰减诱导机制的周期性层状孔隙介质模型基础上,引入了微观尺度硬币型和尖灭型裂隙“喷射流”的影响,构建了周期性层状含孔隙、裂隙介质模型.利用双解耦弹性波动方程的方法数值计算了该模型地震频带的纵波衰减和速度频散并与周期性层状孔隙介质模型做了对比研究.分析了该模型在不同裂隙参数（裂隙密度、裂隙纵横比）及裂隙体积含量下的纵波衰减和频散特征,裂隙密度越高对于纵波衰减和频散的影响越大,裂隙纵横比越小,由裂隙引起的纵波衰减部分向高频段移动,裂隙体积含量越少,纵波衰减先降低后小幅增加再降低,频散速度增加,并逐渐接近于周期性层状孔隙介质模型的纵波衰减和频散速度曲线.最后研究了周期性层状含孔隙、裂隙介质模型有效平面波模量的高低频极限以及流固相对位移在该模型中的分布特征.     

Numerical analysis of bacterial transport in saturated porous media

A mathematical model to describe bacterial transport in saturated porous media is presented. Reversible/irreversible attachment and growth/decay terms were incorporated into the transport model. Additionally, the changes of porosity and permeability due to bacterial deposition and/or growth were accounted for in the model. The predictive model was used to fit the column experimental data from the literature, and the fitting result showed a good match with the data. Based on the parameter values determined from the literature experimental data, numerical experiments were performed to examine bacterial sorption and/or growth during bacterial transport through saturated porous media. In addition, sensitivity analysis was performed to investigate the impact of key model parameters for bacterial transport on the permeability and porosity of porous media. The model results show that the permeability and porosity of porous media could be altered due to bacterial deposition and growth on the solid matrix. However, variation of permeability due to bacterial growth was trivial compared with natural permeability variation. Copyright © 2005 John Wiley & Sons, Ltd.     

Smoothed particle hydrodynamics pore-scale simulations of unstable immiscible flow in porous media

We have conducted a series of high-resolution numerical experiments using the Pair-Wise Force Smoothed Particle Hydrodynamics (PF-SPH) multiphase flow model. First, we derived analytical expressions relating parameters in the PF-SPH model to the surface tension and static contact angle. Next, we used the model to study viscous fingering, capillary fingering, and stable displacement of immiscible fluids in porous media for a wide range of capillary numbers and viscosity ratios. We demonstrated that the steady state saturation profiles and the boundaries of viscous fingering, capillary fingering, and stable displacement regions compare favorably with micromodel laboratory experimental results. For a displacing fluid with low viscosity, we observed that the displacement pattern changes from viscous fingering to stable displacement with increasing injection rate. When a high viscosity fluid is injected, transition behavior from capillary fingering to stable displacement occurred as the flow rate was increased. These observations are also in agreement with the results of the micromodel laboratory experiments.     

An extended stability criterion for density-driven flows in homogeneous porous media

Stability of density-driven flows is a challenging problem with current applications in major areas like energy exploration, water pollution, nuclear and oil industries. The mathematical model for such flows is a system of coupled non linear partial differential equations. To study the physical stability of the system, we consider steady-state flow and perturb the solution of the full system of equations (without Boussinesq approximation) and investigate how it evolves in time: if the solution does not grow indefinitely, the system is called stable. The perturbations are treated as being the result of sub-scale interactions between the velocity field and the solute mass. Making use of a two-scale expansion of the solution, we derived extended stability criteria that include the effects of density, viscosity and flow velocity in flow configurations aligned parallel as well as orthogonal to gravity forces. Numerical simulations with the numerical simulator d³f$d^{3}f$ are presented to test the theoretical stability criteria.     

Impact of geometrical properties on permeability and fluid phase distribution in porous media

To predict fluid phase distribution in porous media, the effect of geometric properties on flow processes must be understood. In this study, we analyze the effect of volume, surface, curvature and connectivity (the four Minkowski functionals) on the hydraulic conductivity and the water retention curve. For that purpose, we generated 12 artificial structures with 800³ voxels (the units of a 3D image) and compared them with a scanned sand sample of the same size. The structures were generated with a Boolean model based on a random distribution of overlapping ellipsoids whose size and shape were chosen to fulfill the criteria of the measured functionals. The pore structure of sand material was mapped with X-rays from synchrotrons.     

A two-scale operator-splitting method for two-phase flow in porous media

In this paper, we develop a two-scale operator-splitting method for the classical two-phase flow model, which handles advective and diffusive processes on different grids. The aim is to reduce computational complexity without loss of accuracy by using the numerical flexibility of operator-splitting techniques. To enhance the stability and the robustness with regards to sharp fronts, an additional slope limiter is introduced as a local post-processing step. For simplicity of notation, we provide the method in one dimension first and then generalize it to higher dimensions. Numerical examples illustrate the effect of the slope-limiting step and show the performance and flexibility of the proposed two-scale method.     

Non-Fickian mass transport in fractured porous media

The paper provides an introduction to fundamental concepts of mathematical modeling of mass transport in fractured porous heterogeneous rocks. Keeping aside many important factors that can affect mass transport in subsurface, our main concern is the multi-scale character of the rock formation, which is constituted by porous domains dissected by the network of fractures. Taking into account the well-documented fact that porous rocks can be considered as a fractal medium and assuming that sizes of pores vary significantly (i.e. have different characteristic scales), the fractional-order differential equations that model the anomalous diffusive mass transport in such type of domains are derived and justified analytically. Analytical solutions of some particular problems of anomalous diffusion in the fractal media of various geometries are obtained. Extending this approach to more complex situation when diffusion is accompanied by advection, solute transport in a fractured porous medium is modeled by the advection-dispersion equation with fractional time derivative. In the case of confined fractured porous aquifer, accounting for anomalous non-Fickian diffusion in the surrounding rock mass, the adopted approach leads to introduction of an additional fractional time derivative in the equation for solute transport. The closed-form solutions for concentrations in the aquifer and surrounding rocks are obtained for the arbitrary time-dependent source of contamination located in the inlet of the aquifer. Based on these solutions, different regimes of contamination of the aquifers with different physical properties can be readily modeled and analyzed.     

Stochastic modeling of solute transport in 3-D heterogeneous porous media with random source condition

During probabilistic analysis of flow and transport in porous media, the uncertainty due to spatial heterogeneity of governing parameters are often taken into account. The randomness in the source conditions also play a major role on the stochastic behavior in distribution of the dependent variable. The present paper is focused on studying the effect of both uncertainty in the governing system parameters as well as the input source conditions. Under such circumstances, a method is proposed which combines with stochastic finite element method (SFEM) and is illustrated for probabilistic analysis of concentration distribution in a 3-D heterogeneous porous media under the influence of random source condition. In the first step SFEM used for probabilistic solution due to spatial heterogeneity of governing parameters for a unit source pulse. Further, the results from the unit source pulse case have been used for the analysis of multiple pulse case using the numerical convolution when the source condition is a random process. The source condition is modeled as a discrete release of random amount of masses at fixed intervals of time. The mean and standard deviation of concentration is compared for the deterministic and the stochastic system scenarios as well as for different values of system parameters. The effect of uncertainty of source condition is also demonstrated in terms of mean and standard deviation of concentration at various locations in the domain.     

Comparison of theory and experiment for solute transport in weakly heterogeneous bimodal porous media

In this work, the influence of non-equilibrium effects on solute transport in a weakly heterogeneous medium is discussed. Three macro-scale models (upscaled via the volume averaging technique) are investigated: (i) the two-equation non-equilibrium model, (ii) the one-equation asymptotic model and (iii) the one-equation local equilibrium model. The relevance of each of these models to the experimental system conditions (duration of the pulse injection, dispersivity values…) is analyzed. The numerical results predicted by these macroscale models are compared directly with the experimental data (breakthrough curves). Our results suggest that the preasymptotic zone (for which a non-Fickian model is required) increases as the solute input pulse time decreases. Beyond this limit, the asymptotic regime is recovered. A comparison with the results issued from the stochastic theory for this regime is performed. Results predicted by both approaches (volume averaging method and stochastic analysis) are found to be consistent.