A low‐dimensional physically based model of hydrologic control of shallow landsliding on complex hillslopes

Hillslopes have complex three‐dimensional shapes that are characterized by their plan shape, profile curvature of surface and bedrock, and soil depth. To investigate the stability of complex hillslopes (with different slope curvatures and plan shapes), we combine the hillslope‐storage Boussinesq (HSB) model with the infinite slope stability method. The HSB model is based on the continuity and Darcy equations expressed in terms of storage along the hillslope. Solutions of the HSB equation account explicitly for plan shape by introducing the hillslope width function and for profile curvature through the bedrock slope angle and the hillslope soil depth function. The presented model is composed of three parts: a topography model conceptualizing three‐dimensional soil mantled landscapes, a dynamic hydrology model for shallow subsurface flow and water table depth (HSB model) and an infinite slope stability method based on the Mohr–Coulomb failure law. The resulting hillslope‐storage Boussinesq stability model (HSB‐SM) is able to simulate rain‐induced shallow landsliding on hillslopes with non‐constant bedrock slope and non‐parallel plan shape. We apply the model to nine characteristic hillslope types with three different profile curvatures (concave, straight, convex) and three different plan shapes (convergent, parallel, divergent). In the presented model, the unsaturated storage has been calculated based on the unit head gradient assumption. To relax this assumption and to investigate the effect of neglecting the variations of unsaturated storage on the assessment of slope stability in the transient case, we also combine a coupled model of saturated and unsaturated storage and the infinite slope stability method. The results show that the variations of the unsaturated zone storage do not play a critical role in hillslope stability. Therefore, it can be concluded that the presented dynamic slope stability model (HSB‐SM) can be used safely for slope stability analysis on complex hillslopes. Our results show that after a certain period of rainfall the convergent hillslopes with concave and straight profiles become unstable more quickly than others, whilst divergent convex hillslopes remain stable (even after intense rainfall). In addition, the relation between subsurface flow and hillslope stability has been investigated. Our analyses show that the minimum safety factor (FS) occurs when the rate of subsurface flow is a maximum. In fact, by increasing the subsurface flow, stability decreases for all hillslope shapes. Copyright © 2008 John Wiley & Sons, Ltd.     

Slope profile variability in first-order drainage basins

If the three-dimensional form of a hillslope is regarded as an infinite population of two-dimensional slope profiles, values for attributes of the form of the hillslope can be determined by sampling and estimating the means of the attributes of the population of slope profiles. Twelve attributes are examined in seven first-order drainage basins located on the South Downs in East Sussex to determine necessary sample sizes. Results show that if spatial randomness is assumed the size of sample required varies considerably from one attribute to another but is commonly very large. Of four alternatives to spatial randomness that are tested, consideration of both hillslope plan form and valley asymmetry are shown not to lead to reduction in sample size. However, results do suggest that maximum slope, percentage of profile length in the range 5° to < 10°, and mean angular difference between adjacent slope segments may all be dependent upon distance of the slope profile down valley. Furthermore, initial tests of the effect of spatial autocorrelation suggest that for some attributes consideration needs to be given to this factor when determining sampling strategies.     

Physically based modelling of sheet erosion (detachment and deposition processes) in complex hillslopes

A one‐dimensional uncoupled model governed by this research is a physics‐based modelling of the rainfall‐runoff induced erosion process. The presented model is composed of three parts of a three‐dimensional (3D) hillslope geometry, a nonlinear storage (kinematic wave) model for hillslope hydrological response, and an unsteady physically based surface erosion model. The 3D hillslope geometry model allows describing of the hillslope morphology by defining their plan shape and profile curvature. By changing these two topographic parameters, nine basic hillslope types are derived. The modelling of hillslope hydrological response is based on a flow continuity equation as the relation of discharge and flow depth is passed on kinematic wave approximation. The erosion model is based on a mass conservation equation for unsteady flow. The model assumes that suspended sediment does not affect flow dynamics. The model also accounts for the effect of flow depth plus loose soil depth on soil detachment. The presented model was run for two different precipitations, slope content, and length, and results were plotted for sediment detachment/deposition rate. Based on the obtained results, in hillslopes with convex and straight profile curvatures, sediment detachment only occurred in the whole length of the hillslope. However, in concave ones, sediment detachment and deposition only occurred together in hillslope. The hillslopes with straight profiles and convergent plans have the highest rate of detachment. Also, results show that most detachment rates occur in convex profile curvatures, which are about 15 times more than in straight profiles. Copyright © 2015 John Wiley & Sons, Ltd.     

Landscape position,surface hydraulic gradients and erosion processes

Different hydraulic gradients, especially due to seepage or drainage, at different locations on a hillslope profile may have a profound effect on the dominant erosion processes. A laboratory study was designed to simulate hillslope processes and quantify effects of surface hydraulic gradients on erosion for a Glynwood clay loam soil (fine, illitic, mesic Aquic Hapludalf). A 5 m long, 1·2 m wide soil pan was used at 5 and 10 per cent slopes with an external watering tube to vary the soil bed's hydrological conditions. Different combinations of slope steepness with seepage or drainage gradients were used to simulate the hydrologic conditions on a 5 m segment of a hillslope profile. Runoff samples were taken during rainfall-only and rainfall with added inflow. Results showed that, under drainage conditions, interrill processes dominated and rilling was limited. The surface contained scattered crescent-shaped pits after the run. Under seepage conditions, rilling processes dominated and the inflow introduced at the top of the soil pan further accelerated the headward erosion of the rills. Erosion rates increased by as much as 60 times under seepage conditions representative of the lower backslope when compared to drainage conditions that generally occur at the upper backslope. This indicated that rills and gullies on backslopes and footslopes may be catalysed or enhanced by seepage conditions rather than form from flow hydraulic shear stress alone. An understanding of spatial and temporal changes that affect both hillslope hydrology and erosional processes is needed to develop accurate process-based erosion prediction models. This knowledge may lead to different management practices on landscape positions where seepage occurs. © 1998 John Wiley & Sons, Ltd.     

The variation of soil erodibility with slope position in a cultivated canadian prairie landscape

This study examined the variation in soil erodibility along hillslopes in a Prairie landscape. The soil loss produced by simulated rainfall on undisturbed soils was used as an index of relative soil erodibility. Relative erodibility, and several soil properties, were measured at the summit, shoulder, midslope footslope and toeslope of 11 slope transects in an area of cultivated grassland soils on hummocky glacial till. The variation of erodibility with slope position was statistically significant, and slope position explained about 40 per cent of the variation in the erodibility measurements. Erodibility was 14 per cent higher on the shoulder and midslope, and 21 per cent lower on the toeslope, than on the summit and footslope. Local variation in erodibility along slopes was considered to be an important control on patterns of soil erosion in the landscape. The variation of erodibility along the slopes reflected soil property trends. The greatest erodibility was associated with upper slope positions where soils tended to be shallow, coarse, poorly leached and low in organic matter, while lower erodibility was found at lower slope positions with deep, organic-rich and leached soils. Of the individual soil properties considered, silt and sand content were the most highly correlated with erodibility. The results, together with results from other studies, also suggest that net erosion and erodibility are positively related.     

Precipitation‐runoff processes in Shimen hillslope micro‐catchment of Taihang Mountain,north China

Precipitation runoff is a critical hillslope hydrological process for downslope streamflow and piedmont/floodplain recharge. Shimen hillslope micro‐catchment is strategically located in the central foothill region of Taihang Mountains, where runoff is crucial for water availability in the piedmont corridors and floodplains of north China. This study analyzes precipitation‐runoff processes in the Shimen hillslope micro‐catchment for 2006–2008 using locally designed runoff collection systems. The study shows that slope length is a critical factor, next only to precipitation, in terms of runoff yield. Regression analysis also shows that runoff is related positively to precipitation, and negatively to slope length. Soil mantle in the study area is generally thin and is therefore not as critical a runoff factor as slope length. The study shows a significant difference between overland and subsurface runoff. However, that between the 0–10 and 10–20 cm subsurfaces is insignificant. Runoff hardly occurs under light rains (<10 mm), but is clearly noticeable under moderate‐to‐rainstorm events. In the hillslope catchment, vertical infiltration (accounting for 42–84% of the precipitation) dominates runoff processes in subsurface soils and weathered granite gneiss bedrock. A weak lateral flow (at even the soil/bedrock interface) and the generally small runoff suggest strong infiltration loss via deep percolation. This is critical for groundwater recharge in the downslope piedmont corridors and floodplains. This may enhance water availability, ease water shortage, avert further environmental degradation, and reduce the risk of drought/flood in the event of extreme weather conditions in the catchment and the wider north China Plain. Copyright © 2011 John Wiley & Sons, Ltd.     

Exploring geochemical controls on weathering and erosion of convex hillslopes: beyond the empirical regolith production function

Landscape curvature evolves in response to physical, chemical, and biological influences that cannot yet be quantified in models. Nonetheless, the simplest models predict the existence of equilibrium hillslope profiles. Here, we develop a model describing steady‐state regolith production caused by mineral dissolution on hillslopes which have attained an equilibrium parabolic profile. When the hillslope lowers at a constant rate, the rate of chemical weathering is highest at the ridgetop where curvature is highest and the ridge develops the thickest regolith. This result derives from inclusion of all the terms in the mathematical definition of curvature. Including these terms shows that the curvature of a parabolic hillslope profile varies with distance from the ridge. The hillslope model (meter‐scale) is similar to models of weathering rind formation (centimeter‐scale) where curvature‐driven solute transport causes development of the thickest rinds at highly curved clast corners. At the clast scale, models fit observations. Here, we similarly explore model predictions of the effect of curvature at the hillslope scale. The hillslope model shows that when erosion rates are small and vertical porefluid infiltration is moderate, the hill weathers at both ridge and valley in the erosive transport‐limited regime. For this regime, the reacting mineral is weathered away before it reaches the land surface: in other words, the model predicts completely developed element‐depth profiles at both ridge and valley. In contrast, when the erosion rate increases or porefluid velocity decreases, denudation occurs in the weathering‐limited regime. In this regime, the reacting mineral does not weather away before it reaches the land surface and simulations predict incompletely developed profiles at both ridge and valley. These predictions are broadly consistent with observations of completely developed element‐depth profiles along hillslopes denuding under erosive transport‐limitation but incompletely developed profiles along hillslopes denuding under weathering limitation in some field settings. Copyright © 2013 John Wiley & Sons, Ltd.     

Study of the importance of splash and wash on cultivated loamy soils of Hesbaye (Belgium)

During 1974 and 1975, measurements of splash and wash were carried out in the cultivated area, in a station installed on 6·5 per cent slope covered with a loess in which a grey-brown podzolic soil has developed. Splash has been measured using an apparatus prepared for this purpose and wash has been measured on plots of standard length (22·13m). The splash is some tens t/ha.year but the splash loss calculated using the results of the measurements of splash is only a few tens kg/ha.year. Splash is positively correlated with the erosion index of the rains and with the structural stability but negatively with the crop cover. As splash, wash is positively correlated with the erosion index of the rains and negatively with the crop cover, but unlike splash it is negatively correlated with structural stability. The mean value of the wash loss is a few t/ha.year but very important differences are observed from one plot to another with regard to the structural stability. However, on a given plot wash loss and splash are positively correlated because the particles of soil detached by splash are easily carried off by runoff, but the relation between wash loss and splash is very different from one plot to another because splash is positively and wash loss negatively correlated with the structural stability of soils.     

Measurement and analysis of depression storage on a hillslope

A precise photogrammetric technique was used to determine the microtopography of seven 2.6 m × 1.2 m experimental units located on a hillslope. Surface elevations were determined with an accuracy of better than 1 mm, from which contours at 2mm intervals were interpolated. These contour plots were then manually interpreted to define depressions and associated storage volumes. Analysis of the results highlighted the complex variability of depression storage over the hillslope, for example there being little relation between depression storage volumes and unit slope. This study also highlights the sampling problem for the measurement of depression storage on natural surfaces, which appears not to have been formally recognized previously, and also emphasizes the practical difficulty of achieving depression storage estimates with coefficients of variation less than ten per cent to 50 per cent, much of this variability being due to problems of interpretation rather than measurement of the surface.     

How well can the subsurface storage–discharge relation be interpreted and predicted using the geometric factors in headwater areas?

Headwater storage–discharge (S–Q) remains one of the least understood processes, and there is renewed interest in the S–Q relation. How well can the S–Q relation be interpreted mechanistically using geometric factors? In this paper, the hillslope storage Boussinesq and hillslope storage kinematic wave equation were adopted to guide the theoretical derivations. Analytical solutions were derived based on the hsKW equation for nine idealized hillslope aquifers, which were subdivided into two groups, i.e. hillslope aquifers with exponential hillslope width function (C1) and hillslope aquifers with Gaussian hillslope width function (C2). We found that analytical expressions of the S–Q relation can be derived for C1 hillslope aquifers. For more compound hillslope aquifers, i.e. C2, no explicit S–Q relation can be obtained. The whole subsurface recession after a rainstorm is simulated by applying the initial saturation condition. We found that the simulated S–Q processes can be characterized by a two‐phase recession, i.e. quick and slow recession. The time (t_b) at the dividing point of the quick and slow recessions depends on the geometric factors, such as the plan and profile curvature. In the quick recession for C1, many of the S–Q curves can be described as linear or quasi‐linear functions, which indicate that linear reservoir models can be applied approximately for recession simulations. However, during the slow recession phase of C1 and during the whole recession of C2, the S–Q relations are highly non‐linear. Finally, we compared the hillslope storage kinematic wave and hillslope storage Boussinesq models for simulating subsurface water recession after a rainstorm event in a real‐world headwater catchment (G5) in China. Through comparison of the recession slope curves, we found that the simulated results of the models employing the Gaussian hillslope width function match the observed hydrograph. The results indicated that appropriate organization of the hillslope geometric factors enhances our ability to make S–Q predictions. Copyright © 2016 John Wiley & Sons, Ltd.     

Interactions between perched and saprolite aquifers on a small,salt-affected and deeply weathered hillslope

A small hillslope was chosen to investigate the role of throughflow as a mechanism responsible for the movement of soil water and solutes towards a saline seep and as a source of recharge to a permanent, regional aquifer at depth. The hydraulic properties, chemical characteristics and physical responses of both systems were studied on a deeply weathered, salt-affected hillslope. Additional data were also obtained from other sites in south-western Australia. Regional groundwater flow occurred in a variably textured, deeply weathered material in which the hydraulic conductivity varied from < 0·001 to 0·14m day^?1. Perched groundwater flow (throughflow) occurred in the higher permeability (? 1·5 m day^?1), near-surface soil materials. Throughflow occurred throughout winter, contributing approximately 530 m³ of fresh (? 160 mg l^?1 Cl) water to a saline seep. By contrast, the deep aquifer discharged approximately 1100 m³ of waters with salt concentrations of 2000–6000 mg l^?1 Cl. Recharge and discharge rates to and from the deep aquifer, were estimated to be of the order of 5–20 mm a^?1 and 50–300 mm a^?1 respectively. Saturated conditions existed throughout winter within the seep and the immediately adjacent non-saline area, with up to 60 per cent of the hillslope soils becoming saturated after major rainfall events ( > 20 mm day^?1). In the mid-slopes, in particular along a central depression, saturation of the shallow soils caused macropore channel recharge to take place through the clay-textured subsoils. Water-level responses suggest that approximately 25–30 per cent of annual recharge occurred from one storm studied in September 1984. Recharge through macropore channels is a significant mechanism in the concave slope areas on the hillslope. Throughflow was found to be a major source of water, but not salt, contributing to the saline seep. In general, the contribution of throughflow was found to decrease further inland at other sites studied. However, at inland sites where perennial, perched aquifers have developed in deep sands, saline areas have been caused by throughflow and not by deep aquifer discharge.     

Stream channel stability and forest drainage in the new forest,hampshire

The Forestry Commission carry out drainage work in the uninclosed portion of the New Forest which includes cutting artificial drains and deepening and straightening natural channels. This paper, based on observation of operations on 53 channels for periods up to eight years between 1962 and 1975, examines the geomorphic results of such work. Of 24,000 metres of channel inspected, 24 per cent by length showed erosion, 40 per cent deposition and 36 per cent no apparent change since the work was executed. The subsequent pattern of erosion and deposition within the channels is governed largely by channel slope, but also by the nature of the superficial material and the width: depth ratio of the excavated channel. It is suggested that the choice of some preferred width: depth ratio for a given channel slope might reduce erosion. Instances are given where erosion might be described as severe, 0.5 cubic metres of material per metre of channel per year in two cases, but it appears that after a few years channels stabilize and erosion is also arrested by growth of vegetation.     

Effects of hillslope topography on runoff response in a small catchment in the Fudoji Experimental Watershed,central Japan

We investigated the role of different hillslope units with different topographic characteristics on runoff generation processes based on field observations at two types of hillslopes (0·1 ha): a valley‐head (a convergent hillslope) and a side slope (a planar hillslope), as well as at three small catchments having two types of slopes with different drainage areas ranging from 1·9 to 49·7 ha in the Tanakami Mountains, central Japan. We found that the contribution of the hillslope unit type to small catchment runoff varied with the magnitude of rainfall. When the total amount of rainfall for a single storm event was < 35 mm, runoff in the small catchment was predominantly generated from the side slope. As the amount of rainfall increased (>35 mm), the valley‐head also began to contribute to the catchment runoff, adding to runoff from the side slope. Although the direct runoff from the valley‐head was greater than that from the side slope, the contribution from the side slope was quantitatively greater than that from the valley‐head due to the proportionally larger area occupied by the side slope in the small catchment. The storm runoff responses of the small catchments reflected the change in the runoff components of each hillslope unit as the amount of rainfall increased and rainfall patterns changed. However, similar runoff responses were found for the small catchments with different areas. The similarity of the runoff responses is attributable to overlay effects of different hillslope units and the similar composition ratios of the valley‐head and side slope in the catchments. This study suggests that the relative roles of the valley‐head and side slope are important in runoff generation and solute transport as the catchment size increases from a hillslope/headwater to a small catchment. Copyright © 2011 John Wiley & Sons, Ltd.     

Effects of hillslope topography on hydrological responses in a weathered granite mountain,Japan: comparison of the runoff response between the valley‐head and the side slope

To evaluate the effects of hillslope topography on storm runoff in a weathered granite mountain, discharge rate, soil pore water pressures, and water chemistry were observed on two types of hillslope: a valley‐head (a concave hillslope) and a side slope (a planar hillslope). Hydrological responses on the valley‐head and side slope reflected their respective topographic characteristics and varied with the rainfall magnitude. During small rainfall events (<35 mm), runoff from the side slope occurred rapidly relative to the valley‐head. The valley‐head showed little response in storm runoff. As rainfall amounts increased (35–60 mm), the valley‐head yielded a higher flow relative to the side slope. For large rainfall events (>60 mm), runoff from both hillslopes increased with rainfall, although that from the valley‐head was larger than that from the side slope. The differences in the runoff responses were caused by differences in the roles of lower‐slope soils and the convergence of the hillslope. During small rainfall events, the side slope could store little water; in contrast, all rainwater could be stored in the soils at the valley‐head hollow. As the amount of rainfall increased, the subsurface saturated area of the valley‐head extended from the bottom to the upper portion of the slope, with the contributions of transient groundwater via lateral preferential flowpaths due to the high concentration of subsurface water. Conversely, saturated subsurface flow did not contribute to runoff responses, and the subsurface saturated area at the side slope did not extend to the upper slope for the same storm size. During large rainfall events, expansion of the subsurface saturated area was observed in both hillslopes. Thus, differences in the concentration of subsurface water, reflecting hillslope topography, may create differences in the extension of the subsurface saturated area, as well as variability in runoff responses. Copyright © 2007 John Wiley & Sons, Ltd.     

Rainfall interception and splash detachment with a Brussels sprouts plant: A laboratory simulation

Rates of splash detachment from a sandy soil of the Cottenham Series, subjected to a five-minute design storm of 50 mm/h with a kinetic energy of 127 J/m² and a median volume drop size of 3·2 mm supplied from a rotating-disc rainfall simulator, are determined without a plant cover and with the cover of a single Brussels sprouts plant. Measurements are made at regular intervals throughout the growing season. Plant canopies of 10 to 25 per cent result in reductions of 10 to 25 per cent in rainfall volume and 10 to 81 per cent in rainfall energy. The volume and energy of the rain beneath the plant are significantly (P < 0·05) correlated with its number of leaves (r = ?0·84 and ?0·92 respectively for n = 49). No reduction was observed in the rate of splash detachment which averages 1·2 kg/m² for the design storm with and without the plant. The detachment rate was found rather surprisingly to be inversely related to the energy of the rainfall under the plant and positively related to the number of leaves. As the number of leaves increases, so does the detachment rate per unit of rainfall energy.     

Groundwater dynamics in a till hillslope: flow directions,gradients and delay

Knowledge of groundwater dynamics is important for the understanding of hydrological controls on chemical processes along the water flow pathways. To increase our knowledge of groundwater dynamics in areas with shallow groundwater, the groundwater dynamics along a hillslope were studied in a boreal catchment in Southern Sweden. The forested hillslope had a 1‐ to 2‐m deep layer of sandy till above bedrock. The groundwater flow direction and slope were calculated under the assumption that the flow followed the slope of the groundwater table, which was computed for different triangles, each defined by three groundwater wells. The flow direction showed considerable variations over time, with a maximum variation of 75°. During periods of high groundwater levels the flow was almost perpendicular to the stream, but as the groundwater level fell, the flow direction became gradually more parallel to the stream, directed in the downstream direction. These findings are of importance for the interpretation of results from hillslope transects, where the flow direction usually is assumed to be invariable and always in the direction of the hillslope. The variations in the groundwater flow direction may also cause an apparent dispersion for groundwater‐based transport. In contrast to findings in several other studies, the groundwater level was most responsive to rainfall and snowmelt in the upper part of the hillslope, while the lower parts of the slope reached their highest groundwater level up to 40 h after the upper parts. This can be explained by the topography with a wetter hollow area in the upper part. Copyright © 2011 John Wiley & Sons, Ltd.     

The design of post‐mining landscapes using geomorphic principles

Nature can provide analogues for post‐mining landscapes in terms of landscape stability and also in terms of the rehabilitated structure ‘blending in’ with the surrounding undisturbed landscape. In soil‐mantled landscapes, hillslopes typically have a characteristic pro?le that has a convex upper hillslope pro?le with a concave pro?le lower down the slope. In this paper hillslope characteristic form is derived using the area–slope relationship from pre‐mining topography at two sites in Western Australia. Using this relationship, concave hillslope pro?les are constructed and compared to linear hillslopes in terms of sediment loss using the SIBERIA erosion model. It is found that concave hillslopes can reduce sediment loss by up to ?ve times that of linear slopes. Concave slopes can therefore provide an alternative method for the construction of post‐mining landscapes. An understanding of landscape geomorphological properties and the use of erosion models can greatly assist in the design of post‐mining landscapes. Copyright © 2003 John Wiley & Sons, Ltd.     

The distribution of solute processes on an acid hillslope and the delivery of solutes to a stream: II. Exchangeable Al3+

In order to identify the distribution of aluminium (Al) within an acid hillslope and its release to a stream, the spatial distribution of acid ammonium oxalate extractable Al (Al_o) and exchangeable Al³⁺ have been investigated on a podzolized hillslope in Bicknoller Combe, Somerset, UK. The eluviated Al from topsoils is mainly deposited in the lower soil horizons forming podzolic B horizons, but some Al flows downslope carried by lateral throughflow. Al oxides may provide the main source of exchangeable Al³⁺ on the study slope due to high soil acidity. Examination of the spatial distribution of exchangeable Al³⁺ suggests that the slope hollow, where active convergent throughflow occurs, and the saturation wedge at the base of the slope are the main delivery routes of dissolved Al³⁺ to the stream. Divalent base cations (Ca²⁺ and Mg²⁺), supplied from atmospheric input and organic decomposition and carried by throughflow, exchange Al³⁺ via cation exchange reactions under high water content. Laterally illuviated Al oxides in the lower hollow adjacent to the saturation wedge probably provide a pool for continuous delivery of Al either as soluble or complexed forms to the stream via the saturated wedge. Copyright © 1999 John Wiley & Sons, Ltd.     

Ten year remeasurement of chemical denudation on a magnesian limestone hillslope

Results from the measurement of microweight loss of Magnesian Limestone rock tablets placed at the soil–bedrock interface on a hillslope over 10 years (1982–1992) gave the same relative pattern of upslope increase in weight loss as did short-term measurements (1979–1981), but the absolute values were an order of magnitude lower (0·01–0·03 per cent per year long term and 0·1–0·6 per cent per year short term). Microweighed rock tablets may therefore give unreliable absolute rates and can only be used to indicate relative spatial differences rather than to give reliable data on temporal changes.     

A simple two‐parameter model for scaling hillslope surface runoff

The objective of this research was to develop and parameterise a physically justified yet low‐parameter model to quantify observed changes in surface runoff ratios with hillslope length. The approach starts with the assumption that a unit of rainfall‐excess runoff generated at a point is a fraction β of precipitation P (m) which travels some variable distance down a slope before reinfiltrating, depending on the local rainfall, climate, soils, etc. If this random distance travelled Y is represented by a distribution, then a survival function will describe the probability of this unit of runoff travelling further than some distance x (m). The total amount of per unit width runoff Q (m²) flowing across the lower boundary of a slope of length λ (m) may be considered the sum of all the proportions of the units of rainfall excess runoff integrated from the lower boundary x = 0 to the upper boundary x = λ of the slope. Using these assumptions we derive a model Q(λ) = βPμλ/(μ + λ), (μ > 0, 0 ≤ β ≤ 1, λ ≥ 0) that describes the change in surface runoff with slope length, where μ (m) is the mean of the random variable Y. Dividing both sides of this equation by Pλ yields a simple two‐parameter equation for the dimensionless hillslope runoff ratio Q_h(λ) = βμ/(μ + λ). The model was parameterised with new rainfall and runoff data collected from three replicates of bounded 2 m wide plots of four different lengths (0.5, 1.0, 2.0 and 4.0 m) for 2 years from a forested SE Australian site, and with 32 slope length–runoff data sets from 12 other published studies undertaken between 1934 and 2010. Using the parameterised model resulted in a Nash and Sutcliffe statistic between observed and predicted runoff ratio (for all data sets combined) of 0.93, compared with –2.1 when the runoff ratio was fixed at the value measured from the shortest plot. Copyright © 2014 John Wiley & Sons, Ltd.