3D inverse modelling of groundwater flow at a fractured site using a stochastic continuum model with multiple statistical populations

 3D groundwater flow at the fractured site of Asp? (Sweden) is simulated. The aim was to characterise the site as adequately as possible and to provide measures on the uncertainty of the estimates. A stochastic continuum model is used to simulate both groundwater flow in the major fracture planes and in the background. However, the positions of the major fracture planes are deterministically incorporated in the model and the statistical distribution of the hydraulic conductivity is modelled by the concept of multiple statistical populations; each fracture plane is an independent statistical population. Multiple equally likely realisations are built that are conditioned to geological information on the positions of the major fracture planes, hydraulic conductivity data, steady state head data and head responses to six different interference tests. The experimental information could be reproduced closely. The results of the conditioning are analysed in terms of ensemble averaged average fracture plane conductivities, the ensemble variance of average fracture plane conductivities and the statistical distribution of the hydraulic conductivity in the fracture planes. These results are evaluated after each conditioning stage. It is found that conditioning to hydraulic head data results in an increase of the hydraulic conductivity variance while the statistical distribution of log hydraulic conductivity, initially Gaussian, becomes more skewed for many of the fracture planes in most of the realisations.     

DESIGN OF MODELS FOR ELECTROMAGNETIC SCALE MODELLING1

Electromagnetic scale modelling for most conductors encountered in prospecting requires models of conductivity in the range 10-1000 S/m. A polyester composite filled with aluminium fillers is used to design the required materials. The fillers are chosen with a high aspect ratio to achieve the desired conductivities without altering mechanical properties. The design process, consisting of a chemical reaction, is simple, repeatable and easy to realize in a geophysical laboratory. The conductivity of the designed material is estimated by an inductive method based upon the variation of the quality factor of a coil in which the samples are inserted. This method is much less influenced by the distribution and orientation of fillers in the resin matrix than a galvanic method. The conductivities obtained fill the major part of the interval 10-1000 S/m without requiring unreasonable mixing ratios. The homogeneity of the material was tested and the anisotropy was low at high conductivities. The electrical properties do not vary with time in free air or when immersed in salt water. We propose standard curves giving conductivities of composites filled with aluminium fibres and flakes. A large cylindrical model of fixed resin-to-filler ratio was made to show the capability of the technique to produce models of useful size. Its measured conductivity agrees well with the expected conductivity from the established standard curves. A scale modelling experiment with the horizontal loop technique was carried out over a horizontal thin sheet made of the material, and again the conductivity obtained from the results agrees reasonably well with the expected value. The noted difference between the inductively and galvanically measured conductivities of the proposed material presently restricts the use of the technique to EM scale experiments for which the host rock is considered very resistive.     

Calibration of base flow separation methods with streamflow conductivity

The conductivity mass-balance (CMB) method can be used to calibrate analytical base flow separation methods. The principal CMB assumptions are base flow conductivity is equal to streamflow conductivity at lowest flows, runoff conductivity is equal to streamflow conductivity at highest flows, and base flow and runoff conductivities are assumed to be constants over the period of record. To test the CMB assumptions, fluid conductivities of ground water, surface runoff, and streamflow were measured during wet and dry conditions in a 12-km(2) stream basin. Ground water conductivities at wells varied an average of 6% from dry to wet conditions, while stream conductivities varied 58%. Shallow ground water conductivity varied significantly with distance from the stream, with lowest conductivities of 87 microS/cm near the divide, a maximum of 520 microS/cm 59 m from the stream, and 215 microS/cm 22 m from the stream. Runoff conductivities measured in three rain events remained nearly constant, with lower conductivities of 35 microS/cm near the divide and 50 microS/cm near the stream. The CMB method was applied to the records from 10 USGS stream-gauging stations in Texas, Kentucky, Georgia, and Florida to calibrate the USGS base flow separation technique, HYSEP, by varying the time parameter 2N*. There is a statistically significant relationship between basin areas and calibrated values of 2N*, expressed as N = 0.46A(0.44), with N in days and A in km(2). The widely accepted relationship N = 0.83A(0.2) is not valid for these basins. Other analytic methods can also be calibrated with the CMB method.     

Diagrammatic theory of effective hydraulic conductivity

This work presents a stochastic diagrammatic theory for the calculation of the effective hydraulic conductivity of heterogeneous media. The theory is based on the mean-flux series expansion of a log-normal hydraulic conductivity medium in terms of diagrammatic representations and leads to certain general results for the effective hydraulic conductivity of three-dimensional media. A selective summation technique is used to improve low-order perturbation analysis by evaluating an infinite set of diagrammatic terms with a specific topological structure that dominates the perturbation series. For stochastically isotropic media the selective summation yeilds the anticipated exponential expression for the effective hydraulic conductivity. This expression is extended to stochastically anisotropic media. It is also shown that in the case of non homogeneous media the uniform effective hydraulic conductivity is replaced by a non-local tensor kernel, for which general diagrammatic expressions are obtained. The non-local kernel leads to the standard exponential behavior for the effective hydraulic conductivity at the homogeneous limit.     

Diagrammatic theory of effective hydraulic conductivity

This work presents a stochastic diagrammatic theory for the calculation of the effective hydraulic conductivity of heterogeneous media. The theory is based on the mean-flux series expansion of a log-normal hydraulic conductivity medium in terms of diagrammatic representations and leads to certain general results for the effective hydraulic conductivity of three-dimensional media. A selective summation technique is used to improve low-order perturbation analysis by evaluating an infinite set of diagrammatic terms with a specific topological structure that dominates the perturbation series. For stochastically isotropic media the selective summation yeilds the anticipated exponential expression for the effective hydraulic conductivity. This expression is extended to stochastically anisotropic media. It is also shown that in the case of non homogeneous media the uniform effective hydraulic conductivity is replaced by a non-local tensor kernel, for which general diagrammatic expressions are obtained. The non-local kernel leads to the standard exponential behavior for the effective hydraulic conductivity at the homogeneous limit.     

Data assimilation methods for estimating a heterogeneous conductivity field by assimilating transient solute transport data via ensemble Kalman filter

An ensemble Kalman filter (EnKF) is developed to identify a hydraulic conductivity distribution in a heterogeneous medium by assimilating solute concentration measurements of solute transport in the field with a steady‐state flow. A synthetic case with the mixed Neumann/Dirichlet boundary conditions is designed to investigate the capacity of the data assimilation methods to identify a conductivity distribution. The developed method is demonstrated in 2‐D transient solute transport with two different initial instant solute injection areas. The influences of the observation error and model error on the updated results are considered in this study. The study results indicate that the EnKF method will significantly improve the estimation of the hydraulic conductivity field by assimilating solute concentration measurements. The larger area of the initial distribution and the more observed data obtained, the better the calculation results. When the standard deviation of the observation error varies from 1% to 30% of the solute concentration measurements, the simulated results by the data assimilation method do not change much, which indicates that assimilation results are not very sensitive to the standard deviation of the observation error in this study. When the inflation factor is more than 1.0 to enlarge the model error by increasing the forecast error covariance matrix, the updated results of the hydraulic conductivity by the data assimilation method are not good at all. Copyright © 2012 John Wiley & Sons, Ltd.     

Unsaturated flows in soils with self-similar hydraulic conductivity distribution

In this article, we are concerned with the statistics of steady unsaturated flow in soils with a fractal hydraulic conductivity distribution. It is assumed that the spatial distribution of log hydraulic conductivity can be described as an isotropic stochastic fractal process. The impact of the fractal dimension of this process, the soil pore-size distribution parameter, and the characteristic length scale on the variances of tension head and the effective conductivity is investigated. Results are obtained for one-dimensional and three-dimensional flows. Our results indicate that the tension head variance is scale-dependent for fractal distribution of hydraulic conductivity. Both tension head variance and effective hydraulic conductivity depend strongly on the fractal dimension. The soil pore-size distribution parameter is important in reducing the variability of the unsaturated hydraulic conductivity and of the fluxes.     

The Application of Combined Seismic and Electrical Measurements to the Determination of the Hydraulic Conductivity of a Lake Bed

The hydraulic properties of lake beds control the interactions between lakes and ground water systems, but these properties are normally difficult to measure directly. The authors'method combines seismic reflection and electrical measurements to map the relative hydraulic conductivity of lake bed sediments. A shipboard seismic profiling system provides sediment thickness, while a towed electrical array yields longitudinal conductance and electrical chargeability. The sediment's leakance (hydraulic conductivity/thickness) can be calculated from the longitudinal conductance data. Leakance may then be converted to relative hydraulic conductivity through the seismically derived sediment thicknesses. Simultaneously acquired electrical chargeability provides an independent measure of clay content. The seismic and electrical systems are computer automated and yield production rates of approximately five line-kilometers/hour or 300 electrical soundings/hour. The systems provide continuous hydraulic information along the ship track rather than the point information derived from coring.
The procedure and systems have been used to map the bed of Lake Michigan offshore from an area of heavy pumpage. This location has been chosen to test the method because lake water has intruded the aquifer in plumes largely controlled by lake bed hydraulics. Mapping these plumes onshore permits the inference of the spatial distribution of offshore hydraulic conductivities. Offshore seepage measurements and numerical, chemical transport modeling of this site have confirmed the reliability of the geophysically derived hydraulic conductivities and have also demonstrated the improvement in numerical results achieved through the availability of spatially determined hydraulic conductivities.     

Stochastic analysis of groundwater flow in aquifers with leakage

 Stochastic analysis of one- and two-dimensional flow through a shallow semi-confined aquifer with spatially variable hydraulic conductivity K represented by a stationary (statistically homogeneous) random process is carried out by using the spectral technique. The hydraulic head covariance functions for flows in a semi-confined aquifer bounded by a leaky layer above and an impervious stratum below are derived by assuming that the randomness forcing the head variation to originate from the hydraulic conductivity field of the aquifer. The head covariance functions are studied using two convenient forms of the logarithmic hydraulic conductivity process. The results demonstrate the significant reduction in the head variances and covariances due to the presence of a leaky layer. The hydraulic head correlation distance is also reduced greatly due to the presence of the leaky layer.     

A method to estimate hydraulic conductivity while drilling

A field test and analysis method has been developed to estimate the vertical distribution of hydraulic conductivity in shallow unconsolidated aquifers. The field method uses fluid injection ports and pressure transducers in a hollow auger that measure the hydraulic head outside the auger at several distances from the injection point. A constant injection rate is maintained for a duration time sufficient for the system to become steady state. Exploiting the analogy between electrical resistivity in geophysics and hydraulic flow two methods are used to estimate conductivity with depth: a half-space model based on spherical flow from a point injection at each measurement site, and a one-dimensional inversion of an entire dataset.

The injection methodology, conducted in three separate drilling operations, was investigated for repeatability, reproducibility, linearity, and for different injection sources. Repeatability tests, conducted at 10 levels, demonstrated standard deviations of generally less than 10%. Reproducibility tests conducted in three, closely spaced drilling operations generally showed a standard deviation of less than 20%, which is probably due to lateral variations in hydraulic conductivity. Linearity tests, made to determine dependency on flow rates, showed no indication of a flow rate bias. In order to obtain estimates of the hydraulic conductivity by an independent means, a series of measurements were made by injecting water through screens installed at two separate depths in a monitoring pipe near the measurement site. These estimates differed from the corresponding estimates obtained by injection in the hollow auger by a factor of less than 3.5, which can be attributed to variations in geology and the inaccurate estimates of the distance between the measurement and the injection sites at depth.     

A comparative study of closed-form hydraulic conductivity prediction models

The Mualem and the Burdine hydraulic conductivity prediction models are considered in combination with the van Genuchten analytical retention curve, as well as the Brooks and Corey prediction model. An equivalence is presented between the retention curves of these models. A comparative study follows between hydraulic conductivities that are based on equivalent retention curves. A unified presentation of prediction models provides a framework for the whole analysis. The treatment of the equivalence problem consists in a minimization procedure characterized by uncoupling of the parameters and analytical evaluation of the objective function. Exact analytical equivalence relations are given for significant parts of the parameter ranges, and, for the remaining parts, analytical approximations are proposed. The comparisons between hydraulic conductivities are carried out via an inequality analysis. It is shown that the hydraulic conductivity of the Burdine model is less than that of the other models for extended ranges of equivalent parameters.     

Simple Hydraulic Conductivity Estimation by the Kalman Filtered Double Constraint Method

This paper presents the Kalman Filtered Double Constraint Method (DCM‐KF) as a technique to estimate the hydraulic conductivities in the grid blocks of a groundwater flow model. The DCM is based on two forward runs with the same initial grid block conductivities, but with alternating flux‐head conditions specified on parts of the boundary and the wells. These two runs are defined as: (1) the flux run, with specified fluxes (recharge and well abstractions), and (2) the head run, with specified heads (measured in piezometers). Conductivities are then estimated as the initial conductivities multiplied by the fluxes obtained from the flux run and divided by the fluxes obtained from the head run. The DCM is easy to implement in combination with existing models (e.g., MODFLOW). Sufficiently accurate conductivities are obtained after a few iterations. Because of errors in the specified head‐flux couples, repeated estimation under varying hydrological conditions results in different conductivities. A time‐independent estimate of the conductivities and their inaccuracy can be obtained by a simple linear KF with modest computational requirements. For the Kleine Nete catchment, Belgium, the DCM‐KF yields sufficiently accurate calibrated conductivities. The method also results in distinguishing regions where the head‐flux observations influence the calibration from areas where it is not able to influence the hydraulic conductivity.     

Numerical simulation of groundwater flow patterns using flux as upper boundary

Since the 1960s, most of the studies on groundwater flow systems by analytical and numerical modelling have been based on given‐head upper boundaries. The disadvantage of the given‐head approach is that the recharge into and discharge from a basin vary with changes in hydraulic conductivity and/or basin geometry. Consequently, flow patterns simulated with given‐head boundaries but with different hydraulic conductivities and/or basin geometry may not reflect the effects of these variables. We conducted, therefore, numerical simulations of groundwater flow in theoretical drainage basins using flux as the upper boundary and realistically positioned fluid‐potential sinks while changing the infiltration intensity, hydraulic conductivities, and geometric configuration of the basin. The simulated results demonstrate that these variables are dominant factors controlling the flow pattern in a laterally closed drainage basin. The ratio of infiltration intensity to hydraulic conductivity (R_ic) has been shown to be an integrated pattern‐parameter in a basin with a given geometric configuration and possible fluid‐potential‐sink distribution. Successively, the changes in flow patterns induced by stepwise reductions in R_ic are identical, regardless of whether the reductions are due to a decrease in infiltration intensity or an increase in hydraulic conductivity. The calculated examples show five sequential flow patterns containing (i) only local, (ii) local–intermediate, (iii) local–intermediate–regional, (iv) local–regional, and (v) just regional flow systems. The R_ic was found to determine also whether a particular sink is active or not as a site of discharge. Flux upper boundary is preferable for numerical simulation when discussing the flow patterns affected by a change of infiltration, the hydraulic conductivity, or the geometry of a basin. Copyright © 2012 John Wiley & Sons, Ltd.     

Estimation of hydraulic conductivity in an alluvial system using temperatures

Well water temperatures are often collected simultaneously with water levels; however, temperature data are generally considered only as a water quality parameter and are not utilized as an environmental tracer. In this paper, water levels and seasonal temperatures are used to estimate hydraulic conductivities in a stream-aquifer system. To demonstrate this method, temperatures and water levels are analyzed from six observation wells along an example study site, the Russian River in Sonoma County, California. The range in seasonal ground water temperatures in these wells varied from <0.2 degrees C in two wells to approximately 8 degrees C in the other four wells from June to October 2000. The temperature probes in the six wells are located at depths between 3.5 and 7.1 m relative to the river channel. Hydraulic conductivities are estimated by matching simulated ground water temperatures to the observed ground water temperatures. An anisotropy of 5 (horizontal to vertical hydraulic conductivity) generally gives the best fit to the observed temperatures. Estimated conductivities vary over an order of magnitude in the six locations analyzed. In some locations, a change in the observed temperature profile occurred during the study, most likely due to deposition of fine-grained sediment and organic matter plugging the streambed. A reasonable fit to this change in the temperature profile is obtained by decreasing the hydraulic conductivity in the simulations. This study demonstrates that seasonal ground water temperatures monitored in observation wells provide an effective means of estimating hydraulic conductivities in alluvial aquifers.     

A partial explanation of the dependency of hydraulic conductivity on positive pore water pressure in peat soils

Part of the relationship between positive pore water pressures and hydraulic conductivity in peat soils may be explained by accumulations of methane bubbles. We show how compression and expansion of gas bubbles with changes in pore water pressure could cause changes in hydraulic conductivity and thus help to explain some observations of dependency of hydraulic conductivity in peats on pore water pressure. Consideration is also given to the effect on hydraulic conductivities of methane gas going into solution with increase in pore water pressure.     

Evaluating the effect of using artificial pore water on the quality of laboratory hydraulic conductivity measurements of peat

Pore dilation, the compaction of humic acids on peat fibres due to the process of flocculation, causes the hydraulic conductivity of peat to increase with increasing pore water electrical conductivity. This is a reversible process and a reduction in the pore water conductivity produces a decrease in the hydraulic conductivity due to the constriction of pores. We verify how this dilation and constriction of pores, resulting from the application of artificial pore water (primarily deionized water), affects laboratory measurements of the hydraulic conductivity of peat. Repeat measurements of the hydraulic conductivity were performed on samples of Sphagnum peat. It is shown that the application of deionized water during constant head permeameter tests causes a significant decrease in the hydraulic conductivity. Between tests, the hydraulic conductivity of the peat continues to decline without an associate decrease in the pore water electrical conductivity because of a lagged pore constriction effect. We suggest that the use of artificially high or low pore water electrical conductivities, during laboratory hydraulic conductivity measurements, is likely to lead to significant errors. Experimental protocols must, therefore, be revised to take better account of the pore water chemistry. The ionic concentrations of the natural pore fluid should be replicated during hydraulic conductivity tests, either by using pore fluid extracted from the study site or by artificially replicating the major ionic composition of the natural pore fluid. In addition, prior to the hydraulic conductivity measurements, peat samples should be flushed with this solution until the hydraulic conductivity stabilizes and the samples subsequently allowed to equilibrate. Copyright © 2010 John Wiley & Sons, Ltd.     

The relationship of total dissolved solids measurements to bulk electrical conductivity in an aquifer contaminated with hydrocarbon

A recent conceptual model links higher bulk conductivities at hydrocarbon impacted sites to higher total dissolved solids (TDS) resulting from enhanced mineral weathering due to acids produced during biodegradation. In this study, we evaluated the above model by investigating the vertical distribution of bulk conductivity, TDS, and specific conductance in groundwater. The results showed higher TDS at contaminated locations consistent with the above model. Further, steep vertical gradients in bulk conductivity and TDS suggest vertical and spatial heterogeneity at the site. We observed that at fluid conductivities <40 mS/m, bulk conductivity was inversely related to fluid conductivity, but at fluid conductivities >40 mS/m, bulk conductivity increased with increasing fluid conductivity. However, at fluid conductivities >80 mS/m, bulk conductivities increased without a corresponding increase in fluid conductivity, resulting in a poor correlation between bulk conductivity and fluid conductivity for the contaminated samples. This suggests that electrolytic conductivity was not completely responsible for the observed variability in bulk conductivity. We suggest two possible reasons for the inverse relationship at low fluid conductivity and poor positive correlation at high fluid conductivity: (1) geochemical heterogeneity due to biological processes not captured at a scale comparable to the bulk conductivity measurement and (2) variability in the surface conductivity, consistent with a simple petrophysical model that suggests higher surface conductivity for contaminated sediments. We conclude that biodegradation processes can impact both electrolytic and surface conduction properties of contaminated sediments and these two factors can account for the higher bulk conductivities observed in sediments impacted by hydrocarbon.     

Determining hydraulic resistance in gravel‐bed rivers from the dynamics of their water surfaces

Traditionally, approaches to account for the effect of the boundary roughness of a gravel‐bed river have used a grain‐size index of the bed surface as a surrogate for hydraulic resistance. The use of a single grain‐size does not take into account the spatial heterogeneity in the bed surface and how this heterogeneity imparts resistance on the flow, nor the way in which this relationship changes with variables such as flow stage. A new technique to remotely quantify hydraulic resistance is proposed. It is based on measuring the dynamics of a river's water surface and relating this to the actual hydraulic resistance created by a rough sediment boundary. The water surface dynamics are measured using a new acoustic technique, grazing angle sound propagation (GRASP). This proposed method to measure hydraulic resistance is based on a greater degree of physical reasoning, and this is discussed in the letter. By measuring acoustically the temporal dynamics of turbulent water surfaces over a water‐worked gravel bed in a laboratory flume, a dependency is demonstrated between the temporal variation in the reflected acoustic pressure and measured hydraulic resistance. It is shown that the standard deviation in acoustic pressure decreases with increasing hydraulic resistance. This is shown to apply for a range of relative submergences and bed slopes that are typical of gravel‐bed rivers. This remote sensing technique is both rapid and inexpensive, and has the potential to be applied to natural river channels and to other environmental turbulent flows, such as overland flows. A whole new class of low‐cost, remote and non‐intrusive instruments could be developed as a result and used in a wide range of hydraulic and hydrological applications. Copyright © 2006 John Wiley & Sons, Ltd.     

In situ Mixing of Organic Matter Decreases Hydraulic Conductivity of Denitrification Walls in Sand Aquifers

In a previous study, a denitrification wall was constructed in a sand aquifer using sawdust as the carbon substrate. Ground water bypassed around this sawdust wall due to reduced hydraulic conductivity. We investigated potential reasons for this by testing two new walls and conducting laboratory studies. The first wall was constructed by mixing aquifer material in situ without substrate addition to investigate the effects of the construction technique (mixed wall). A second, biochip wall, was constructed using coarse wood chips to determine the effect of size of the particles in the amendment on hydraulic conductivity. The aquifer hydraulic conductivity was 35.4 m/d, while in the mixed wall it was 2.8 m/d and in the biochip wall 3.4 m/d. This indicated that the mixing of the aquifer sands below the water table allowed the particles to re-sort themselves into a matrix with a significantly lower hydraulic conductivity than the process that originally formed the aquifer. The addition of a coarser substrate in the biochip wall significantly increased total porosity and decreased bulk density, but hydraulic conductivity remained low compared to the aquifer. Laboratory cores of aquifer sand mixed under dry and wet conditions mimicked the reduction in hydraulic conductivity observed in the field within the mixed wall. The addition of sawdust to the laboratory cores resulted in a significantly higher hydraulic conductivity when mixed dry compared to cores mixed wet. This reduction in the hydraulic conductivity of the sand/sawdust cores mixed under saturated conditions repeated what occurred in the field in the original sawdust wall. This indicated that laboratory investigations can be a useful tool to highlight potential reductions in field hydraulic conductivities that may occur when differing materials are mixed under field conditions.     

Identification of Groundwater Parameters Using an Adaptative Multiscale Method

The identification of groundwater parameters in heterogeneous systems is a major challenge in groundwater modeling. Flexible parameterization methods are needed to assess the complexity of the spatial distributions of these parameters in real aquifers. In this article, we introduce an adaptative parameterization to identify the distribution of hydraulic conductivity within the large‐scale (4400 km²) Upper Rhine aquifer. The method is based on adaptative multiscale triangulation (AMT) coupled with an inverse problem procedure that identifies the parameters' distributions by reducing the error between measured and simulated heads. The AMT method has the advantage of combining both zonation and interpolation approaches. The AMT method uses area‐based interpolation rather than an interpolation based on stochastic features. The method is applied to a standard 2D groundwater model that takes into account the interactions between the aquifer and surface water bodies, groundwater recharge, and pumping wells. The simulation period covers 204 months, from January 1986 to December 2002. Recordings at 109 piezometers are used for model calibration. The simulated heads are globally quite accurate and reproduce the main dynamics of the system. The local hydraulic conductivities resulting from the AMT method agree qualitatively with existing local experimental observations across the Rhine aquifer.