Stochastic modelling of groundwater flow and solute transport in aquifers

This paper presents an introductory overview of recently developed stochastic theories for tackling spatial variability problems in predicting groundwater flow and solute transport. Advantages and limitations of the theories are discussed. Lastly, strategies based on the stochastic approaches to predict solute transport in aquifers are recommended.     

Comparison of laboratory-scale solute transport visualization experiments with numerical simulation using cross-bedded sandstone

Using a slab of Massillon Sandstone, laboratory-scale solute tracer experiments were carried out to test numerical simulations using the Advection–Dispersion Equation (ADE). While studies of a similar nature exist, our work differs in that we combine: (1) experimentation in naturally complex geologic media, (2) X-ray absorption imaging to visualize and quantify two-dimensional solute transport, (3) high resolution transport property characterization, with (4) numerical simulation. The simulations use permeability, porosity, and solute concentration measured to sub-centimeter resolution. While bulk breakthrough curve characteristics were adequately matched, large discrepancies exist between the experimental and simulated solute concentration fields. Investigation of potential experimental errors suggests that the failure to fit solute concentration fields may lie in loss of intricate connectivity within the cross-bedded sandstone occurring at scales finer than our property characterization measurements (i.e., sub-centimeter).     

Characterizing solute transport with transient storage across a range of flow rates: The evidence of repeated tracer experiments in Austrian and Italian streams

Solute transport in rivers and streams with hyporheic zone exchange and/or in-stream storage is typically affected by the prevailing flow rate. The research reported here focuses on stream tracer experiments repeated many times along the same Austrian (Mödlingbach) and Italian (Torrente Lura) channel reaches to characterize parameter dependency on flow rate. Both groups of data sets showed an increase of storage zone area and main stream area with discharge. In either case, a strong negative correlation was obtained between storage zone residence time and flow rate. From the Mödlingbach data, no clear relationship with Q emerged for the dispersion coefficient and the dead zone ratio, whereas Torrente Lura showed a clear positive correlation of the dispersion coefficient with the flow rate and a slightly negative Q-dependency for the dead zone ratio. Mödlingbach and Torrente Lura results are presented against the background of other repeat experiments reported in literature.     

Evaluating different GCMs for predicting spatial recharge in an irrigated arid region

Groundwater systems in arid regions will be particularly sensitive to climate change owing to the strong dependence of rates of evapotranspiration on temperature, and shifts in the precipitation regimes. Irrigation use in these arid regions is typically a large component of the water budget, and may increase due to changes in soil moisture resulting from higher temperatures and changes in the timing of precipitation events. In this study, future predicted climate change scenarios from three global climate models (CGCM1 GHG+A1, CGCM3.1 A2, and HadCM3 A2) are used to determine the sensitivity of recharge to different climate models in an irrigated agricultural region. The arid Oliver region (annual precipitation 300 mm) in the Okanagan Basin, British Columbia, is used to demonstrate the approach. Irrigation return flow, as a contribution to total diffuse recharge, is simulated by calculating the daily applied irrigation based on estimates of seasonal crop water demand and the forecasted precipitation and evaporation data. The relative contribution of irrigation return flow to groundwater recharge under current and future climate conditions is modelled. Temperature data were downscaled using Statistical Downscaling Model (SDSM), while precipitation and solar radiation changes were estimated directly from the GCM data. Shifts in climate, from present to future predicted, were applied to a stochastic weather generator, and used to force a one-dimensional hydrologic model, HELP 3.80D. Results were applied spatially, according to different soil profiles, slope and vegetation, over a 22.5 km by 8.6 km region. Changes to recharge in future time periods for each GCM result in modest increases of recharge with the peak recharge shifting from March to February. Lower recharge rates and higher potential evapotranspiration rates are similarly predicted by all three models for the summer months. All scenarios show that the potential growing season will expand between 3 and 4 weeks due to increases in temperature. However, the magnitude of the change varies considerably between models. CGCM3.1 has the largest increases of recharge rates, CGCM1 has very minor increases, and HadCM3 is relatively stable (as indicated by the near-zero changes between climate states). The significant differences between these three models indicate that prediction of future recharge is highly dependent on the model selected. The minor increase of annual recharge in future predicted climate states is due the shift of peak recharge from increased temperature. Irrigation rates dominate total recharge during the summer months in this arid area. Recharge in irrigated areas is significantly higher than natural recharge, with irrigation return flow between 25% and 58%. A comparison of recharge results for the least efficient and the most efficient irrigation systems indicates that the latter are more sensitive to choice of GCM.     

Numerical simulations of non-ergodic solute transport in three-dimensional heterogeneous porous media

Numerical simulations of non-ergodic transport of a non-reactive solute plume by steady-state groundwater flow under a uniform mean velocity, , were conducted in a three-dimensional heterogeneous and statistically isotropic aquifer. The hydraulic conductivity, K(x), is modeled as a random field which is assumed to be log-normally distributed with an exponential covariance. Significant efforts are made to reduce the simulation uncertainties. Ensemble averages of the second spatial moments of the plume and the plume centroid variances were simulated with 1600 Monte Carlo (MC) runs for three variances of log K, _Y²=0.09, 0.23, and 0.46, and a square source normal to of three dimensionless lengths. It is showed that 1600 MC runs are needed to obtain stabilized results in mildly heterogeneous aquifers of _Y²0.5 and that large uncertainty may exist in the simulated results if less MC runs are used, especially for the transverse second spatial moments and the plume centroid variance in transverse directions. The simulated longitudinal second spatial moment and the plume centroid variance in longitudinal direction fit well to the first-order theoretical results while the simulated transverse moments are generally larger than the first-order values. The ergodic condition for the second spatial moments is far from reaching in all cases simulated and transport in transverse directions may reach ergodic condition much slower than that in longitudinal direction.     

A 3D ERT study of solute transport in a large experimental tank

A high resolution, cross-borehole, 3D electrical resistivity tomography (ERT) study of solute transport was conducted in a large experimental tank. ERT voxels comprising the time sequence of electrical images were converted into a 3D array of ERT estimated fluid conductivity breakthrough curves and compared with direct measurements of fluid conductivity breakthrough made in wells. The 3D ERT images of solute transport behaviour were also compared with predictions based on a 3D finite-element, coupled flow and transport model, accounting for gravity induced flow caused by concentration differences.The tank (dimensions 185×245×186 cm) was filled with medium sand, with a gravel channel and a fine sand layer installed. This heterogeneous system was designed to complicate solute transport behaviour relative to a homogeneous sand tank, and to thus provide a challenging but insightful analysis of the ability of 3D ERT to resolve transport phenomena. Four ERT arrays and 20 piezometers were installed during filling. A NaCl tracer (conductivity 1.34 S/m) was injected and intensively monitored with 3D ERT and direct sampling of fluid chemistry in piezometers.We converted the bulk conductivity estimate for 250 voxels in the ERT imaged volume into ERT estimated voxel fluid conductivity by assuming that matrix conduction in the tank is negligible. In general, the ERT voxel response is in reasonable agreement with the shape of fluid conductivity breakthrough observed in six wells in which direct measurements of fluid conductivity were made. However, discrepancies occur, particularly at early times, which we attribute to differences between the scale of the image voxels and the fluid conductivity measurement, measurement errors mapped into the electrical inversion and artificial image roughness resulting from the inversion.ERT images revealed the 3D tracer distribution at 15 times after tracer injection. The general pattern and timing of solute breakthrough observed with ERT agreed with that predicted from the flow/transport modelling. However, the ERT images indicate a vertical component of tracer transport and preferential flow paths in the medium sand. We attribute this to transient vertical gradients established during tracer injection, and heterogeneity caused by sorting of the sand resulting from the filling procedure. In this study, ERT provided a unique dataset of 250 voxel breakthrough curves in 1.04 m³. The use of 3D ERT to generate an array of densely sampled estimated fluid conductivity breakthrough curves is a potentially powerful tool for quantifying solute transport processes.     

Describing near surface, transient flow processes in unconfined aquifers below irrigated lands: A deforming finite element model for heterogeneous aquifers

Hydrologic models of irrigated lands generally adopt either a basin-scale or a root-zone perspective. While basin-wide macro-scale models rely on the aggregation of important spatial and temporal data across large areas, micro-scale root-zone models depend on the definition of rigid boundaries around the zone of plant–soil–water interaction. In reality, irrigation management decisions are made on a field by field basis and can interact across field boundaries. This paper first describes a shallow water table model, based on deforming finite element (DFE) framework, to characterize the near-surface field-to-field hydrologic response to various irrigation and drainage management regimes along a gently sloping alluvial fan. The model is then enhanced through changing geometry of a fluctuating water table below a series of irrigated fields. Such an enhancement also offers computational flexibility relative to the saturated–unsaturated models commonly used in micro-scale studies. The model is designed with the alluvial fan aquifers of California’s western San Joaquin Valley as reference systems.     

Conditions when anisotropy is negligible for solute transfer in sediment beds of lakes or streams

To reduce the complexity and save computation time, an isotropic and a scalar dispersion model are explored and compared to the anisotropic advection/dispersion model to study the interstitial flow in a stream and lake sediment induced by a periodic pressure wave. In these systems, the solute transport is controlled by the ratio (R = a/(LS)) of the pressure wave steepness (a/L) to the stream slope (S), and the dispersivity ratio (λ = α_L/L) that measures the longitudinal dispersivity (α_L) relative to the pressure wave length (L). Through a series of numerical experiments, the conclusion is reached that a scalar dispersion model can be applied with satisfactory results for advection-dominated transport, i.e. when R ?  0.1 and λ ? 0.01, or λ ? 0.0001, i.e. Peclet number (Pe) ? 10000; an isotropic dispersion model can be applied when R ? 10 or λ ? 0.001, and the full anisotropic advection/dispersion model has to be applied when R > 10 and λ > 0.001.     

Multi-block lattice Boltzmann simulations of solute transport in shallow water flows

We report a two-dimensional multi-block lattice Boltzmann model for solute transport in shallow water flows, which is developed based on the advection–diffusion equation for mass transport and the shallow water equations for the flows. A weighting factor is included in the centered scheme for improved accuracy. The model is firstly verified by simulating three benchmark tests: wind-driven circulation in a dish-shaped lake, jet-forced flow in a circular basin, and flow formed by two parallel streams containing different uniform concentrations at the same constant velocity; and then it is applied to a practical wind-induced flow, Baiyangdian Lake, which is characterized by irregular geometries and complex bathymetries. The numerical results have shown that the model is able to produce accurate and detailed results for both water flows and solute transport, which is attractive, especially for flows in narrow zones of practical terrains and certain areas with largely varying pollutant concentrations.     

Comparison of theory and experiment for solute transport in weakly heterogeneous bimodal porous media

In this work, the influence of non-equilibrium effects on solute transport in a weakly heterogeneous medium is discussed. Three macro-scale models (upscaled via the volume averaging technique) are investigated: (i) the two-equation non-equilibrium model, (ii) the one-equation asymptotic model and (iii) the one-equation local equilibrium model. The relevance of each of these models to the experimental system conditions (duration of the pulse injection, dispersivity values…) is analyzed. The numerical results predicted by these macroscale models are compared directly with the experimental data (breakthrough curves). Our results suggest that the preasymptotic zone (for which a non-Fickian model is required) increases as the solute input pulse time decreases. Beyond this limit, the asymptotic regime is recovered. A comparison with the results issued from the stochastic theory for this regime is performed. Results predicted by both approaches (volume averaging method and stochastic analysis) are found to be consistent.     

Stochastic modeling of solute transport in 3-D heterogeneous porous media with random source condition

During probabilistic analysis of flow and transport in porous media, the uncertainty due to spatial heterogeneity of governing parameters are often taken into account. The randomness in the source conditions also play a major role on the stochastic behavior in distribution of the dependent variable. The present paper is focused on studying the effect of both uncertainty in the governing system parameters as well as the input source conditions. Under such circumstances, a method is proposed which combines with stochastic finite element method (SFEM) and is illustrated for probabilistic analysis of concentration distribution in a 3-D heterogeneous porous media under the influence of random source condition. In the first step SFEM used for probabilistic solution due to spatial heterogeneity of governing parameters for a unit source pulse. Further, the results from the unit source pulse case have been used for the analysis of multiple pulse case using the numerical convolution when the source condition is a random process. The source condition is modeled as a discrete release of random amount of masses at fixed intervals of time. The mean and standard deviation of concentration is compared for the deterministic and the stochastic system scenarios as well as for different values of system parameters. The effect of uncertainty of source condition is also demonstrated in terms of mean and standard deviation of concentration at various locations in the domain.     

Effect of permeability variations on solute transport in highly heterogeneous porous media

The effect of aquifer heterogeneity on flow and solute transport in two-dimensional isotropic porous media was analyzed using the Monte Carlo method. The two-dimensional logarithmic permeability (ln K) was assumed to be a non-stationary random field with its increments being a truncated fractional Lévy motion (fLm). The permeability fields were generated using the modified successive random additions (SRA) algorithm code SRA3DC [http://www.iamg.org/CGEditor/index.htm]. The velocity and concentration fields were computed respectively for two-dimensional flow and transport with a pulse input using the finite difference codes of MODFLOW 2000 and MT3DMS. Two fLm control parameters, namely the width parameter (C) and the Lévy index (α), were varied systematically to examine their effect on the resulting permeability, flow velocity and concentration fields. We also computed the first- and second-spatial moments, the dilution index, as well as the breakthrough curves at different control planes with the corresponding concentration fields. In addition, the derived breakthrough curves were fitted using the continuous time random walk (CTRW) and the traditional advection-dispersion equation (ADE). Results indicated that larger C and smaller α both led to more heterogeneous permeability and velocity fields. The Lévy-stable distribution of increments in ln K resulted in a Lévy-stable distribution of increments in logarithm of the velocity (ln v). Both larger C and smaller α created sharper leading edges and wider tailing edges of solute plumes. Furthermore, a relatively larger amount of solute still remained in the domain after a relatively longer time transport for smaller α values. The dilution indices were smaller than unity and increased as C increased and α decreased. The solute plume and its second-spatial moments increased as C increased and α decreased, while the first-spatial moments of the solute plume were independent of C and α values. The longitudinal macrodispersivity was scale-dependent and increased as a power law function of time. Increasing C and decreasing α both resulted in an increase in longitudinal macrodispersivity. The transport in such highly heterogeneous media was slightly non-Gaussian with its derived breakthrough curves being slightly better fitted by the CTRW than the ADE, especially in the early arrivals and late-time tails.     

Models of sub-grid variability in numerical simulations of solute transport in heterogeneous porous formations: three-dimensional flow and effect of pore-scale dispersion

Modeling transport of contaminants in the earths subsurface relies on numerical solutions over grids with blocks larger than Darcys scale. The hydraulic conductivity is homogenized over the grid blocks and the plumes spreading is reduced as a consequence of the wiped-out variability. To compensate for this loss Rubin et al. (1999) proposed to augment mixing by block-effective dispersion coefficients, and Rubin et al. (2003) showed, by means of two dimensional simulations, how this concept can be applied in practice. In this paper, we present new solutions of the block-effective dispersion tensor for an axisymmetric exponential covariance model. In addition, we discuss the influence of pore-scale dispersion in both two- and three-dimensional applications.     

低频检波器频谱补偿算法建模及仿真

采用数值仿真的方法建立了动圈式检波器的仿真模型,推导检波器传递函数,通过对实际动圈式检波器频率特性的测试,验证了动圈式检波器数值仿真模型的正确性,分析动圈检波器频率响应特性,分别对动圈式检波器的自然频率、灵敏度、阻尼比进行仿真,对不同自然频率、灵敏度、阻尼比对检波器输出频率特性的影响进行了分析,使用二阶二次型反馈补偿方法,设计检波器低频补偿结构,结果表明：数值仿真模型与实际检波器的频率响应一致,补偿结构使检波器自然频率由2 Hz降低至0.2 Hz.

     

Modeling the effects of nonlinear equilibrium sorption on the transport of solute plumes in saturated heterogeneous porous media

Transport of sorbing solutes in 2D steady and heterogeneous flow fields is modeled using a particle tracking random walk technique. The solute is injected as an instantaneous pulse over a finite area. Cases of linear and Freundlich sorption isotherms are considered. Local pore velocity and mechanical dispersion are used to describe the solute transport mechanisms at the local scale. This paper addresses the impact of the degree of heterogeneity and correlation lengths of the log-hydraulic conductivity field as well as negative correlation between the log-hydraulic conductivity field and the log-sorption affinity field on the behavior of the plume of a sorbing chemical. Behavior of the plume is quantified in terms of longitudinal spatial moments: center-of-mass displacement, variance, 95% range, and skewness. The range appears to be a better measure of the spread in the plumes with Freundlich sorption because of plume asymmetry. It has been found that the range varied linearly with the travelled distance, regardless of the sorption isotherm. This linear relationship is important for extrapolation of results to predict behavior beyond simulated times and distances. It was observed that the flow domain heterogeneity slightly enhanced the spreading of nonlinearly sorbing solutes in comparison to that which occurred for the homogeneous flow domain, whereas the spreading enhancement in the case of linear sorption was much more pronounced. In the case of Freundlich sorption, this enhancement led to further deceleration of the solute plume movement as a result of increased retardation coefficients produced by smaller concentrations. It was also observed that, except for plumes with linear sorption, correlation between the hydraulic conductivity and the sorption affinity fields had minimal effect on the spatial moments of solute plumes with nonlinear sorption.