A Note on the Apparent Conductivity of Stratified Porous Media in Unsaturated Steady Flow Above a Water Table

We aim at deriving the apparent unsaturated conductivity (AUC) K ^(ap), defined as the ratio between the mean flux and the mean head gradient in a stratified vadose zone above the water table. This is achieved for steady flow generated by a constant infiltrating flux applied at the soil surface. By adopting the first-order approximation in the two parameters of the conductivity curve, and under a few additional simplifying assumptions, we were able to analytically compute K ^(ap). It is shown that this latter varies between K ^(ap) ≡ K _H (the harmonic mean) at the water table, and K ^(ap) ≡ K _ef (the effective conductivity in gravitational flow) far above the water table. Profiles of the AUC are illustrated, and the impact of parameters values is discussed.     

An overdeterministic approach for measuringK I using strain gages

An overdeterministic method for determining the opening-mode stress-intensity factor,K _I , from many measurements of the radial strain,∈ _rr , is described. The method was verified with an experimental study of a compact-tension specimen where strains along the 0, 45 and 90-degree lines were measured using strip gages with ten strain sensors per strip gage. The results indicated errors in the range of one to three percent with three or four parameter models of the strain field in the region near the crack tip. Paper was presented at the 1986 SEM Spring Conference on Experimental Mechanics which was held in New Orleans, LA on June 8–13.     

Steady-State Source Flow in Heterogeneous Porous Media

The flow of fluids in heterogeneous porous media is modelled by regarding the hydraulic conductivity as a stationary random space function. The flow variables, the pressure head and velocity field are random functions as well and we are interested primarily in calculating their mean values. The latter had been intensively studied in the past for flows uniform in the average. It has been shown that the average Darcy's law, which relates the mean pressure head gradient to the mean velocity, is given by a local linear relationship. As a result, the mean head and velocity satisfy the local flow equations in a fictitious homogeneous medium of effective conductivity. However, recent analysis has shown that for nonuniform flows the effective Darcy's law is determined by a nonlocal relationship of a convolution type. Hence, the average flow equations for the mean head are expressed as a linear integro-differential operator. Due to the linearity of the problem, it is useful to derive the mean head distribution for a flow by a source of unit discharge. This distribution represents a fundamental solution of the average flow equations and is called the mean Green function G _d (x). The mean head G _d(x) is derived here at first order in the logconductivity variance for an arbitrary correlation function (x) and for any dimensionality d of the flow. It is obtained as a product of the solution G _d ⁽⁰⁾(x) for source flow in unbounded domain of the mean conductivity K _A and the correction _d (x) which depends on the medium heterogeneous structure. The correction _d is evaluated for a few cases of interest.Simple one-quadrature expressions of _d are derived for isotropic two- and three-dimensional media. The quadratures can be calculated analytically after specifying (x) and closed form expressions are derived for exponential and Gaussian correlations. The flow toward a source in a three-dimensional heterogeneous medium of axisymmetric anisotropy is studied in detail by deriving ₃ as function of the distance from the source x and of the azimuthal angle . Its dependence on x, on the particular (x) and on the anisotropy ratio is illustrated in the plane of isotropy (=0) and along the anisotropy axis ( = /2).The head factor k ^* is defined as a ratio of the head in the homogeneous medium to the mean head, k ^*=G _d ⁽⁰⁾/G _d= _d ^–1. It is shown that for isotropic conductivity and for any dimensionality of the flow the medium behaves as a one-dimensional and as an effective one close and far from the source, respectively, that is, lim _x0 k ^*(x) = K _H/K _A and lim _x k ^*(x) = K ^efu/K _A, where K _A and K _H are the arithmetic and harmonic conductivity means and K ^efu is the effective conductivity for uniform flow. For axisymmetric heterogeneity the far-distance limit depends on the direction. Thus, in the coordinate system of (x) principal directions the limit values of k ^* are obtained as . These values differ from the corresponding components of the effective conductivities tensor for uniform flow for = 0 and /2, respectively. The results of the study are applied to solving the problem of the dipole well flow. The dependence of the mean head drop between the injection and production chambers on the anisotropy of the conductivity and the distance between the chambers is analyzed.     

Fractional Dispersion,Lévy Motion,and the MADE Tracer Tests

The macrodispersion experiments (MADE) at the Columbus Air Force Base in Mississippi were conducted in a highly heterogeneous aquifer that violates the basic assumptions of local second-order theories. A governing equation that describes particles that undergo Lévy motion, rather than Brownian motion, readily describes the highly skewed and heavy-tailed plume development at the MADE site. The new governing equation is based on a fractional, rather than integer, order of differentiation. This order (), based on MADE plume measurements, is approximately 1.1. The hydraulic conductivity (K) increments also follow a power law of order =1.1. We conjecture that the heavy-tailed K distribution gives rise to a heavy-tailed velocity field that directly implies the fractional-order governing equation derived herein. Simple arguments lead to accurate estimates of the velocity and dispersion constants based only on the aquifer hydraulic properties. This supports the idea that the correct governing equation can be accurately determined before, or after, a contamination event. While the traditional ADE fails to model a conservative tracer in the MADE aquifer, the fractional equation predicts tritium concentration profiles with remarkable accuracy over all spatial and temporal scales.     

Upscaling of conductivity of heterogeneous formations: General approach and application to isotropic media

The numerical simulation of flow through heterogeneous formations requires the assignment of the conductivity value to each numerical block. The conductivity is subjected to uncertainty and is modeled as a stationary random space function. In this study a methodology is proposed to relate the statistical moments of the block conductivity to the given moments of the continuously distributed conductivity and to the size of the numerical blocks. After formulating the necessary conditions to be satisfied by the flow in the upscaled medium, it is found that they are obeyed if the mean and the two-point covariance of the space averaged energy disspation function over numerical elements in the two media, of point value and of upscaled conductivity, are identical. This general approach leads to a systematic upscaling procedure for uniform average flow in an unbounded domain. It yields the statistical moments of upscaled logconductivity that depend only on those of the original one and on the size and shape of the numerical elements.The approach is applied to formations of isotropic heterogeneity and to isotropic partition elements. After a general discussion based on dimensional analysis, the procedure is illustrated by using a first-order approximation in the logconductivity variance. The upscaled logconductivity moments (mean, two-point covariance) are computed for two and three dimensional flows, isotropic heterogeneous media and elements of circular or spherical shape. The asymptotic cases of elements of small size, which preserve the point value conductivity structure on one hand, and of large blocks for which the medium can be replaced by one of deterministic effective properties, on the other hand, are analyzed in detail. The results can be used in order to generate the conductivity of numerical elements in Monte Carlo simulations.Nomenclature C covariance - e rate of dissipation of mechanical energy per unit weight of fluid - E total rate of energy dissipation in the flow domain - H overlap function - K hydraulic conductivity - K _G geometrical mean of conductivity - I integral scale - J=P mean head gradient - L characteristic size of - l characteristic size of also diameter of circle and sphere - n number of dimensions - P pressure head - Q total fluid discharge - S _A ,S _B inlet and outlet boundaries of flow domain - v velocity - Y logconductivity - characteristic scale of flow nonuniformity - autocorrelation function - ² variance - flow domain - partition element Overlining space averaged over - Ã upscaled quantity - â Fourier transform ofa     

Effects of magnetic field,porosity, and wall properties for anisotropically elastic multi-stenosis arteries on blood flow characteristics

A mathematical model for blood flow through an elastic artery with multistenosis under the effect of a magnetic field in a porous medium is presented. The considered arterial segment is simulated by an anisotropically elastic cylindrical tube filled with a viscous incompressible electrically conducting fluid representing blood. An artery with mild local narrowing in its lumen forming a stenosis is analyzed. The effects of arterial wall parameters represent viscoelastic stresses along the longitudinal and circumferential directions T _t and T _θ , respectively. The degree of anisotropy of the vessel wall γ, total mass of the vessel, and surrounding tissues M and contributions of the viscous and elastic constraints to the total tethering C and K respectively on resistance impedance, wall shear stress distribution, and radial and axial velocities are illustrated. Also, the effects of the stenosis shape m, the constant of permeability X, the Hartmann number H _α and the maximum height of the stenosis size δ on the fluid flow characteristics are investigated. The results show that the flow is appreciably influenced by surrounding connective tissues of the arterial wall motion, and the degree of anisotropy of the vessel wall plays an important role in determining the material of the artery. Further, the wall shear stress distribution increases with increasing T _t and γ while decreases with increasing T _θ , M, C, and K. Transmission of the wall shear stress distribution and resistance impedance at the wall surface through a tethered tube are substantially lower than those through a free tube, while the shearing stress distribution at the stenosis throat has inverse characteristic through totally tethered and free tubes. The trapping bolus increases in size toward the line center of the tube as the permeability constant X increases and decreases with the Hartmann number Ha increased. Finally, the trapping bolus appears, gradually in the case of non-symmetric stenosis, and disappears in the case of symmetric stenosis. The size of trapped bolus for the stream lines in a free isotropic tube (i.e., a tube initially unstressed) is smaller than those in a tethered tube.     

Dynamic fracture of a kane-mindlin plate

We study dynamic crack problems for an elastic plate by using Kane-Mindlin's kinematic assumptions. The general solutions of the Laplace transformed displacements and stresses are first derived. Path independent integrals for stationary cracks subjected to transient loads and steadily growing cracks are deduced. For a stationary crack in a very thin plate subjected to impact loads, the crack tip dynamic stress intensity factor (DSIF), K₁(t), is related to the far field plane stress one, K₁⁰(t), by where ν is Poisson's ratio. For a crack steadily growing with speed V, the crack tip DSIF, K₁(V), is given by where K₁⁰(V) is the plane stress DSIF and A(V) and B(V) are known functions of V. These results are applied to compute the DSIF for a semi-infinite stationary crack in an unbounded plate subjected to impact pressure on the crack faces. The results of DSIF for a finite crack in an infinite plate under uniform impact pressure on the crack surfaces show that for each plate thickness, the maximum DSIF is higher than that for the plane stress case.     

NUMERICAL STUDY ON STEADY STATE THREE-DIMENSIONAL ATMOSPHERIC DIFFUSION OF SULFUR DIOXIDE AND SULFATE DISPERSION WITH NON-LINEAR KINETICS

The aim of the present work is to study the three dimensional, steady state atmospheric diffusion equation for sulfate with its formation by conversion of sulfur dioxide (SO₂) and removal by wet and dry depositions under the general Michaelis—Mcnlen process. The equations governing the non-linear diffusion process are numerically solved using an alternating direction implicit finite difference scheme. It has been shown that in the case of wet deposition, the percentage decrease in sulfate level is rapid over the range 0 < X ≤ 0.67 (X = dimensionless longitudinal distance) as compared to that of the case of dry deposition. Further, sulfate levels remain higher at ground level for X = 1.0 but its levels remain lower at ground level for X = 0.3 and 0.1. Wet removal is found to be more dominant than dry removal in the far off regions of the point source. The fallout of deposits is significant near the source and at the far off lateral distance.     

Accuracy of the Renormalization Method for Computing Effective Conductivities of Heterogeneous Media

The accuracy of the renormalization method for upscaling two-dimensional hydraulic conductivity fields is investigated, using two canonical 2 × 2 blocks: a checkerboard geometry and a geometry in which three of the cells have conductivity K ₁ and the other has conductivity K ₂. The predictions of the renormalization algorithm are compared to the arithmetic, harmonic and geometric means, as well as to theoretical predictions and finite element calculations. For the latter geometry renormalization works well over the entire range of the conductivity ratio K ₂/K ₁, but for the checkerboard geometry the error becomes unbounded as the conductivity ratio grows.     

Transport of a Passive Scalar in a Stratified Porous Medium

A uniform and horizontal head gradient J is applied to a stratified formation whose given random conductivity K is function of the vertical coordinate x ₃ only. K is assumed to be stationary and of finite integral scale I _v. By Darcy's law, the velocity field V ₁(x ₃)=JK depicts a fluctuating shear flow. A solute body is injected instantaneously in the formation. In a Lagrangean framework, the second spatial moment of the mean concentration C(x,t) can be related to the one-particle trajectories variance X ₁₁(t,Pe) where Pe = V₁I_v/D _dT and _dT is the transverse pore-scale dispersion coefficient. X ₁₁ was determined in the past by Matheron and de Marsily (1980). The present study is concerned with determining the local concentration variance _C ² , that depends on the two-particles trajectories covariance Z ₁₁(t). The latter is derived exactly and langle Crangle and _C ² are determined by assuming normal or lognormal probability distribution of trajectories. The results are illustrated for small and very large (ergodic) solute plumes. For large travel time the concentration coefficient of variation at the center of the plume tends asymptotically to a constant value, unlike formations with finite horizontal correlation length of the hydraulic conductivity. The results may serve for benchmarking of numerical codes and in applications for short travel distances in highly anisotropic formations.     

A theory of homogeneous isotropic turbulence

Summary Homogeneous and isotropic turbulence has been discussed in the present paper. An attempt has been made to find the simplifying hypothesis for connecting the higher order correlation tensor with the lower ones. Starting from the Navier-Stokes equations of motion for an incompressible fluid and following the usual method of taking the averages, a differential equation in Q and X, the defining scalar of the second order correlation tensor Q _x and the defining scalar of a third order isotropic tensor X _ijk , has been derived. The tensor X _ijk stands for a tensorial expression containing the derivatives of the third and the fourth order tensors. Then the hypothesis is used that X=F(Q), where F is an unknown function. To find the forms of F, Kolmogoroff's similarity principles have been used, and thus two forms for F(Q) corresponding to two regions of the validity of these principles have been deduced.     

Optimal anisotropic three-phase conducting composites: Plane problem

The paper establishes tight lower bound for effective conductivity tensor K₁ of two-dimensional three-phase conducting anisotropic composites and defines optimal microstructures. It is assumed that three materials are mixed with fixed volume fractions and that the conductivity of one of the materials is infinite. The bound expands the Hashin–Shtrikman and translation bounds to multiphase structures, it is derived using a combination of translation method and additional inequalities on the fields in the materials; similar technique was used by Nesi, 1995, Cherkaev, 2009 for isotropic multiphase composites. This paper expands the bounds to the anisotropic composites with effective conductivity tensor K₁. The lower bound of conductivity (G-closure) is a piece-wise analytic function of eigenvalues of K₁, that depends only on conductivities of components and their volume fractions. Also, we find optimal microstructures that realize the bounds, developing the technique suggested earlier by Albin et al., 2007a, Cherkaev, 2009. The optimal microstructures are laminates of some rank for all regions. The found structures match the bounds in all but one region of parameters; we discuss the reason for the gap and numerically estimate it.     

The effect of magnetic field on two-phase liquid metal flow

In this paper, the basic equations of two-phase liquid metal flow in a magnetic field are derived, and specifically, two-phase liquid metal MHD flow in a rectangular channel is studied, and the expressions of velocity distribution of liquid and gas phases and the ratioK ₀ of the pressure drop in two-phase MHD flow to that in single-phase are derived. Results of calculation show that the ratioK ₀ is smaller than unity and decreases with increasing void fraction and Hartmann number because the effective electrical conductivity in the two-phase case decreases. The Project is supported by the National Natural Science Foundation of China.     

Mixed Convection Heat Transfer in the Annulus Between Two Concentric Vertical Cylinders Using Porous Layers

A numerical investigation of the steady-state, laminar, axi-symmetric, mixed convection heat transfer in the annulus between two concentric vertical cylinders using porous inserts is carried out. The inner cylinder is subjected to constant heat flux and the outer cylinder is insulated. A finite volume code is used to numerically solve the sets of governing equations. The Darcy–Brinkman–Forchheimer model along with Boussinesq approximation is used to solve the flow in the porous region. The Navier–Stokes equation is used to describe the flow in the clear flow region. The dependence of the average Nusselt number on several flow and geometric parameters is investigated. These include: convective parameter, λ, Darcy number, Da, thermal conductivity ratio, K _r, and porous-insert thickness to gap ratio (H/D). It is found that, in general, the heat transfer enhances by the presence of porous layers of high thermal conductivity ratios. It is also found that there is a critical thermal conductivity ratio on which if the values of Kr are higher than the critical value the average Nusselt number starts to decrease. Also, it found that at low thermal conductivity ratio (K _r ≈ 1) and for all values of λ the porous material acts as thermal insulation.     

Experimental investigation of dynamic punch tests on isotropic and composite materials

An experimental investigation is conducted on the two-dimensional punch problem for isotropic materials and unidirectional fiber-reinforced composite materials under quasi-static and impact loading. Singular stresses are generated in the specimen near the punch corners, and the stress intensity factorK _Iis introduced to describe the singular stress field. Laser interferometry is used to measure in-plane stresses (transmission mode) and out-of-plane displacements (reflection mode) and then estimate the stress intensity factor. In the dynamic case, a high-speed photography technique was employed to capture the transient response of the specimen and measureK(t) just after the impact. In all the cases, a good agreement between the measurements ofK and theoretical predictions was found.     

Pressure hole errors in the annular flow measurement of the second normal stress difference

Summary For viscometric, axial, annular flow, the second normal stress differenceN ₂ is related to the difference in normal thrust across the annular space,T _rr . Past attempts using this method have yielded values ofN ₂ for polymer solutions which are different in magnitudeand opposite in sign from those obtained in other experiments. This inconsistency is attributed to errors resulting from the use of pressure holes in the measurement ofT _rr , and is supported by a second-order fluid analysis.The present work focuses on the measurement of the effect of pressure hole errors on the determination ofN ₂ with aqueous polymer solutions. In the measurement ofT _rr , simultaneous use is made of both pressure holes and miniature pressure transducers to measure and account for pressure hole errors. Results indicate that hole errors are sufficiently large to reverse the sign of the computedN ₂. This technique is therefore suggested as a preferred method for determiningN ₂, especially at high shear rates.With 6 figures     

Numerical simulation of stress intensity factors via the thermoelastic technique

The purpose of this study is to investigate the accuracy of the least squares method for finding the in-plane stress intensity factorsK _I andK _II using thermoelastic data from isotropic materials. To fully understand the idealized condition ofK _I andK _II calculated from thermoelastic experiments, the total stress field calculated from finite element analysis is used to take the place of data obtained from real thermoelastic experiments. In the finite element analysis, theJ-integral is also calculated to compare with (K _I ² +K _II ² )/E evaluated by the least squares method. The stress fields near the crack tip are dominated by the two stress intensity factors; however, the edge effect will cause inaccuracy of the thermoelastic data near the crack tip. Furthermore, the scan area of thermoelastic experiments cannot be too small. Therefore, we suggest that three or four terms of stress function be included in the least squares method for evaluating stress intensity factors via the thermoelastic technique. In the idealized condition, the error can be smaller than 3 percent from our numerical simulations. If only ther ^–1/2 term (K _I andK _II ) is included in the least squares method, even in the idealized case the error can be up to 20 percent.     

Analytical Solutions for Macrodispersion in a 3D Heterogeneous Porous Medium with Random Hydraulic Conductivity and Dispersivity

A stochastic analysis of macrodispersion for conservative solute transport in three-dimension (3D) heterogeneous statistically isotropic and anisotropic porous media when both hydraulic conductivity and local dispersivity are random is presented. Analytical expressions of macrodispersivity are derived using Laplace and Fourier transforms. The effects of various parameters such as ratio of transverse to longitudinal local dispersivity, correlation length ratio, correlation coefficient and direction of flow on asymptotic macrodispersion are studied. The behaviour of growth of macrodispersivity in preasymptotic stage is also shown in this paper. The variation in local dispersion coefficient causes change in transverse macrodispersivity. The consideration of random dispersivity along with random hydraulic conductivity indicates that the total dispersion is affected and important in the case when the hydraulic conductivity and dispersivity are correlated. It is observed that the pre-asymptotic behavior of the macrodispersivity is not sensitive to the choice of spectral density functions.     

The Effect of varying correlation lengths on Anomalous Transport

Conventional concepts for transport in porous media assume that the heterogeneous distribution of hydraulic conductivities is the source for the contaminant temporal and spatial heavy tail. This tailing, known as anomalous or non-Fickian transport, can be captured by the β parameter in the continuous-time random walk framework. This study shows that with the increase in spatial correlation length between these heterogeneous distributions of hydraulic conductivities, the transport’s anomaly reduces; yet, the β value is unchanged, suggesting a topological component of the conductivity field, captured by the β. This finding is verified by an analysis of the solute transport, showing that the changing conductivity values have a moderate effect on the transport shape.

     

Experimental technique for measurement of stress-acoustic coefficients of Rayleigh waves

Two stress-acoustic coefficients,K ₁ andK ₂, are required to determine the state of biaxial surface stress from ultrasonic Rayleigh-wave velocity or time of flight measurements in elastic, initially isotropic solids. An experimental technique is described for the precise measurement of these two coefficients in uniaxially stressed test specimens. The technique is applied to aluminum 2024-T351 and 6061-T651 alloys. The influence on measurement results of various parameters such as material anisotropy and temperature is considered.