A one-dimensional dual-porosity model has been developed for the purpose of studying variably saturated water flow and solute transport in structured soils or fractured rocks. The model involves two overlaying continua at the macroscopic level: a macropore or fracture pore system and a less permeable matrix pore system. Water in both pore systems is assumed to be mobile. Variably saturated water flow in the matrix as well as in the fracture pore system is described with the Richards' equation, and solute transport is described with the convection-dispersion equation. Transfer of water and solutes between the two pore regions is simulated by means of first-order rate equations. The mass transfer term for solute transport includes both convective and diffusive components. The formulation leads to two coupled systems of nonlinear partial differential equations which were solved numerically using the Galerkin finite element method. Simulation results demonstrate the complicated nature of solute leaching in structured, unsaturated porous media during transient water flow. Sensitivity studies show the importance of having accurate estimates of the hydraulic conductivity near the surface of soil aggregates or rock matrix blocks. The proposed model is capable of simulating preferential flow situations using parameters which can be related to physical and chemical properties of the medium.

A dual-porosity model for simulating the preferential movement of water and solutes in structured porous media

http://www.pc-progress.com/Documents/Jirka/Simunek_et_al_MacroFl_2002.pdf

Review and comparison of models for describing non-equilibrium and preferential flow and transport in the vadose zone

Progress in the synthesis and engineering of advanced porous materials demands better pore structure characterization. The analysis of pore structure is complicated by (1) the wide range in pore sizes observed, from molecular ( 1 mm) dimensions, (2) complex pore shapes and connectivities, (3) chemical and physical heterogeneities, and (4) pore structure changes that can occur during characterization.The required pore structure information varies with application. Bulk density and the pore-size distribution are needed for thermal insulation. In this case, the dimension of interest is the so-called hydraulic radius since, for small pores, the gas-phase conductivity is proportional to the mean hydraulic radius to the mean free path. A few large but isolated pores will significantly affect conductivity but will go undetected in typical gas-absorption methods. In contrast, for separations, bottlenecks control performance. For transport, such as migration through geologic formations, both the pore-size distribution and pore connectivity are important. For adsorption, surface area and pore size are the relevant factors. Finally, the conventional concepts of pore structure lose meaning as the pore size approaches molecular dimensions, typical of adsorbents and gas-separation membranes.

Characterization of Porous Solids

Among the current problems that hydrogeologists face, perhaps there is none as challenging as the characterization of fractured rock (Faybishenko and Benson 2000). This paper discusses issues associated with the quantification of flow and transport through fractured rocks on scales not exceeding those typically associated with single- and multi-well pressure (or flow) and tracer tests. As much of the corresponding literature has focused on fractured crystalline rocks and hard sedimentary rocks such as sandstones, limestones (karst is excluded) and chalk, so by default does this paper. Direct quantification of flow and transport in such rocks is commonly done on the basis of fracture geometric data coupled with pressure (or flow) and tracer tests, which therefore form the main focus. Geological, geophysical and geochemical (including isotope) data are critical for the qualitative conceptualization of flow and transport in fractured rocks, and are being gradually incorporated in quantitative flow and transport models, in ways that this paper unfortunately cannot describe but in passing. The hydrogeology of fractured aquifers and other earth science aspects of fractured rock hydrology merit separate treatments. All evidence suggests that rarely can one model flow and transport in a fractured rock consistently by treating it as a uniform or mildly nonuniform isotropic continuum. Instead, one must generally account for the highly erratic heterogeneity, directional dependence, dual or multicomponent nature and multiscale behavior of fractured rocks. One way is to depict the rock as a network of discrete fractures (with permeable or impermeable matrix blocks) and another as a nonuniform (single, dual or multiple) continuum. A third way is to combine these into a hybrid model of a nonuniform continuum containing a relatively small number of discrete dominant features. In either case the description can be deterministic or stochastic. The paper contains a brief assessment of these trends in light of recent experimental and theoretical findings, ending with a short list of prospects and challenges for the future.

Trends, prospects and challenges in quantifying flow and transport through fractured rocks

The effect of pore structure and gas pressure upon the transport properties of coal: a laboratory and modeling study. 1. Isotherms and pore volume distributions

NMR measurements of the spin—lattice relaxation decay time for a fluid in a porous solid have been previously demonstrated as a significant tool for pore size/structure analysis. The analysis requires the solution of a Fredholm integral equation of the first kind to extract the desired T1, and therefore, pore size distribution from the relaxation measurements. Earlier work successfully used a nonnegative least-squares (NNLS) technique to solve this problem. However, for stability to be achieved with that approach, the problem must be significantly overconstrained (i.e., more relaxation measurements than pore size distribution points) which limits resolution. In addition, broad distributions are represented as several narrow peaks. In this work, the method of regularization is applied to extracting continuous pore volume distributions from relaxation measurements. The validity of the approach used to calculate the optimum value of the regularization smoothing parameter given an expected noise level and the data is demonstrated using “data sets” generated from known T1 distributions. Varying levels of random noise are added to these data sets and the T1 distribution is calculated for comparison to the starting distribution. For error levels in the range of 0.001 to 0.01, agreement between the starting and the calculated distributions is good. The method has also been applied to relaxation data obtained for water in several controlled pore glasses and “random packing of sphere” solids. Agreement to NMR/regularization and mercury porosimetry-derived pore size distributions was excellent with the only deviations being the result of porosimetry measuring pore neck size and NMR measuring the true pore volume to surface area ratio.

A NMR technique for the analysis of pore structure: Determination of continuous pore size distributions

The application of NMR spin—lattice relaxation measurements of water contained in porous solids is investigated as a possible tool for the determination of pore size distributions. NMR methods have several potential advantages over conventional porosimetry/adsorption techniques including the study of wet porous solids, a wide range of sample sizes that may be studied, and the fact that no pore shape assumption is required. In principle, water contained in a pore will relax faster than bulk water. This decrease in the observed relaxation rate decay constant, T1, is directly related to the pore volume to surface area ratio (i.e., hydraulic radius) according to the “two-fraction, fast-exchange model.” However, the suitability of this model and NMR spin—lattice relaxation measurements for the determination of pore size distributions has not been previously demonstrated. Pore size distributions have been determined for two series of porous solids by mercury porosimetry and a NMR method. A series of four controlled-pore glasses with very narrow pore size distributions and a mean radius in the range 3.4 to 17.6 nm were studied at a frequency of 20 MHz and at four temperatures. By using four samples with similar surface chemistry/ pore shape, the validity of the two-fraction, fast-exchange model was assessed. Experiments were conducted using a 180°–τ-90° pulse sequence and the distribution of T1 was calculated using a nonnegative leastsquares routine (NNLS) presented in a previous paper. The two-fraction model described the change of mean T1 with pore radius at a given temperature. The change of the surface relaxation decay constant, T1s, fit an Arrhenius expression with an activation energy of 1.8 kcal/mole. Agreement between the mercury porosimetry and NMR derived pore size distributions is excellent. T1 measurements at both 20 and 300 MHz and 303 K were also made on a series of porous solids fabricated by pelleting submicron silica spheres. These solids exhibit afairly wide pore size distribution (⋍ one order of magnitude). The two-fraction, fast-exchange model satisfactorily described the relationship between pore size and T1. Agreement between mercury porosimetry and NMR pore size distributions is good. However, the NMR derived distributions consisted of a series of narrow peaks overlapping the single broad porosimetry peak. The surface relaxation effect, and, hence, the overall pore size sensitivity, is increased by a factor of 3 when the proton frequency is decreased from 300 to 20 MHz.

A NMR technique for the analysis of pore structure: Application to materials with well-defined pore structure

Groundwater flow in unsaturated, fractured rock is often assumed to be dominated by the porous matrix component. This is frequently the result of reasoning that water flowing in the fractures is rapidly imbibed into the rock matrix by capillary suction forces with negligible resistance to uptake at the fracture/matrix interface. However, the existence of a low-permeability mineralized layer or coating at this interface may substantially reduce matrix imbibition and could consequently result in fracture-dominated flow. To test this idea, four tuff samples containing natural fractures were obtained from tuff formations in southern Nevada. By performing imbibition experiments into the matrix rock, across a mineralized fracture face and then across a fresh uncoated fracture face, water uptake as a function of time and coating was measured. A model combining Darcy's law and the Washburn equation has been used to describe the imbibition behavior. Before testing the tuff samples, the experimental technique and imbibition model were first tested using ordered sphere packings with known pore size and structure. From the tuff imbibition data, the ratio of the permeability through the mineralized face to the permeability through the uncoated face was found to be 0.3 (relatively little effect of the mineralized layer) for two samples, 0.01 for one sample, and 10−7 (uptake strongly restricted by the mineralized layer) for one sample. Using a simple fracture flow model and the imbibition model mentioned above to simulate matrix uptake from the fracture, numerical simulations indicate that the existence of fracture coatings could significantly decrease the travel time and increase the travel depth of water flowing through fractures.

David P. Gallegos

Papers

A NMR technique for the analysis of pore structure: Determination of continuous pore size distributions

A NMR technique for the analysis of pore structure: Application to materials with well-defined pore structure

Impact of fracture coatings on fracture/matrix flow interactions in unsaturated, porous media

An NMR technique for the analysis of pore structure: Application to mesopores and micropores

Rapid surface area determination via NMR spin-lattice relaxation measurements