Explicit infiltration equations and the Lambert W-function

The Green and Ampt infiltration formula, as well as the Talsma and Parlange formula, are two-parameter equations that are both expressible in terms of Lambert W-functions. These representations are used to derive explicit, simple and accurate approximations for each case. The two infiltration formulas are limiting cases that can be deduced from an existing three-parameter infiltration equation, the third parameter allowing for interpolation between the limiting cases. Besides the limiting cases, there is another case for which the three-parameter infiltration equation yields an exact solution. The three-parameter equation can be solved by fixed-point iteration, a scheme which can be exploited to obtain a sequence of increasingly complex explicit infiltration equations. For routine use, a simple, explicit approximation to the three-parameter infiltration equation is derived. This approximation eliminates the need to iterate for most practical circumstances.     

Robert E. Horton's perceptual model of infiltration processes

Robert E. Horton is best known as the originator of the infiltration excess overland flow concept for storm hydrograph analysis and prediction, which, in conjunction with the unit hydrograph concept, provided the foundation for engineering hydrology for several decades. Although these concepts, at least in their simplest form, have been largely superseded, a study of Horton's archived scientific papers reveals that his perceptual model of infiltration processes and appreciation of scale problems in modelling were far more sophisticated and complete than normally presented in hydrological texts. His understanding of surface controls on infiltration remain relevant today. Copyright © 2004 John Wiley & Sons, Ltd.     

A study of the infiltration characteristics of undisturbed soil under simulated rainfall

Infiltration is the single most important parameter in deriving the net quick response rainfall which contributes to stream flood discharges. Rainfall simulation is used to study the infiltration characteristics in a typical catchment, the Six Mile Water in N. Ireland. The design of the simulator was such that it could be easily moved from one test area to another within the catchment to examine the effect of soil and slope variation. The simulator was first calibrated in controlled laboratory conditions and later the calibration was checked in the field. The simulator was mounted over an undisturbed plot of 37 m² and the surface runoff from the area measured by means of a collecting channel located along a lower edge of the plot. Soil moisture variations were monitored using a soil moisture neutron probe. Soil classification tests and gravimetric moisture contents were carried out on each plot. The field tests were carried out with variations in rainfall intensity, initial conditions, changing seasons, and for different plots within the catchment area. The results obtained are unique in that they present data obtained under field conditions for undisturbed soil within a natural catchment. The infiltration behaviour was found to depend upon rainfall intensity, initial conditions of the plot under consideration, seasonal temperature, and a slope of the plot. The data showed that while a classical Horton type equation for infiltration was suitable for the later stages of each test result when significant surface runoff was taking place, the model failed to represent early response adequately due to storage effects being omitted in the equation. A modified form of Horton equation is proposed, which models more accurately the infiltration characteristics of the full period of each test run.     

Flood routing model incorporating intensive streambed infiltration

Flood routing models are critical to flood forecasting and confluence calculations. In the streams that dry up and disconnect from groundwater, the streambed infiltration is intensive and has a significant effect on flood wave movement. Streambed infiltration should be considered in flood routing. A flood routing model incorporating intensive streambed infiltration is proposed. In the model a streambed infiltration simulation method based on soil infiltration theory is developed. In this method the Horton equation is used to calculate infiltration capacity. A trial-and-error method is developed to calculate infiltration rate and determine whether the flood wave can travel downstream. A formula is derived to calculate infiltration flow per unit length. The Muskingum-Cunge method with streambed infiltration flow as lateral outflow is used for flood routing. The proposed model is applied to the stream from the downstream of the Yuecheng Reservoir to the Caixiaozhuang Hydrometric Station in the Zhangwei River of the Haihe River Basin. Simulation results show that the accuracy of the model is high, and the infiltration simulation method can represent infiltration processes well. The proposed model is simple and practical for flood simulation and forecasting, and can be used in river confluence calculations in a rainfall-runoff model for arid and semiarid regions.     

A new baseflow separation method based on analytical solutions of the Horton infiltration capacity curve

The separation of baseflow is an important issue in hydrology. The objective of this paper is to develop a new baseflow separation method based on the Horton infiltration capacity curve. For this purpose, the mathematical expressions of three parameters for the Horton infiltration capacity curve were derived in terms of rainfall and runoff data, and the lag time of the effective rainfall in the unsaturated zone and the groundwater flow routing equation are also presented. With these equations, the baseflow hydrographs at the outlet of the basin can be separated. The flow chart of the proposed method for baseflow hydrograph separation is given. Three recent baseflow separation methods, i.e. digital filter, non‐linear reservoir and the Boussinesq equation, were chosen as parallel schemes to compare with the proposed method. Rainfall‐runoff data from four watersheds located in different climatic regions in China were selected and used as case studies. Test and application results indicate that the proposed baseflow hydrograph separation method is in accordance with the hydrological physical process. The proposed method is comparable with current available methods and reduces some of the subjective aspects for the rising limb of the baseflow hydrograph, and it is useful for unit hydrograph analysis and for the study of the rainfall‐runoff relationship. Copyright © 2006 John Wiley & Sons, Ltd.     

Comparison of infiltration models

Fourteen popular, representative infiltration models, some physically based, some semi‐empirical and some empirical, were selected for a comparative evaluation. Using the Nash and Sutcliffe efficiency criterion, the models were evaluated and compared for 243 sets of infiltration data collected from field and laboratory tests conducted in India and the USA on soils ranging from coarse sand to fine clay. Based on a relative grading scale, the semi‐empirical Singh–Yu general model, Holtan model and Horton model were graded respectively as 6·52, 5·57 and 5·48 out of 10. The empirical Huggins and Monke model, modified Kostiakov and Kostiakov model were graded as 5·57, 5·30 and 5·22, respectively. The physically based non‐linear and linear models of Smith–Parlange were graded as 5·48 and 5·22, respectively. Other models were ranked lower than these models. All the models generally performed poorly in field tests on Georgia's sandy soils, except the Robertsdale loamy sand. Copyright © 2003 John Wiley & Sons, Ltd.     

Similarity analysis and scaling: An application to soil infiltration properties

Similarity and scaling theory are applied to soil physics, specifically to several parameters of unsaturated soil water movement. Following a dimensional analysis of Richards' equation, a mechanical similarity criterion of the hydraulic parameters is developed. Dimensionless factors which conform to assumptions of kinematic and dynamic mechanical similarity in the flow system are converted to relations using the scale factor a derived under the assumption of geometric similarity. As an example, the infiltration process is assessed through scaling of the parameters in Philip's equation, using experimental data from double-ring infiltration measurements at 54 locations in a study catchment as the scaling test.     

An approach to analysing plot scale infiltration and runoff responses to rainfall of fluctuating intensity

Simulated rainfall of fluctuating intensity was applied to runoff plots on bare dryland soils in order to explore a new method for analysing the non‐steady‐state responses of infiltration and overland flow. The rainfall events all averaged 10 mm/h but included intensity bursts of up to 70 mm/h and lasting 5–15 min, as well as periods of low intensity and intermittency of up to 25 min. Results were compared with traditional steady‐state estimates of infiltrability made under simulated rainfall sustained at a fixed intensity of 10 mm/h. Mean event infiltration rate averaged 13.6% higher under fluctuating intensities, while runoff ratios averaged only 63% of those seen under constant intensity. In order to understand the changing soil infiltrability, up to three affine Horton infiltration equations were fitted to segments of each experiment. All equations had the same final infiltrability f_c, but adjusted values for coefficients f₀ (initial infiltrability) and K_f (exponential decay constant) were fitted for periods of rainfall that followed significant hiatuses in rainfall, during which subsurface redistribution allowed near‐surface soil suction to recover. According to the fitted Horton equations, soil infiltrability recovered by up 10–24 mm/h during intra‐event rainfall hiatuses of 15 to 20‐min duration, contributing to higher overall event infiltration rates and to reduced runoff ratios. The recovery of infiltrability also reduced the size of runoff peaks following periods of low intensity rainfall, compared with the predictions based on single Horton infiltration equations, and in some cases, no runoff at all was recorded from late intensity peaks. The principal finding of this study is that, using a set of affine equations, the intra‐event time variation of soil infiltrability can be tracked through multiple intensity bursts and hiatuses, despite the lack of steady‐state conditions. Copyright © 2016 John Wiley & Sons, Ltd.     

Modified models for better prediction of infiltration rates in trapezoidal permeable stormwater channels

ABSTRACT

In stormwater management, it is important to accurately quantify the infiltration rates to solve urban runoff-related problems. This study proposes a method to improve estimates of the infiltration rate in permeable stormwater channels. As part of the analysis, five infiltration models were evaluated: the Kostiakov, Horton, modified Kostiakov, Philip and SCS (Soil Conservation Service) models. Infiltration tests with various initial water levels were performed on channel models with differing base width and side slopes. The results show that the addition of three parameters that describe the trapezoidal cross-sectional area, i.e. the depth, side slope and base width, in the infiltration models yielded better estimates of the infiltration rate. A comparison of the infiltration capacity values obtained from the models after the three parameters were added with those that were experimentally obtained, shows that the improved modified Kostiakov model is the most suitable model to predict infiltration rates in trapezoidal permeable stormwater channels.     

Towards using pedotransfer functions for estimating infiltration parameters

ABSTRACT

Water infiltration into soils is an important component of hydrological processes. Direct measurement of infiltration is time consuming, expensive and often involves large spatial and temporal variability. The objective of this study was to develop and verify parametric pedotransfer functions (PTFs) to predict infiltration parameters. Consequently, 119 double-ring infiltration data were collected. The parameters of Philip, Kostiakov, Kostiakov-Lewis and Horton models were obtained, using the sum of squares error optimization method. Some parametric PTFs were then derived to predict the parameters of the infiltration models, using stepwise regression analysis. The results indicated a reasonable estimation of infiltration parameters by the derived PTFs. These results were more accurate when the land use of the studied area was considered. Overall results of this study suggest infiltration-based PTFs could be established as a reasonable indirect method for estimating infiltration parameters.

Editor M.C. Acreman; Associate editor N. Verhoest     

Analytical approximation to the solutions of Richards' equation with applications to infiltration,ponding, and time compression approximation

A recent approach to solve Richards' equation is further improved. This approach brings understanding into the physical processes of infiltration and ponding. In particular we apply it to analyze the standard hydrologic tool of Time Compression Approximation (TCA). We also suggest that the new approach provides a more reliable alternative to TCA, e.g. for predicting ponding time.     

Applicability of some infiltration formulae to rangeland infiltrometer data

Three algebraic infiltration equations (Kostiakov's, Horton's and Philip's) were examined to determine which one would best fit infiltrometer data collected from a variety of mostly semi-arid rangeland plant communities from both Australia and the United States. Approximately 1,100 infiltrometer plots were included in the analysis. Results indicated that, in every instance, Horton's equation best fit the infiltrometer data. Variability of “point” measures of short-term infiltration rates were never satisfactorily accounted for by using either Kostiakov's or Philip's equation. Though Horton's equation provided a best fit to the overall infiltration data, R² values indicated a potential usefulness of this equation only under the certain conditions that were sampled in several rangeland plant communities in the Northern Territory, Australia. The equation could not be considered consistently useful under conditions sampled on rangelands in the United States.     

Exponential data fitting applied to infiltration, hydrograph separation, and variogram fitting

Most lumped rainfall-runoff models separate the interflow and groundwater components from the measured runoff hydrograph in an attempt to model these as hydrologic reservoir units. Similarly, rainfall losses due to infiltration as well as other abstractions are separated from the measured rainfall hyetograph, which are then used as inputs to the various hydrologic reservoir units. This data pre-processing is necessary in order to use the linear unit hydrograph theory, as well as for maintaining a hydrologic budget between the surface and subsurface flow processes. Since infiltration determines the shape of the runoff hydrograph, it must be estimated as accurately as possible. When measured infiltration data is available, Horton’s exponential infiltration model is preferable due to its simplicity. However, estimating the parameters from Horton’s model constitutes a nonlinear least squares fitting problem. Hence, an iterative procedure that requires initialization is subject to convergence. In a similar context, the separation of direct runoff, interflow, and baseflow from the total hydrograph is typically done in an ad hoc manner. However, many practitioners use exponential models in a rather “layer peeling” fashion to perform this separation. In essence, this also constitutes an exponential data fitting problem. Likewise, certain variogram functions can be fitted using exponential data fitting techniques. In this paper we show that fitting a Hortonian model to experimental data, as well as performing hydrograph separation, and total hydrograph and variogram fitting can all be formulated as a system identification problem using Hankel-based realization algorithms. The main advantage is that the parameters can be estimated in a noniterative fashion, using robust numerical linear algebra techniques. As such, the system identification algorithms overcome the problem of convergence inherent in iterative techniques. In addition, the algorithms are robust to noise in the data since they optimally separate the signal and noise subspaces from the observed noisy data. The algorithms are tested with real data from field experiments performed in Surinam, as well as with real hydrograph data from a watershed in Louisiana. The system identification techniques presented herein can also be used with any other type of exponential data such as exponential decays from nuclear experiments, tracer studies, and compartmental analysis studies.     

Comparison of infiltration models with regard to design of rectangular infiltration trenches

ABSTRACT

Guidelines for the design of infiltration trenches of rectangular cross-section are compared. Four concepts of infiltration into a native soil from linear soakaways of rectangular cross-section are discussed. The results of calculations using analytical equations are compared with numerical two-dimensional (2D) simulations using HYDRUS computer code and field measurements. Satisfactory agreement is achieved for Kozeny’s concept of infiltration and the free-surface approach, applied to a trench in a highly saturated sandy loam. Theoretical solutions, neglecting matric potential, suggest that the mean infiltration rate along the wetted perimeter of a rectangular trench varies in the range 1.0 to 1.5 of the value of saturated hydraulic conductivity.     

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.     

Non-equilibrium concentration profile formulas of suspended sediment and their preliminary applications

The vertical concentration profiles in non-equilibrium sediment transport processes generally deviate from the equilibrium concentration distribution of suspended sediment. The non-equilibrium concentration profile formulas currently available are those of Han and Brown, respectively. However, the complexity of these formulas limits their use in practical calculations. To improve the usefulness of these formulas, the unknown parameters in Han’s formula are reduced from three to two, and the thre...     

Reduced complexity models for probabilistic forecasting of infiltration rates

Soil heterogeneity and data sparsity combine to render estimates of infiltration rates uncertain. We develop reduced complexity models for the probabilistic forecasting of infiltration rates in heterogeneous soils during surface runoff and/or flooding events. These models yield closed-form semi-analytical expressions for the single- and multi-point infiltration-rate PDFs (probability density functions), which quantify predictive uncertainty stemming from uncertainty in soil properties. These solutions enable us to investigate the relative importance of uncertainty in various hydraulic parameters and the effects of their cross-correlation. At early times, the infiltration-rate PDFs computed with the reduced complexity models are in close agreement with their counterparts obtained from a full infiltration model based on the Richards equation. At all times, the reduced complexity models provide conservative estimates of predictive uncertainty.     

Soil infiltration processes of different underlying surfaces in the permafrost region on the Tibetan Plateau

ABSTRACT

Soil infiltration processes were evaluated under field conditions by double-ring infiltrometers with different underlying surfaces in permafrost regions of the Tibetan Plateau. The results show that initial infiltration rates, stable soil infiltration rates and cumulative soil infiltration are strongly dependent on the underlying surface types, with the highest initial and stable soil infiltration rates in the alpine desert steppe, and the lowest in alpine meadow. The effects of soil moisture and texture on infiltration processes were also assessed. Within the same underlying surfaces, the values of infiltration parameters increased with the amount of vegetation cover, while soil moisture and soil infiltration rates displayed opposing trends, with fitting slopes of ?0.03 and ?0.01 for the initial and stable soil infiltration rates, respectively. The accuracies of the five models in simulating soil infiltration rates and seven models in predicting cumulative infiltration rates were evaluated against data generated from field experiments at four sites. Based on a comparative analysis, the Horton model provided the most complete understanding of the underlying surface effects on soil infiltration processes. Altogether, these findings show that different underlying surfaces can alter soil infiltration processes. This study provides a useful reference for understanding the parameterization of land surface processes for simulating changes in hydrological processes under global warming conditions in the permafrost region on the Tibetan Plateau.     

Effect of a thermokarst lake on soil physical properties and infiltration processes in the permafrost region of the Qinghai-Tibet Plateau,China

Changes in the hydrological processes in alpine soil constitute one of the several key problems encountered with studying watershed hydrology and ecosystem stability against the background of global warming. A typically developing thermokarst lake was chosen as a subject for a study using model simulation based on observations of soil physical properties, infiltration processes, and soil moisture. The results showed that the selected thermokarst lake imposed certain changes on the soil infiltration processes and, with the degree of impact intensifying, the initial infiltration rate decreased. The greatest reduction was achieved in the area of moderate impact. However, the stable infiltration rate and cumulative infiltration gradually increased in the surface layer at a depth of 10 and 20 cm, both decreasing initially and then increasing, which is correlated significantly with soil textures. Moreover, the cumulative infiltration changed in line with steady infiltration rate. Based on a comparative analysis, the Horton model helps better understand the effect on the soil infiltration processes of the cold alpine meadow close to the chosen thermokarst lake. In conclusion, the formation of the thermokarst lake reduced the water holding capacity of the alpine meadow soil and caused the hydraulic conductivity to increase, resulting in the reduction of runoff capacity in the area of the thermokarst lake.     

A new surface runoff parameterization with subgrid-scale soil heterogeneity for land surface models

Soil heterogeneity plays an important role in determining surface runoff generation mechanisms. At the spatial scales represented by land surface models used in regional climate model and/or global general circulation models (GCMs) for numerical weather prediction and climate studies, both infiltration excess (Horton) and saturation excess (Dunne) runoff may be present within a studied area or a model grid cell. Proper modeling of surface runoff is essential to a reasonable representation of feedbacks in the land–atmosphere system. In this paper, a new surface runoff parameterization that dynamically represents both Horton and Dunne runoff generation mechanisms within a model grid cell is presented. The new parameterization takes into account of effects of soil heterogeneity on Horton and Dunne runoff. A series of numerical experiments are conducted to study the effects of soil heterogeneity on Horton and Dunne runoff and on soil moisture storage under different soil and precipitation conditions. The new parameterization is implemented into the current version of the hydrologically based variable infiltration capacity (VIC) land surface model and tested over three watersheds in Pennsylvania. Results show that the new parameterization plays a very important role in partitioning the water budget between surface runoff and soil moisture in the atmosphere–land coupling system. Significant underestimation of the surface runoff and overestimation of subsurface runoff and soil moisture could be resulted if the Horton runoff mechanism were not taken into account. Also, the results show that the Horton runoff mechanism should be considered within the context of subgrid-scale spatial variability of soil properties and precipitation. An assumption of time-invariant spatial distribution of potential infiltration rate may result in large errors in surface runoff and soil moisture. In addition, the total surface runoff from the new parameterization is less sensitive to the choice of the soil moisture shape parameter of the distribution.