Karst spring responses examined by process-based modeling

Ground water in karst terrains is highly vulnerable to contamination due to the rapid transport of contaminants through the highly conductive conduit system. For contamination risk assessment purposes, information about hydraulic and geometric characteristics of the conduits and their hydraulic interaction with the fissured porous rock is an important prerequisite. The relationship between aquifer characteristics and short-term responses to recharge events of both spring discharge and physicochemical parameters of the discharged water was examined using a process-based flow and transport model. In the respective software, a pipe-network model, representing fast conduit flow, is coupled to MODFLOW, which simulates flow in the fissured porous rock. This hybrid flow model was extended to include modules simulating heat and reactive solute transport in conduits. The application of this modeling tool demonstrates that variations of physicochemical parameters, such as solute concentration and water temperature, depend to a large extent on the intensity and duration of recharge events and provide information about the structure and geometry of the conduit system as well as about the interaction between conduits and fissured porous rock. Moreover, the responses of solute concentration and temperature of spring discharge appear to reflect different processes, thus complementing each other in the aquifer characterization.     

Time–space fractional governing equations of transient groundwater flow in confined aquifers: Numerical investigation

In order to model non‐Fickian transport behaviour in groundwater aquifers, various forms of the time–space fractional advection–dispersion equation have been developed and used by several researchers in the last decade. The solute transport in groundwater aquifers in fractional time–space takes place by means of an underlying groundwater flow field. However, the governing equations for such groundwater flow in fractional time–space are yet to be developed in a comprehensive framework. In this study, a finite difference numerical scheme based on Caputo fractional derivative is proposed to investigate the properties of a newly developed time–space fractional governing equations of transient groundwater flow in confined aquifers in terms of the time–space fractional mass conservation equation and the time–space fractional water flux equation. Here, we apply these time–space fractional governing equations numerically to transient groundwater flow in a confined aquifer for different boundary conditions to explore their behaviour in modelling groundwater flow in fractional time–space. The numerical results demonstrate that the proposed time–space fractional governing equation for groundwater flow in confined aquifers may provide a new perspective on modelling groundwater flow and on interpreting the dynamics of groundwater level fluctuations. Additionally, the numerical results may imply that the newly derived fractional groundwater governing equation may help explain the observed heavy‐tailed solute transport behaviour in groundwater flow by incorporating nonlocal or long‐range dependence of the underlying groundwater flow field.     

Semi-analytical solutions for solute transport in fractured porous media using a strip source of finite width

Transient and steady-state analytical solutions are derived to investigate solute transport in a fractured porous medium consisting of evenly spaced, parallel discrete fractures. The solutions incorporate a finite width strip source, longitudinal and transverse dispersion in the fractures, source decay, aqueous phase decay, one-dimensional diffusion into the matrix, sorption to fracture walls, and sorption within the matrix. The solutions are derived using Laplace and Fourier transforms, and inverted by interchanging the order of integration and utilizing a numerical Laplace inversion algorithm. The solutions are verified for simplified cases by comparison to solutions derived by Batu [Batu V. A generalized two-dimensional analytical solution for hydrodynamic dispersion in bounded media with the first-type condition at the source. Wat Resour Res 1989;25(6):1125] and Sudicky and Frind [Sudicky EA, Frind EO. Contaminant transport in fractured porous media: analytical solutions for a system of parallel fractures. Wat Resour Res 1982;18(6):1634]. The application of the solutions to a fractured sandstone demonstrates that narrower source widths and larger values of transverse dispersivity both lead to lower downstream concentrations in the fractures and shorter steady-state plumes. The incorporation of aqueous phase decay and source concentration decay both lead to lower concentrations and shorter plumes, with even moderate amounts of decay significantly shortening the persistence of contamination.     

Shear dispersion in a fracture with porous walls

We present an analytical expression for the shear dispersion during solute transport in a coupled fracture–matrix system. The dispersion coefficient is obtained in a fracture with porous walls by taking into account an accurate boundary condition at the interface between the matrix and fracture, and the results were compared with those in a non-coupled system. The analysis presented identifies three regimes: diffusion-dominated, transition, and advection-dominated. The results showed that it is important to consider the exchange of solute between the fracture and matrix in development of the shear dispersion coefficient for the transition and advection-dominated regimes. The new dispersion coefficient is obtained by imposing the continuity of concentrations and mass fluxes along the porous walls. The resulting equivalent transport equation revealed that the effective velocity in a fracture increases while the dispersion coefficient decreases due to mass transfer between the matrix and fracture. A larger effective advection term leads to greater storage of mass in the matrix as compared with the classical double-porosity model with a non-coupled dispersion coefficient. The findings of this study can be used for modeling of tracer tests as well as fate, transport, and remediation of groundwater contaminants in fractured rocks.     

Magnetic resonance microscopy of biofouling induced scale dependent transport in porous media

Non-invasive magnetic resonance microscopy (MRM) methods are applied to study biofouling of a homogeneous model porous media. MRM of the biofilm biomass using magnetic relaxation weighting shows the heterogeneous nature of the spatial distribution of the biomass as a function of growth. Spatially resolved MRM velocity maps indicate a strong variation in the pore scale velocity as a function of biofilm growth. The hydrodynamic dispersion dynamics for flow through the porous media is quantitatively characterized using a pulsed gradient spin echo technique to measure the propagator of the motion. The propagator indicates a transition in transport dynamics from a Gaussian normal diffusion process following a normal advection diffusion equation to anomalous transport as a function of biofilm growth. Continuous time random walk models resulting in a time fractional advection diffusion equation are shown to model the transition from normal to anomalous transport in the context of a conceptual model for the biofouling. The initially homogeneous porous media is transformed into a more complex heterogeneous porous media by the biofilm growth.     

Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications

We investigate the spatiotemporal nonlocality underlying fractional-derivative models as a possible explanation for regional-scale anomalous dispersion with heavy tails. Properties of four fractional-order advection–dispersion equation (fADE) models were analyzed and compared systematically, including the space fADEs with either maximally positive or negative skewness, the time fADE with a temporal fractional-derivative 0<γ<1$0<\gamma <1$, and the extension of the time fADE with 1<γ<2$1<\gamma <2$. Space fADEs describe the dependence of local concentration change on a wide range of spatial zones (i.e., the space nonlocality), while time fADEs describe dynamic mass exchange between mobile and multiple immobile phases and therefore record the temporal history of concentration “loading” (i.e., the time-nonlocality). We then applied the fADEs as models of anomalous dispersion to four extensively-studied, regional-scale, natural systems, including a hillslope composed of fractured soils, a river with simultaneous active flow zones and various dead-zones, a relatively homogeneous glaciofluvial aquifer dominated by stratified sand and gravel, and a highly heterogeneous alluvial aquifer containing both preferential flowpaths and abundant aquitards. We find that the anomalous dispersion observed at each site might not be characterized reasonably or sufficiently by previous studies. In particular, the use of the space fADE with less than maximally positive skewness implies a spatial dependence on downstream concentrations that may not be physically realistic for solute transport in watershed catchments and rivers (where the influence of dead-zones on solute transport can be described by a temporal, not spatial, fractional model). Field-scale transport studies show that large ranges of solute displacement can be described by a space nonlocal, fractional-derivative model, and long waiting times can be described efficiently by a time-nonlocal, fractional model. The unknown quantitative relationship between the nonlocal parameters and the heterogeneity, and the similarity in concentration profiles that are solutions to the different nonlocal transport models, all demonstrate the importance of distinguishing the representative nonlocality (time and/or space) for any given regional-scale anomalous dispersion process.     

Lattice Boltzmann Models for Flow and Transport in Saturated Karst

Flow and transport simulation in karst aquifers remains a significant challenge for the ground water modeling community. Darcy's law–based models cannot simulate the inertial flows characteristic of many karst aquifers. Eddies in these flows can strongly affect solute transport. The simple two-region conduit/matrix paradigm is inadequate for many purposes because it considers only a capacitance rather than a physical domain. Relatively new lattice Boltzmann methods (LBMs) are capable of solving inertial flows and associated solute transport in geometrically complex domains involving karst conduits and heterogeneous matrix rock. LBMs for flow and transport in heterogeneous porous media, which are needed to make the models applicable to large-scale problems, are still under development. Here we explore aspects of these future LBMs, present simple examples illustrating some of the processes that can be simulated, and compare the results with available analytical solutions. Simulations are contrived to mimic simple capacitance-based two-region models involving conduit (mobile) and matrix (immobile) regions and are compared against the analytical solution. There is a high correlation between LBM simulations and the analytical solution for two different mobile region fractions. In more realistic conduit/matrix simulation, the breakthrough curve showed classic features and the two-region model fit slightly better than the advection-dispersion equation (ADE). An LBM-based anisotropic dispersion solver is applied to simulate breakthrough curves from a heterogeneous porous medium, which fit the ADE solution. Finally, breakthrough from a karst-like system consisting of a conduit with inertial regime flow in a heterogeneous aquifer is compared with the advection-dispersion and two-region analytical solutions.     

Heat Tracing in a Fractured Aquifer with Injection of Hot and Cold Water

Heat as a tracer in fractured porous aquifers is more sensitive to fracture-matrix processes than a solute tracer. Temperature evolution as a function of time can be used to differentiate fracture and matrix characteristics. Experimental hot (50 °C) and cold (10 °C) water injections were performed in a weathered and fractured granite aquifer where the natural background temperature is 30 °C. The tailing of the hot and cold breakthrough curves, observed under different hydraulic conditions, was characterized in a log–log plot of time vs. normalized temperature difference, also converted to a residence time distribution (normalized). Dimensionless tail slopes close to 1.5 were observed for hot and cold breakthrough curves, compared to solute tracer tests showing slopes between 2 and 3. This stronger thermal diffusive behavior is explained by heat conduction. Using a process-based numerical model, the impact of heat conduction toward and from the porous rock matrix on groundwater heat transport was explored. Fracture aperture was adjusted depending on the actual hydraulic conditions. Water density and viscosity were considered temperature dependent. The model simulated the increase or reduction of the energy level in the fracture-matrix system and satisfactorily reproduced breakthrough curves tail slopes. This study shows the feasibility and utility of cold water tracer tests in hot fractured aquifers to boost and characterize the thermal matrix diffusion from the matrix toward the flowing groundwater in the fractures. This can be used as complementary information to solute tracer tests that are largely influenced by strong advection in the fractures.     

Nonlocal transport models for capturing solute transport in one-dimensional sand columns: Model review,applicability, limitations and improvement

Modelling pollutant transport in water is one of the core tasks of computational hydrology, and various physical models including especially the widely used nonlocal transport models have been developed and applied in the last three decades. No studies, however, have been conducted to systematically assess the applicability, limitations and improvement of these nonlocal transport models. To fill this knowledge gap, this study reviewed, tested and improved the state-of-the-art nonlocal transport models, including their physical background, mathematical formula and especially the capability to quantify conservative tracers moving in one-dimensional sand columns, which represents perhaps the simplest real-world application. Applications showed that, surprisingly, neither the popular time-nonlocal transport models (including the multi-rate mass transfer model, the continuous time random walk framework and the time fractional advection-dispersion equation), nor the spatiotemporally nonlocal transport model (ST-fADE) can accurately fit passive tracers moving through a 15-m-long heterogeneous sand column documented in literature, if a constant dispersion coefficient or dispersivity is used. This is because pollutant transport in heterogeneous media can be scale-dependent (represented by a dispersion coefficient or dispersivity increasing with spatiotemporal scales), non-Fickian (where plume variance increases nonlinearly in time) and/or pre-asymptotic (with transition between non-Fickian and Fickian transport). These different properties cannot be simultaneously and accurately modelled by any of the transport models reviewed by this study. To bypass this limitation, five possible corrections were proposed, and two of them were tested successfully, including a time fractional and space Hausdorff fractal model which minimizes the scale-dependency of the dispersion coefficient in the non-Euclidean space, and a two-region time fractional advection-dispersion equation which accounts for the spatial mixing of solute particles from different mobile domains. Therefore, more efforts are still needed to accurately model transport in non-ideal porous media, and the five model corrections proposed by this study may shed light on these indispensable modelling efforts.     

Laplace transform solutions for solute transport in fissured aquifers

The main processes affecting the migration of a solute in a fissured aquifer will be advection and dispersion in the fissures, diffusion into the porous matrix; and adsorption. This paper considers solute transport in an idealized fissured aquifer consisting of slabs of saturated rock-matrix separated by equally spaced, planar fissures. The solution of the transport equations is developed as far as Laplace transforms of the solute concentrations in the fissure and matrix water. Numerical inversion of the transforms is used to investigate characteristic behaviour of the model for a number of special cases.     

Equilibration Times of Gas-Filled Diffusion Samplers in Slow-Moving Ground Water Systems

The installation of gas-filled diffusion samplers into small-diameter boreholes results in a significant reduction of the dissolved gas concentration around the sampler. In aquifers where the diffusive flux of solutes outpaces advective transport, the process that governs the equilibration time of a sampler is the resupply of solutes by diffusion from the aquifer. We have derived a solution that can be used to estimate the time required for a diffusion sampler to reach equilibrium with the dissolved gas concentration in the aquifer, where diffusion is the only solute transport mechanism. Thus the solutions provide equilibration times for cells placed in aquifers where diffusion dominates and maximum equilibration times for cells placed in aquifers where advection can also resupply solutes. The solutions are generic and are functions of nondimensionalized variables, therefore providing estimates of equilibration times for any type of solute, sampler volume, bore dimensions, and aquifer porosity. Examples are given for various sized gas-filled helium samplers placed in boreholes of different radii.     

Analytical solutions for solute transport in saturated porous media with semi-infinite or finite thickness

Three-dimensional analytical solutions for solute transport in saturated, homogeneous porous media are developed. The models account for three-dimensional dispersion in a uniform flow field, first-order decay of aqueous phase and sorbed solutes with different decay rates, and nonequilibrium solute sorption onto the solid matrix of the porous formation. The governing solute transport equations are solved analytically by employing Laplace, Fourier and finite Fourier cosine transform techniques. Porous media with either semi-infinite or finite thickness are considered. Furthermore, continuous as well as periodic source loadings from either a point or an elliptic source geometry are examined. The effect of aquifer boundary conditions as well as the source geometry on solute transport in subsurface porous formations is investigated.     

Incorporating Super‐Diffusion due to Sub‐Grid Heterogeneity to Capture Non‐Fickian Transport

Numerical transport models based on the advection‐dispersion equation (ADE) are built on the assumption that sub‐grid cell transport is Fickian such that dispersive spreading around the average velocity is symmetric and without significant tailing on the front edge of a solute plume. However, anomalous diffusion in the form of super‐diffusion due to preferential pathways in an aquifer has been observed in field data, challenging the assumption of Fickian dispersion at the local scale. This study develops a fully Lagrangian method to simulate sub‐grid super‐diffusion in a multidimensional regional‐scale transport model by using a recent mathematical model allowing super‐diffusion along the flow direction given by the regional model. Here, the time randomizing procedure known as subordination is applied to flow field output from MODFLOW simulations. Numerical tests check the applicability of the novel method in mapping regional‐scale super‐diffusive transport conditioned on local properties of multidimensional heterogeneous media.     

The impact of boundary on the fractional advection–dispersion equation for solute transport in soil: Defining the fractional dispersive flux with the Caputo derivatives

The inherent heterogeneity of geological media often results in anomalous dispersion for solute transport through them, and how to model it has been an interest over the past few decades. One promising approach that has been increasingly used to simulate the anomalous transport in surface and subsurface water is the fractional advection–dispersion equation (FADE), derived as a special case of the more general continuous time random walk or the stochastic continuum model. In FADE, the dispersion is not local and the solutes have appreciable probability to move long distances, and thus reach the boundary faster than predicted by the classical advection–dispersion equation (ADE). How to deal with different boundaries associated with FADE and their consequent impact is an issue that has not been thoroughly explored. In this paper we address this by taking one-dimensional solute movement in soil columns as an example. We show that the commonly used FADE with its fractional derivatives defined by the Riemann–Liouville definition is problematic and could result in unphysical results for solute transport in bounded domains; a modified method with the fractional dispersive flux defined by the Caputo derivatives is presented to overcome this problem. A finite volume approach is given to numerically solve the modified FADE and its associated boundaries. With the numerical model, we analyse the inlet-boundary treatment in displacement experiments in soil columns, and find that, as in ADE, treating the inlet as a prescribed concentration boundary gives rise to mass-balance errors and such errors could be more significant in FADE because of its non-local dispersion. We also discuss a less-documented but important issue in hydrology: how to treat the upstream boundary in analysing the lateral movement of tracer in an aquifer when the tracer is injected as a pulse. It is shown that the use of an infinite domain, as commonly assumed in literature, leads to unphysical backward dispersion, which has a significant impact on data interpretation. To avoid this, the upstream boundary should be flux-prescribed and located at the upstream edge of the injecting point. We apply the model to simulate the movement of Cl⁻ in a tracer experiment conducted in a saturated hillslope, and analyse in details the significance of upstream-boundary treatments in parameter estimation.     

广东省地方防震减灾工作思路与实践

防震减灾的四个环节,包括地震重点监视防御区的工作和实现全省防震减灾十年目标的任务,最终实施都得依赖市,县地地震工作机构的落实。根据广东省的政治,经济,社会发展以有震情特点,介绍了近两个来地方防震减灾工作的思路与实践,为今后的工作有益的借鉴。     

A fractal Richards’ equation to capture the non-Boltzmann scaling of water transport in unsaturated media

The traditional Richards’ equation implies that the wetting front in unsaturated soil follows Boltzmann scaling, with travel distance growing as the square root of time. This study proposes a fractal Richards’ equation (FRE), replacing the integer-order time derivative of water content by a fractal derivative, using a power law ruler in time. FRE solutions exhibit anomalous non-Boltzmann scaling, attributed to the fractal nature of heterogeneous media. Several applications are presented, fitting the FRE to water content curves from previous literature.     

Convergent Radial Tracing of Viral and Solute Transport in Gneiss Saprolite

Deeply weathered crystalline rock aquifer systems comprising unconsolidated saprolite and underlying fractured bedrock (saprock) underlie 40% of sub-Saharan Africa. The vulnerability of this aquifer system to contamination, particularly in rapidly urbanizing areas, remains poorly understood. In order to assess solute and viral transport in saprolite derived from Precambrian gneiss, forced-gradient tracer experiments using chloride and Escherichia coli phage ΦX174 were conducted in southeastern Uganda. The bacteriophage tracer was largely unrecovered; adsorption to the weathered crystalline rock matrix is inferred and enabled by the low pH (5.7) of site ground water and the bacteriophage's relatively high isoelectric point (pI = 6.6). Detection of the applied ΦX174 phage in the pumping well discharge at early times during the experiment traces showed, however, that average ground water flow velocities exceed that of the inert solute tracer, chloride. This latter finding is consistent with observations in other hydrogeological environments where statistically extreme sets of microscopic flow velocities are considered to transport low numbers of fecal pathogens and their proxies along a selected range of linked ground water pathways. Application of a radial advection-dispersion model with an exponentially decaying source term to the recovered chloride tracer estimates a dispersivity (α) of 0.8 ± 0.1 m over a distance of 4.15 m. Specific yield (S_y) is estimated to be 0.02 from volume balance calculations based on tracer experiments. As single-site observations, our estimates of saprolite S_y and α are tentative but provide a starting point for assessing the vulnerability of saprolite aquifers in sub-Saharan Africa to contamination and estimating quantitatively the impact of climate and abstraction on ground water storage.     

Plume Detachment and Recession Times in Fractured Rock

The influence of source zone concentration reduction on solute plume detachment and recession times in fractured rock was investigated using new semianalytical solutions to transient solute transport in the presence of advection, dispersion, sorption, matrix diffusion, and first-order decay. Novel aspects of these solutions are: (1) the source zone concentration behavior is simulated using a constant concentration with the option for either an instantaneous reduction to zero concentration or an exponentially decaying source zone concentration initiated at some time (t^*) after the source is introduced, and (2) different biodegradation rates in the fracture and rock matrix. These solutions were applied for sandstone bedrock and revealed that biodegradation in the matrix, not the fracture, may be the most significant attenuation mechanism and therefore may dictate remediation time scales. Also, instantaneous and complete source concentration reduction in aged plumes may not be beneficial with respect to plume response because back-diffusion can sustain plume migration for long periods of time. Moderate source zone concentration reduction has a similar impact on the rate of advance of the leading edge of the plume as aggressive concentration reduction. If the source zone concentration reduction half-life is less than the plume decay half-life, then volatile organic compound (VOC) mass sequestered in the rock matrix will ultimately dictate plume persistence and not the presence of the source zone.     

Effects of compound-specific dilution on transient transport and solute breakthrough: A pore-scale analysis

This pore-scale modeling study in saturated porous media shows that compound-specific effects are important not only at steady-state and for the lateral displacement of solutes with different diffusivities but also for transient transport and solute breakthrough. We performed flow and transport simulations in two-dimensional pore-scale domains with different arrangement of the solid grains leading to distinct characteristics of flow variability and connectivity, representing mildly and highly heterogeneous porous media, respectively. The results obtained for a range of average velocities representative of groundwater flow (0.1–10 m/day), show significant effects of aqueous diffusion on solute breakthrough curves. However, the magnitude of such effects can be masked by the flux-averaging approach used to measure solute breakthrough and can hinder the correct interpretation of the true dilution of different solutes. We propose, as a metric of mixing, a transient flux-related dilution index that allows quantifying the evolution of solute dilution at a given position along the main flow direction. For the different solute transport scenarios we obtained dilution breakthrough curves that complement and add important information to traditional solute breakthrough curves. Such dilution breakthrough curves allow capturing the compound-specific mixing of the different solutes and provide useful insights on the interplay between advective and diffusive processes, mass transfer limitations, and incomplete mixing in the heterogeneous pore-scale domains. The quantification of dilution for conservative solutes is in good agreement with the outcomes of mixing-controlled reactive transport simulations, in which the mass and concentration breakthrough curves of the product of an instantaneous transformation of two initially segregated reactants were used as measures of reactive mixing.     

Macroscopic representation of structural geometry for simulating water and solute movement in dual-porosity media

The structure of macroporous or aggregated soils and fractured rocks is generally so complex that it is impractical to measure the geometry at the microscale (i.e., the size and the shape of soil aggregates or rock matrix blocks, and the myriad of fissures or fractures), and use such data in geometry-dependent macroscale flow and transport models. This paper analyzes a first-order type dual-porosity model which contains a geometry-dependent coefficient, β, in the mass transfer term to macroscopically represent the size and shape of soil or rock matrix blocks. As a reference, one- and two-dimensional geometry-based diffusion models were used to simulate mass transport into and out of porous blocks of defined shapes. Estimates for β were obtained analytically for four different matrix block geometries. Values for β were also calculated by directly matching analytical solutions of the diffusion models for a number of selected matrix block geometries to results obtained with the first-order model assuming standard boundary conditions. Direct matching improved previous results for cylindrical macropore geometries, especially when relatively small ratios between the outer soil mantle and the radius of the inner cylinder were used. Results of our analysis show that β is closely related to the ratio of the effective surface area available for mass transfer, and the soil matrix volume normalized by the effective characteristic length of the matrix system. Using values of β obtained by direct matching, an empirical function is derived to estimate macroscopic geometry coefficients from medium properties which in principle are measurable. The method permits independent estimates of β, thus allowing the dual-porosity approach eventually to be applied to media with complex and mixed types of structural geometry.