Fluid Transport In Porous Media Research Articles

Minimization principles for the coupled problem of Darcy–Biot-type fluid transport in porous media linked to phase field modeling of fracture

This work develops new minimization and saddle point principles for the coupled problem of Darcy–Biot-type fluid transport in porous media at fracture. It shows that the quasi-static problem of elastically deforming, fluid-saturated porous media is related to a minimization principle for the evolution problem. This two-field principle determines the rate of deformation and the fluid mass flux vector. It provides a canonically compact model structure, where the stress equilibrium and the inverse Darcy's law appear as the Euler equations of a variational statement. A Legendre transformation of the dissipation potential relates the minimization principle to a characteristic three field saddle point principle, whose Euler equations determine the evolutions of deformation and fluid content as well as Darcy's law. A further geometric assumption results in modified variational principles for a simplified theory, where the fluid content is linked to the volumetric deformation. The existence of these variational principles underlines inherent symmetries of Darcy–Biot theories of porous media. This can be exploited in the numerical implementation by the construction of time- and space-discrete variational principles, which fully determine the update problems of typical time stepping schemes. Here, the proposed minimization principle for the coupled problem is advantageous with regard to a new unconstrained stable finite element design, while space discretizations of the saddle point principles are constrained by the LBB condition. The variational principles developed provide the most fundamental approach to the discretization of nonlinear fluid–structure interactions, showing symmetric systems in algebraic update procedures. They also provide an excellent starting point for extensions towards more complex problems. This is demonstrated by developing a minimization principle for a phase field description of fracture in fluid-saturated porous media. It is designed for an incorporation of alternative crack driving forces, such as a convenient criterion in terms of the effective stress. The proposed setting provides a modeling framework for the analysis of complex problems such as hydraulic fracture. This is demonstrated by a spectrum of model simulations.

Variational Phase Field Modeling of Fracture Caused by Fluid Transport in Poroelastic Solids

AbstractThe numerical modeling of failure mechanisms due to fracture based on sharp crack discontinuities is extremely demanding and suffers in situations with complex crack topologies. This drawback can be overcome by recently developed diffusive crack modeling concepts, which are based on the introduction of a crack phase field. Such an approach is conceptually in line with gradient‐extended continuum damage models which include internal length scales. In this paper, we extend our recently outlined mechanical framework [1–3] towards the phase field modeling of fracture in the coupled problem of fluid transport in deforming porous media. Here, extremely complex crack patterns may occur due to drying or hydraulic induced fracture, the so called fracking. We develop new variational potentials for Biot‐type fluid transport in porous media at finite deformations coupled with phase field fracture. It is shown, that this complex coupled multi‐field problem is related to an intrinsic mixed variational principle for the evolution problem. This principle determines the rates of deformation, fracture phase field and fluid content along with the fluid potential. We develop a robust computational implementation of the coupled problem based on the potentials mentioned above and demonstrate its performance by the numerical simulation of complex fracture patterns. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A Resistance Model for Newtonian and Power-Law Non-Newtonian Fluid Transport in Porous Media

A theoretical model based on the force balance between pressure, viscous force, and inertia force is proposed to predict the flow resistance of Newtonian and power-law non-Newtonian fluids through porous packed beds. The present model takes inertia effect into consideration, and the flow regime can be extended from Darcy flow to non-Darcy flow. It is demonstrated that the present model can predict most available experimental data well. The present results are also compared to the Ergun equation and other drag correlations.

Spatially resolved D–T2 correlation NMR of porous media

Within the past decade, 2D Laplace nuclear magnetic resonance (NMR) has been developed to analyze pore geometry and diffusion of fluids in porous media on the micrometer scale. Many objects like rocks and concrete are heterogeneous on the macroscopic scale, and an integral analysis of microscopic properties provides volume-averaged information. Magnetic resonance imaging (MRI) resolves this spatial average on the contrast scale set by the particular MRI technique. Desirable contrast parameters for studies of fluid transport in porous media derive from the pore-size distribution and the pore connectivity. These microscopic parameters are accessed by 1D and 2D Laplace NMR techniques. It is therefore desirable to combine MRI and 2D Laplace NMR to image functional information on fluid transport in porous media. Because 2D Laplace resolved MRI demands excessive measuring time, this study investigates the possibility to restrict the 2D Laplace analysis to the sum signals from low-resolution pixels, which correspond to pixels of similar amplitude in high-resolution images. In this exploratory study spatially resolved D–T2 correlation maps from glass beads and mortar are analyzed. Regions of similar contrast are first identified in high-resolution images to locate corresponding pixels in low-resolution images generated with D–T2 resolved MRI for subsequent pixel summation to improve the signal-to-noise ratio of contrast-specific D–T2 maps. This method is expected to contribute valuable information on correlated sample heterogeneity from the macroscopic and the microscopic scales in various types of porous materials including building materials and rock.

On the challenges of measuring interfacial characteristics of three‐phase fluid flow with x‐ray microtomography

Synchrotron-based x-ray computed microtomography contributes high-resolution, three-dimensional observations to investigations of multiphase fluid transport in porous media. Pore-scale observations are valuable to the development and validation of new theory, as well as numerical models. Computed microtomography has been used previously to measure fluid content and interfacial areas in systems containing two fluids (air-water, oil-water) and to a limited extent to measure fluid content and entrapped fluid morphology in systems containing three fluids (air-oil-water). This study addresses challenges that arise when imaging three-phase flow in spreading systems. The first challenge is related to wettability alteration. Observations reported herein suggest that the wettability of solid surfaces changed over the course of a three-fluid phase flow experiment, a phenomenon that has not been observed in similar, previously conducted two-fluid phase experiments. Follow-up experiments showed that wettability alteration is significant when oil-solid contact is combined with x-ray exposure, and is not reversed with a conventional cleaning procedure. The second challenge arises in segmenting three-phase images, and thereby obtaining data from which various measures can be quantified with sufficient accuracy. Partial volume effects and blur often cause the grey-scale values of different fluids to overlap and appropriate steps must be taken to avoid ambiguity at phase boundaries. A comparison of images collected at standard resolution (10.6 micronsvoxel(-1) ) to those collected at a higher resolution (5.3 micronsvoxel(-1) ) showed that saturation measurements are within 5% of each other, but interfacial areas for three-phase systems may be underestimated at standard resolution by as much as 25%.

A peridynamic formulation of pressure driven convective fluid transport in porous media

A general state-based peridynamic formulation is presented for convective single-phase flow of a liquid of small and constant compressibility in heterogeneous porous media. In addition to local fluid transport, possible anomalous diffusion due to non-local fluid transport is considered and simulated. The governing integral equations of the peridynamic formulation are computationally easier to solve in domains with discontinuities than the traditional conservation models containing spatial derivatives. A bond-based peridynamic formulation is also developed and demonstrated to be a special case of the state-based formulation. The non-local model does not assume continuity in the field variables, satisfies mass conservation over an arbitrary bounded body and approaches the corresponding local model as the non-local region goes to zero. The exact solution of the local model closely matches the non-local model for a classical two-dimensional flow problem with fluid sources and sinks and for both Neumann and Dirichlet boundary conditions. The model is shown to capture arbitrary flow discontinuities/heterogeneities without any fundamental changes to the model and with small incremental computational costs.

Neutron tomography of axisymmetric flow fields in porous media

A significant problem in the study of fluid transport in porous media is the ability to visualize the structure of the flow field when moisture contents vary rapidly in space and time. Here we present a method for determining the radial and vertical saturation profiles within axisymmetric preferential flow fields using neutron radiography. Flow fields such as these are surprisingly common in nature and determining the three-dimensional structure of their wetting front region has proven difficult. In this work, the moisture profiles are determined using a simple algorithm for algebraic computed tomography, which gives the three-dimensional structure of the moisture profile with a temporal resolution that is limited only by the desired noise level. The algorithm presented can be translated to radiography done using X-rays or light and is applicable to any rotationally symmetric object.

Computing the apparent permeability of an array of staggered square rods using volume-penalization

In the present paper we uncover through numerical simulation the velocity and pressure fields inside a model porous medium composed of an infinitely extending array of staggered square rods. These microtransport simulations allow for the prediction of macrotransport parameters that are of value to the volume-averaged description of fluid motion in porous media. We focus on computing the macroscopic apparent permeability and investigate its dependence on the Reynolds number, the porosity, and the flow direction. For the microtransport simulations a volume-penalizing immersed boundary method is presented that facilitates the computation of fluid transport in porous media, accounting for the full geometrical complexity of the porous medium. We represent porous media on uniform Cartesian grids and separate solid from fluid domains using a binary phase-indicator function. The effect of solid bodies on the fluid motion is modeled using a source term in the momentum equation. This source term approximates the no-slip condition at the solid–fluid interface.

A stochastic mixed finite element heterogeneous multiscale method for flow in porous media

A computational methodology is developed to efficiently perform uncertainty quantification for fluid transport in porous media in the presence of both stochastic permeability and multiple scales. In order to capture the small scale heterogeneity, a new mixed multiscale finite element method is developed within the framework of the heterogeneous multiscale method (HMM) in the spatial domain. This new method ensures both local and global mass conservation. Starting from a specified covariance function, the stochastic log-permeability is discretized in the stochastic space using a truncated Karhunen–Loève expansion with several random variables. Due to the small correlation length of the covariance function, this often results in a high stochastic dimensionality. Therefore, a newly developed adaptive high dimensional stochastic model representation technique (HDMR) is used in the stochastic space. This results in a set of low stochastic dimensional subproblems which are efficiently solved using the adaptive sparse grid collocation method (ASGC). Numerical examples are presented for both deterministic and stochastic permeability to show the accuracy and efficiency of the developed stochastic multiscale method.

A two-scale modeling of motion-induced fluid release from thin fibrous porous media

In this work, a two-scale two-phase modeling methodology is presented for studying fluid release from saturated/unsaturated thin fibrous media when brought in contact with a moving solid surface. Our macroscale model is based on the Richards’ equation for two-phase fluid transport in porous media. The required constitutive relationships, capillary pressure and relative permeability as functions of medium's saturation, are obtained through microscale modeling. At microscales, a 3-D model based on fiber diameter, fiber orientation, and medium's solid volume fraction (SVF), is generated to resemble the internal structure of the fibrous sheets and be used in full-morphology analysis as well as microscale permeability simulation. A mass convection boundary condition is considered here to model the fluid transport at the boundary in contact with the target surface. It was shown that the mass convection coefficient, k f , plays a significant role in determining the release rate and is expected to be in the range of 10 - 6 < k f < 10 - 9 , depending on the properties of the fluid, fibrous sheet, the target surface as well as the speed of the relative motion, and remains to be determined experimentally.

On the flow of a dusty gas with constant number density through granular porous media

In this work, we apply the method of intrinsic volume averaging to Saffman’s dusty gas equations in an attempt to develop a dusty gas model of flow through isotropic porous media. We assume the particle distribution to be uniform and the porous medium to be of the granular type. The effects of the porous microstructure on the flow mixture are analyzed using the pioneering results of Du Plessis and his coworkers, [J.P. Du Plessis, Analytical quantification of coefficients in the Ergun equation for fluid friction in a packed bed, Transp. Porous Media 16 (1994) 189–207; J.P. Du Plessis J.H. Masliyah, Flow through isotropic granular porous media, Transp. Porous Media 6 (1991) 207–221; J.P. Du Plessis, G.P.J. Diedericks, Pore-scale modeling of interstitial phenomena, in: J.P. Du Plessis (Ed.), Fluid Transport in Porous Media, Computational Mechanics Publications, 1997, pp. 61–104], which facilitate classification of the developed model into two sub-models: one for low speed, and one for high speed flow through granular media. A dust-phase partial pressure is introduced to render the governing equations determinate. This work provides an extension to a previously developed model, [F.M. Allan, M.H. Hamdan, Fluid-particle model of flow through porous media: the case of uniform particle distribution and parallel velocity fields, Appl. Math. Comput. 183 (2) (2006) 1208–1213], in which the velocity fields of the phases involved were assumed to be parallel.

Anomalous Fluid Transport in Porous Media Induced by Biofilm Growth

Magnetic resonance measurements of the transition from normal to anomalous hydrodynamic dispersion in porous media due to biological activity are presented. Fractional advection-diffusion equations are shown to provide models for the measured impact of biofilm growth on porous media transport dynamics.

Computational Technique Facilitates Modeling of Fluid Transport in Porous Media

Computational Technique Facilitates Modeling of Fluid Transport in Porous Media

Multi-gradient pulse investigations of fluid transport in porous media

NMR pulsed field gradient (PFG) experiments employing the application of n gradient pulses k 1 … k n are discussed in a general way as an n-fold encoding of position at successive times. The experiments are then represented by a sampling of n-dimensional k-space, K n . Various parameters of motion can be derived from the evolution of correlations within the n-dimensional position ( r-)space, R n , which is the Fourier conjugate space to K n . A wide class of NMR experiments may be described by this formalism, where the dimension of the experiment is often reduced by imposing conditions to the free variables. This is demonstrated for the case of displacement measurements where the condition ∑ k i = 0 is met. The two simplest pulse sequences which allow one to correlate displacements at two different times with each other are presented. While the three-pulse version of SERPENT encodes displacements in two interleaved time intervals Δ 1 and Δ 2, the four-pulse VEXSY experiment includes a mixing time τ m in between both encoding intervals Δ. The behaviour of fluid transport subject to external pressure through a model porous system is demonstrated by means of numerical simulations of SERPENT and VEXSY experiments for water flowing through a packed bed of monosized spherical particles. Displacements parallel ( Z) and perpendicular ( X) to the main flow direction are determined and the 2-D joint probability densities and the conditional probabilities are discussed along with the correlation coefficients related to the displacements at different encoding times. It is shown that all possible correlations between Z and X 2 in VEXSY decay with time constants comparable to the average time needed for a fluid molecule to cover one bead diameter, while a negative correlation is observed between transverse ( X) displacements which is explained by molecules flowing along streamlines which follow the circumference of the spherical particles. Correlations for displacements during the different times in SERPENT generally decay much slower and provide complementary information about the evolution of displacements with time.

Review of: “Fluid Transport in Porous Media Edited by Prieur de Plessis” Published by Computational Mechanics Inc. 25 Bridge St.Billerica,MA 01821,US A

Review of: “Fluid Transport in Porous Media Edited by Prieur de Plessis” Published by Computational Mechanics Inc. 25 Bridge St.Billerica,MA 01821,US A

Effective conductivity of porous silicon: A theoretical approach

From a theoretical point of view, linear transport in homogeneous systems is well understood. However, transport phenomena through porous materials are far more complicated. In fact, there are some materials like porous silicon with a very promising technological impact, where mainly experimental evidence is available and theoretical models have scarcely been developed. In a previous work, we obtained an analytical expression for the axial effective electrical conductivity of a model simulating porous silicon (PS). We used the averaging volume method that has proven to be successful in treating fluid transport in porous media. The PS model was based on experimental information about the structure of this new material and experimental data for crystalline Si and hydrogenated a-Si thin films. With this method we can calculate the bulk and the surface contribution to the tensorial effective conductivity. Here we present the transverse effective conductivity results where percolation effects can be observed, using the same method and model. However, in the XY plane the calculation is more complicated and it can only be performed numerically. The understanding of the effective electrical conductivity behavior as a whole may be important for electronic applications of PS.

Transverse component of the electrical conductivity of porous silicon — I

The study of transport in porous materials is a very challenging problem. In particular, the understanding of electric transport in porous silicon is fundamental for its promising technological impact. In a previous work we obtained an analytical expression for the axial effective electrical conductivity of a model simulating porous silicon (PS). We used the averaging volume method that has proven to be successful in treating fluid transport in porous media. The PS model was based on experimental information about this new material's structure and experimental data for crystalline Si and hydrogenated a-Si thin films. With this method we can calculate the bulk and the surface contribution to the tensorial effective conductivity. Here we present the transverse effective conductivity results where percolation effects can be observed, using the same method and model. However, in the XY plane the calculation is more complicated and it can only be solved numerically. We find that there is a range at low porosities where the effective electrical conductivity can be enhanced due to the surface contribution.

Effective Dielectric Function of Porous Silicon: the Transverse Component

ABSTRACTIn previous works we have obtained an expression for the effective electrical conductivity of a columnar model simulating porous silicon. We used the averaging volume method that has proven to be successful in treating fluid transport in porous media. With this method we can calculate the bulk and the surface contribution to an effective transport property. The axial component can be solved analytically, but in the xY plane the calculation can only be performed numerically. However there is a certain approximation called Chang's cell (valid for high porosities) where the transverse component is also analytical. The extension of our original approach to find the axial component of the effective dielectric function is relatively simple, but the transverse component calculation presents interesting features.

Effective electrical conductivity of porous silicon: A novel theoretical approach

From a theoretical point of view, linear transport is well understood. However, transport phenomena through porous materials are far more complicated. In this paper we obtain an analytical expression for the effective electrical conductivity of a model simulating porous silicon (PS). We use the averaging volume method that has proven to be successfull in treating fluid transport in porous media. The PS model is based on experimental information about this new material's structure and experimental data for crystalline Si and hydrogenated a-Si thin films. With this method we can calculate the bulk and the surface contribution to the effective conductivity. We find that the effective electrical conductivity increases to a maximum value at an optimum porosity; beyond this value, it decreases to zero. The behavior of the effective electrical conductivity may be important for electronic applications of PS.

Characteristics of immiscible fluid transport in porous media with anisotropic phase permeabilities

An average description of the transport of nonuniform fluids in porous media nonuniform with respect to their permeability leads to a system of equations of multiphase flow with tensor absolute and phase permeability functions [1], The solution of problems of flow through porous media when the relative phase permeabilities are tensor functions involves considerable difficulties and requires special analysis. Below, the system of equations of two-dimensional two-phase flow in a porous medium anisotropic with respect to the absolute and relative permeabilities is investigated. Some simple examples are considered.