Saturated Porous Materials Research Articles

Axial conduction effect in flow through circular porous passages with prescribed wall heat flux

This study presents a series solution for computation of the steady state temperature field in circular ducts with prescribed wall heat flux. The ducts are filled with fluid saturated porous materials. The developed methodology includes a simple transformation that improves the convergence of this series solution. The acquired solution includes the contribution of axial conduction that leads to a modified Graetz-type solution for these fluid passages. Finally, this solution is augmented by the contribution of frictional heating.

Scattered fracture of porous materials with brittle skeleton

A model of damage accumulation in a porous medium with a brittle skeleton saturated with a compressible fluid is formulated in the isothermal approximation. The model takes account of the skeleton elastic energy transformation into the surface energy of microcracks. In the case of arbitrary deformations of an anisotropic material, constitutive equations are obtained in a general form that is necessary and sufficient for the objectivity and thermodynamic consistency principles to be satisfied. We also formulate the kinetics equation ensuring that the scattered fracture dissipation is nonnegative for any loading history. For small deviations from the initial state, we propose an elastic potential which permits describing the principal characteristics of the behavior of a saturated porous medium with a brittle skeleton. We study the acoustic properties of the material under study and find their relationship with the strength criterion depending on the accumulated damage and the material current deformation. We consider the problem of scattered fracture of a saturated porous material in a neighborhood of a spherical cavity. We show that the cavity failure occurs if the Hadamard condition is violated.

A two-scale approach for propagating cracks in a fluid-saturated porous material

An extension to a finite strain framework of a two-scale numerical model for propagating crack in porous material is proposed to model the fracture in intervertebral discs. In the model, a crack is described as a propagating cohesive zone by exploiting the partition-of-unity property of finite element shape functions. At the micro-scale, the flow in the cohesive crack is modelled as viscous fluid using Stokes' equations which are averaged over the cross section of the cavity. At the macro-scale, identities are derived to couple the local momentum and the mass balance to the governing equations for a saturated porous material. The resulting discrete equations are nonlinear due to the cohesive constitutive equations and the geometrically nonlinear kinematic relations. A Newton-Raphson iterative procedure is used to consistently linearise the derived system while a Crank-Nicholson scheme takes care of the time integration of the system. The derived model is used to analyse a quasi-static crack growth in confined compression under tensile loading.

Comparison between the Gauss’ law method and the zero current method to calculate multi-species ionic diffusion in saturated uncharged porous materials

A novel numerical method based on the finite element approach is established for the zero current method approach for calculating multi-species ionic diffusion. The proposed numerical method uses the direct calculation of the coupled set of equations in favor of the staggering approach. A one-step truly implicit time stepping scheme is adopted together with an implementation of a modified Newton–Raphson iteration scheme for search of equilibrium at each considered time step calculation. Results from the zero current case are compared with existing results from the solutions of the more general Gauss’ law method.

Determination of the transport properties of a blended concrete from its electrical properties measured during a migration test

The Nernst–Planck equation describes ionic movements in saturated porous materials subject to chemical concentration gradients in the pore fluid and applied electric fields. When applied to a macroscopic system the equation supposes that the flux of each ion is independent of every other one. However, owing to the ionic exchange among different or similar species, there are ionic potentials that affect the final flux. These will distort the applied electric field and thus keep the electroneutrality of the sum of all the ionic species involved. The applied potential will fall linearly across the sample but an additional ‘membrane' potential will change with position and time. This paper summarises a theoretical and experimental investigation into the application of the non-linear electric membrane potential to the simulation of the migration of chlorides in concrete. A new electrochemical test has been developed and carried out with different samples of concrete blended with pulverised fuel ash (PFA) and ground granulated blastfurnace slag (GGBS). During the tests the transient current and the electrical membrane potential were measured. A numerical model developed previously using the classical equations, but including changes in the concrete membrane potential, was optimised using an artificial neural network (ANN). Based on the experimental results and the simulations, the intrinsic diffusion coefficients of chloride, hydroxide, sodium and potassium were obtained. Also, the initial hydroxide composition of the pore solution, the porosity, and the chloride binding capacity were determined. The results showed good agreement with the theory and can help to explain the complex phenomena that occur during a concrete migration test.

An Enriched Biphasic Model for Solute Driven Degradation

AbstractTransport of solutes in porous materials plays an important role in many kinds of materials such as biological tissues, porous implants or even soils. In most of the cases the liquid phase in the pores acts as a solvent for one or more solutions. The motion of the solutions is driven by both, the advective and convective transport. The former is related to the fluid phase velocity whereas the letter follows the concentration gradient. The interactions between the solutes and the solid and liquid phase may influence the overall material behavior. Although the solutes often carry electrical charges this paper is focused on neutrally charged solutions. In this contribution the model to describe the solute transport in a fluid saturated porous material is based on the well founded Theory of Porous Media. We will present the basic framework and the governing equations. Finally, we will show a three dimensional numerical example of the solute driven degradation of a skull implant. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Heat transfer with upstream thermal penetration in flow through porous plate passages

The objective of this study is to develop a series solution for determination of the temperature field in parallel-plate ducts with prescribed wall heat flux. Consideration is also given to ducts filled with fluid saturated porous materials. A simple transformation is used to improve the convergence of this series solution. Temperature variations and heat transfer coefficients are determined for infinitely long parallel-plate ducts when the walls undergo a step change in the applied wall heat flux. Axial thermal penetration near the applied wall heat flux location is studied. The solutions with the contribution of axial conduction in these fluid passages are acquired using a modified Graetz-type solution. Finally, this technique is augmented by the contribution of frictional heating.

Experimental Determination of Some Micro-Structural and Hydrous Properties of a Terra Cotta and Modelisation of Hygrothermal Transfers

This experimental study present some micro-structural and thermo-hydrous properties of terra cotta. A tube, built from this porous material, bathes in a tank filled with water. The internal surface of the wall carries a water film which evaporates by forced convection of hot and dry air, allowing the storage of negative kilocalories. The form of the lube used is a factor that allows neglecting the variations of the thermo-physical properties of the air between the entry and the exit of the tube. A computer code (based on the equation of Darcy for fluid crossing saturated porous material, the equation of Laplace on pressures and the equation of heat in polar co-ordinates) was established to describe the thermo-hydrous behavior of the material. The code makes it possible to determine the zones of intrinsic permeability favorable to a continuous evaporation without draining off the water film and it also allows the calculation of the average temperature of bath.

Effect of the volume of the drainage system on the measurement of undrained thermo-poro-elastic parameters

For evaluation of the undrained thermo-poro-elastic properties of saturated porous materials in conventional triaxial cells, it is important to take into account the effect of the dead volume of the drainage system. The compressibility and the thermal expansion of the drainage system, along with the dead volume of the fluid filling this system, influence the measured pore pressure and volumetric strain during an undrained thermal or mechanical loading in a triaxial cell. The correction methods previously presented by Wissa [Pore pressure measurement in saturated stiff soils. ASCE J Soil Mech Found Div 1969;95(SM 4):1063–73], Bishop [Influence of system compressibility on observed pore pressure response to an undrained change in stress in saturated rock. Geotechnique 1976;26(2):371–5] and Ghabezloo and Sulem [Stress dependent thermal pressurization of a fluid-saturated rock. Rock Mech Rock Eng 2009;42(1):1–24], only permit one to correct the measured pore pressures during an undrained isotropic compression test or an undrained heating test. An extension of these methods is presented in this paper to correct the measured volumetric strain, and consequently the measured undrained bulk compressibility and undrained thermal expansion coefficients, during these tests. Two examples of application of the proposed correction method are presented for an undrained isotropic compression test and an undrained heating test performed on a fluid-saturated granular rock. A parametric study has demonstrated that the porosity and the drained compressibility of the tested material, and the ratio of the volume of the drainage system to the one of the tested sample, are the key parameters that most influence the error induced by the drainage system.

Numerical modelling of a slope stability test by means of porous media mechanics

PurposeThe purpose of this paper is to present a finite‐element analysis of the initiation of a slope failure in a small‐scale laboratory test due to pore pressure variation. To this aim, a fully coupled multiphase model for saturated/partially saturated solid porous materials based on porous media mechanics is used.Design/methodology/approachThe slope is described as a three‐phase deforming porous continuum where heat, water and gas flow are taken into account. The gas phase is modelled as an ideal gas composed of dry air and water vapour. Phase changes of water, heat transfer through conduction and convection and latent heat transfer are considered. The independent variables are: solid displacements, capillary pressure, gas pressure and temperature. The effective stress state is limited by Drucker‐Prager yield surface for the sake of simplicity. Small strains and quasi‐static loading conditions are assumed.FindingsThe paper shows that the multiphase modelling is able to capture the main experimental observations such as the local failure zone at the onset of slope failure and the outflow appeared in that zone. It also allows understanding of the triggering mechanisms of the failure zone.Research limitations/implicationsThis work can be considered as a step towards a further development of a suitable numerical model for the simulation of non‐isothermal geo‐environmental engineering problems.Practical implicationsThe multiphysics approach looks promising for the analysis of the onset of landslides, provided that the constitutive models for the multiphase porous media in saturated/unsaturated conditions and the related mechanical and hydraulic properties are described with sufficient accuracy.Originality/valueElasto‐plastic thermo‐hydro‐mechanical modelling of the initiation of slope failure subjected to variation in pore pressure boundary condition.

Fabric dependence of poroelastic wave propagation.

The governing equations for wave motion in the linear theory of anisotropic poroelastic materials are developed and extended to include the dependence of the constitutive relations upon fabric. Fabric is a quantitative stereological measure of the degree of structural anisotropy in the pore architecture of a porous medium. With the addition of the second order symmetric tensor fabric variable, the formulation of wave motions in the poroelastic theory is consistent with the presentations of Biot and later authors. The dependence of all the material tensors in the Biot theory on the fabric tensor aligns the material tensors, so when a single direction that is a plane of material symmetry is selected, the equations simplify considerably. The customary polynomial of the sixth deg for the wave speeds of the anisotropic Biot theory in a selected direction reduces to three quadratic equations to solve for the wave speeds and attenuation. While the theory is applicable to any saturated porous material, the two longitudinal waves predicted by this model are measured in cancellous bone and used to derive the corresponding anisotropic elastic constants. Representative examples of bone loss are analyzed as a function of the porosity, tissue density, and fabric.

Performance of the theta probe ML2 in the presence of nonuniform soil water profiles

An attempt is made to investigate the soil permittivity regime which is predicted by ML2 theta probe in cases where a nonuniform soil moisture profile prevails. Our investigation was performed in laboratory columns, under temperature-controlled conditions for a number of different porous media. Five distinct cases, where the moisture regime could be established in a predesigned manner were tested. These are: (a) Air over a water or ethanol layer of changing thickness. (b) Air over a saturated porous material of changing thickness and the same in the reverse order. (c) Dry porous material over the same but saturated porous material of changing thickness and vice versa. (d) Gradual wetting from the top surface of initially dry soils. (e) Initially saturated vertical soil profiles, left to gradually dry by evaporation from both open surfaces (top and bottom). From our experimental investigations it is shown that the bulk dielectric constant predicted by the ML2 probe in nonuniformly wet profiles irrespective of their layer thickness is closely described by the arithmetic averaging scheme. This fact makes the water content predictions from the manufacturer calibration equations for these cases, seriously overestimated. A comparison of the actual water contents with those predicted with the calibration equations of Robinson et al. [Robinson, D.A., Jones, S.B., Blonquist, J.M., Friedman, S.P., 2005. A physical derived water content/permittivity calibration model for coarse–textured layered soils. Soil Sci. Soc. Am. J. 69 (5), 1372–1377] gives very satisfactory results.

Coupled Problems of Dynamics in Materially Incompressible Saturated Porous Media

AbstractThe mechanical behavior of saturated porous materials is largely governed by the interaction between the solid skeleton and the pore fluid. This interaction is particularly strong in dynamic problems and leads to numerical challenges especially in the case of incompressible constituents. In fact, the permeability plays a significant role in this coupling and influences the choice of a proper time integration scheme. Proceeding from the macroscopic Theory of Porous Media (TPM) within the isothermal and geometrical linear regime, the governing balance equations of the dynamic binary solid–fluid model are the solid and fluid momentum balances, and the overall volume balance of the biphasic mixture. This set of coupled partial differential equations (PDEs) is solved within the framework of the mixed Finite Element Method (FEM) applying two different time solution methods, viz., a monolithic implicit and a splitted implicit–explicit scheme. The time stepping algorithms are implemented into the FE program PANDAS and a Scilab FE routine and compared on a one–dimensional wave propagation example. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

An artificial compressibility method for buoyancy‐driven flow in heterogeneous saturated packed beds

PurposeThe purpose of this paper is to focus on modeling buoyancy driven viscous flow and heat transfer through saturated packed pebble‐beds via a set of homogeneous volume‐averaged conservation equations in which local thermal disequilibrium is accounted for.Design/methodology/approachThe local thermal disequilibrium accounted for refers to the solid and liquid phases differing in temperature in a volume‐averaged sense, which is modeled by describing each phase with its own governing equation. The partial differential equations are discretized and solved via a vertex‐centered edge‐based dual‐mesh finite volume algorithm. A compact stencil is used for viscous terms, as this offers improved accuracy compared to the standard finite volume formulation. A locally preconditioned artificial compressibility solution strategy is employed to deal with pressure incompressibility, whilst stabilisation is achieved via a scalar‐valued artificial dissipation scheme.FindingsThe developed technology is demonstrated via the solution of natural convective flow inside a heated porous axisymmetric cavity. Predicted results were in general within 10 per cent of experimental measurements.Originality/valueThis is the first instance in which both artificial compressibility and artificial dissipation is employed to model flow through saturated porous materials.

Numerical study of cooling water via an evaporatively cooled immersed porous tube

SYNOPSIS We present a numerical study of the cooling of a volume of water through the wall of a cylindrical terracotta tube, the pores of which are saturated with water. The internal surface of the wall carries a water film which evaporates by forced convection of hot and dry air. Taking into account the physical properties of the material and that the aspect ratio of the tube used has a maximum value (D/L = 0.32), variations of the thermophysical properties of the air between the entry and the exit of the tube are ignored. The mathematical model describing the physical processes is based on the Darcy equation of fluid flow through a saturated porous material, the equation of Laplace related to pressure and the equation of heat in polar coordinates. One can then describe the space-time evolution of the field of temperature inside the wall as well as the temporal evolution of the average temperature of water.

Are Deicing Salts Necessary to Promote Scaling in Concrete?

The main purpose of the present study is to investigate the role of the material parameters such as permeability, thermal diffusivity, and pore size distribution on the mechanical behavior of cementitious structures submitted to frost action, such as surface scaling. An experimental device, in which a cement paste specimen is exposed to freezing-thawing cycles under a thermal gradient, has been developed. The experimental results show that under high thermal gradient (up to 1.5°C/mm), skin damage can occur without a saline layer in contact with the frozen surface. This can be explained and quantified in the framework of poromechanics. The model is based on the coupling between liquid-ice crystal thermodynamic equilibrium, liquid water transport, thermal conduction, and elastic properties of the different phases that form the saturated porous material. It eventually predicts that a less permeable sample is more susceptible to damage by surface defacement, which explains the observed experimental result.

Stress dependent thermal pressurization of a fluid-saturated rock

Temperature increase in saturated porous materials under undrained conditions leads to thermal pressurization of the pore fluid due to the discrepancy between the thermal expansion coefficients of the pore fluid and of the solid matrix. This increase in the pore fluid pressure induces a reduction of the effective mean stress and can lead to shear failure or hydraulic fracturing. The equations governing the phenomenon of thermal pressurization are presented and this phenomenon is studied experimentally for a saturated granular rock in an undrained heating test under constant isotropic stress. Careful analysis of the effect of mechanical and thermal deformation of the drainage and pressure measurement system is performed and a correction of the measured pore pressure is introduced. The test results are modelled using a non-linear thermo-poro-elastic constitutive model of the granular rock with emphasis on the stress-dependent character of the rock compressibility. The effects of stress and temperature on thermal pressurization observed in the tests are correctly reproduced by the model.

Multi-objective optimization in variably saturated fluid flow

A general purpose multi-objective inverse modeling strategy was developed and implemented to quantify fluid flow parameters in variably saturated porous materials. The strategy combines a robust and mass-conservative numerical simulator of the flow equation with an optimization algorithm and an experimental data set to estimate the parameters. The numerical simulator of the direct problem shows excellent agreement with a reference solution and conserves global mass with near perfection. An adaptive method was proposed in which the sensitivity matrix was calculated by one-sided finite difference approximation at the early stages of the optimization and the more accurate two-sided differentiation as the search approaches the minimum. A combined termination criterium was developed to stop the inverse code at the solution. The results of the multi-objective optimization were compared with those of single-objective minimization. While single-objective optimization generates reasonable results for either the fluid pressure head profile or the fluid content data, the proposed multi-objective optimization shows excellent agreement with both profiles simultaneously.

Derivation of a Darcy’s Law for a Porous Medium Composed of Two Solid Phases Saturated by a Single- Phase Fluid: A Homogenization Approach

The objective of this article is to derive a macroscopic Darcy’s law for a fluid-saturated moving porous medium whose matrix is composed of two solid phases which are not in direct contact with each other (weakly coupled solid phases). An example of this composite medium is the case of a solid matrix, unfrozen water, and an ice matrix within the pore space. The macroscopic equations for this type of saturated porous material are obtained using two-space homogenization techniques from microscopic periodic structures. The pore size is assumed to be small compared to the macroscopic scale under consideration. At the microscopic scale the two weakly coupled solids are described by the linear elastic equations, and the fluid by the linearized Navier–Stokes equations with appropriate boundary conditions at the solid–fluid interfaces. The derived Darcy’s law contains three permeability tensors whose properties are analyzed. Also, a formal relation with a previous macroscopic fluid flow equation obtained using a phenomenological approach is given. Moreover, a constructive proof of the existence of the three permeability tensors allows for their explicit computation employing finite elements or analogous numerical procedures.

Quasi-static deformation due to two-dimensional seismic sources embedded in an elastic half-space in welded contact with a poroelastic half-space

The Biot linearized theory of fluid saturated porous materials is used to study the plane strain deformation of a two-phase medium consisting of a homogeneous, isotropic, poroelastic half-space in welded contact with a homogeneous, isotropic, perfectly elastic half-space caused by a two-dimensional source in the elastic half-space. The integral expressions for the displacements and stresses in the two half-spaces in welded contact are obtained from the corresponding expressions for an unbounded elastic medium by applying suitable boundary conditions at the interface. The case of a long dip-slip fault is discussed in detail. The integrals for this source are solved analytically for two limiting cases: (i) undrained conditions in the high frequency limit, and (ii) steady state drained conditions as the frequency approaches zero. It has been verified that the solution for the drained case (ω → 0) coincides with the known elastic solution. The drained and undrained displacements and stresses are compared graphically. Diffusion of the pore pressure with time is also studied.