One-dimensional Analytical Solution Research Articles

Large deformation elastic electro-osmosis consolidation of clays

This paper presents the theoretical background of an elastic electro-osmosis consolidation model for saturated soils experiencing large strains, which considers volumetric strains induced by changes in both the hydraulic and electric driven pore water flows. Three fully coupled governing equations, considering the soil mechanical behaviour, pore water transport and electrical field, and their numerical implementation within an updated Lagrangian finite element formulation, are presented. The proposed model is first verified against a classical one-dimensional analytical solution for electro-osmosis consolidation to demonstrate its accuracy and efficiency. Then, various numerical examples are investigated to study the deformation characteristics and time dependent evolution of excess pore pressure. Finally, the importance of considering large strains in a consistent and proper way is demonstrated, and differences compared to models based on small strain theory are highlighted.

Investigation of the Dynamic Response of Functionally Graded Materials Using Smoothed Particle Hydrodynamics

In the present study, the problem of functionally graded materials (FGMs) under a stress pulse is analyzed based on smoothed particle hydrodynamics (SPH) using the formulation for large deformation. First, the formulation of SPH for this problem is described, and a benchmark calculation is performed and compared to one-dimensional analytical solutions. The behavior of FGMs subjected to a stress pulse is then investigated for several cases, including various distributions of inhomogeneous materials and two-dimensional problems with different boundary conditions. It is found that in the two-dimensional case, if there is a free boundary not parallel to the direction of the external force, the influence from this boundary cannot be ignored.

Seepage velocities derived from thermal records using wavelet analysis

Summary The main objective of this study was to determine if wavelet analysis can be used as a band-pass filter for non-stationary time series of temperature records. It was demonstrated that this new processing technique is efficient in calculating seepage velocities by using a virtual experiment and field data measured in the Mess Stream, Grand Duchy of Luxembourg, Europe. Temperature time series were measured continuously over a 2-year period in the streambed sediments. The seepage velocity was derived from the one-dimensional analytical solution of Stallman. However, the ubiquitous property of the temperature signal is that it is non-stationary, i.e. the frequency component of the signal becomes a function of time. Instead of applying traditional filtering techniques based on the Fourier transform, we introduce the continuous wavelet transform (CWT) as a band-pass filter to extract the daily component from raw temperature data. For the analysed time series, the estimated seepage velocities based on the amplitude ratio were always negative and within the range of −0.7 to −2.5 m d −1 , indicating gaining conditions prevailing in the Mess Stream. The uncertainty associated with the seepage velocities was calculated by the Monte Carlo analysis allowing several physical parameters of the model to vary over pre-defined intervals with normal distribution. Auxiliary hydraulic measurements in the stream and in the riverbank confirm the dominance of gaining stream conditions. However, the seepage rates based on the thermal records were much higher than the calculated Darcy velocities. This study demonstrates the applicability of continuous wavelet transform as an alternative method to Fourier transform for the analysis of non-stationary temperature time series.

Thermal wave model of bioheat transfer with modified Riemann–Liouville fractional derivative

In this paper a new fractional thermal wave model of the bioheat transfer (FTWMBT) caused by spatial heating is built using Taylor’s series expansion of modified Riemann–Liouville fractional derivatives. A one-dimensional analytical solution of the FTWMBT in a finite medium is obtained. The FTWMBT in the case (α = 1) interpolates the standard thermal wave model of bioheat transfer and the well-known Pennes’ bioheat equation (τ = 0). Finally, numerical results are presented graphically for various values of different parameters. This study demonstrates that fractional models can provide a unified approach to examine the heat transfer in biological tissue.

Investigation of the Dynamic Response of Functionally Graded Materials Using Smoothed Particle Hydrodynamics

In the present study, the problem of functionally graded materials (FGMs) under a stress pulse is analyzed based on smoothed particle hydrodynamics (SPH) using the formulation for large deformation. First, the formulation of SPH for this problem is described, and a benchmark calculation is performed and compared to one-dimensional analytical solutions. The behavior of FGMs subjected to a stress pulse is then investigated for several cases, including various distributions of inhomogeneous materials and two-dimensional problems with different boundary conditions. It is found that in the two-dimensional case, if there is a free boundary not parallel to the direction of the external force, the influence from this boundary cannot be ignored.

Implications of Hyporheic Flow on Temperature-Based Estimates of Groundwater/Surface Water Interactions

AbstractThe hyporheic zone has received significant attention in recent years due to its role in regulating the physical, chemical, and biological processes that buffer fluvial systems at a variety of scales. The exchange of water through the hyporheic zone is also important for the regulation of stream and streambed temperatures. Recent research has utilized stream and streambed temperatures to quantify groundwater discharge to streams using a variety of methods, including one-dimensional analytical solutions for vertical flux across the streambed interface. The presence of lateral flows, including hyporheic flow, will cause uncertainty in these analytical solution results. In this study, HydroGeoSphere, a three-dimensional, fully integrated surface/subsurface hydrologic model, is used to simulate the manner in which streambed heterogeneity influences groundwater/surface water interactions along a stream section, and the uncertainty of groundwater/surface water flux estimates based upon streambed tempera...

Influence of non-conducting pore inclusions on phase change behavior of porous media with constant heat flux boundary

A one-dimensional analytical solution was presented for the temporal variation of phase interface in two-phase materials having randomly distributed stagnant low conducting pores with constant heat flux boundary. For validation, numerical simulations based on the finite difference method were conducted. It was demonstrated that the phase interface propagates faster with pore inclusion than that associated with constant wall temperature boundary. Such faster movement of the phase interface is attributed to the decreased effective density of the medium as a result of the included low conducting pores. The decreased effective density reduces the amount of latent heat of the medium whilst the amount of heat removed at the boundary remains constant. Therefore, at a given period of time, more latent heat is liberated. In addition, for a given porosity, the influence of pore shape upon phase change interface location is negligible since the effective conductivity, although varying considerably with pore shape, has a marginal effect on the behavior of phase change interface.

Closed‐Form Approximations for Two‐Dimensional Groundwater Age Patterns in a Fresh Water Lens

Simple closed-form approximations are presented for calculating the steady-state groundwater age distribution in two-dimensional vertical cross sections of idealized fresh water lenses overlying salt water, for aquifers that are vertically semi-infinite and of finite thickness. The approximations are developed on the basis of existing one-dimensional analytical solutions for travel-time calculation in fresh water lenses and approximate streamline formulations. The two-dimensional age distributions based on the closed-form solutions match convincingly with numerical simulations. As expected, notable deviations from the numerical solution are encountered at the groundwater flow divide and when submarine groundwater discharge occurs. Ratios of recharge over hydraulic conductivities are varied to explore how the magnitude of the deviations changes, and it is found that the approximate closed-form solutions perform well over a range of conditions found in natural systems.

Defect detection and characterization in composite materials using square pulse thermography coupled with singular value decomposition analysis and thermal quadrupole modeling

In this study, square pulse thermography is used to detect and characterize defects in carbon/epoxy composite plates. Defects are polymeric disks inserted between plies at different depths in the samples. A practical experimental set-up is developed: temperature measurements are only required on the front face of the samples and the heating source consists in using standard halogen lamps. A transient thermal modeling based on thermal quadrupoles and several adimensional parameters provides a one-dimensional analytical solution. Singular value decomposition analysis is used to differentiate sound and defect areas. An inversion procedure is carried out to estimate unknown parameters without any prior knowledge on sample properties or defect locations. Especially it is possible to estimate both depth and thermal resistance of the defect with reduced uncertainties. Results on the influence of the analysis and heating durations are presented. A comparison of the thermal contact quality relative to the different defects is finally performed.

Numerical simulation of tracer transport in the Altmark gas field

The paper presents a modeling strategy as well as simulation results of a designed conservative tracer test in the depleted Altensalzwedel natural gas reservoir in order to know the tracer concentration and breakthrough time at the production wells. Krypton is considered as a suitable tracer. The production wells are located in several hundred meters to a few kilometers away from the injection well. The numerical simulation has been performed by using the newly implemented compositional gas flow module of OpenGeoSys (OGS), a scientific open-source simulator for calculation of coupled geohydraulic (single and multiphase flow), transport (heat and mass transfer), geomechanical and geochemical processes. The tracer breakthrough curves at the production wells show that the breakthrough times vary between 2 and 3 years. The model verification is presented by comparing the OGS results with a one-dimensional analytical solution. The numerical results have been verified by code comparison, additionally.

Equilibrium form of horizontally retreating, soil-mantled hillslopes: Model development and application to a groundwater sapping landscape

[1] We present analytical solutions for the steady state topographic profile of a soil-mantled hillslope retreating into a level plain in response to a horizontally migrating base level. This model applies to several scenarios that commonly arise in landscapes, including widening valleys, eroding channel banks, and retreating scarps. For a sediment transport law in which sediment flux is linearly proportional to the topographic slope, the steady state profile is exponential, with an e-folding length, L, proportional to the ratio of the sediment transport coefficient to the base level migration speed. For the case in which sediment flux increases nonlinearly with slope, the solution has a similar form that converges to the linear case as L increases. We use a numerical model to explore the effects of different base level geometries and find that the one-dimensional analytical solution is a close approximation for the hillslope profile above an advancing channel tip. We then compare the analytical model with hillslope profiles above the tips of a groundwater sapping channel network in the Florida Panhandle. The model agrees closely with hillslope profiles measured from airborne laser altimetry, and we use a predicted log linear relationship between topographic slope and horizontal distance to estimate L for the measured profiles. Mapping 1/L over channel tips throughout the landscape reveals that adjacent channel networks may be growing at different rates and that south facing slopes experience more efficient hillslope transport.

COMBINED TWO-FLUX APPROXIMATION AND MONTE CARLO MODEL FOR IDENTIFICATION OF RADIATIVE PROPERTIES OF HIGHLY SCATTERING DISPERSED MATERIALS

An identification procedure is developed for obtaining spectral radiative properties of highly scattering dispersed materials such as porous ceramics. Traditional techniques based on measurements of the directional-hemispherical reflectance and transmittance are of limited use because of difficulties in fabricating sufficiently thin and mechanically stable samples to obtain reliable values of directional-hemispherical transmittance. However, one can use the directional-hemispherical reflectance measurements for optically thick samples to obtain the transport scattering albedo. A one-dimensional analytical solution employs the modified two-flux approximation for the identification of transport scattering albedo. An additional transmittance measurement is required to identify the transport extinction coefficient. Binormal narrow cone transmittance is measured for this purpose. Because the one-dimensional analytical solution is not applicable to model the binormal narrow cone transmittance, the Monte Carlo ray-tracing technique is used to identify the transport extinction coefficient. The identification procedure is applied to obtain near-infrared radiative properties of porous ceria ceramics used in solar thermochemical reactors. The identified transport scattering coefficient is shown to be in good agreement with theoretical estimates based on the Mie theory for polydisperse pores and grains. This verifies the applicability of a model based on independent scattering and Mie theory for theoretical predictions of radiative properties of two types of ceria ceramics with porosity of 0.08 and 0.72, and for extrapolating the properties of both ceramics in a limited near-infrared range to the range of significant absorption.

UNDERSTANDING THE IMPACT OF RETURN-CURRENT LOSSES ON THE X-RAY EMISSION FROM SOLAR FLARES

I obtain and examine the implications of one-dimensional analytic solutions for return-current losses on an initially power-law distribution of energetic electrons with a sharp low-energy cutoff in flare plasma with classical (collisional) resistivity. These solutions show, for example, that return-current losses are not sensitive to plasma density, but are sensitive to plasma temperature and the low energy cutoff of the injected nonthermal electron distribution. A characteristic distance from the electron injection site, x(sub rc), is derived. At distances less than x(sub rc) the electron flux density is not reduced by return-current losses, but plasma heating can be substantial in this region, in the upper, coronal part of the flare loop. Before the electrons reach the collisional thick-target region of the flare loop, an injected power-law electron distribution with a low-energy cutoff maintains that structure, but with a flat energy distribution below the cutoff energy, which is now determined by the total potential drop experienced by the electrons. Modifications due to the presence of collisional losses are discussed. I compare these results with earlier analytical results and with more recent numerical simulations. Emslie's 1980 conjecture that there is a maximum integrated X-ray source brightness on the order of 10(exp -15) photons per square centimeter per second per square centimeter is examined. I find that this is not actually a maximum brightness and its value is parameter dependent, but it is nevertheless a valuable benchmark for identifying return-current losses in hard X-ray spectra. I discuss an observational approach to identifying return-current losses in flare data, including identification of a return-current bump in X-ray light curves at low photon energies.

Heat transfer in a multi-layered thermal protection system under aerodynamic heating

Future generation reusable re-entry vehicles must be capable of sustaining consistent repeated aero-thermal loads without damage or deterioration. This means that such structures should be able to withstand high temperatures engendered by aero-thermal re-entry fluxes due to the establishment of a hypersonic regime over the body. Thermal Protection Systems (TPS) are used to maintain a reusable launch vehicle structural temperature within acceptable limits during re-entry flights; that is internal temperature should not be higher than the limit required by the internal structure. TPS are usually composed of several layers of different materials. Heat transfer through a multilayer insulation during atmospheric re-entry involves combined modes of heat transfer: heat conduction and radiation through the solid, heat radiation to the outer space etc. In the framework of the TPS design activities, a procedure based on one-dimensional analytical solutions of transient non-linear analysis has been developed in order to estimate the temperature variation with time and space of a multi-layered body subjected to aerodynamic heating inside a radiating space. Furthermore a semi-analytical procedure, able to take also into account non-linear thermal properties, has been conceived. Since internal temperature values of TPS of re-entry vehicles cannot exceed certain values, these procedures allow to quickly evaluate these temperature values and the preliminary size layer thicknesses without resorting to numerical solutions.

On the biases affecting water ages inferred from isotopic data

Groundwater age has become a fundamental concept in groundwater hydrology, but ages originating from isotopic analyses are still identified with a lack of clarity and using models that occasionally are unrealistic. If the effect of advection and dispersion on water ages has already been extensively identified, very few studies address the reliability of using radiometric ages as derived from isotopic data to estimate aquifer properties such as average velocities. Using simple one-dimensional and two-dimensional analytical solutions for single-site and two-sites mobile–immobile systems, we compare the radiometric ages to the mean ages (or residence times) as deduced from a direct, physically-based simulation approach (using the mean age equation), and show that the competition between isotope decay rate and dispersion coefficient can generate important discrepancies between the two types of ages. A correction for the average apparent velocity originating from apparent isotopic ages is additionally provided. The particular case of the tritium age dating method is also addressed, and a numerical example is finally given for illustrating the analysis considering a more complex and heterogeneous aquifer system. Our results suggest that age definitions based on the radioactivity of isotopes may not be representative for the mean age of the sample or for the groundwater velocity at given locations, and may not always be suitable for constraining the calibration of hydrogeological models

Determination of spatially-resolved porosity, tracer distributions and diffusion coefficients in porous media using MRI measurements and numerical simulations

This work is focused on measuring the concentration distribution of a conservative tracer in a homogeneous synthetic porous material and in heterogeneous natural sandstone using MRI techniques, and on the use of spatially resolved porosity data to define spatially variable diffusion coefficients in heterogeneous media. The measurements are made by employing SPRITE, a fast MRI method that yields quantitative, spatially-resolved tracer concentrations in porous media. Diffusion experiments involving the migration of H(2)O into D(2)O-saturated porous media are conducted. One-dimensional spatial distributions of H(2)O-tracer concentrations acquired from experiments with the homogeneous synthetic calcium silicate are fitted with the one-dimensional analytical solution of Fick's second law to confirm that the experimental method provides results that are consistent with expectations for Fickian diffusion in porous media. The MRI-measured concentration profiles match well with the solution for Fick's second law and provide a pore-water diffusion coefficient of 1.75×10(-9)m(2)s(-1). The experimental approach was then extended to evaluate diffusion in a heterogeneous natural sandstone in three dimensions. The relatively high hydraulic conductivity of the sandstone, and the contrast in fluid density between the H(2)O tracer and the D(2)O pore fluid, lead to solute transport by a combination of diffusion and density-driven advection. The MRI measurements of spatially distributed tracer concentration, combined with numerical simulations allow for the identification of the respective influences of advection and diffusion. The experimental data are interpreted with the aid of MIN3P-D - a multicomponent reactive transport code that includes the coupled processes of diffusion and density-driven advection. The model defines local diffusion coefficients as a function of spatially resolved porosity measurements. The D(e) values calculated for the heterogeneous sandstone and used to simulate diffusive and advective transport range from 5.4×10(-12) to 1.0×10(-10)m(2)s(-1). These methods have broad applicability to studies of contaminant migration in geological materials.

Uncertainty in 1D Heat-Flow Analysis to Estimate Groundwater Discharge to a Stream

Temperature measurements have been used by a variety of researchers to gain insight into groundwater discharge patterns. However, much of this research has reduced the problem to heat and fluid flow in one dimension for ease of analysis. This approach is seemingly at odds with the goal of determining spatial variability in specific discharge, which implies that the temperature field will vary in more than one dimension. However, it is unclear how important the resulting discrepancies are in the context of determining groundwater discharge to surface water bodies. In this study, the importance of these variations is examined by testing two popular one-dimensional analytical solutions with stochastic models of heat and fluid flow in a two-dimensional porous medium. For cases with low degrees of heterogeneity in hydraulic conductivity, acceptable results are possible for specific discharges between 10(-7) and 10(-5) m/s. However, conduction into areas with specific discharges less than 10(-7) m/s from adjacent areas can lead to significant errors. In some of these cases, the one-dimensional solutions produced estimates of specific discharge of nearly 10(-6) m/s. This phenomenon is more likely in situations with greater degrees of heterogeneity.

Numerical simulation of non-isothermal compositional gas flow: Application to carbon dioxide injection into gas reservoirs

This paper examines non-isothermal effects during carbon dioxide injection into a depleted gas reservoir for storage and EGR (enhanced gas recovery). We illustrate Joule–Thomson behavior with VHD (viscous heat dissipation) effects during reservoir repressurization. The equations of state for density (empirically extended ideal gas equation) and dynamic viscosity of the real gaseous mixture are tested with numerical and experimental data. We consider the real behavior of the mixture by using energy and distance parameters. The energy conservation equation is solved to account for gas expansion with convection and conduction processes. Additionally, we present temperature profiles of the reservoir column during gas expansion and discover that the Joule–Thomson process and VHD decrease reservoir temperature, with a decrement in temperature of the reservoir depending strongly on medium porosity. Isothermal pressure and temperature calculations are compared with existing one-dimensional analytical solutions for model verification.

DYNAMICAL FINITE ELEMENT MODELING OF SOFT TISSUES AS CHEMOELECTRIC POROUS MEDIA

An implicit mixed finite element formulation of hydrated soft biological tissues, based on the Simon model, is presented that incorporates the coupling of solid, fluid, and ion phases as well as the viscoelasticity of soft tissue in the dynamical process. The tissues are modeled as a multi-field viscoelastic body subject to finite deformation. In addition to a three-field (u-w-p) modeling of the porous matrix, the study also includes an ion phase for the ionic solution. After presenting the formulation, an efficient staggered solution scheme is presented: within each time step, the ion charge equation is solved first to give the distribution of the charge concentration, the charge induced osmotic water pressure is then employed in solving the u-w-p equations. The resulting u field becomes a forcing term to the solution of the ion charge concentration equations for iteration. This methodology and codes developed for the study have been verified with one-dimensional (1D) analytical solutions. A 2D chemical electric swelling model illustrates the important role of viscoelasticity. A brain tissue impact example demonstrates the potential application of the model.

Comparative Study of Analytical Solutions for Time-Dependent Solute Transport Along Unsteady Groundwater Flow in Semi-infinite Aquifer

A comparative study is made among Laplace Transform Technique (LTT) and Fourier Transform Technique (FTT) to obtain one-dimensional analytical solution for conservative solute transport along unsteady groundwater flow in semi-infinite aquifer. The time-dependent source of contaminant concentration is considered at the origin and at the other end of the aquifer is supposed to be zero. Initially, aquifer is not solute free which means that the solute concentration exits in groundwater system and it is assumed as a uniform concentration. The aquifer is considered homogeneous and semi-infinite. The time-dependent velocity expressions are considered. The result may be used as preliminary predictive tools in groundwater management and benchmark the numerical code and solutions.