New equations for binary gas transport in porous media,Part 2: experimental validation

A previous study [Water Resour Res 39 (3) (2003) doi:10.1029/2002WR001338] questioned the validity of the traditional advection–dispersion equation for describing gas flow in porous media. In an original mathematical derivation presented in Part 1 [Adv Water Resour, this issue] we have demonstrated the theoretical existence of two novel physical phenomena which govern the macroscopic transport of gases in porous media. In this work we utilize laboratory experiments and numerical modeling in order to ascertain the importance of these novel theoretical terms. Numerical modeling results indicate that the newly derived sorptive velocity, arising from closure level coupling effects, does not contribute noticeably to the overall flux, under the conditions explored in this work. We demonstrate that the newly discovered “slip coupling” phenomenon in the mass conservation equation plays an important role in describing the physics of gas flow through porous solids for flow regimes of both environmental and industrial interest.     

Analysis of tracer tomography using temporal moments of tracer breakthrough curves

Hydraulic/partitioning tracer tomography (HPTT) was recently developed by Yeh and Zhu [Yeh T-CJ, Zhu J. Hydraulic/partitioning tracer tomography for characterization of dense nonaqueous phase liquid source zones, Water Resour Res 2007;43:W06435. doi:10.1029/2006WR004877.] for estimating spatial distribution of dense nonaqueous phase liquids (DNAPLs) in the subsurface. Since discrete tracer concentration data are directly utilized for the estimation of DNAPLs, this approach solves the hyperbolic convection–dispersion equation. Solution to the convection–dispersion equation however demands fine temporal and spatial discretization, resulting in high computational cost for an HPTT analysis. In this work, we use temporal moments of tracer breakthrough curves instead of discrete concentration data to estimate DNAPL distribution. This approach solves time independent partial differential equations of the temporal moments, and therefore avoids solving the convection–dispersion equation using a time marching scheme, resulting in a dramatic reduction of computational cost. To reduce numerical oscillations associated with convection dominated transport problems such as in inter-well tracer tests, the approach uses a finite element solver adopting the streamline upwind Petrov–Galerkin method to calculate moments and sensitivities. We test the temporal moment approach through numerical simulations. Comparing the computational costs between utilizing moments and discrete concentrations, we find that temporal moments significantly reduce the computation time. We also find that tracer moment data collected through a tomographic survey alone are able to yield reasonable estimates of hydraulic conductivity, as indicated by a correlation of 0.588 between estimated and true hydraulic conductivity fields in the synthetic case study.     

Exploring the nature of non-Fickian transport in laboratory experiments

Observations of non-Fickian transport in sandbox experiments [Levy M, Berkowitz B. Measurement and analysis of non-Fickian dispersion in heterogeneous porous media. J Contam Hydrol 2003;64:203–26] were analyzed previously using a power law tail ψ(t) ∼ t^−1−β with 0 < β < 2 for the spectrum of transition times comprising a tracer plume migration. For each sandbox medium a choice of β resulted in an excellent fit to the breakthrough curve (BTC) data, and the value of β decreased slowly with increasing flow velocity. Here, the data are reanalyzed with the full spectrum of ψ(t) gleaned from analytical calculations [Cortis A, Chen Y, Scher H, Berkowitz B. Quantitative characterization of pore-scale disorder effects on transport in “homogeneous” granular media. Phys Rev E 2004;10(70):041108. doi: 10.1103/PhysRevE.70.041108], numerical simulations [Bijeljic B, Blunt MJ. Pore-scale modeling and continuous time random walk analysis of dispersion in porous media. Water Resour Res 2006;42:W01202. doi: 10.1029/2005WR004578] and permeability fields [Di Donato G, Obi E-O, Blunt MJ. Anomalous transport in heterogeneous media demonstrated by streamline-based simulation. Geophys Res Lett 2003;30:1608–12s. doi: 10.1029/2003GL017196]. We represent the main features of the full spectrum of transition times with a truncated power law (TPL), ψ(t) ∼ (t₁ + t)^−1−βexp(−t/t₂), where t₁ and t₂ are the limits of the power law spectrum. An excellent fit to the entire BTC data set, including the changes in flow velocity, for each sandbox medium is obtained with a single set of values of t₁, β, t₂. The influence of the cutoff time t₂ is apparent even in the regime t < t₂. Significantly, we demonstrate that the previous apparent velocity dependence of β is a result of choosing a pure power law tail for ψ(t). The key is the change in the log–log slope of the TPL form of ψ(t) with a shifting observational time window caused by the change in the mean velocity. Hence, the use of the full spectrum of ψ(t) is not only necessary for the transition to Fickian behavior, but also to account for the dynamics of these laboratory observations of non-Fickian transport.     

Predicting annual and long-term flow-duration curves in ungauged basins

The construction of flow-duration curves is a fundamental task for several activities related to water resources management. The scarcity of observed streamflow data is a diffuse problem in the real world, and flow-duration curves often need to be constructed for ungauged basins. We address this problem by regionalising the stochastic index-flow model of flow-duration curves proposed by Castellarin et al. [Castellarin A, Vogel RM, Brath A. A stochastic index flow model of flow-duration curves. Water Resour Res 2004;40:W03104. doi:10.1029/2003WR002524]. The index-flow model differs from any other stochastic model of flow-duration curves proposed in the literature because it can be used for deriving long-term as well as annual flow-duration curves. The former are constructed on the basis of several years of streamflow data, whereas the latter refer to a given water or calendar year (a typical hydrologic year or a particularly wet or dry year). We apply an extensive cross-validation procedure to quantify the uncertainty of the proposed regional model and to compare it with the uncertainty of traditional regional models of flow-duration curves proposed in the literature. The results of the study indicate that the regional index-flow model is as reliable as or more reliable than traditional regional models for estimating long-term flow-duration curves. Also, the proposed model is more versatile than previous regional models as it can be used for estimating long-term and annual flow-duration curves and for reproducing the variance of annual flow-duration curves.     

Tide-induced airflow in a two-layered coastal land with atmospheric pressure fluctuations

Tide-induced airflow is commonly seen in coastal lands and affects ground stability especially with a less permeable pavement on the ground surface. A tide-induced airflow model in a two-layered unsaturated zone consisting of a highly permeable layer underneath a less permeable layer was established by Li and Jiao [Li HL, JJ Jiao. One-dimensional airflow in unsaturated zone induced by periodic water table fluctuation. Water Resour Res 2005;41:W04007. doi:10.1029/2004WR003916] to describe the one-dimensional airflow with constant atmospheric pressure at the ground surface. In this study, we expand the Li and Jiao model by considering the realistic atmospheric pressure fluctuations and the initial condition. A new transient solution to the airflow model is developed for an initial boundary value problem (IBVP). The transient solution can be used not only to calculate the subsurface air pressure at a future time with a known initial condition, but also to evaluate the asymptotic air pressure variations when time becomes long. The amplitude ratio and phase lag of the subsurface air pressure relative to the tide-induced hydraulic head variations inside the unconfined aquifer below the unsaturated zone are investigated. The results reveal that effect on the subsurface pressure due to changes of atmospheric pressure amplitude depends on the configurations of air resistance in the less permeable layer and the air-filled porosity difference in the two layers. The introduction of atmospheric pressure fluctuations into the airflow model leads to insignificant influence on water table level. A field application of the new solution at Hong Kong International Airport in Hong Kong, China is demonstrated. It indicates that the new transient solution can be conveniently used to evaluate the subsurface air pressure with discrete atmospheric pressure data at the ground surface.     

Pore-scale imbibition experiments in dry and prewetted porous media

Two-phase imbibition behavior of immiscible fluids was studied in dry and prewetted porous media using a laser-induced fluorescence technique. Imbibition was first investigated in two-dimensional (2-D) systems under conditions comparable to those for a study of drainage [Ovdat H, Berkowitz B. Pore-scale study of drainage displacement under combined capillary and gravity effects in index-matched porous media. Water Resources Research 2006;42:W06411. doi:10.1029/2005WR004553] in the capillary-dominated regime. The effect of initial wetting saturation (IWS) was then explored in 2-D and 3-D porous media under the combined effect of gravity, capillary and viscous forces, within and outside the capillary-dominated regime. Parameters that describe maximum vertical advance, volumetric fraction, total surface area and specific surface area of the invading fluid were used to quantify the behavior. Comparison of 2-D drainage and imbibition patterns demonstrates significant qualitative differences under analogous viscosity ratio, buoyancy number, and capillary number values. However, quantitative analyses show strong pore-scale similarities between these patterns. Invasion structures in 3-D, prewetted (IWS ≈ 8% of the pore volume) porous media are ramified, with lateral branching and regions containing trapped residual fluid. These structures are qualitatively and quantitatively different from the compact, branchless structures that develop in dry (IWS = 0) porous media.     

Gas exsolution and flow during supersaturated water injection in porous media: II. Column experiments and continuum modeling

Degassing and in situ development of a mobile gas phase takes place when an aqueous phase equilibrated with a gas at a pressure higher than the subsurface pressure is injected in water-saturated porous media. This process, which has been termed supersaturated water injection (SWI), is a novel and hitherto unexplored means of introducing a gas phase in the subsurface. We give herein a first macroscopic account of the SWI process on the basis of continuum scale simulations and column experiments with CO₂ as the dissolved gas. A published empirical mass transfer correlation [Nambi IM, Powers SE. Mass transfer correlations for nonaqueous phase liquid dissolution from regions with high initial saturations. Water Resour Res 2003;39(2):1030. doi:10.1029/2001WR000667] is found to adequately describe non-equilibrium transfer of CO₂ between the aqueous and gas phases. Remarkably, the dynamics of gas-water two-phase flow, observed in a series of SWI experiments in homogeneous columns packed with silica sand or glass beads, are accurately predicted by traditional two-phase flow theory and the corresponding gas relative permeability is determined. A key consequence of this finding, namely that the displacement of the aqueous phase by gas is compact at the macroscopic scale, is consistent with pore scale simulations of repeated mobilization, fragmentation and coalescence of large gas clusters (i.e., large ganglion dynamics) driven entirely by mass transfer. The significance of this finding for the efficient delivery of a gas phase below the water table is discussed in connection to the alternative process of in situ air sparging, and potential advantages of SWI are highlighted.     

An overview of research on Eulerian–Lagrangian localized adjoint methods (ELLAM)

For problems of convection–diffusion type, Eulerian–Lagrangian localized adjoint methods provide a methodology that maintains the accuracy and efficiency of Eulerian–Lagrangian methods, while also conserving mass and systematically treating any type of boundary condition. In groundwater hydrology, this framework is useful for solute transport, as well as vadose-zone transport, multiphase transport, and reactive flows. The formulation was originated around 1990 by the authors, Herrera and Ewing, in a paper that appeared in Advances in Water Resources [Adv. Water Resour. 13 (1990) 187]. This paper reviews the progress in the development, analysis, and application of these methods since 1990, and suggests topics for future work.     

Conditional simulations of water–oil flow in heterogeneous porous media

This study is an extension of the stochastic analysis of transient two-phase flow in randomly heterogeneous porous media (Chen et al. in Water Resour Res 42:W03425, 2006), by incorporating direct measurements of the random soil properties. The log-transformed intrinsic permeability, soil pore size distribution parameter, and van Genuchten fitting parameter are treated as stochastic variables that are normally distributed with a separable exponential covariance model. These three random variables conditioned on given measurements are decomposed via Karhunen–Loève decomposition. Combined with the conditional eigenvalues and eigenfunctions of random variables, we conduct a series of numerical simulations using stochastic transient water–oil flow model (Chen et al. in Water Resour Res 42:W03425, 2006) based on the KLME approach to investigate how the number and location of measurement points, different random soil properties, as well as the correlation length of the random soil properties, affect the stochastic behavior of water and oil flow in heterogeneous porous media.     

An example of solute spreading in nonstationary, bounded geological formations

We analyze the impact of a linear trend in the mean log-conductivity on the transport of a conservative tracer in a bounded domain. The effects of such a linear trend on solute transport were analyzed in depth for unbounded domains (Rubin and Seong, Water Resour Res 30(11):2901–2911, 1994; Indelman and Rubin, Water Resour Res 31(5):1257–1265, 1995; Water Resour Res 32(5):1257–1265, 1996), whereas studies concerning this special case of medium nonstationarity in finite domains usually focus on head or flow statistics (Guadagnini et al., Stoch Environ Res Risk Assess, 17:394–407, 2003). In this study both ensemble and effective plume moments are provided for an instantaneous release of a solute through a linear source normal to the mean flow direction, by taking into account different sizes of the source. The analysis involving a steady velocity field spatially nonstationary is developed by using the stochastic finite element method. Results show that ensemble moments are affected by increasing trends both parallel and normal to the mean flow direction, but the impact on effective plume moments is very different. A parallel trend does not seem to influence the effective second moments; while a normal trend, although modifies the transverse effective moment only weakly, strongly increases the longitudinal one, especially for large initial sizes of the source. Furthermore, the increase of the particle displacement variance produced by a parallel trend in the finite domain disagrees with the results obtained in an unbounded domain, due to the boundary conditions here considered making both head and velocity moments nonstationary and nonsymmetric.     

A finite analytic method for solving the 2-D time-dependent advection–diffusion equation with time-invariant coefficients

Difficulty in solving the transient advection–diffusion equation (ADE) stems from the relationship between the advection derivatives and the time derivative. For a solution method to be viable, it must account for this relationship by being accurate in both space and time. This research presents a unique method for solving the time-dependent ADE that does not discretize the derivative terms but rather solves the equation analytically in the space–time domain. The method is computationally efficient and numerically accurate and addresses the common limitations of numerical dispersion and spurious oscillations that can be prevalent in other solution methods. The method is based on the improved finite analytic (IFA) solution method [Lowry TS, Li S-G. A characteristic based finite analytic method for solving the two-dimensional steady-state advection–diffusion equation. Water Resour Res 38 (7), 10.1029/2001WR000518] in space coupled with a Laplace transformation in time. In this way, the method has no Courant condition and maintains accuracy in space and time, performing well even at high Peclet numbers. The method is compared to a hybrid method of characteristics, a random walk particle tracking method, and an Eulerian–Lagrangian Localized Adjoint Method using various degrees of flow-field heterogeneity across multiple Peclet numbers. Results show the IFALT method to be computationally more efficient while producing similar or better accuracy than the other methods.     

Analytical solutions of tidal groundwater flow in coastal two-aquifer system

This paper presents a complete analytical solution to describe tidal groundwater level fluctuations in a coastal subsurface system. The system consists of two aquifers and a leaky layer between them. Previous solutions of Jacob [Flow of groundwater, in: H. Rouse (Ed.), Engineering Hydraulics, Wiley, New York, 1950, pp. 321–386], Jiao and Tang [Water Resour. Res. 35 (3) (1999) 747], Li and Jiao [Adv. Water Resour. 24 (5) (2001a) 565], Li et al. [Water Resour. Res. 37 (2001) 1095] and Jeng et al. [Adv. Water Resour. (in press)] are special cases of the new solution. The present solution differs from previous work in that both the effects of the leaky layer's elastic storage and the tidal wave interference between the two aquifers are considered. If the upper and lower aquifers have the same storativities and transimissivities, the system can be simplified into an equivalent double-layered, aquifer–aquitard system bounded by impermeable layers from up and down. It is found that the leaky layer's elastic storage behaves as a buffer to the tidal wave interference between the two aquifers. The buffer capacity increases with the leaky layer's thickness, specific storage, and decreases with the leaky layer's vertical permeability. Great buffer capacity can result in negligible tidal wave interference between the upper and lower aquifers so that the Li and Jiao (loc. cit.) solution applies.     

Spatial moments for colloid-enhanced radionuclide transport in heterogeneous aquifers

We consider colloid facilitated radionuclide transport by steady groundwater flow in a heterogeneous porous formation. Radionuclide binding on colloids and soil-matrix is assumed to be kinetically/equilibrium controlled. All reactive parameters are regarded as uniform, whereas the hydraulic log-conductivity is modelled as a stationary random space function (RSF). Colloid-enhanced radionuclide transport is studied by means of spatial moments pertaining to both the dissolved and colloid-bounded concentration. The general expressions of spatial moments for a colloid-bounded plume are presented for the first time, and are discussed in order to show the combined impact of sorption processes as well as aquifer heterogeneity upon the plume migration. For the general case, spatial moments are defined by the aid of two characteristic reaction functions which cannot be expressed analytically. By adopting the approximation for the longitudinal fluid trajectory covariance valid for a flow parallel to the formation bedding suggested by Dagan and Cvetkovic [Dagan G, Cvetkovic V. Spatial Moments of Kinetically Sorbing Plume in a Heterogeneous Aquifers. Water Resour Res 1993;29:4053], we obtain closed form solutions.     

Stochastic analysis of bounded unsaturated flow in heterogeneous aquifers: Spectral/perturbation approach

This paper describes a stochastic analysis of steady state flow in a bounded, partially saturated heterogeneous porous medium subject to distributed infiltration. The presence of boundary conditions leads to non-uniformity in the mean unsaturated flow, which in turn causes non-stationarity in the statistics of velocity fields. Motivated by this, our aim is to investigate the impact of boundary conditions on the behavior of field-scale unsaturated flow. Within the framework of spectral theory based on Fourier–Stieltjes representations for the perturbed quantities, the general expressions for the pressure head variance, variance of log unsaturated hydraulic conductivity and variance of the specific discharge are presented in the wave number domain. Closed-form expressions are developed for the simplified case of statistical isotropy of the log hydraulic conductivity field with a constant soil pore-size distribution parameter. These expressions allow us to investigate the impact of the boundary conditions, namely the vertical infiltration from the soil surface and a prescribed pressure head at a certain depth below the soil surface. It is found that the boundary conditions are critical in predicting uncertainty in bounded unsaturated flow. Our analytical expression for the pressure head variance in a one-dimensional, heterogeneous flow domain, developed using a nonstationary spectral representation approach [Li S-G, McLaughlin D. A nonstationary spectral method for solving stochastic groundwater problems: unconditional analysis. Water Resour Res 1991;27(7):1589–605; Li S-G, McLaughlin D. Using the nonstationary spectral method to analyze flow through heterogeneous trending media. Water Resour Res 1995; 31(3):541–51], is precisely equivalent to the published result of Lu et al. [Lu Z, Zhang D. Analytical solutions to steady state unsaturated flow in layered, randomly heterogeneous soils via Kirchhoff transformation. Adv Water Resour 2004;27:775–84].     

Analytical solutions for 2D topography-driven flow in stratified and syncline aquifers

Two analytical solution methods are presented for regional steady-state groundwater flow in a two-dimensional stratified aquifer cross section where the water table is approximated by the topographic surface. For the first solution, the surficial aquifer is represented as a set of dipping parallel layers with different, but piecewise constant, anisotropic hydraulic conductivities, where the anisotropy is aligned with the dip of the layered formation. The model may be viewed as a generalization of the solutions developed by [Tóth JA. A theoretical analysis of groundwater flows in small drainage basins. J Geophys Res 1963;68(16):4795–812; Freeze R, Witherspoon P. Theoretical analysis of regional groundwater flow 1) analytical and numerical solution to the mathematical model, water resources research. Water Resour Res 1966;2(4):641–56; Selim HM. Water flow through multilayered stratified hillside. Water Resour Res 1975;11:949–57] to an multi-layer aquifer with general anisotropy, layer orientation, and a topographic surface that may intersect multiple layers. The second solution presumes curved (syncline) layer stratification with layer-dependent anisotropy aligned with the polar coordinate system. Both solutions are exact everywhere in the domain except at the topographic surface, where a Dirichlet condition is met in a least-squared sense at a set of control points; the governing equation and no-flow/continuity conditions are met exactly. The solutions are derived and demonstrated on multiple test cases. The error incurred at the location where the layer boundaries intersect the surface is assessed.     

Cluster-based fuzzy models for groundwater flow in the unsaturated zone

In this paper fuzzy models are used as an alternative to describe groundwater flow in the unsaturated zone. The core of these models consists of a fuzzy rule-based model of the Takagi–Sugeno type. Various fuzzy clustering algorithms are compared in the data-driven identification of these Takagi–Sugeno models. The performance of the resulting fuzzy models is evaluated on the training surface on which they were identified, and on time series measurements of water content values obtained through an experiment carried out by the non-vegetated terrain (NVT) workgroup of the European Microwave Signature Laboratory (EMSL) (see [Mancini M, Hoeben R, Troch PA. Multifrequency radar observations of bare surface soil moisture content: a laboratory experiment. Water Resour Res 1999;35(6):1827–38] and [Hoeben R, Troch PA. Assimilation of active microwave observation data for soil moisture profile estimation. Water Resour Res 2000;36(10):2805–19]). Despite higher errors at the borders of high water content values in the training surface, good results are obtained on the simulation of the time series.     

Subsurface Hydrology of the Lake Chad Basin from Convection Modelling and Observations

In the Lake Chad basin, the quaternary phreatic aquifer (named hereafter QPA) presents large piezometric anomalies referred to as domes and depressions whose depths are ~15 and ~60 m, respectively. A previous study (Leblanc et al. in Geophys Res Lett, 2003, doi: 10.1029/2003GL018094) noticed that brightness temperatures from METEOSAT infrared images of the Lake Chad basin are correlated with the QPA piezometry. Indeed, at the same latitude, domes are ~4–5 K warmer than the depressions. Leblanc et al. (Geophys Res Lett, 2003, doi: 10.1029/2003GL018094) suggested that such a thermal behaviour results from an evapotranspiration excess above the piezometric depressions, an interpretation implicitly assuming that the QPA is separated from the other aquifers by the clay-rich Pliocene formation. Based on satellite visible images, here we find evidence of giant polygons, an observation that suggests instead a local vertical connectivity between the different aquifers. We developed a numerical water convective model giving an alternative explanation for the development of QPA depressions and domes. Beneath the depressions, a cold descending water convective current sucks down the overlying QPA, while, beneath the dome, a warm ascending current produces overpressure. Such a basin-wide circulation is consistent with the water geochemistry. We further propose that the thermal diurnal and evaporation/condensation cycles specific to the water ascending current explain why domes are warmer. We finally discuss the possible influence of the inferred convective circulation on the transient variations of the QPA reported from observations of piezometric levels and GRACE-based water mass change over the region.     

Analytical solutions to the transient,unsaturated transport of water and contaminants through horizontal porous media

We present a range of analytical solutions to the combined transient water and solute transport for horizontal flow. We adopt the concept of a scale and time dependent dispersivity used for contaminant transport in aquifers and apply it to transient, unsaturated horizontal flow to develop similarity solutions for both constant solute concentration and solute flux boundary conditions. Through the use of a specific form of the water profile as used by Brutsaert [Water Resour Res 1968:4;785], the solute profiles can be reduced to a simple quadrature. We also derive a solution for the instantaneous injection of water and solute into a horizontal media for an arbitrary dispersivity. It is found that the solute concentration remains constant in both space and time as the water redistributes, suggesting that the solute does not disperse relative to the water.