Average Erodibility Research Articles

Mapping and evaluating potential Italian traditional and Seawater Pumped Hydro Storage Plants: siltation impact and new GIS approaches

Abstract The development of a significant amount of economically and technically efficient storage infrastructure to enable the integration of non-programmable renewable energy sources into the electricity grid is essential to address challenges such as load balancing, frequency regulation and the need for flexible generation of energy reserves. Pumped Hydro Energy Storage (PHES) plants, a long tradition in Italy, can be a promising solution if new plants suitable for this purpose were to be built (traditional PHES and Seawater PHES plants). To investigate which possible actions can be concretely implemented a Geographic Information System (GIS) procedure was developed and applied across the entire Italian territory. This procedure was further refined using advanced GIS object-based image analysis software to pinpoint areas with ideal topographical features for constructing new reservoirs. An innovative method was used to consider the impacts of the present reservoir silting, which forms part of the potential PHES taking into account the average erodibility of the soils and the sediment contributions from the reservoir basin. All costs were estimated to propose a future plant economic indicator, which also considered siltation conditions. Finally, the most promising plants, identified using this indicator, were proposed. Following an initial screening of 574 currently existing reservoirs, 29 potential PHES were identified deemed suitable, based on considerations of cost/power ratio and siltation incidence. The total Italian potential installable power amounts to approximately 10 GW, with Sardinia emerging as a key region with 4 GW of potential capacity distributed across 11 selected sites. Concerning the Seawater PHES an enormous seawater pumped potential of approximately 20 GW, with 39 possible configurations with and specific cost value per unit of power lower than 2,5 M€/MW could be realized.

Diagnosis and Qualitative Estimation of the Hydric Erosion in Watershed of Talangaï (Brazzaville) by the Application of Geographical Information System (GIS)

This research work exposes the results of our study about the cartographic and the qualitative estimation of the hydric erosion of the watershed of talangaï (Brazzaville) following the PAP/RAC approach, tele detection’s data and the geographical information system in order to elaborate thematic cards by enlightening erosion phenomenon. In this watershed of talangaï, this phenomenon of erosion became recurrent and causes more and more spectacular and worrings. As a matter of fact, except climatic aspects of the study zone, this watershed is characterized by a large irregularity of precipitations, of steep slope, and a weak vegetable coverage; which renders it very vulnerable to the erosion. Thus, after the analysis and the treatment of collected data, from the geographical information system (GIS), the study shows that 65% of zone of study present a weak and average erodibility and 35% of a high and the highest erodibility. Likewise, the descriptive cartography of different forms of erosions shows that ravine surface and the erosion in table-cloth are the most current. Also, the consolidated card of erosion states shows in a qualitative way that soil loss in the zone of study is in relationship with its geomorphology. In short, after combining different cards of principle erosive states, a synthesis has been done producing the consolidated card of these states (Integration approach). All these unfavorable conditions; slopes; geomorphology; erodibility; friability of soils, vegetal covert …), triggering of the catastrophe about erosive phenomenon are fulfilled. Thus, the obtained results allow to identify zones which are the most concerned, and the most vulnerable to hydric erosions.

Determining the effects of land use on soil erodibility in the Mediterranean highland regions of Turkey: a case study of the Korsulu stream watershed

Sustainable soil management can be concisely defined as using soil without impairing soil function. It has become crucial due to soil degradation, especially that caused by soil erosion, throughout the world. In this context, this study aimed to determine the erodibility and some soil properties to evaluate the actual state of soil resources in a watershed located in the Mediterranean highland of Turkey. A total of 180, 90 disturbed and 90 undisturbed, soil samples were collected from different land-use types, namely, forest, pasture, and agriculture. Erodibility and soil properties such as texture, soil organic matter, permeability, particle density, bulk density, porosity, pH, electrical conductivity, field capacity, permanent wilting point, and water holding capacity were determined. A soil erodibility map was also produced using ArcGIS software. According to the results, the average soil organic matter was 6.27%, 4.56%, and 2.05% in forest, pasture, and agriculture, respectively, and the differences among land-use types were significantly different. The average erodibility (USLE-K) value was 0.09 for forest, while it was 0.12 and 0.22 for pasture and agriculture, respectively. The difference between agriculture and forest and pasture was statistically significant, while no statistically significant difference was found between forest and pasture in the study area. Forest was included in the slightly erodible class, while pasture and agriculture were included in the moderately and highly erodible classes, respectively. The erodibility map also revealed that a major part of the study area is susceptible to erosion. The study clearly showed that sustainable soil management is a necessity, especially for agricultural lands.

Estimation of riverbank soil erodibility parameters using genetic algorithm

Determination of the erodibility parameters, such as critical shear stress and erodibility coefficient, are necessary before estimating the annual bank erosion (or bank retreat) at river reaches. However, in many cases, the river site is inaccessible making it difficult to assess the soil parameters either by in situ tests or by laboratory experiments. In this study, Genetic Algorithm (GA)-based optimisation technique was used to estimate the erodibility parameters of middle reaches of the Brahmaputra River in India. Two approaches were followed. At first, erodibility parameters were estimated using daily stage records at a selected site. Secondly, based on the annual observed bank erosions (bank retreat) from satellite images, erodibility parameters were estimated in three different river reaches. All these results were compared with that from a previous study using in situ jet tests. Annual bank erosions (bank retreat) were estimated using the median values of the erodibility parameters. The results agree well with the average observed annual bank erosion of these river reaches. In addition, the effects of measurement errors and optimisation algorithms on the parameter estimation were analysed. Sensitivity analysis of the parameters in GA was evaluated and it was found that GA can be utilised in the data-scarce regions to estimate the average erodibility parameters.

Soil properties influencing erodibility of soils in the Ntabelanga area, Eastern Cape Province, South Africa

ABSTRACTSoil erosion has serious off-site impacts caused by increased mobilization of sediment and delivery to water bodies causing siltation and pollution. To evaluate factors influencing soil erodibility at a proposed dam site, 21 soil samples collected were characterized. The soils were analyzed for soil organic carbon (SOC), exchangeable bases, exchangeable acidity, pH, electrical conductivities, mean weight diameter and soil particles’ size distribution. Cation exchange capacity, exchangeable sodium percentage, sodium adsorption ratio, dispersion ratio (DR), clay flocculation index (CFI), clay dispersion ratio (CDR) and Ca:Mg ratio were then calculated. Soil erodibility (K-factor) estimates were determined using SOC content and surface soil properties. Soil loss rates by splashing were determined under rainfall simulations at 360 mmh−1 rainfall intensity. Soil loss was correlated to the measured chemical and physical soil properties. There were variations in soil form properties and erodibility indices showing influence on soil loss. The average soil erodibility and SOC values were 0.0734 t MJ−1 mm−1 and 0.81%, respectively. SOC decreased with depth and soil loss increased with a decrease in SOC content. SOC significantly influenced soil loss, CDR, CFI and DR (P < .05). The soil loss rate was 5.60 t/ha per 8 minute rainstorm of 360 mmh−1. Addition of organic matter stabilize the soils against erosion.

Implications of Texture and Erodibility for Sediment Retention in Receiving Basins of Coastal Louisiana Diversions

Although the Mississippi River deltaic plain has been the subject of abundant research over recent decades, there is a paucity of data concerning field measurement of sediment erodibility in Louisiana estuaries. Two contrasting receiving basins for active diversions were studied: West Bay on the western part of Mississippi River Delta and Big Mar, which is the receiving basin for the Caernarvon freshwater diversion. Push cores and water samples were collected at six stations in West Bay and six stations in Big Mar. The average erodibility of Big Mar sediment was similar to that of Louisiana shelf sediment, but was higher than that of West Bay. Critical shear stress to suspend sediment in both West Bay and Big Mar receiving basins was around 0.2 Pa. A synthesis of 1191 laser grain size data from surficial and down-core sediment reveals that silt (4–63 μm) is the largest fraction of retained sediment in receiving basins, larger than the total of sand (&gt;63 μm) and clay (&lt;4 μm). It is suggested that preferential delivery of fine grained sediment to more landward and protected receiving basins would enhance mud retention. In addition, small fetch sizes and fragmentation of large receiving basins are favorable for sediment retention.

Spatial variability of soil erodibility and its correlation with soil properties in semi-arid mountainous watershed, Saudi Arabia

Soil erodibility values are best estimated from long-term direct measurements on runoff-plots; however, in lack of field tests, these values can be estimated using relationships based on physico-chemical soil properties. The study objective was to assess the erodibility and its correlation with soil properties. The average erodibility value was estimated 0.043 t ha h ha−1 MJ−1 mm−1. The areas with heavy textured soil and low organic matter content had the lowest values of erodibility. The erodibility decreases as the sand content increases, whereas silt showed a positive correlation. The erodibility factors and its relation to soil properties were evaluated using multiple regression analysis. Results revealed that sand and organic matter content of soil combinedly explained 78% of variation. Altitudinal increases also seem to affect the soil texture. This study has demonstrated that soil properties and erodibility values can be used as assistance for soil conservation practices and modelling of landscape processes.

STANOVENÍ INTENZITY VĚTRNÉ EROZE NA TĚŽKÝCH PŮDÁCH

Wind erosion, common problem of light-textured soils, was determined on heavy clay soils in the foothills of Bílé Karpaty Mountains, Czech Republic. Soil erodibility by wind was determined from the Map of potential erodibility of soil by wind and from the calculation of potential and real soil loss by wind. All the determinations show underestimation of soil erodibility by wind on heavy clay soils, because methods that are used for this are based above all on the assessment of clay particles content and the presumption the more clay particles soil contains, the less vulnerable to wind erosion is. The potential erodibility of soil by wind is 0,09 t . ha−1 per year. The determined value does not exceed the tolerable soil loss limit 10 t . ha−1 per year for deep soils. The real average erodibility of soil by wind has the highest value 1,47 g . m−2 on November 30th, 2008. Other soil losses that do not exceed the tolerable soil loss limit 1,4 g . m−2, were determined on March 18th and 28th, 2008. Big difficulties come with the assessment of the erodibility of heavy clay soils in the areas, where soil erosion ve­ri­fia­bly exists, but it is not assessable by objective calculating methods. Evident necessity of new know­ledge concerning the determination of wind erosion intensity follows from the results.

Using universal soil loss equation and soil erodibility factor to assess soil erosion in Tshesebe village, north east Botswana

Soil erodibility (K) factor in the Universal Soil Loss Equation (USLE), defines the resistance of soil to detachment by rainfall impact and/or surface flow force. Whilst there are a number of factors of erosion, this study aims to use erodibility factor and related length slope factor to assess soil erosional loss at field scale. To quantify soil erodibility the following properties were measured; texture, organic matter content and structural properties of the soil samples in eroded and non-eroded sites. Sub sampling was conducted in both eroded and non-eroded site and a total of six samples were collected in each site. In addition, slope length and slope angle were determined to evaluate the slope effect on the degree of soil loss associated with the K-factor. The measured or estimated K-factor value compared with the USLE K-based nomograph. The average soil erodibility (K-factor) was 0.031 and (t ha h ha-1 MJ-1mm-1 ) for eroded and non-eroded area, respectively. The high K-factor value in eroded area (almost doubled) was associated with low organic matter content (0.75%) compared to high organic matter in non-eroded (1.18%) as well as the significant slope (3°) in eroded than non-eroded areas (1°). The results also show that K-factor significantly (P<0.05) correlates with soil texture and organic matter due to their strong binding effect on aggregate stability and water infiltration hence enhanced particles’ resistant to detachment. Interestingly there was no significant difference in K- factor values between eroded and non-eroded areas. Further, the K-factor based nomograph over-predicted the measured K-factor value by 10 times in eroded and 19 times in non-eroded soil, with a strong correlation in eroded (r2 =0.77) than in non-eroded (r2 =0.10). Key words: Universal soil loss equation (USLE), soil erodibility, soil erosion, soil properties, eroded and noneroded areas.

Evaluating SEDCAD Model Performance on Reclaimed Coal Mine Lands in East Tennessee

AbstractThe Sediment, Erosion, Discharge by Computer Aided Design (SEDCAD) program is extensively used in the mining industry for engineering site layout plans with best management practices (BMPs) for erosion control. Although SEDCAD is the primary BMP design tool used in this industry, very limited published information is available on its performance estimating site runoff and sediment yields. This study compared sediment yields from three surface coal mining sites in east Tennessee with SEDCAD modeled outputs. Study sites included active mining operations on steep slopes (>20°) where after mining, approximate natural hillslope contours were reconstructed by using loose spoil materials on top of slope, at shallow depths of 1–2 m, following the Forest Reclamation Approach. The SEDCAD model inputs included the site-derived hydrologic curve number (CN) of 59 and average erodibility K factors ranging from 0.001–0.034 Mg·ha·h·ha−1·MJ−1·mm−1 varying on the basis of pre- and postrill development periods and ...

Assessment of soil losses and siltation of the K’sob hydrological system (semiarid area—East Algeria)

Soil losses and siltation of the hydrological system (watershed–dam) of K’sob were obtained using direct and indirect methods. The Wadi K’sob watershed of 1,484 km2, average slope of 0.14, and average elevation of 1,060 m is located in a semiarid climate. The average annual rainfall is 341 mm and the mean annual water discharge is 0.89 m3/s. Data from the Medjez gauging station located 6 km upstream of the dam, are the daily liquid flow and instantaneous concentrations of suspended sediments. Over a time period from 1973 to 2010, the relationship between water and sediment discharges is quantified by the equation: Qs = 5.6 Q1.31. Thus, in view of the availability data on a daily scale, the assessment of soil erodibility of the K’sob watershed was used to estimate specific soil losses of 203 t km−2 year−1or 301,000 t eroded annually from the K’sob basin. The bathymetric measurements of the sediment volumes deposited in the K’sob dam, has quantified the annual siltation of 0.8 hm3, corresponding to an average erodibility of the K’sob watershed of 809 t km−2 year−1. However, when adding the volume of sediment removed by the dredging operation and de-silting by the valves during heavy floods, the value of soil losses is 2,780 t km−2 year−1. The indirect assessment of soil erodibility of the basin was obtained by applying two models: the quantitative geomorphological analysis (QGA) and PISA model (prediction of silting in the artificial reservoirs, in Italian: Previsioni dell’Interimento nei Serbatoi Artificiali) using physical and climatic factors in the watershed. The obtained results by QGA method underestimate specific soil losses of 524 t km−2 year−1. The PISA model gives a value of 2,915 t km−2 year−1, which is close to the value obtained by bathymetric measurements. This study concludes that PISA model is most suitable to estimate soil loss and siltation of the K’sob hydrological system.

Erodibility of fine soil from the composite river bank of Brahmaputra in India

AbstractThe erosion of a composite river bank critically depends on the erodibility of its fine soils, as the fine soil has higher resistance against erosion. Therefore, for the estimation of the bank erosion in the case of a composite river bank, it is important to determine the critical shear stress and erodibility coefficients of the bank soil and their spatial distribution. In the present study, erodibility parameters of the river bank of Brahmaputra in India have been estimated through 58 in situ submerged jet tests. The significance of spatial and layer‐wise distribution of the erodibility parameters was tested through analysis of variance (ANOVA). Results indicate that the spatial variation of erodibility parameters is highly significant, but layer‐wise variations of the erodibility parameters are not significant. Therefore, the erodibility of the riverbank depends on the particular location, whereas layer‐wise average erodibility parameters can be lumped for the estimation of the bank erosion for the specific site. Using the measured erodibility parameters, yearly river bank erosions at the study locations were computed and found to fall within the reported range of the bank erosion in the Brahmaputra River. Copyright © 2010 John Wiley &amp; Sons, Ltd.

Impact of Vertical Hydraulic Gradient on Rill Erodibility and Critical Shear Stress

Field observations and subsequent support from controlled lab experiments have shown the linkage between vertical hydraulic gradients and initiation of rills or incised channels. Nevertheless, current erosion prediction models do not include the hydraulic gradient effect in the erodibility parameters. The objectives of this study were to: (i) quantify the impact of a subsurface hydraulic gradient on rill erodibility and critical shear stress; and (ii) determine the intrinsic critical shear stress of the soil, which is the critical shear stress of the soil without the influence of the vertical hydraulic gradient. Two series of concentrated flow experiments were conducted on a silt loam soil subjected to hydraulic gradients varying from −2 to 2 m m−1 The first series measured soil loss at a given shear stress, while the second was a direct measurement of the critical shear stress by calculating the shear stress at the point of incipient motion of particles. The intrinsic critical shear stress was determined using a fluidized bed approach. We found that the average erodibility under an upward or positive hydraulic gradient was 5.64 times larger than that under a downward hydraulic gradient. The critical shear stress decreased from 1.0 to 0.2 Pa as the hydraulic gradient was increased from −2 to 2 m m−1 Using the fluidized bed approach, we obtained an inherent critical shear stress value of 1.61 Pa. We concluded that the presence of positive pore pressure near the soil surface was a major factor in reducing soil cohesion. We also concluded that with some improvements, the fluidized bed concept can be applied to determine the critical shear stress independent of the hydraulic gradient effect.

Spatial modeling of soil erosion potential in a tropical watershed of the Colombian Andes

Soil erosion potential of a 58 km 2 watershed in the coffee growing region of the Colombian Andes was assessed using the Revised Universal Soil Loss Equation (RUSLE) in a GIS environment. The RUSLE factors were developed from local rainfall, topographic, soil and land use data. Seasonal erosivity factors ( R) were calculated for six pluviographic stations (1987–1997) located within 22 km of the basin. Two regression models, one for the wet and one for the dry seasons, were created and used to estimate seasonal erosivity for 10 additional stations with pluviometric data. Erosivity was on average higher in the wet seasons (4686 MJ mm ha − 1 h − 1 season − 1 ) than the dry ones (2599 MJ mm ha − 1 h − 1 season − 1 ). Seasonal erosivity surfaces were generated using the local polynomial interpolation method, and showed increases from west to east in accordance with regional elevation. Soil erodibility was calculated from field measurements of water stable aggregates (> 2 mm) and infiltration, which were influenced by land use. Three erodibility scenarios were considered (high, average and low) to represent the variability in infiltration measurements within each land use. The topographic and land cover factors were developed from existing contour and land use data. Model results indicated that in the dry seasons, and under the average erodibility scenario, 534 ha (11%) of the basin's rural area were within the extreme erosion potential category (above 3.5 t ha − 1 season − 1 ). During the wet seasons, this area increased to 1348 ha (28%). In general, areas under forest and shrub had low erosion potential values, while those under coffee and pasture varied according to topography. Modeling of probable land use change scenarios indicated that the erosion potential of the basin would decrease as a result of coffee conversion to pasture.

Erodibilidade de um cambissolo húmico alumínico léptico, determinada sob chuva natural entre 1989 e 1998 em Lages (SC)

O fator de erodibilidade do solo (fator K) da Equação Universal de Perda de Solo (EUPS) refere-se à susceptibilidade natural do solo à erosão e representa a quantidade de solo perdida por unidade de erosividade da chuva (fator R), sendo o seu conhecimento importante no planejamento conservacionista. Utilizando dados de perda de solo, obtidos sob condições de chuva natural, em tanques coletores de escoamento superficial, e de erosividade (EI30) das chuvas naturais, no período de 1989 a 1998, em Lages (SC), calculou-se, pelo quociente entre essas variáveis e por regressão linear simples entre elas, o fator de erodibilidade do solo para um Cambissolo Húmico alumínico léptico com 0,102 m m-1 de declividade média. Com este objetivo, foram utilizados valores de erosividade (EI30) de 437 eventos de chuva e de perdas de solo, obtidas em parcelas de 3,5 x 22,1 m desprovidas de vegetação e crosta superficial. O preparo do solo, executado no sentido paralelo ao declive duas vezes ao ano, consistiu de uma aração e duas gradagens. Os valores de erodibilidade médios anuais, estimados pelo quociente e por regressão linear simples entre as perdas de solo e as erosividades, foram de 0,0115 e 0,0151 Mg ha h ha-1 MJ-1 mm-1, respectivamente. Pelos respectivos modos de obtenção, os valores de erodibilidade médios estacionais estimados foram de 0,0105 e 0,0121 Mg ha h ha-1 MJ-1 mm-1, para a primavera-verão, e de 0,0132 e 0,0220 Mg ha h ha-1 MJ-1 mm-1, para o outono-inverno.

Non-dimensional analysis of a physically based rainfall-runoff-erosion model over steep slopes

A non-dimensional analysis is undertaken to study the rainfall-runoff-erosion process over steep slopes. This analysis leads to a reduction in the number of parameters required for modeling erosion by overland flow. Two important parameters are identified: the dimensionless erodibility σ ∗ denoting the average erodibility over the whole slope, and G which partitions the sediment transport dynamics into detachment, deposition and equilibrium regimes. Analytical solutions are presented for the non-dimensional equations. It is observed that G remains practically time-invariant but has a marked spatial variation along the slope. The dimensionless sediment discharge along the slope shows strong dependence on both σ ∗ and G. Some asymptotic results are also discussed along with practical applications of this work.

ERODIBILITY AND SEDIMENT YIELD BY NATURAL RAINFALL FROM RECONSTRUCTED MINE SOILS

Standard erosion plots, 4.6 m × 22.1 m with a 9% slope, were established at the Alston Surface Mine in Ohio County, Kentucky. Topsoil, subsoil, and mine spoil were reconstructed according to contemporary reclamation techniques using bulldozers and scraper pans. The following parameters were evaluated under natural rainfall conditions: runoff, erodibility, and sediment yield. Mine spoil yielded significantly greater runoff volumes, compared with reconstructed topsoil and subsoil. Average curve numbers (CN) were 89.1, 90.1, and 94.7 for bare topsoil, subsoil, and mine spoil, respectively. The subsoil plot generated the largest amount of sediment and was significantly more erodible than topsoil and mine spoil. The average annual erodibility (K) factor was .046 for topsoil, .067 for subsoil, and .051 for mine spoil, with units of Mg ha h (ha MJ mm)−1. Erodibility of the mine spoil in this study was more than twice that measured from an earlier rainfall simulation study at this same site. Differences in the degree of physical weathering were believed to be the primary contributing factor and, therefore, have important implications when assessing the erodibility of highly weathered spoil materials (e.g., abandoned mine lands). Compared with the soil interpretation record's K factor estimate for the soil series evaluated in this study, the reconstructed topsoil was not inherently more erodible than when existing in its natural, undisturbed condition. Therefore, factors that the mine operator can control (e.g., slope, vegetative cover, management practices) will likely have a greater impact on whether erosion losses from reconstructed soils are significantly greater than losses that occur under pre-mining conditions.

A STUDY OF RILL DEVELOPMENT PROCESS BASED ON FIELD EXPERIMENTS

Two rectangular experimental slopes are set on a bare slope in order to study the development process of rills (Figs. 1 and 2, Photo. 1). One slope is aimed to observe mainly the development pattern of rills (A-slope in Fig. 2) and another to measure the hydraulic conditions (B-slope in Fig. 2). Both have concrete partition walls except the lower end to eliminate the inflowing of water and sediment from outside the experimental slopes. The electrical resistivity survey, the cone penetration test and the grain-size analysis are also carried out to survey the soil conditions (Figs. 3, 4, 5 and 7). The results of them show that the soil conditions are nearly homogeneous in the traverse and change remarkably from the middle part to the lower part along the longitudinal profile. As for the hydraulic conditions, iL is recognized that the surface erosion occurrs when the rainfall intensity is above 1 mm/10 min (Fig. 8) and surface discharge is more closely related to the rainfall in 30 min than the rainfall in 60 min (Figs. 9 and 10). It is also observed that the sediment discharge per unit width at the end of the slope is nearly proportional to the square of the surface discharge (Fig. 13). Therefore, it can be considered that the sediment discharge per unit area at a certain place in such slopes is proportional to the length from the upper end to the place in a given rainfall intensity if the surface run-off coefficient is constant on the whole slope. Next, we shall discuss the erodibility of the slope by using _??_ where _??_ is the average erodibility coefficient, _??_the sediment discharge and the surface discharge at the end of the slope respectively and S the gradient of the slope. Using the values of _??_ and S obtained from the field experiment for the equation, we can obtain values of the coefficient key (Fig. 15). The figure shows that the value (_??_) decreases almost exponentially with the passing of time. This means that the erosible material of the slope decreases in the course of time. Then, let us investigate the relationship between the slope of the experiment plot and the model slope in order to examine the theoretical model which can validly apply to the development of rills in the model slope (Kashiwaya, 1979). The relationship between variables which play important roles for the rill morphology can be expressed as follows ;_??_(8-2) with D; drainage density, υ; flow velocity, ke; erosion proportionality factor, w; width of rill, ρ; density fluid, μ, viscosity of fluid and g; acceleration of gravity. Assuming that p, p and g take the same values both in the prototype and in the model, we can obtain_??_(9-1) and_??_ with RA; stream area number and Ho; Morton number, from eq. (8-2) by employing dimensional analysis. Therefore, it is said that the law of similarity can be established by using the above two equations. As for the number of rills in the steady state, let us introduce the next equation which is valid for the model slope; _??_ with k; the number of rills, p ; the coefficient expressing the relationship between the number of rills and the total width of rills, r; the ratio of the branching coefficient to the joining coefficient and N; the number of initial rills. It is derived from the stochastic differential equation based on the two hypotheses; the joining probability is proportional to the number of rills and the branching probability is proportional to the relative width (width/depth). In the present slope, assuming that the number of initial rills N is proportional to the slope length and r is in inverse to N (Fig. 17) and using the value of p given in the field experiment (Fig. 16), we can obtain the theoretical number of rills in the steady state of this slope.