Local Transport Parameters Research Articles

Drying of capillary porous media simulated by coupling of continuum-scale and micro-scale models

In this work, we recall an isothermal version of the broadly accepted macroscopic continuum model of drying, and we impose flux boundary conditions at two interfaces, i.e., at the interface between the (gas-side) boundary layer and the (medium-side) dry region, as well as the interface between the dry and unsaturated regions which evolves freely during drying. Local relative humidity, local saturation as well as local transport parameters are computed from microscopic pore network simulations. This dataset is then employed to compute fluxes that couple the internal and external mass transfer in the continuum model. Decisive advantages of this approach over the classical method are that the continuity of the mass flux at the drying front and the porous medium surface is ensured and that the continuum model parameters are computed directly from pore network simulations – no need for any empirical correlation. Derivation of the macroscopic parameter functions from the pore network simulations for use in the continuum model raises issues of averaging intervals and treatment of the dataset. Operation of the continuum model is very sensitive upon the moisture transport coefficient in the totally or partially saturated zone of the porous medium. A hybrid method is introduced to control the effect of the sensitivity of the continuum model on the macroscopic parameters. By punctually adjusting the dataset in the high saturation period, the continuum model provides a stronger agreement with pore network simulations, which shows that the underlying transport phenomena are better preserved in the scattered dataset that the new hybrid method provides.

Gas Phase Measurements of Matrix Diffusion in Rock Samples from Olkiluoto Bedrock, Finland

The Finnish plan for the final deposition of nuclear waste is deposition deep in the crystalline bedrock. In the safety case of the final deposition, matrix diffusion, along with sorption, is considered the most important retarding factor for radionuclide transport in the geosphere. We set out to measure matrix diffusion in the gas phase for two 80 cm long drill-core samples from Olkiluoto, Finland. One sample consisted of veined gneiss and the other of pegmatitic granite. The measurements were taken in the laboratory with an advection–diffusion measurement setup that uses nitrogen as the carrier gas and helium as the tracer. The measured breakthrough curve for the veined gneiss sample could be interpreted with a homogeneous mathematical model, and the result for the effective diffusion coefficient (\(D_{\mathrm{e}}\)), \(4.2 \pm 0.9 \times 10^{-10}\,\frac{\mathrm {m}^2}{\mathrm {s}}\), agreed with previous results for veined gneiss from through-diffusion experiments in the gas phase. The breakthrough curve for the pegmatitic granite sample required a three-component model because of high variations in the local transport parameters (porosity, diffusion coefficient). The spatial distributions of porosities were characterized with the \(^{14}\)C-polymethylmethacrylate-autoradiography technique. The result, \(D_{\mathrm{e}} = \left( 32 \pm 7\right) \times 10^{-10}\,\frac{\mathrm {m}^2}{\mathrm {s}}\), for the most abundant component was also in accordance with the previous through-diffusion results. Comparison of the two measurements showed that diffusion phenomena can be very sensitive to the microstructure of rock even when macroscopic properties such as porosity remain the same. The retarding effect of matrix diffusion from flow was clearly observed for both samples, and knowledge of the sample structure, rock types, and mineralogy were concluded to be vital in analyzing and interpreting the results.

Using the nonstationary spectral method to analyze asymptotic macrodispersion in uniformly recharged heterogeneous aquifers

This paper describes an investigation of the influence of uniformly distributed groundwater recharge on asymptotic macrodispersion in two-dimensional heterogeneous media. This is performed using a nonstationary spectral approach [Li, S.-G., McLaughlin, D., 1991. A nonstationary spectral method for solving stochastic groundwater problems: unconditional analysis. Water Resour. Res. 27 (7), 1589–1605; Li, S.-G., McLaughlin, D., 1995. Using the nonstationary spectral method to analyze flow through heterogeneous trending media. Water Resour. Res. 31 (3), 541–551] based on Fourier–Stieltjes representations for the perturbed quantities. To solve the problem analytically, focus is placed on the case where the local longitudinal dispersivity α L is much smaller than the integral scale of log transmissivity λ (i.e., α L / λ ≪ 1). The closed-form expressions are obtained for describing the spectrum of flow velocity, the variability of flow velocity and asymptotic macrodispersion, in terms of the statistical properties and the integral scale of log transmissivity, local transport parameters and a parameter β [Rubin, Y., Bellin, A., 1994. The effects of recharge on flow nonuniformity and macrodispersion. Water Resour. Res. 30 (4), 939–948] used to characterize the degree of flow nonuniformity due to the groundwater recharge. The impact of β on these results is examined.

Surface effects on nanowire transport: a numerical investigation using the Boltzmannequation

A direct numerical solution of the steady-state Boltzmann equation in a cylindricalgeometry is reported. Finite-size effects are investigated in large semiconducting nanowiresusing the relaxation-time approximation. A nanowire is modelled as a combination of aninterior with local transport parameters identical to those in the bulk, and a finite surfaceregion across whose width the carrier density decays radially to zero. The roughness of thesurface is incorporated by using lower relaxation times there than in the interior.An argument supported by our numerical results challenges a commonly used zero-widthparametrization of the surface layer (Chambers 1950 Proc. R. Soc. A 202 378). In thenon-degenerate limit, appropriate for moderately doped semiconductors, a finite surfacewidth model does produce a positive longitudinal magneto-conductance, in agreementwith existing theory of Chambers. However, the effect is seen to be quite small(a few per cent) for realistic values of the wire parameters even at the highestpractical magnetic fields. Physical insights emerging from the results are discussed.

Distributed modeling of diffusive solute transport in peritoneal dialysis.

The diffusive transport between blood and an ex-tissue medium (dialysis fluid) is evaluated using a mathematical model that takes into account the (quasicontinuous) distribution of capillaries within the tissue at various distances from the tissue surface, and includes diffusive-convective transport through the capillary wall and lymphatic absorption from the tissue. General formulas for solute penetration depth, lambda, and for the diffusive mass transport coefficient for the transport between blood and dialysis fluid, K(BD), are provided in terms of local transport coefficients for capillary wall, tissue, and lymphatic absorption. For pure diffusive transport between blood and dialysis fluid and thick tissue layers (i.e., if the solute penetration depth is much lower than the tissue thickness) these formulas yield previously known expressions. It is shown that apparent tissue layers, with widths lambdaTBL and lambdaT, respectively, may be defined according to the values of local transport parameters in such a way that K(BD) is equal to the solute clearance K(TBL) from the tissue by blood and lymph for a layer with width lambdaTBL or to the solute clearance K(T) from blood to dialysate by diffusion through the tissue layer with width lambdaT. For tissue layers with width much higher than the penetration depth: lambdaT approximately = lambdaTBL approximately = lambda. These characteristic width lengths depend on the transport parameters (and thus on the size) of solutes. Effective blood flow, which may be related to the exchange of the solute between blood and dialysate, is defined using an analogy to the extraction/absorption coefficients for blood-tissue exchange. Various approximations for the distributed model formula for diffusive mass transport coefficient (K(BD)) are possible. The appropriate range for their application is obtained from the general formula.

Ion transport through unsaturated soils: field experiments and regional simulations

This paper describes the movement of anions and cations through soils at the regional scale using block-scale and regional simulations of one-dimensional ion transport through cultivated soils. The simulations were based on field experiments in a region of about 10 km2 in Lower Saxony, Germany. Transport was modelled with the convection–dispersion equation, and the cation exchange was described using the Gapon equation. We evaluated the spatial variation of cation exchange parameters, obtained estimates valid at the block scale, and simulated the one-dimensional transport of anions and cations. The movement of anions and cations was simulated over blocks using effective transport parameters calculated from local transport parameters. The approach led to a good agreement between measured and predicted concentrations of Br–, Na+, K+, Ca2+ and Mg2+ on four different 1 ha blocks. However, the mean concentrations of K+ in the soil solution in the uppermost horizons could not be described satisfactorily by the model. For the regional simulations, transport and exchange parameters were estimated by block kriging. All variograms of the exchange parameters were spatially structured with correlation lengths varying from 100 m to 300 m. Results of the regional simulations imply that Cl– and K+ were transported substantially deeper in the southern part than in the northern part of the area. The transport depth of the ions strongly depended on the pore water velocities. The simulation of solute transport to the water table showed the influence of the depth of water table on the estimated travel times, superimposing the influence of the transport parameters in the region. The results of the regional simulations also emphasize the importance of careful fertilization, especially in regions with shallow water tables such as in the north of the area.

Macrodispersion in a radially diverging flow field with finite Peclet Numbers: 2. Homogenization theory approach

We study the transport behavior of a tracer in a radially diverging heterogeneous velocity field. Making use of homogenization theory, we derive effective transport equations. These effective transport equations are very similar to those defined on the local scale. However, the local transport parameters such as local dispersion coefficients are replaced by effective dispersion coefficients. For smoothly varying heterogeneous media, explicit results for effective radial dispersion coefficients are derived. Starting with the purely advective transport behavior (infinite Peclet numbers), we extend our calculations to transport with finite Peclet numbers. We find that the impact of molecular diffusion on the effective dispersivity differs from the impact of local dispersion: Including local dispersion leads to effective dispersivities which are constant and equivalent to the effective dispersivities found in uniform flow configurations. In contrast, effective dispersivities including diffusion are not constant but depend on the radial distance. We compare the results found by homogenization theory with those derived by Neuweiler et al. (this issue) by standard method of moments.

Comparison of three stream tube models predicting field-scale solute transport

Abstract. In this paper the relation between local- and field-scale solute transport parameters in an unsaturated soil profile is investigated. At two experimental sites, local-scale steady-state solute transport was measured in-situ using 120 horizontally installed TDR probes at 5 depths. Local-scale solute transport parameters determined from BTCs were used to predict field-scale solute transport using stochastic stream tube models (STM). Local-scale solute transport was described by two transport models: (1) the convection-dispersion transport model (CDE), and (2) the stochastic convective lognormat transfer model (CLT). The parameters of the CDE-model were found to be lognormally distributed, whereas the parameters of the CLT model were normally distributed. Local-scale solute transport heterogeneity within the measurement volume of a TDR-probe was an important factor causing field-scale solute dispersion. The study of the horizontal scale-dependency revealed that the variability in the solute transport parameters contributes more to the field-scale dispersion at deeper depths than at depths near the surface. Three STMs were used to upscale the local transport parameters: (i) the stochastic piston flow STM-I assuming local piston flow transport, (ii) the convective-dispersive STM-II assuming local CDE transport, and (iii) the stochastic convective lognormal STM-III assuming local CLT. The STM-I considerably underpredicted the field-scale solute dispersion indicating that local-scale dispersion processes, which are captured within the measurement volume of the TDR-probe, are important to predict field-scale solute transport. STM-II and STM-III both described the field-scale breakthrough curves (BTC) accurately if depth dependent parameters were used. In addition, a reasonable description of the horizontal variance of the local BTCs was found. STM-III was (more) superior to STM-II if only one set of parameters from one depth is used to predict the field-scale solute BTCs at several depths. This indicates that the local-scale solute transport process, as measured with TDR in this study, is in agreement with the CLT-hypothesis.