Comparison of Hydraulic Tomography with Traditional Methods at a Highly Heterogeneous Site

Over the past several decades, different groundwater modeling approaches of various complexities and data use have been developed. A recently developed approach for mapping hydraulic conductivity (K) and specific storage (S_s) heterogeneity is hydraulic tomography, the performance of which has not been compared to other more “traditional” methods that have been utilized over the past several decades. In this study, we compare seven methods of modeling heterogeneity which are (1) kriging, (2) effective parameter models, (3) transition probability/Markov Chain geostatistics models, (4) geological models, (5) stochastic inverse models conditioned to local K data, (6) hydraulic tomography, and (7) hydraulic tomography conditioned to local K data using data collected in five boreholes at a field site on the University of Waterloo (UW) campus, in Waterloo, Ontario, Canada. The performance of each heterogeneity model is first assessed during model calibration. In particular, the correspondence between simulated and observed drawdowns is assessed using the mean absolute error norm, (L₁), mean square error norm (L₂), and correlation coefficient (R) as well as through scatterplots. We also assess the various models on their ability to predict drawdown data not used in the calibration effort from nine pumping tests. Results reveal that hydraulic tomography is best able to reproduce these tests in terms of the smallest discrepancy and highest correlation between simulated and observed drawdowns. However, conditioning of hydraulic tomography results with permeameter K data caused a slight deterioration in accuracy of drawdown predictions which suggests that data integration may need to be conducted carefully.     

A Type-Curve Method for the Analysis of Pumping Tests with Piecewise-Linear Pumping Rates

Aquifer hydraulic parameters are commonly inferred from constant-rate pumping tests, while variable pumping rates are frequently encountered in actual field conditions. In this study, we propose a generally applicable dimensionless form of the analytical solution for variable-rate pumping tests in confined aquifers. In particular, we adopt a piecewise-linear fitting of variable pumping rates and propose a new type-curve method for estimating the hydraulic conductivity (K ) and specific storage (S_s ) of the investigated confined aquifer. For each test, a series of type curves, which depend on the variable pumping rates, the location of observation wells and the introduced first dimensionless inflection time, need to be provided for matching the observed drawdown data on a log-log graph. We first demonstrate the applicability and robustness of this method through a synthetic pumping test. Subsequently, we apply this method to analyze drawdown data from four pumping tests conducted within a multilayered aquifer/aquitard system in Wuxi city, Jiangsu Province, China. The parameter estimates are then compared with those reported by PEST. The K and S_s values estimated by the new type-curve method are found to be quite close to PEST-based estimates. Parameter estimation results demonstrate the difference in K and S_s values between observation wells. The difference could be attributed to the spatial heterogeneity in K and S_s . A future research topic may focus on the characterization of K and S_s heterogeneity with the currently available drawdown data from variable-rate pumping tests.     

Field Study of Subsurface Heterogeneity with Steady‐State Hydraulic Tomography

Remediation of subsurface contamination requires an understanding of the contaminant (history, source location, plume extent and concentration, etc.), and, knowledge of the spatial distribution of hydraulic conductivity (K) that governs groundwater flow and solute transport. Many methods exist for characterizing K heterogeneity, but most if not all methods require the collection of a large number of small‐scale data and its interpolation. In this study, we conduct a hydraulic tomography survey at a highly heterogeneous glaciofluvial deposit at the North Campus Research Site (NCRS) located at the University of Waterloo, Waterloo, Ontario, Canada to sequentially interpret four pumping tests using the steady‐state form of the Sequential Successive Linear Estimator (SSLE) ( Yeh and Liu 2000 ). The resulting three‐dimensional (3D) K distribution (or K‐tomogram) is compared against: ( 1 ) K distributions obtained through the inverse modeling of individual pumping tests using SSLE, and ( 2 ) effective hydraulic conductivity (K_eff) estimates obtained by automatically calibrating a groundwater flow model while treating the medium to be homogeneous. Such a K_eff is often used for designing remediation operations, and thus is used as the basis for comparison with the K‐tomogram. Our results clearly show that hydraulic tomography is superior to the inversions of single pumping tests or K_eff estimates. This is particularly significant for contaminated sites where an accurate representation of the flow field is critical for simulating contaminant transport and injection of chemical and biological agents used for active remediation of contaminant source zones and plumes.     

Evaluation of Hydraulic Conductivity Estimates from Various Approaches with Groundwater Flow Models

Significant efforts have been expended for improved characterization of hydraulic conductivity (K) and specific storage (S_s) to better understand groundwater flow and contaminant transport processes. Conventional methods including grain size analyses (GSA), permeameter, slug, and pumping tests have been utilized extensively, while Direct Push-based Hydraulic Profiling Tool (HPT) surveys have been developed to obtain high-resolution K estimates. Moreover, inverse modeling approaches based on geology-based zonations, and highly parameterized Hydraulic Tomography (HT) have also been advanced to map spatial variations of K and S_s between and beyond boreholes. While different methods are available, it is unclear which one yields K estimates that are most useful for high resolution predictions of groundwater flow. Therefore, the main objective of this study is to evaluate various K estimates at a highly heterogeneous field site obtained with three categories of characterization techniques including: (1) conventional methods (GSA, permeameter, and slug tests); (2) HPT surveys; and (3) inverse modeling based on geology-based zonations and highly parameterized approaches. The performance of each approach is first qualitatively analyzed by comparing K estimates to site geology. Then, steady-state and transient groundwater flow models are employed to quantitatively assess various K estimates by simulating pumping tests not used for parameter estimation. Results reveal that inverse modeling approaches yield the best drawdown predictions under both steady and transient conditions. In contrast, conventional methods and HPT surveys yield biased predictions. Based on our research, it appears that inverse modeling and data fusion are necessary steps in predicting accurate groundwater flow behavior.     

Hydraulic Conductivity Calibration of Logging NMR in a Granite Aquifer,Laramie Range,Wyoming

In granite aquifers, fractures can provide both storage volume and conduits for groundwater. Characterization of fracture hydraulic conductivity (K) in such aquifers is important for predicting flow rate and calibrating models. Nuclear magnetic resonance (NMR) well logging is a method to quickly obtain near-borehole hydraulic conductivity (i.e., K_NMR) at high-vertical resolution. On the other hand, FLUTe flexible liner technology can produce a K profile at comparable resolution but requires a fluid driving force between borehole and formation. For three boreholes completed in a fractured granite, we jointly interpreted logging NMR data and FLUTe K estimates to calibrate an empirical equation for translating borehole NMR data to K estimates. For over 90% of the depth intervals investigated from these boreholes, the estimated K_NMR are within one order of magnitude of K_FLUTe. The empirical parameters obtained from calibrating the NMR data suggest that “intermediate diffusion” and/or “slow diffusion” during the NMR relaxation time may occur in the flowing fractures when hydraulic aperture are sufficiently large. For each borehole, “intermediate diffusion” dominates the relaxation time, therefore assuming “fast diffusion” in the interpretation of NMR data from fractured rock may lead to inaccurate K_NMR estimates. We also compare calibrations using inexpensive slug tests that suggest reliable K_NMR estimates for fractured rock may be achieved using limited calibration against borehole hydraulic measurements.     

A New Randomized Binary Prior Model for Hydraulic Tomography in Fractured Aquifers

We present a novel pilot-point-based hydraulic tomography (HT) inversion procedure to delineate preferential flow paths and estimate hydraulic properties in a fractured aquifer. Our procedure considers a binary prior model developed using a randomized algorithm. The randomized algorithm involves discretizing the domain into grid cells, assigning a binary label to each cell, traversing the grid randomly, and choosing the optimal grid configuration cell-by-cell. This binary prior model is used to guide the placement of pilot points and to constrain aquifer parameters during pilot-point-based HT inversion. A two-dimensional fractured granite rock block was considered to test our methodology under controlled laboratory conditions. Multiple pumping tests were conducted at selected ports and the pressure responses were monitored. The pumping datasets thus obtained were preprocessed using median filters to remove random noise, and then analyzed using the proposed procedure. The proposed binary prior algorithm was implemented in C++ by supplying the forward groundwater model, HydroGeoSphere (HGS). Pilot-point-assisted HT inversion was performed using the parameter-estimation tool, coupled to HGS. The resulting parameter distributions were assessed by: (1) a visual comparison of the K- and S_s-tomograms with the known topology of the fractures and (2) comparing model predictions with measurements made at two validation ports that were not used in calibration. The performance assessment revealed that HT with the proposed randomized binary prior could be used to recover fracture-connectivity and to predict drawdowns in fractured aquifers with reasonable accuracy, when compared to a conventional pilot-point inversion scheme.     

Performance of Gradient and Gradient-Free Optimizers in Transient Hydraulic Tomography

Sub-surface characterization in fractured aquifers is challenging due to the co-existence of contrasting materials namely matrix and fractures. Transient hydraulic tomography (THT) is proved to be an efficient and robust technique to estimate hydraulic (K_m, K_f) and storage (S_m, S_f) properties in such complex hydrogeologic settings. However, performance of THT is governed by data quality and optimization technique used in inversion. We assessed the performance of gradient and gradient-free optimizers with THT inversion. Laboratory experiments were performed on a two-dimensional, granite rock (80 cm × 45 cm × 5 cm) with known fracture pattern. Cross-hole pumping experiments were conducted at 10 ports (located on fractures), and time-drawdown responses were monitored at 25 ports (located on matrix and fractures). Pumping ports were ranked based on weighted signal-to-noise ratio (SNR) computed at each observation port. Noise-free, good quality (SNR > 100) datasets were inverted using Levenberg–Marquardt: LM (gradient) and Nelder–Mead: NM (gradient-free) methods. All simulations were performed using a coupled simulation-optimization model. Performance of the two optimizers is evaluated by comparing model predictions with observations made at two validation ports that were not used in simulation. Both LM and NM algorithms have broadly captured the preferential flow paths (fracture network) via K and S tomograms, however LM has outperformed NM during validation ( $$). Our results conclude that, while method of optimization has a trivial effect on model predictions, exclusion of low quality (SNR ≤ 100) datasets can significantly improve the model performance.     

Anisotropy and depth‐related heterogeneity of hydraulic conductivity in a bog peat. II: modelling the effects on groundwater flow

Anisotropy and heterogeneity of hydraulic conductivity (K) are seldom considered in models of mire hydrology. We investigated the effect of anisotropy and heterogeneity on groundwater flow in bog peat using a steady‐state groundwater model. In five model simulations, four sets of K data were used. The first set comprised measured K values from an anisotropic and heterogeneous bog peat. These data were aggregated to produce the following simplified data sets: an isotropic and heterogeneous distribution of K; an isotropic and homogeneous distribution; and an anisotropic and homogeneous distribution. We demonstrate that, where anisotropy and heterogeneity exist, groundwater flow in bog peat is complex. Fine‐scale variations in K have the potential to influence patterns and rates of groundwater flow. However, for our data at least, it is heterogeneity and not anisotropy that has the greater influence on producing complex patterns of groundwater flow. We also demonstrate that patterns and rates of groundwater flow are simplified and reduced when measured K values are aggregated to create a more uniform distribution of K. For example, when measured K values are aggregated to produce isotropy and homogeneity, the rate of modelled seepage is reduced by 28%. We also show that when measured K values are used, the presence of a drainage ditch can increase seepage through a modelled cross‐section. Our work has implications for the accurate interpretation of hydraulic head data obtained from peat soils, and also the understanding of the effect of drainage ditches on patterns and rates of groundwater flow. Copyright © 2002 John Wiley & Sons, Ltd.     

Cost Comparisons of Aquifer Heterogeneity Characterization Methods

The characterization of heterogeneity in hydraulic conductivity (K) is a major challenge for subsurface remediation projects. There are a number of field studies that compare the K estimates obtained using various techniques, but to our knowledge, no field‐based studies exists that compare the performance of estimated K heterogeneity fields or the associated characterization costs. In this paper, we compare the costs of characterizing the three‐dimensional K heterogeneity and its uncertainty estimates of a glaciofluvial aquifer‐aquitard sequence at a 15 m × 15 m × 18 m field site situated on the University of Waterloo campus. We compare geostatistical analysis of high resolution permeameter K data obtained from repacked core samples in five boreholes and hydraulic tomography analysis of four pumping tests consisting of up to 41 monitoring points per test. Aside from the comparison of costs, we also assess the performance of each method by predicting several pumping tests. Our analysis reveals that hydraulic tomography is somewhat more costly than the geostatistical analysis of high resolution permeameter K data due to the higher capital costs associated with the method. However, the equipment may be reused at other sites; hence these costs may be recovered over the life of the equipment. More significantly, hydraulic tomography is able to capture the most important features of the aquifer‐aquitard sequence leading to more accurate predictions of independent pumping tests. This suggests that more robust remediation systems may be designed if site characterization is performed with hydraulic tomography.     

Inverse modelling using PS‐InSAR data for improved land subsidence simulation in Las Vegas Valley,Nevada

Las Vegas Valley has had a long history of groundwater development and subsequent surface deformation. InSAR interferograms have revealed detailed and complex spatial patterns of subsidence in the Las Vegas Valley area that do not coincide with major pumping regions. This research represents the first effort to use high spatial and temporal resolution subsidence observations from InSAR and hydraulic head data to inversely calibrate transmissivities (T), elastic and inelastic skeletal storage coefficients (S_ke and S_kv) of the developed‐zone aquifer and conductance (CR) of the basin‐fill faults for the entire Las Vegas basin. The results indicate that the subsidence observations from InSAR are extremely beneficial for accurately quantifying hydraulic parameters, and the model calibration results are far more accurate than when using only groundwater levels as observations, and just a limited number of subsidence observations. The discrepancy between distributions of pumping and greatest levels of subsidence is found to be attributed to spatial variations in clay thickness. The Eglington fault separates thicker interbeds to the northwest from thinner interbeds to the southeast and the fault may act as a groundwater‐flow barrier and/or subsidence boundary, although the influence of the groundwater barrier to this area is found to be insignificant. Copyright © 2016 John Wiley & Sons, Ltd.     

Transient Recharge Estimability Through Field‐Scale Groundwater Model Calibration

The estimation of recharge through groundwater model calibration is hampered by the nonuniqueness of recharge and aquifer parameter values. It has been shown recently that the estimability of spatially distributed recharge through calibration of steady‐state models for practical situations (i.e., real‐world, field‐scale aquifer settings) is limited by the need for excessive amounts of hydraulic‐parameter and groundwater‐level data. However, the extent to which temporal recharge variability can be informed through transient model calibration, which involves larger water‐level datasets, but requires the additional consideration of storage parameters, is presently unknown for practical situations. In this study, time‐varying recharge estimates, inferred through calibration of a field‐scale highly parameterized groundwater model, are systematically investigated subject to changes in (1) the degree to which hydraulic parameters including hydraulic conductivity (K) and specific yield (S_y) are constrained, (2) the number of water‐level calibration targets, and (3) the temporal resolution (up to monthly time steps) at which recharge is estimated. The analysis involves the use of a synthetic reality (a reference model) based on a groundwater model of Uley South Basin, South Australia. Identifiability statistics are used to evaluate the ability of recharge and hydraulic parameters to be estimated uniquely. Results show that reasonable estimates of monthly recharge (<30% recharge root‐mean‐squared error) require a considerable amount of transient water‐level data, and that the spatial distribution of K is known. Joint estimation of recharge, S_y and K, however, precludes reasonable inference of recharge and hydraulic parameter values. We conclude that the estimation of temporal recharge variability through calibration may be impractical for real‐world settings.     

Determining hydraulic properties of a loam soil by alternative infiltrometer techniques

Testing infiltrometer techniques to determine soil hydraulic properties is necessary for specific soils. For a loam soil, the water retention and hydraulic conductivity predicted by the BEST (Beerkan Estimation of Soil Transfer parameters) procedure of soil hydraulic characterization was compared with data collected by more standard laboratory and field techniques. Six infiltrometer techniques were also compared in terms of saturated soil hydraulic conductivity, K_s. BEST yielded water retention values statistically similar to those obtained in the laboratory and K_s values practically coinciding with those determined in the field with the pressure infiltrometer (PI). The unsaturated soil hydraulic conductivity measured with the tension infiltrometer (TI) was reproduced satisfactorily by BEST only close to saturation. BEST, the PI, one‐potential experiments with both the TI and the mini disk infiltrometer (MDI), the simplified falling head (SFH) technique and the bottomless bucket (BB) method yielded statistically similar estimates of K_s, differing at the most by a factor of three. Smaller values were obtained with longer and more soil‐disturbing infiltration runs. Any of the tested infiltration techniques appears usable to obtain the order of magnitude of K_s at the field site, but the BEST, BB and PI data appear more appropriate to characterize the soil at some stage during a rainfall event. Additional investigations on both similar and different soils would allow development of more general procedures to apply infiltrometer techniques for soil hydraulic characterization. Copyright © 2015 John Wiley & Sons, Ltd.     

Changes in Entrapped Gas Content and Hydraulic Conductivity with Pressure

Water table fluctuations continuously introduce entrapped air bubbles into the otherwise saturated capillary fringe and groundwater zone, which reduces the effective (quasi‐saturated) hydraulic conductivity, K_quasi, thus impacting groundwater flow, aquifer recharge and solute and contaminant transport. These entrapped gases will be susceptible to compression or expansion with changes in water pressure, as would be expected with water table (and barometric pressure) fluctuations. Here we undertake laboratory experiments using sand‐packed columns to quantify the effect of water table changes of up to 250 cm on the entrapped gas content and the quasi‐saturated hydraulic conductivity, and discuss our ability to account for these mechanisms in ground water models. Initial entrapped air contents ranged between 0.080 and 0.158, with a corresponding K_quasi ranging between 2 and 6 times lower compared to the K_s value. The application of 250 cm of water pressure caused an 18% to 26% reduction in the entrapped air content, resulting in an increase in K_quasi by 1.16 to 1.57 times compared to its initial (0 cm water pressure) value. The change in entrapped air content measured at pressure step intervals of 50 cm, was essentially linear, and could be modeled according to the ideal gas law. Meanwhile, the changes in K_quasi with compression–expansion of the bubbles because of pressure changes could be adequately captured with several current hydraulic conductivity models.     

Two-Dimensional Probabilistic Infiltration Analysis in a Hillslope Using First-Order Moment Approach

A first-order moment analysis method is introduced to evaluate the pore-water pressure variability within a hillslope due to spatial variability in saturated hydraulic conductivity (K_s) during rainfall. The influences of the variance of the natural logarithm of K_s(ln K_s), spatial structure anisotropy of ln K_s, and normalized vertical infiltration flux (q) on the evaluations of the pore-water pressure uncertainty are investigated. Results indicate different responses of pressure head variability in the unsaturated region and the saturated region. In the unsaturated region, a larger variance of ln K_s, a higher spatial structure anisotropy, and a smaller q lead to a larger variability in pressure head, while in the saturated region, the variability in pressure head increases with the increase of variance of ln K_s, the decrease of spatial structure anisotropy, or the increase of q. These variables have great impacts on the range of fluctuation of the phreatic surface within the hillslope. The influences of these three variables on the variance of pressure head within the saturated region are greater than those within the unsaturated region, and the variance of ln K_s has the greatest impact. These results yield useful insight into the effects of heterogeneity on pressure head and uncertainty associated with predicted flow field.     

Quasi-Saturated Layer: Implications for Estimating Recharge and Groundwater Modeling

This study presents an extension of the concept of “quasi-saturation” to a quasi-saturated layer, defined as the uppermost dynamic portion of the saturated zone subject to water table fluctuations. Entrapped air here may cause substantial reductions in the hydraulic conductivity (K) and fillable pore water. Air entrapment is caused by a rising water table, usually as a result of groundwater recharge. The most significant effects of entrapped air are recharge overestimation based on methods that use specific yield (S_y), such as the water table fluctuation method (WTF), and reductions in K values. These effects impact estimation of fluid flow velocities and contaminant migration rates in groundwater. In order to quantify actual groundwater recharge rates and the effects of entrapped air, numerical simulations with the FEFLOW (Version 7.0) groundwater flow model were carried out using a quasi-saturated layer for a pilot area in Rio Claro, Brazil. The calculated recharge rate represented 16% of the average precipitation over an 8-year period, approximately half of estimates using the WTF method. Air entrapment amounted to a fillable porosity of 0.07, significant lower that the value of 0.17 obtained experimentally for S_y. Numerical results showed that the entrapped air volume in the quasi-saturated layer can be very significant (0.58 of the air fraction) and hence can significantly affect estimates of groundwater recharge and groundwater flow rates near the water table.     

Application of 1D paleo‐fluvial process modelling at a basin scale to augment sparse borehole data: example of a Permian formation in the Galilee Basin,Australia

The heterogeneous hydraulic conductivity (K) in water‐bearing formations controls subsurface flow and solute transport processes. Geostatistical techniques are often employed to characterize the K distribution in space based on the correlation between K measurements. However, at the basin scale, there are often insufficient measurements for inferring the spatial correlation. This is a widespread problem that we address in this study using the example of the Betts Creek Beds (BCB) in the Galilee Basin, Australia. To address the lack of data, we use a 1D stochastic fluvial process‐based model (SFPM) to quantify the total sediment thickness, Z( x ), and the sandstone proportion over the total thickness, P_s( x ), in the BCB. The semivariograms of Z( x ) and P_s( x ) are then extracted and used in sequential Gaussian simulation to construct the 2D spatial distribution of Z( x ) and P_s( x ). P_s( x ) can be converted to a K distribution based on classical averaging methods. The results demonstrate that the combination of SFPM and geostatistical simulation allows for the evaluation of upscaled K distribution with a limited number of K measurements. Copyright © 2015 John Wiley & Sons, Ltd.     

Effects of variations of river stage and hydraulic conductivity on temporal scaling of groundwater levels: numerical simulations

It was found in previous studies that groundwater levels may fluctuate as a temporal fractal. In this study numerical simulations of groundwater level fluctuations in an unconfined aquifer near a river were conducted to investigate the effects of aquifer heterogeneity and river stage variations on the fractal behavior of the water levels, h(t). Groundwater recharge was taken to be a white-noise process. The aquifer heterogeneity was simulated with a second-order stationary field of hydraulic conductivity (K) with an exponential variogram model. The results showed that groundwater levels fluctuate as a temporal fractal in both homogeneous and heterogeneous aquifers as long as K is less than 10 m/d. Most aquifers may indeed act as a fractal filter which takes a random non-fractal recharge inputs and produces a fractal responses of groundwater level fluctuations. A crossover in temporal scaling of h(t) may appear in more permeable aquifers. Fluctuations of the groundwater level in a homogeneous aquifer are dominated by the recharge process when the river stage is constant or by the river stage variations when the river stage varies in highly permeable aquifers. Heterogeneity plays an important role in the temporal scaling of h(t) in more permeable aquifers: the stronger the heterogeneity, the stronger the temporal scaling of h(t).     

Effect of anisotropy on solute transport in degraded fen peat soils

Peat soils are heterogeneous, anisotropic porous media. Compared to mineral soils, there is still limited understanding of physical and solute transport properties of fen peat soils. In this study, we aimed to explore the effect of soil anisotropy on solute transport in degraded fen peat. Undisturbed soil cores, taken in vertical and horizontal direction, were collected from one drained and one restored fen peatland both in a comparable state of soil degradation. Saturated hydraulic conductivity (K _s) and chemical properties of peat were determined for all soil cores. Miscible displacement experiments were conducted under saturated steady state conditions using potassium bromide as a conservative tracer. The results showed that (1) the K _s in vertical direction (K _sv) was significantly higher than that in horizontal direction (K_sh), indicating that K _s of degraded fen peat behaves anisotropically; (2) pronounced preferential flow occurred in vertical direction with a higher immobile water fraction and a higher pore water velocity; (3) the 5% arrival time (a proxy for the strength of preferential flow) was affected by soil anisotropy as well as study site. A strong correlation was found between 5% arrival time and dispersivity, K _s and mobile water fraction; (4) phosphate release was observed from drained peat only. The impact of soil heterogeneity on phosphate leaching was more pronounced than soil anisotropy. The soil core with the strongest preferential flow released the highest amount of phosphate. We conclude that soil anisotropy is crucial in peatland hydrology but additional research is required to fully understand anisotropy effects on solute transport.     

Variation in soil saturated hydraulic conductivity along the hillslope of collapsing granite gullies

Saturated hydraulic conductivity (K_s) affects the soil hydrological process and is influenced by many factors that exhibit strong spatial variations. To accurately measure K_s and its scale, spatial variability and relationship with collapsing gullies, we analysed four double-ring infiltrometer diameters in three soil layers during in situ experiments designed to measure K_s in two typical collapsing gullies (three slope sites) in Tongcheng County of China. The results showed that K_s increased with increasing inner ring diameter, but no significant difference existed between inner diameters of 30 and 40 cm. The K_s in red soil layers was higher than that in sandy soil layers, the transition layers had the lowest values. K_s also varied with slope position, gradually decreasing with distance from the gully head. The suggestion is that the spatial variation in K_s is affected not only by the intrinsic soil properties but also by the interaction with the collapsing gully.