The acoustic signature of fluid flow in complex porous media

Effective medium approximations for the frequency-dependent and complex-valued effective stiffness tensors of cracked/ porous rocks with multiple solid constituents are developed on the basis of the T-matrix approach (based on integral equation methods for quasi-static composites), the elastic - viscoelastic correspondence principle, and a unified treatment of the local and global flow mechanisms, which is consistent with the principle of fluid mass conservation. The main advantage of using the T-matrix approach, rather than the first-order approach of Eshelby or the second-order approach of Hudson, is that it produces physically plausible results even when the volume concentrations of inclusions or cavities are no longer small. The new formulae, which operates with an arbitrary homogeneous (anisotropic) reference medium and contains terms of all order in the volume concentrations of solid particles and communicating cavities, take explicitly account of inclusion shape and spatial distribution independently. We show analytically that an expansion of the T-matrix formulae to first order in the volume concentration of cavities (in agreement with the dilute estimate of Eshelby) has the correct dependence on the properties of the saturating fluid, in the sense that it is consistent with the Brown-Korringa relation, when the frequency is sufficiently low. We present numerical results for the (anisotropic) effective viscoelastic properties of a cracked permeable medium with finite storage porosity, indicating that the complete T-matrix formulae (including the higher-order terms) are generally consistent with the Brown-Korringa relation, at least if we assume the spatial distribution of cavities to be the same for all cavity pairs. We have found an efficient way to treat statistical correlations in the shapes and orientations of the communicating cavities, and also obtained a reasonable match between theoretical predictions (based on a dual porosity model for quartz-clay mixtures, involving relatively flat clay-related pores and more rounded quartz-related pores) and laboratory results for the ultrasonic velocity and attenuation spectra of a suite of typical reservoir rocks. (C) 2003 Elsevier B.V. All rights reserved.

Comparative study of different formulations and solution methods for two phase flow in saturated porous media

Six models (Simulators) are formulated and developed with all possible combinations of pressure and saturation of the phases as primary variables. A comparative study between six simulators with two numerical methods, conventional simultaneous and modified sequential methods are carried out. The results of the numerical models are compared with the laboratory experimental results to study the accuracy of the model especially in heterogeneous porous media. From the study it is observed that the simulator using pressure and saturation of the wetting fluid (PW, SW formulation) is the best among the models tested. Many simulators with nonwetting phase as one of the primary variables did not converge when used along with simultaneous method. Based on simulator 1 (PW, SW formulation), a comparison of different solution methods such as simultaneous method, modified sequential and adaptive solution modified sequential method are carried out on 4 test problems including heterogeneous and randomly heterogeneous problems. It is found that the modified sequential and adaptive solution modified sequential methods could save the memory by half and as also the CPU time required by these methods is very less when compared with that using simultaneous method. It is also found that the simulator with PNW and PW as the primary variable which had problem of convergence using the simultaneous method, converged using both the modified sequential method and also using adaptive solution modified sequential method. The present study indicates that pressure and saturation formulation along with adaptive solution modified sequential method is the best among the different simulators and methods tested.

肝脏中生物流体流动的多重介质渗流模型和数值模拟

本文采用生物渗流理论，建立了肝脏内不同生物流体流动的多重介质渗流模型，采用有限元法求解这种特殊的渗流问题，根据数值计算结果揭示了肝内血液、组织液以及胆汁等的流动规律，并探讨了肝脏血流动力学的一些问题。论文将肝脏内部与生物代谢功能有关的肝血窦和窦周间隙当作两重并存的多孔介质，血液在肝血窦中，以及组织液在窦周间隙中的流动均当作渗流处理，通过Starling公式考虑了两重介质之间的流量交换，从而建立了肝血窦-窦周间隙的双重介质模型。针对肝脏胆汁分泌功能，将肝脏内密布的毛血肝管网当作多孔介质，以受静压及渗透压驱动的流体跨壁流动表示肝汁从肝细胞向毛细肝管的分泌，肝汁在毛细胆管网中的流动作为渗流处理，从而建立了肝汁分泌与输运的双重介质模型。采用有限元法求解了生物流体的双重介质渗流问题，针对非牛顿渗流和两重介质的相互作用，本文发展了一种嵌套迭代方法，即采用直接迭代求解血液在肝血窦中的非线性渗流，采用交替迭代解决双重介质渗流中由跨壁流支引起的相互流体交换，直接迭代嵌套于交替迭代中。这种算法比较有效的解决了包含非牛顿渗流的双重介质渗流问题。根据生物多孔介质中微细管系统的构筑方式以及不同微细管系统之间的联系方式，论文提出将生物多孔介质划分为分级多孔介质和多重多孔介质两种主要类型。基于多相混合物的平均化的理论，论文推导了双重多孔介质中的动量守恒方程、质量守恒方程以及相应的渗流方程，建立了双重多孔介质渗流的平均化模型。基于分级多孔介质渗流的理论，论文将脏器中的血管树按管径分为不同级别的多孔介质，各级血管中和血液流动均作为渗流处理，从而提出了计算脏器整体血流的一种渗流方法。采用这种方法，在论文提出的肝血窦 - 窦周间隙双重介质渗流流模型的基础之上，初步研究了肝脏门静脉系统的血液动力学规律。采用本文提出的肝血窦 - 窦周间隙双重介质模型和胆汁分泌 - 流动的双重介质模型，得到了血液、组织液和胆汁在肝小叶中的压力分布和速度分布，并分析了肝血窦壁的跨壁流动模式，胆汁流量的影响因素，以及窦周间隙中组织液流量与肝血窦中血液流动及肝血窦壁渗透系数等因素的关系，揭示了肝脏内血液、组织液及胆汁等生物流体流动的一般规律。

Modeling the co-transport of viruses and colloids in unsaturated porous media

A mathematical model is developed to simulate the co-transport of viruses and colloids in unsaturated porous media under steady-state flow conditions. The virus attachment to the mobile and immobile colloids is described using a linear reversible kinetic model. Colloid transport is assumed to be decoupled from virus transport; that is, we assume that colloids are not affected by the presence of attached viruses on their surface. The governing equations,are solved numerically using an alternating three-step operator splitting approach. The model is verified by fitting three sets of experimental data published in the literature: (1) Syngouna and Chrysikopoulos (2013) and (2) Walshe et al. (2010), both on the co-transport of viruses and clay colloids under saturated conditions, and (3) Syngouna and Cluysikopoulos (2015) for the co-transport of viruses and clay colloids under unsaturated conditions. We found a good agreement between observed and fitted breakthrough curves (BTCs) under both saturated and unsaturated conditions. Then, the developed model was used to simulate the co-transport of viruses and colloids in porous media under unsaturated conditions, with the aim of understanding the relative importance of various processes on the co-transport of viruses and colloids in unsaturated porous media. The virus retention in porous media in the presence of colloids is greater during unsaturated conditions as compared to the saturated conditions due to: (1) virus attachment to the air-water interface (AWI), and (2) co-deposition of colloids with attached viruses on its surface to the AWL A sensitivity analysis of the model to various parameters showed that the virus attachment to AWI is the most sensitive parameter affecting the BTCs of both free viruses and total mobile viruses and has a significant effect on all parts of the BTC. The free and the total mobile viruses BTCs are mainly influenced by parameters describing virus attachment to the AIM, virus interaction with mobile and immobile colloids, virus attachment to solid-water interface (SWI), and colloid interaction with SWI and AWL The virus BTC is relatively insensitive to parameters describing the maximum adsorption capacity of the AWI for colloids, inlet colloid concentration, virus detachment rate coefficient from the SW!, maximum adsorption capacity of the AWI for viruses and inlet virus concentration. (C) 2015 Elsevier B.V. All rights reserved.

Natural convection flow in a porous enclosure with localized heating from below

Natural convection flow in a two-dimensional fluid saturated porous enclosure with localized heating from below, symmetrical cooling from the sides and the top and rest of the bottom walls are insulated, has been investigated numerically. Darcy’s law for porous media along with the energy equation based on the 1st law of thermodynamics has been considered. Implicit finite volume method with TDMA solver is used to solve the governing equations. Localized heating is simulated by a centrally located isothermal heat source on the bottom wall, and four different values of the dimensionless heat source length, 1/5, 2/5, 3/5 and 4/5 are considered. The effect of heat source length and the Rayleigh number on streamlines and isotherms are presented, as well as the variation of the local rate of heat transfer in terms of the local Nusselt number from the heated wall. Finally, the average Nusselt number at the heated part of the bottom wall has been shown against Rayleigh number for the non-dimensional heat source length.

Non-darcy double-diffusive mixed convection from heated vertical and horizontal plates in saturated porous media

The non-darcy mixed convection flows from heated vertical and horizontal plates in saturated porous media have been considered using boundary layer approximations. The flows are considered to be driven by multiple buoyancy forces. The similarity solutions for both vertical and horizontal plates have been obtained. The governing equations have been solved numerically using a shooting method. The heat transfer, mass transfer and skin friction are reduced due to inertial forces. Also, they increase with the buoyancy parameter for aiding flow and decrease for the opposing flow. For aiding flow, the heat and mass transfer coefficients are found to approach asymptotically the forced or free convection values as the buoyancy parameter approaches zero or infinity.

Experimental Investigation of Transient Saltwater Intrusion in Heterogeneous Porous Media

A 2D sandbox style experiment was developed to compare the results of numerical modelling to physical testing for saltwater intrusion in homogeneous and heterogeneous aquifers. The sandbox consisted of a thin central viewing chamber filled with glass beads of varying diameters (780μm, 1090μm and 1325μm) under fully saturated conditions. Dyed saltwater (SW) was introduced at the side boundary and a head difference imposed across the porous media. Images of the SW wedge were recorded at intervals in order to assess the suitability of the numerical models predictions of transient SW intrusion. Numerical modelling of the experimental cases were simulated using SUTRA. Two main parameters were chosen to express the condition of the intruding SW wedge at each recorded time step; the toe penetration length (TL) and the width of the mixing zone (WMZ). The WMZ was larger under transient conditions in the heterogeneous case, while the TL was longer for the homogeneous case. The increased variability in the flow field fo the heterogeneous case resulted in increased dispersion, and thus, increased WMZ.

Superfast non-linear diffusion: capillary transport in particulate porous media

The migration of liquids in porous media, such as sand, has been commonly considered at high saturation levels with liquid pathways at pore dimensions. In this letter we reveal a low saturation regime observed in our experiments with droplets of extremely low volatility liquids deposited on sand. In this regime the liquid is mostly found within the grain surface roughness and in the capillary bridges formed at the contacts between the grains. The bridges act as variable-volume reservoirs and the flow is driven by the capillary pressure arising at the wetting front according to the roughness length scales. We propose that this migration (spreading) is the result of interplay between the bridge volume adjustment to this pressure distribution and viscous losses of a creeping flow within the roughness. The net macroscopic result is a special case of non-linear diffusion described by a superfast diffusion equation (SFDE) for saturation with distinctive mathematical character. We obtain solutions to a moving boundary problem defined by SFDE that robustly convey a time power law of spreading as seen in our experiments.

Fractal Behaviour and Irregularity of Wetting Fronts in Heterogeneous Porous Media

The wetting front is the zone where water invades and advances into an initially dry porous material and it plays a crucial role in solute transport through the unsaturated zone. Water is an essential part of the physiological process of all plants. Through water, necessary minerals are moved from the roots to the parts of the plants that require them. Water moves chemicals from one part of the plant to another. It is also required for photosynthesis, for metabolism and for transpiration. The leaching of chemicals by wetting fronts is influenced by two major factors, namely: the irregularity of the fronts and heterogeneity in the distribution of chemicals, both of which have been described by using fractal techniques. Soil structure can significantly modify infiltration rates and flow pathways in soils. Relations between features of soil structure and features of infiltration could be elucidated from the velocities and the structure of wetting fronts. When rainwater falls onto soil, it doesn?t just pool on surfaces. Water ?or another fluid- acts differently on porous surfaces. If the surface is permeable (porous) it seeps down through layers of soil, filling that layer to capacity. Once that layer is filled, it moves down into the next layer. In sandy soil, water moves quickly, while it moves much slower through clay soil. The movement of water through soil layers is called the the wetting front. Our research concerns the motion of a liquid into an initially dry porous medium. Our work presents a theoretical framework for studying the physical interplay between a stationary wetting front of fractal dimension D with different porous materials. The aim was to model the mass geometry interplay by using the fractal dimension D of a stationary wetting front. The plane corresponding to the image is divided in several squares (the minimum correspond to the pixel size) of size length ". We acknowledge the help of Prof. M. García Velarde and the facilities offered by the Pluri-Disciplinary Institute of the Complutense University of Madrid. We also acknowledge the help of European Community under project Multi-scale complex fluid flows and interfacial phenomena (PITN-GA-2008-214919). Thanks are also due to ERCOFTAC (PELNoT, SIG 14)

Very intense pulse in the groundwater flow in fissurized-porous stratum

An asymptotic solution is obtained corresponding to a very intense pulse: a sudden strong increase and fast subsequent decrease of the water level at the boundary of semi-infinite fissurized-porous stratum. This flow is of practical interest: it gives a model of a groundwater flow after a high water period or after a failure of a dam around a collector of liquid waste. It is demonstrated that the fissures have a dramatic influence on the groundwater flow, increasing the penetration depth and speed of fluid penetration into the stratum. A characteristic property of the flow in fissurized-porous stratum is the rapid breakthrough of the fluid at the first stage deeply into the stratum via a system of cracks, feeding of porous blocks by the fluid in cracks, and at a later stage feeding of advancing fluid flow in fissures by the fluid, accumulated in porous blocks.

Micropolar flow in a porous channel with high mass transfer

Two dimensional flow of a micropolar fluid in a porous channel is investigated. The flow is driven by suction or injection at the channel walls, and the micropolar model due to Eringen is used to describe the working fluid. An extension of Berman's similarity transform is used to reduce the governing equations to a set of non-linear coupled ordinary differential equations. The latter are solved for large mass transfer via a perturbation analysis where the inverse of the cross-flow Reynolds number is used as the perturbing parameter. Complementary numerical solutions for strong injection are also obtained using a quasilinearisation scheme, and good agreement is observed between the solutions obtained from the perturbation analysis and the computations.

A mass-conservative control volume-finite element method for solving Richards’ equation in heterogeneous porous media

We present a mass-conservative vertex-centred finite volume method for efficiently solving the mixed form of Richards’ equation in heterogeneous porous media. The spatial discretisation is particularly well-suited to heterogeneous media because it produces consistent flux approximations at quadrature points where material properties are continuous. Combined with the method of lines, the spatial discretisation gives a set of differential algebraic equations amenable to solution using higher-order implicit solvers. We investigate the solution of the mixed form using a Jacobian-free inexact Newton solver, which requires the solution of an extra variable for each node in the mesh compared to the pressure-head form. By exploiting the structure of the Jacobian for the mixed form, the size of the preconditioner is reduced to that for the pressure-head form, and there is minimal computational overhead for solving the mixed form. The proposed formulation is tested on two challenging test problems. The solutions from the new formulation offer conservation of mass at least one order of magnitude more accurate than a pressure head formulation, and the higher-order temporal integration significantly improves both the mass balance and computational efficiency of the solution.

A variable-stepsize Jacobian-free exponential integrator for simulating transport in heterogeneous porous media : application to wood drying

A Jacobian-free variable-stepsize method is developed for the numerical integration of the large, stiff systems of differential equations encountered when simulating transport in heterogeneous porous media. Our method utilises the exponential Rosenbrock-Euler method, which is explicit in nature and requires a matrix-vector product involving the exponential of the Jacobian matrix at each step of the integration process. These products can be approximated using Krylov subspace methods, which permit a large integration stepsize to be utilised without having to precondition the iterations. This means that our method is truly "Jacobian-free" - the Jacobian need never be formed or factored during the simulation. We assess the performance of the new algorithm for simulating the drying of softwood. Numerical experiments conducted for both low and high temperature drying demonstrates that the new approach outperforms (in terms of accuracy and efficiency) existing simulation codes that utilise the backward Euler method via a preconditioned Newton-Krylov strategy.