Variable power‐law scaling of hillslope Hortonian rainfall–runoff processes

Hydrological studies focused on Hortonian rainfall–run‐off scaling have found that the run‐off depth generally declines with the plot length in power‐law scaling. Both the power‐law proportional coefficient and the scaling exponent show great variability for specific conditions, but why and how they vary remain unclear. In the present study, the scaling of hillslope Hortonian rainfall–run‐off processes is investigated for different rainfall, soil infiltration, and hillslope surface characteristics using the physically based cell‐based rainfall‐infiltration‐run‐off model. The results show that both temporally intermittent and steady rainfalls can result in prominent power‐law scaling at the initial stage of run‐off generation. Then, the magnitude of the power‐law scaling decreases gradually due to the decreasing run‐on effect. The power‐law scaling is most sensitive to the rainfall and soil infiltration parameters. When the ratio of rainfall to infiltration exceeds a critical value, the magnitude of the power‐law scaling tends to decrease notably. For different intermittent rainfall patterns, the power‐law exponent varies in the range of ?1.0 to ?0.113, which shows an approximately logarithmic increasing trend for the proportional coefficient as a function of the run‐off coefficient. The scaling is also sensitive to the surface roughness, soil sealing, slope angle, and hillslope geometry because these factors control the run‐off routing and run‐on infiltration processes. These results provide insights into the variable scaling of the Hortonian rainfall–run‐off process, which are expected to benefit modelling of large‐scale hydrological and ecological processes.     

A probabilistic investigation of infiltration in the vadose zone: proposal for a new formula of infiltration rate

The infiltration rate in the unsaturated zone is analysed from a probabilistic point of view. It is shown that the empirical formulas of Horton and Kostiakov, without apparent physical basis, are explained in a probabilistic approach. Horton's formula reflects a Markovian process contrary to Kostiakov's formula. This approach made it possible to explain why Kostiakov's formula is more powerful than that of Horton. A new equation of infiltration is proposed. The three formulas were compared, for four types of soil, with the model of van Genuchten based on the Richards equation. Copyright © 2006 John Wiley & Sons, Ltd.     

LISEM: A SINGLE-EVENT PHYSICALLY BASED HYDROLOGICAL AND SOIL EROSION MODEL FOR DRAINAGE BASINS. I: THEORY,INPUT AND OUTPUT

A new physically based hydrological and soil erosion model has been developed, which can be used for planning and conservation purposes: the LImburg Soil Erosion Model (LISEM). The LISEM model is one of the first examples of a physically based model that is completely incorporated in a raster Geographical Information System. This incorporation facilitates easy application in larger catchments, improves the user friendliness by avoiding conversion routines and allows remotely sensed data to be used. Processes incorporated in the model are rainfall, interception, surface storage in micro-depressions, infiltration and vertical movement of water in the soil, overland flow, channel flow, detachment by rainfall and throughfall, detachment by overland flow and transport capacity of the flow. Special attention has been given to the influence of tractor wheelings, small roads and surface sealing. Vertical movement of water in the soil is simulated using the Richard's equation. Optionally, the user can choose the Holtan or the Green–Ampt infiltration model. For the distribution flow routing, a four-point finite-difference solution of the kinematic wave is used together with Manning's equation.     

A new saturated/unsaturated model for stormwater infiltration systems

Infiltration systems are widely used as an effective urban stormwater control measure. Most design methods and models roughly approximate the complex physical flow processes in these systems using empirical equations and fixed infiltration rates to calculate emptying times from full. Sophisticated variably saturated flow models are available, but rarely applied owing to their complexity. This paper describes the development and testing of an integrated one‐dimensional model of flow through the porous storage of a typical infiltration system and surrounding soils. The model accounts for the depth in the storage, surrounding soil moisture conditions and the interaction between the storage and surrounding soil. It is a front‐tracking model that innovatively combines a soil‐moisture‐based solution of Richard's equation for unsaturated flow with piston flow through a saturated zone as well as a reservoir equation for flow through a porous storage. This allows the use of a simple non‐iterative numerical solution that can handle ponded infiltration into dry soils. The model is more rigorous than approximate stormwater infiltration system models and could therefore be valuable in everyday practice. A range of test cases commonly used to test soil water flow models for infiltration in unsaturated conditions, drainage from saturation and infiltration under ponded conditions were used to test the model along with an experiment with variable depth in a porous storage over saturated conditions. Results show that the model produces a good fit to the observed data, analytical solutions and Hydrus. Copyright © 2008 John Wiley & Sons, Ltd.     

Rainfall runoff and erosion in Napa Valley vineyards: effects of slope,cover and surface roughness

The effects of slope, cover and surface roughness on rainfall runoff, infiltration and erosion were determined at two sites on a hillside vineyard in Napa County, California, using a portable rainfall simulator. Rainfall simulation experiments were carried out at two sites, with five replications of three slope treatments (5%, 10% and 15%) in a randomized block design at each site (0%bsol;64 m² plots). Prior to initiation of the rainfall simulations, detailed assessments, not considered in previous vineyard studies, of soil slope, cover and surface roughness were conducted. Significant correlations (at the 95% confidence level) between the physical characteristics of slope, cover and surface roughness, with total infiltration, runoff, sediment discharge and average sediment concentration were obtained. The extent of soil cracking, a physical characteristic not directly measured, also affected analysis of the rainfall–runoff–erosion process. Average cumulative runoff and cumulative sediment discharge from site A was 87% and 242% greater, respectively, than at site B. This difference was linked to the greater cover, extent of soil cracking and bulk density at site B than at site A. The extent of soil cover was the dominant factor limiting soil loss when soil cracking was not present. Field slopes within the range of 4–16%, although a statistically significant factor affecting soil losses, had only a minor impact on the amount of soil loss. The Horton infiltration equation fit field data better than the modified Philip's equation. Owing to the variability in the ‘treatment’ parameters affecting the rainfall–runoff–erosion process, use of ANOVA methods were found to be inappropriate; multiple‐factor regression analysis was more useful for identifying significant parameters. Overall, we obtained similar values for soil erosion parameters as those obtained from vineyard erosion studies in Europe. In addition, it appears that results from the small plot studies may be adequately scaled up one to two orders of magnitude in terms of land areas considered. Copyright © 2000 John Wiley & Sons, Ltd.     

Infiltration at the Tai rain forest (Ivory Coast): Measurements and modelling

Infiltration experiments have been performed at three sites along a well-known catena under virgin tropical rain forest using a portable sprinkling infiltrometer. Experimentally determined infiltration curves are presented. Infiltration curves are also simulated on the basis of the Mein-Larson equation. The parameters for this model have been obtained from the infiltration curves (saturated conductivity) and simple soil moisture determinations (fillable porosity). The agreement between experimentally determined and modelled infiltration is reasonable, provided (a) saturated conductivity as derived from the experimental data is corrected, (b) a storage parameter, also derived from the experimental data, is added to the Mein-Larson model, and (c) the decline in soil porosity with depth is either small or occurs abruptly at shallow depth. Comparison of observed infiltration rates with rainfall intensity shows that Horton Overland Flow has to occur naturally at least on the middle and lower section of the catena. Despite the fact that most parameters can be estimated in principle from basic soil data, it remains advisable to obtain sprinkling infiltrometer field measurements, because of soil variability due to dynamic surface conditions, macroporosity, air entrapment, and irregularity of the wetting front.     

Modelling to analyse the impacts of animal treading effects on soil infiltration

Animal treading can change soil physical properties, and thus is an important factor in hydrological modelling. We investigated the impacts of animal treading on infiltration by using a series of rainfall simulation experiments at Whatawhata Research Center, Waikato, New Zealand. The study identified significant variables for estimating soil steady‐state infiltration at a micro‐site (0·5 m²) and fitted the Green and Ampt equation by modifying or including variables for soil and water parameters and animal activities on grazing paddocks. A regression function for estimating steady‐state infiltration rate was created for each of four scenarios: between tracks (inter‐track), track, easy slope with ash soil, and easy slope with clay soil. Significant variables included the number of days after treading, antecedent soil moisture, field capacity, percentage of bare ground, bulk density, and the high degree of soil damage (damage not compacted). Regression models explained more than 71% of the variance in steady‐state infiltration for three scenarios, but only 53% for the easy slope with clay soil. The remodified Green and Ampt equation provided satisfactory estimation of infiltration for all scenarios (accuracy > 80%), and thus enables us to use the modified model for Waikato hill country pastures of different topography, soil physical condition, season and grazing management. Copyright © 2006 John Wiley & Sons, Ltd.     

Hydrological simulation of a conservation bench terrace system in a subhumid climate

Abstract

A finite element model to simulate runoff and soil erosion from agricultural lands has been developed. The sequential solutions of the governing differential equations were found: Richards' equation with a sink term for infiltration and soil water dynamics under cropped conditions; St Venant equation with kinematic wave approximation for overland and channel flow; and sediment continuity equation, for soil erosion. The model developed earlier has been improved to simulate erosion/deposition in impoundments and predicted and observed soil loss values were in reasonably good agreement when the model was tested for a conservation bench terrace (CBT) system. The finite element model was extensively applied to study the hydrological behaviour of a CBT system vis-à-vis the conventional system of sloping borders. The model estimates runoff and soil loss reasonably well, under varying conditions of rainfall and at different crop growth stages. The probable reasons for discrepancies between observation and simulation are reported and discussed. Sensitivity analysis was carried out to study the effect of various hydrological, soil and topographical parameters, such as ratio of contributing to receiving areas, weir length, depth of impoundment, slope of contributing area, etc. on the flow behaviour in a CBT system.     

Capturing the essential spatial variability in distributed hydrological modelling: Infiltration parameters

Selecting the correct resolution in distributed hydrological modelling at the watershed scale is essential in reducing scale-related errors. The work presented herein uses information content (entropy) to identify the resolution which captures the essential variability, at the watershed scale, of the infiltration parameters in the Green and Ampt infiltration equation. A soil map of the Little Washita watershed in south-west Oklahoma, USA was used to investigate the effects of grid cell resolution on the distributed modelling of infiltration. Soil-derived parameters and infiltration exhibit decreased entropy as resolutions become coarser. This is reflected in a decrease in the maximum entropy value for the reclassified/derived parameters vis a vis the original data. Moreover, the entropy curve, when plotted against resolution, shows two distinct segments: a constant section where no entropy was lost with decreasing resolution and another part which is characterized by a sharp decrease in entropy after a critical resolution of 1209 m is reached. This methodology offers a technique for assessing the largest cell size that captures the spatial variability of infiltration parameters for a particular basin. A geographical information system (GIS) based rainfall-runoff model is used to simulate storm hydrographs using infiltration parameter maps at different resolutions as inputs. Model results up to the critical resolution are reproducible and errors are small. However, at resolutions beyond the critical resolution the results are erratic with large errors. A major finding of this study is that a large resolution (1209 m for this basin) yields reproducible model results. When modelling a river basin using a distributed model, the resolution (grid cell size) can drastically affect the model results and calibration. The error structure attributable to grid cell resolution using entropy as a spatial variability measure is shown.     

Modelling soil solute release and transport in run‐off on a loessial slope with and without surface stones

Stone covers on loessial slopes can increase the time of infiltration by slowing the velocity of the overland flow, which reduces the transport of solutes, but few mechanistic models have been tested under water‐scouring conditions. We carried out field experiments to test a previously proposed, physically based model of water and solute transport. The area of soil infiltration was calculated from the uncovered surface area, and Richards' equation and the kinematic wave equation were used to describe water infiltration and flow along slopes with stone covers. The transport of chemicals into the run‐off from the surface soil, presumably by diffusion, and their movement in the soil profile could be described by the convection–diffusion equations of the model. The simulated and measured data correlated well. The stones on the soil surface reduced the area available for infiltration but increased the Manning coefficient, eventually leading to increased water infiltration and decreased solute loss with run‐off. Our results indicated that the traditional model of water movement and solute migration could be used to simulate water transport and solute migration for stone‐covered soil on loessial slopes.     

Curve‐Number/Green–Ampt mixed procedure for streamflow predictions in ungauged basins: Parameter sensitivity analysis

A procedure combining the Soil Conservation Service‐Curve Number (SCS‐CN) method and the Green–Ampt (GA) infiltration equation was recently developed to overcome some of the drawbacks of the classic SCS‐CN approach when estimating the volume of surface runoff at a sub‐daily time resolution. The rationale of this mixed procedure, named Curve Number for Green–Ampt (CN4GA), is to use the GA infiltration model to distribute the total volume of the net hyetograph (rainfall excess) provided by the SCS‐CN method over time. The initial abstraction and the total volume of rainfall given by the SCS‐CN method are used to identify the ponding time and to quantify the hydraulic conductivity parameter of the GA equation. In this paper, a sensitivity analysis of the mixed CN4GA parameters is presented with the aim to identify conditions where the mixed procedure can be effectively used within the Prediction in Ungauged Basin perspective. The effects exerted by changes in selected input parameters on the outputs are evaluated using rectangular and triangular synthetic hyetographs as well as 100 maximum annual storms selected from synthetic rainfall time series. When applied to extreme precipitation events, which are characterized by predominant peaks of rainfall, the CN4GA appears to be rather insensitive to the input hydraulic parameters of the soil, which is an interesting feature of the CN4GA approach and makes it an ideal candidate for the rainfall excess estimation at sub‐daily temporal resolution at ungauged sites. Copyright © 2013 John Wiley & Sons, Ltd.     

Effects of Coal Gangue Content on Water Movement and Solute Transport in a China Loess Plateau Soil

The mining industry has grown strongly in China in recent decades, resulting in large amounts of coal gangues, which cause water and soil pollution, soil erosion, and various other environmental problems. They are often used in reclamation projects in attempts to restore land damaged by mining, hence they are frequently present (in widely varying proportions) in the topsoil in areas around mines. Their presence can strongly affect key soil variables, including its bulk density, structure, water retention, water movement, and solute transport rates. In the study presented here, the effects of gangue contents on infiltration, saturated hydraulic conductivity, and solute transport parameters of a Chinese Loess plateau soil were examined. The results show that infiltration rates and saturated hydraulic conductivity decreased with increasing gangue content. The Peck–Watson equation modeled these relationships well, but Bouwer–Rice equations provided poorer matches with the acquired data. Cumulative infiltration over time was described well by both the Philip equation and Kostiakov equation. Both the simplified convection–dispersion equation and a two‐region model described the solute transport processes well. In addition, the dispersion increased, while both the Peclet number and mobile water fraction decreased, with increases in gangue contents.     

On estimating the hydraulic properties of soil,Part 3. Parameters of the Philip infiltration equation

Using a large number of experimental data, the paper estimates the parameters of the Philip infiltration equation. The values of these parameters are obtained using the Philip approach, which employs the hydraulic properties of soil. In the absence of extensive measurements, the tabulated results can be used in modelling infiltration. The paper introduces approximate sorptivity expressions that use a limited number of parameters. Sorptivity may be estimated reasonably well from a knowledge of moisture deficit, a representative capillary pressure, and saturated hydraulic conductivity. The accuracy of prediction is highest for coarser materials. For coarse materials (sands), second-power sorptivity can be approximated as a linear function of initial effective saturation. The proposed expressions for sorptivity can be used as a substitute for the lengthy Philip procedure.     

A two‐dimensional model of hydraulic performance of stormwater infiltration systems

Stormwater infiltration systems are a popular method for urban stormwater control. They are often designed using an assumption of one‐dimensional saturated outflow, although this is not very accurate for many typical designs where two‐dimensional (2D) flows into unsaturated soils occur. Available 2D variably saturated flow models are not commonly used for design because of their complexity and difficulties with the required boundary conditions. A purpose‐built stormwater infiltration system model was thus developed for the simulation of 2D flow from a porous storage. The model combines a soil moisture–based model for unsaturated soils with a ponded storage model and uses a wetting front‐tracking approach for saturated flows. The model represents the main physical processes while minimizing input data requirements. The model was calibrated and validated using data from laboratory 2D stormwater infiltration trench experiments. Calibrations were undertaken using five different combinations of calibration data to examine calibration data requirements. It was found that storage water levels could be satisfactorily predicted using parameters calibrated with either data from laboratory soils tests or observed water level data, whereas the prediction of soil moistures was improved through the addition of observed soil moisture data to the calibration data set. Copyright © 2012 John Wiley & Sons, Ltd.     

Polynomial approximate solutions to the Boussinesq equation

We provide closed-form approximate solutions to models of horizontal infiltration described by the Boussinesq equation in a semi-infinite aquifer that is initially dry. The approximations preserve such important qualitative properties as scaling and wetting fronts. They are applicable to four types of boundary conditions, two on head and two on flux, enumerated in the paper. All the considered problems admit self-similar variables that allow reduction to boundary value problems for a nonlinear ordinary differential equation. This work extends recent results by Lockington et al. [Lockington DA, Parlange J-Y, Parlange MB, Selker J. Similarity solution of the Boussinesq equation. Adv Water Resour 2000;23(7):725–9] and Telyakovskiy et al. [Telyakovskiy AS, Braga GA, Furtado F. Approximate similarity solutions to the Boussinesq equation. Adv Water Resour 2002;25(2):191–4], with new approximations developed for two of the four cases and a new extension of a previously existing method for a third case. Numerical results extending the work of Shampine [Shampine LF. Some singular concentration dependent diffusion problems. ZAMM 1973;53:421–2] provide a basis for assessing the accuracy of the new methods.     

Assessing the impact of the hydraulic properties of a crusted soil on overland flow modelling at the field scale

Soil surface crusts are widely reported to favour Hortonian runoff, but are not explicitly represented in most rainfall‐runoff models. The aim of this paper is to assess the impact of soil surface crusts on infiltration and runoff modelling at two spatial scales, i.e. the local scale and the plot scale. At the local scale, two separate single ring infiltration experiments are undertaken. The first is performed on the undisturbed soil, whereas the second is done after removal of the soil surface crust. The HYDRUS 2D two‐dimensional vertical infiltration model is then used in an inverse modelling approach, first to estimate the soil hydraulic properties of the crust and the subsoil, and then the effective hydraulic properties of the soil represented as a single uniform layer. The results show that the crust hydraulic conductivity is 10 times lower than that of the subsoil, thus illustrating the limiting role the crust has on infiltration. Moving up to the plot scale, a rainfall‐runoff model coupling the Richards equation to a transfer function is used to simulate Hortonian overland flow hydrographs. The previously calculated hydraulic properties are used, and a comparison is undertaken between a single‐layer and a double‐layer representation of the crusted soil. The results of the rainfall‐runoff model show that the soil hydraulic properties calculated at the local scale give acceptable results when used to model runoff at the plot scale directly, without any numerical calibration. Also, at the plot scale, no clear improvement of the results can be seen when using a double‐layer representation of the soil in comparison with a single homogeneous layer. This is due to the hydrological characteristics of Hortonian runoff, which is triggered by a rainfall intensity exceeding the saturated hydraulic conductivity of the soil surface. Consequently, the rainfall‐runoff model is more sensitive to rainfall than to the subsoil's hydrodynamic properties. Therefore, the use of a double‐layer soil model to represent runoff on a crusted soil does not seem necessary, as the increase of precision in the soil discretization is not justified by a better performance of the model. Copyright © 2005 John Wiley & Sons, Ltd.     

ESTIMATING INFILTRATION PARAMETERS FROM BASIC SOIL PROPERTIES

Infiltration data were collected on two rectangular grids with 25 sampling points each. Both experimental grids were located in tropical rain forest (Guyana), the first in an Arenosol area and the second in a Ferralsol field. Four different infiltration models were evaluated based on their performance in describing the infiltration data. The model parameters were estimated using non-linear optimization techniques. The infiltration behaviour in the Ferralsol was equally well described by the equations of Philip, Green–Ampt, Kostiakov and Horton. For the Arenosol, the equations of Philip, Green–Ampt and Horton were significantly better than the Kostiakov model. Basic soil properties such as textural composition (percentage sand, silt and clay), organic carbon content, dry bulk density, porosity, initial soil water content and root content were also determined for each sampling point of the two grids. The fitted infiltration parameters were then estimated based on other soil properties using multiple regression. Prior to the regression analysis, all predictor variables were transformed to normality. The regression analysis was performed using two information levels. The first information level contained only three texture fractions for the Ferralsol (sand, silt and clay) and four fractions for the Arenosol (coarse, medium and fine sand, and silt and clay). At the first information level the regression models explained up to 60% of the variability of some of the infiltration parameters for the Ferralsol field plot. At the second information level the complete textural analysis was used (nine fractions for the Ferralsol and six for the Arenosol). At the second information level a principal components analysis (PCA) was performed prior to the regression analysis to overcome the problem of multicollinearity among the predictor variables. Regression analysis was then carried out using the orthogonally transformed soil properties as the independent variables. Results for the Ferralsol data show that the parameters of the Green–Ampt and Kostiakov model were estimated relatively accurately (maximum R² = 0.76). For the Arenosol, use of the second information level together with PCA produced regression models with an R² value ranging from 0.38 to 0.68. For the Ferralsol, most of the variance was explained by the root content and organic matter content. In the Arenosol plot, the fractions medium and fine sand explained most of the observed variance.     

The errors in surface runoff prediction by neglecting the relationship between infiltration rate and overland flow depth

Many simplifications are used in modeling surface runoff over a uniform slope. A very common simplification is to determine the infiltration rate independent of the overland flow depth and to combine it afterward with the kinematic-wave equation to determine the overland flow depth. Another simplication is to replace the spatially variable infiltration rates along the slope i(x, t) due to the water depth variations h(x,t) with an infiltration rate that is determined at a certain location along the slope. The aim of this study is to evaluate the errors induced by these simplications on predicted infiltration rates, overland flow depths, and total runoff volume. The error analysis is accomplished by comparing a simplified model with a model where the interaction between the overland flow depth and infiltration rate is counted. In this model, the infiltration rate is assumed to vary along the slope with the overland flow depth, even for homogeneous soil profiles. The kinematic-wave equation with interactive infiltration rate, calculated along the slopy by Richard's equation, are then solved by a finite difference scheme for a 100-m-long uniform slope. In the first error analysis, we study the effect of combining an ‘exact’ and ‘approximate’ one-dimensional infiltration rate with the kinematic-wave equation for three different soil surface roughness coefficients. The terms ‘exact’ and ‘approximate’ stand for the solution of Richard's equation with and without using the overland flow depth in the boundary condition, respectively. The simulations showed that higher infiltration rates and lower overland flow depths are obtained during the rising stage of the hydrograph when overland flow depth is used in the upper boundary condition of the one-dimensional Richard's equation. During the recession period, the simplified model predicts lower infiltration rates and higher overland flow depths. The absolute relative errors between the ‘exact’ and ‘approximate’ solutions are positively correlated to the overland flow depths which increase with the soil surface roughness coefficient. For this error analysis, the relative errors in surface runoff volume per unit slope width throughout the storm are much smaller than the relative errors in momentary overland flow depths and discharges due to the alternate signs of the deviations along the rising and falling stages. In the second error analysis, when the spatially variable infiltration rate along the slope i(x, t) is replaced in the kinematic-wave equation by i(t), calculated at the slope outlet, the overland flow depth is underestimated during the rising stage of the hydrograph and overestimated during the falling stage. The deviations during the rising stage are much smaller than the deviations during the falling stage, but they are of a longer duration. This occurs because the solution with i(x, t) recognizes that part of the slope becomes dry after rainfall stops, while overland flow still exists with i(t) determined at the slope outlet. As obtained for the first error analysis, the relative errors in surface runoff volume per unit slope width are also much smaller than the relative errors in momentary overland flow depths and discharges. The relation between the errors in overland flow depth and discharge to different mathematical simplifications enables to evaluate whether certain simplifications are justified or more computational efforts should be used.     

A sensitivity analysis of Hortonian flow

We present a sensitivity analysis for infiltration excess (Hortonian) overland flow based on a classic laboratory experiment by Smith and Woolhiser [Smith RE, Woolhiser DA. Overland flow on an infiltrating surface. Water Resour Res 1971;7(4):899–913]. The model components of the compartment approach are comprised of a diffusive wave approximation to the Saint–Venant equations for overland flow, a Richards model for flow in the variably saturated zone, and an interface coupling concept that combines the two components. In the coupling scheme a hydraulic interface is introduced to allow the definition of an exchange flux between the surface and the unsaturated zone. The effects of friction processes, soil capillarity, hydraulic interface, and vertical soil discretization on both infiltration and runoff prediction are investigated in detail. The corresponding sensitivity analysis is conducted using a small-perturbation method. As a result the importance of the hydraulic processes and related parameters are evaluated for the coupled hydrosystem.