Study of surface runoff usingphysical modelsM. Amin Haque

Abstract The profiles of overland water flow as afunction of space and time obtained by applyingkinematic wave approximations combined with theDarcy-Weisbach resistance formula to laminar floware presented. Rainfall-excess is assumed to remainconstant during a certain period of time. Runofffrom surfaces of constant slope, with uniform sur-face texture, and the effects of different parameterson overland flow have been studied. Comparisons ofrunoff using Darcy-Weisbach, Manning, and Chezyresistance formulas have been made. It was foundthat the lower the rainfall-excess rates, the longer thesurface runoff starting time, peak time, and smallerthe peak runoff value at any distance. It was alsofound that the overland flow increases rapidly to thepeak value, followed by a rapid decline which beginsat the moment the rainfall-excess ceases, and thenapproaches zero slowly. Comparison of the theo-retical calculations of runoff with the measured dataof the Los Angeles field tests on concrete-pavedsurface shows good agreement.

Keywords Surface runoff Æ Kinematic waveapproximations Æ Darcy-Weisbach resistanceformula Æ Laminar flow Æ Constant slope Æ Constantrainfall-excess

Introduction

Overland flow at a point on a soil surface starts aftersurface ponding takes place. Ponding occurs when surface

soil layers become saturated, and the saturation zone be-gins to move downward into the soil. The infiltration rate,i(x,t), becomes smaller than the rainfall rate, R(x,t), andthe difference of the two, the rainfall-excess rate, r(x,t),becomes available for overland flow. In arid and semi-aridand upslope areas, surface saturation occurs when therainfall intensity, R(x,t), is greater than saturated hydraulicconductivity, Ks, of the surface soil, and rainfall duration islonger than the ponding time for a given initial soilmoisture profile (Horton 1933, 1945; Rubin and Steinhardt1963). In humid areas, where surface hydraulic conduc-tivities are low, or where a shallow water table exists,runoff is produced for R(x,t)<Ks, whose rate is approxi-mately equal to the rainfall intensity, i.e., i(x,t)�0 andr(x,t)�R(x,t) (Dunne 1970, 1978). According to Horton(1933, 1945), runoff from partial areas of the hillslope isproduced where surface hydraulic conductivities are low-est. According to Dunne (1970, 1978), runoff from partialareas of the hillslope is produced where water tables areshallowest. Starting times of overland flow from wetlandsare very short as compared to arid or semi-arid lands, butthe contribution due to the Dunne mechanism is small ascompared to that of the Horton mechanism (Betson 1964;Dunne 1970). Woolheiser (1982) has discussed the over-land flow domain, showing regions of both the Hortonianand Dunne surface runoff generating mechanisms. Larsenand others (1994) have studied both the surface runoffmechanisms in small experimental catchments. Koivusaloand Karvonen (1995) found that the quantity of runoffdepends considerably on the antecedent conditions in soilmoisture and groundwater level. They found that even thelargest summer storms generated hardly any runoff whenthe soil was dry and well-cracked. They found that afterthe groundwater level reached the topsoil layers, theamount of surface runoff increased. There are interactionsamong the varying infiltration properties as water flowsover the surface, which in turn affect the production ofHortonian runoff (Freeze 1972; Smith and Hebbert 1979;Freeze 1980; Hawkins 1981; Woolheiser and Goodrich1988). Infiltration into soil occurs due to rainfall and tosurface water after rainfall ceases.The most important parameters which govern overlandflow are topography, surface roughness, soil infiltrationcharacteristics – initial soil moisture content, pressurehead, and saturated hydraulic conductivity, as well as thedistribution, duration, and intensity of rainfall. Theintensity of rainfall varies continuously in space and time.

Received: 29 June 2001 / Accepted: 3 October 2001Published online: 24 November 2001ª Springer-Verlag 2001

M. Amin HaqueDepartment of Chemistry and Physics,Alcorn State University, Alcorn State, Lorman,Mississippi 39096, USAE-mail: amin@lorman.alcorn.eduTel.: +1-601-6385727Fax: +1-601-8773989

Original article

DOI 10.1007/s00254-001-0455-1 Environmental Geology (2002) 41:797–805 797

Surface roughness and soil characteristics are spacedependent. The dependence of surface runoff on theseparameters has been recognized and studied by severalresearchers (Wilson and others 1979; Wu and others 1983;Troutman 1983, 1985a, 1985b; Woolheiser and Osborne1985; Sivapalan and Wood 1986; Wood and others 1988;Logue and Gander 1990).

Methodology

Analytical solutions of equations of motionof overland flow of rainfall-excess

The rainfall-excess rate, r(x,t), is the difference betweenrainfall and the infiltration rates on a surface soil. In thisstudy, the author assumed uniform rainfall-excess rate,and uniform soil properties across the surface. The rain-fall-excess generated at any point contributes to infiltra-tion supply in addition to rainfall at all points on the flowpath below it. Runoff from the upstream would cause anearlier runoff from a downstream point if it had beenexpected to pond much later by itself. A flow path withdecreasing infiltrability should have no effects frominteraction of surface flow and ponding time.The analysis of overland flow of rainfall-excess consists of(1) calculating infiltration and rainfall-excess rates for agiven soil and rainfall rate, and (2) solving the kinematicwave equations of motion for the rainfall-excess analyticallyor numerically by applying appropriate surface resistanceformulas. Overland flow has been studied extensively usingboth physical and analytical models by a number of re-searchers (Henderson and Wooding 1964; Morgali andLinsley 1965; Wooding 1965a, 1965b; Woolheiser and Ligget1967; Foster and others 1968; Kibler and Woolheiser 1970;Smith and Woolheiser 1971; Woolheiser 1975; Singh andWoolheiser 1976; Chery and others 1979; Smith and Hebbert1979; Freeze 1980; Parlange and others 1981; Woolheiser1982; Rose and others 1983; Campbell and others 1985; Limaand Van der Molen 1988; Woolheiser and Goodrich 1988;Pearson 1989; Julien and Moglen 1990; Logue 1990; Dunne

and others 1991; Ogden and Julien 1993; Van der Molen andothers 1995; Koivusala and Karvonen 1995; Kuchment andothers 1996; Woolheiser and others 1996).A typical rainfall-excess flow on a surface of width W,length L, and constant slope S is shown in Fig. 1. Applyingthe principles of conservation of mass, momentum andenergy, and appropriate surface resistance formula to themotion of rainfall-excess, and solving the equations, rela-tionships are obtained which describe the characteristicsof the overland water flow profile (Henderson and Woo-ding 1964; Wooding 1965a, 1965b; Woolheiser and Ligget1967; Woolheiser 1975; Parlange and others 1981; Wool-heiser 1982; Rose and others 1983; Lima and Van derMolen 1988; Julien and Moglen 1990). The one-dimen-sional partial differential equation for conservation ofmass is

@h=@t þ @q=@x ¼ rðx; tÞ ð1Þ

The volumetric water flux q(x,t) is related to the waterlayer thickness h(x,t), surface roughness parameter a, andmomentum equation coefficient m, and is given by thekinematic flow approximation of Lighthill and Whitham(1955) by the momentum equation

q ¼ vh ¼ ahm ð2Þ

v ¼ dx=dt ¼ ahm�1 ð3Þ

Whether analytically or numerically, Eqs. (1) and (2) arebest solved by the method of characteristics (Hendersonand Wooding 1964; Wooding 1965a, 1965b; Woolheiserand Ligget 1967; Woolheiser 1975; Parlange and others1981; Woolheiser 1982; Rose and others 1983; Lima andVan der Molen 1988). Applying appropriate initial andboundary conditions, the differential equations of motion(Eqs. 1 and 2) are solved analytically.The solutions, Eqs. (5), (6), (7), (8), and (9), describe thecharacteristics of the overland water flow profile underdifferent conditions.

Fig. 1Rainfall-excess flow on a surface of widthW, length L, and constant slope S

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798 Environmental Geology (2002) 41:797–805

@h=@t þ c@h=@x ¼ r ¼ dh=dt ¼ dq=dx ð4Þ

h ¼ rt and q ¼ rx ¼ ahm ¼ armtm ð5Þ

dq=dh ¼ c ¼ amhm�1 ¼ mv ð6Þ

x ¼ arm�1tm ð7Þ

q ¼ vh ¼ ahm ¼ armtm ¼ rx ð8Þ

h ¼ ðr=aÞ1=mxm ð9Þ

where x is any distance (m) from top of the field (the pointof initiation of rainfall-excess, t time (s), q outflow ordischarge per unit width (m2/s), v mean flow velocity(m/s), and c characteristic velocity.The outflow flux q (surface runoff) as a function of time tis calculated using Eq. (8) under different situations.

Case 1. Duration of rainfall-excess t0 > steadystate time ts

Increasing Outflow Range : 0 � t � ts :

q ¼ armtm ð10Þ

Constant Outflow Range : ts � t � t0 : q ¼ rL ð11Þ

Decreasing Outflow Range : t0 � t � tm :

q � rL þ rma1=mqðm�1Þ=mðt � t0Þ ¼ 0 ð12Þ

where tm is some specified time >t0

Case 2. Duration of rainfall-excess t0 < steadystate time ts

Increasing Outflow Range : 0 � t � t0 :

q ¼ armtm ð13Þ

Constant Outflow Range : t0 � t � ðt0 þ tcÞ :q ¼ armtm

0 ð14Þ

Decreasing Outflow Range : t > ðt0 þ tcÞ :q � rL þ rma1=2qðm�1Þ=mðt � t0Þ ¼ 0 ð15Þ

where t0 þ tc ¼ t0 þ Lðrt0Þðm�1Þ=ðamÞ � t0=m ¼ t1 ð16Þ

and

ts ¼ ðL=arm�1Þ1=m ð17Þ

Surface resistance laws and parameters a and mDue to fluctuations in surface water layer thickness h andthe surface roughness, the flow may vary betweenlaminar and turbulent. The surface roughness parametera and the momentum equation coefficient m are semi-empirical constants, although they are usually given aphysical or quasi-physical interpretation by relating themto the soil surface and hydraulic characteristics. Theparameter a is generally considered to represent theeffects of surface slope and roughness on the flow ofwater layer thickness h.The exponent m is considered to be a measure of liquidturbulence, which characterizes the flow regime as lami-nar, turbulent, transitional, or disturbed due to raindropimpact. The value of m indicates the linearity or nonlin-earity of the surface runoff q and h relationship. The

Fig. 2Water layer thickness h plotted as afunction of distance x

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nonlinearity relationship between q and h has long beenrecognized.The Darcy-Weisbach resistance law is appropriate forlaminar flow, and the Manning and Chezy formulas areused for turbulent flow (Woolheiser 1975).

a ¼ ð8g SÞ=ðFmÞm ¼ 3 Darcy � Weisbach Law Laminar Flow ð18Þ

a ¼ 1:49S1=2=n

m ¼ 1:67 Manning Formula Turbulent Flow ð19Þ

a ¼ CS1=2

m ¼ 1:5 Chezy Formula Turbulent Flow ð20Þ

where g is the acceleration due to gravity (9.8 m/s2), F thesurface characteristic parameter (24 for concrete to 40,000for bluegrass sod), n the Manning roughness coefficient(s/m1/3), m soil water viscosity (0.01 cm2/s), and C theChezy coefficient (m1/2/s).Most of overland flow problems can be modeled as kineticflow and can adequately be described by the Darcy-Weisbach resistance law (Woolheiser 1975). The values ofresistance parameters F, n, and C for overland laminar and

Fig. 3Variation of surface runoff q with thedistance x

Fig. 4Partial steady hydrographs as a function oftime t for three different values of rainfall-excess rates

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turbulent flows on different types of surfaces are given byWoolheiser (1975).

Results and discussion

The results of these studies of surface runoff are presentedin the following figures. The runoff using the Darcy-Weisbach resistance law assuming laminar flow (m=3)from surfaces of constant slope, and its dependence on thenature of the surface and the nature of flow were studied.

The rainfall-excess rate r is assumed to be constant duringa certain period of time t0‡t‡0, i.e., r is considered to be astep function rather than a continuous function of time t.In Fig. 2, the water layer thickness h is plotted as a func-tion of distance x. The variation of runoff q with x isshown in Fig. 3. Figure 4 shows that the partial steadyhydrographs are nonlinear because the peak value of qdepends on both the rate of rainfall-excess r and its du-ration t0. The duration of peak for peak value of q dependson a complex function of t0 and r. Figure 5 shows that as t0

becomes larger than ts, the complete hydrographs becomeasymptotically linear, because the peak value of q increases

Fig. 5Full hydrographs as a function of time t fortwo different values of duration of rainfall-excess

Fig. 6Variation of surface runoff with time tusing Darcy-Weisbach, Manning, and Chezysurface resistance laws

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linearly with r, and duration of peak q becomes asymp-totically equal to t0 when t0>>ts. As can be seen from theinspection of the surface runoff graphs, the starting timefor the runoff, steady state onset time ts, and the peakvalue of flow are all dependent on the rainfall-excess rate r.The lower the value of r, the longer the runoff startingtime, the peak time ts, and smaller the peak value of q. Thevalue of q decreases very rapidly from the moment r be-comes zero, and then approaches 0 slowly. Figures 6 and 7indicate that the nature of outflow as described by dif-ferent resistance laws is different. It is observed that thepeak value of q is reached almost instantly according to

the Manning and Chezy formulas, but much slower ac-cording to the Darcy-Weisbach formula. Figure 8 showsthat the surface runoff increases linearly with surface slopeS. The slope shape of surfaces affects the magnitude andtime of occurrence of the peak value of q (Lane andWoolheiser 1977). Lima and Van der Molen (1988) havestudied overland flow over an infiltrating parabolic-shapedsurface and found that it is strongly affected by the shapesof the slopes. Figure 9 shows strong dependence of q onthe surface parameter F, decreasing sharply with the in-crease of F. Resistance to overland flow over a naturalsurface is affected by several factors (Lane and Woolheiser

Fig. 7Variation of surface runoff with water layerthickness h using Darcy-Weisbach, Manning,and Chezy surface resistance laws

Fig. 8Variation of surface runoff with surfaceslope S

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1977; Emmett 1978). At high rates of flow the boundarymay change in time and distance because of erosion orbending of vegetation. On non-vegetated surfaces, rain-drop impact exerts a significant retarding effect.In Fig. 10, the calculated results of q versus t based on theDarcy-Weisbach resistance formula and m=3 have beencompared with the measured data of the Los Angeles field

tests on concrete-paved surface (Yu and McNown 1964).The measured data are: slope S=0.02, L=1.52·104 cm,rainfall intensity R=18.9 cm/h (�r), and t0=480 s. Thesurface resistance F is assumed 90 which gives the value ofa=174.222. The comparison shows good agreement.However, the calculated increase and decrease of q arefaster than the measured values. The calculated steadystate time is 180 s, whereas the measured data give 240 s.The reason for the disagreement could be a combination ofseveral factors, including variation in rainfall-excess r in

Fig. 9Variation of surface runoff with surfaceroughness parameter F

Fig. 10Comparison of the calculated results with the measured data of theLos Angeles field tests

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space and time, variation in surface slope and slope shapeof the surface, and non-uniformity in surface resistance.Also, water-drop impact exerts a significant retardingeffect. Overland flow is an unsteady and spatially variableprocess, because rainfall and infiltration are time- andspace-dependent. The surface runoff may either be lami-nar or turbulent, or a combination of both.

Conclusion

Surface runoff is a complicated process because it dependson so many factors, including the rate and duration ofrainfall, the type of soil, the antecedent soil water, thesaturated hydraulic conductivity of the soil, the densityand type of vegetation cover, topographic features, slope,and length of slope. Physical models based on kinematictheory describe satisfactorily the profiles of runoff on soilsurface under different conditions. Woolheiser and Liggett(1967) and other authors (Wooding 1965a, 1965b; Parlangeand others 1981; Woolheiser 1982; Rose and others 1983;Lima and van der Molen 1988; Pearson 1989; Julien andMoglen 1990; Saghafian and others 1995; Van der Molenand others 1995) have shown that kinematic approxima-tion is valid on almost all overland flow planes. Over thelength of slope, most overland flow can be considered aslaminar flow, and the Darcy-Weisbach resistance law canbe applied for predicting surface runoff. Manning andChezy formulas are used for turbulent flow. Comparisonof the results of this study of surface runoff based on theDarcy-Weisbach resistance law and laminar flow showsreasonably good agreement with the measured data of theLos Angeles field tests on concrete-paved surface (Yu andMcNown 1964).

Acknowledgements The tests and the resulting data presentedherein, unless otherwise noted, were from research conducted atU.S. Army Engineer Waterways Experiment Station, Vicksburg,Mississippi.

References

Betson RP (1964) What is watershed runoff? J Geophys Res69(8):1541

Campbell SY, Van der Molen WH, Rose CW, Parlange JY (1985) Anew method for obtaining a spatially averaged infiltration ratefrom rainfall and runoff rates. J Hydrol 82:57

Chery DL Jr, Clyde CG, Smith RE (1979) An application runoffmodel strategy for ungaged watersheds. Water Resources Res15(4):1126

Dunne T (1970) An experimental investigation of runoff pro-duction in permeable soils. Water Resour Res 6(2):478

Dunne T (1978) Field studies of hillslope flow processes.In: Kirkby MJ (ed) Hillslope hydrology. Wiley, Chichester

Dunne T, Zhang W, Aubry BF (1991) Effects of rainfall, vegeta-tion, and microtopography on infiltration and runoff. WaterResour Res 27(9):2271–2285

Emmett WW (1978) Overland flow. In: Kirkby MJ (ed) Hillslopehydrology. Wiley, Chichester

Foster GR, Huggins LF, Meyer LD (1968) Simulation of overlandflow on short filed plots. Water Resour Res 4(6):1179

Freeze RA (1972) Role of subsurface flow in generating surfacerunoff, 2. Upstream source areas. Water Resour Res 8(5):1272–1283

Freeze RA (1980) A stochastic-conceptual analysis of rainfall-runoff processes on a hillslope. Water Resour Res 16(2):391–408

Hawkins RH (1981) Interpretations of source area variability inrainfall-runoff relations. In: Singh VP (ed) Rainfall-runoffrelationships. Water Resources Publ, Fort Collins, Colorado,pp 303–324

Henderson FM, Wooding RA (1964) Overland flow and ground-water flow from a steady rainfall of finite duration. J GeophysRes 69:1531

Horton RE (1933) The role of infiltration in the hydrologicalcycle. Trans Am Geophys Union Part 2, 393

Horton RE (1945) Erosional development of streams and theirdrainage basins; hydrophysical approach to quantitativemorphology. Bull Geol Soc Am 56:275

Julien PY, Moglen GE (1990) Similarity and length scale forspatially varied overland flow. Water Resour Res 26(8):1819–1832

Kibler DF, Woolheiser DA (1970) The kinematic cascade as ahydrologic model. Colorado State University, Fort Collins,Hydrol Pap 39

Koivusalo H, Karvonen T (1995) Modeling surface runoff: A casestudy of a cultivated field in Southern Finland. Nordic Hydrol26:205

Kuchment LS, Demidov VN, Naden PS, Cooper DM, Broadhurst P(1996) Rainfall-runoff modelling of the Ouse Basin, NorthYorkshire: An application of a physically based distributedmodel. J Hydrol 181:323

Lane LJ, Woolheise DA (1977) Simplifications of watershed ge-ometry affecting simulation of surface runoff. J Hydrol 35:173

Larsen JE, Sivapalan M, Coles NA, Linnet PE (1994) Similarityanalysis of runoff generation processes in real-world catch-ments. Water Resour Res 30(6):1641

Lighthill FRS, Whitham GB (1955) On kinematic waves, 1. Floodmovement in long rivers. Proc R Soc Lond (A)229, 28

Lima JLMP, Van der Molen WH (1988) An analytical kinematicmodel for the rising limb of overland flow on infiltratingparabolic shaped surfaces. J Hydrol 104:363

Logue KM (1990) R-5 revisited, 2. Reevaluation of a quasi-physically based rainfall-runoff model with supplementalinformation Water Resour Res 26:973–988

Logue KM, Gander GA (1990) R-5 revisited, 1. Spatial variabilityof infiltration on a small rangeland catchment. Water ResourRes 26:957–972

Morgali JR, Linsley RK (1965) Computer analysis of overlandflow. J Hydraul ASCE 91, HY3, 81

Ogden FL, Julien PY (1993) Runoff sensitivity to temporal andspatial rainfall variability at runoff plane and small basin scales.Water Resour Res 29(8):2589–2597

Parlange JY, Rose CW, Sander G (1981) Kinematic flow approx-imation of runoff on a plane: An exact analytical solution.J Hydrol 52:171

Pearson CP (1989) One-dimensional flow over a plane : Criteriafor kinematic wave modeling. J Hydrol 111:39

Rose CW, Parlange JY, Sander GC, Campbell SY, Barry DA (1983)Kinematic flow approximation to runoff on a plane: An ap-proximate analytic solution. J Hydrol 62:363

Rubin J, Steinhardt R (1963) Soil water relations during raininfiltration, 1. Theory. Proc Soil Sci Soc Am 27:246

Saghafian B, Julien PY, Ogden FL (1995) Similarity in catchmentresponse, 1. Stationary rainstorms. Water Resour Res31(6):1533–1541

Singh VP, Woolheiser DA (1976) A nonlinear kinematic wavemodel for watershed surface runoff. J Hydrol 31:221

Original article

804 Environmental Geology (2002) 41:797–805

Sivapalan M, Wood EF (1986) Spatial heterogeneity andscale in the infiltration response of catchments. In: Scaleproblems in hydrology. Reidel, Hingham, Massachusetts,pp 81–106

Smith RE, Hebbert RHB (1979) A Monte Carlo analysis of thehydrologic effects of spatial variability of infiltration. WaterResour Res 15(2):419–429

Smith RE, Woolheiser DA (1971) Overland flow on an infiltratingsurface. Water Resour Res 7(4):899

Troutman BM (1983) Runoff prediction errors and bias in pa-rameter estimation induced by spatial variability of precipita-tion. Water Resour Res 19(3):791–810

Troutman BM (1985a) Errors and parameter estimation inprecipitation-runoff modeling, Theory. Water Resour Res21(8):1195–1213

Troutman BM (1985b) Case study. Water Resour Res 21(8):1214–1222

Van der Molen WH, Torfs PJJF, de Lima JLMP (1995) Waterdepths at upper boundary for overland flow on small gradients.J Hydrol 171:93

Wilson CB, Valdes JB, Rodriguez-Itrube (1979) On the influenceof the spatial distribution of rainfall on storm runoff. WaterResour Res 15(2):321-328

Wood EF, Sivapalan M, Beven KJ, Band L (1988) Effects of spatialvariability and scale with implications to hydrologic modeling.J Hydrol 102:29–47

Wooding RA (1965a) A hydraulic model for the catchment-stream problem. I. Kinematic wave theory. J Hydrol 3:254

Wooding RA (1965b) A hydraulic model for the catchment-stream problem. II. Numerical solutions. J Hydrol 3:268

Woolheiser DA (1975) Simulation of unsteady overland flow. In:Mahmood K, Yevjevich V (eds) Unsteady flow in open channels,vol 3, Chap 12. Water Resources Publ, Fort Collins, Colorado

Woolheiser DA (1982) Physically based models of watershedrunoff. In: Singh VP (ed) Rainfall-runoff relationships. WaterResources Publ, Fort Collins, Colorado

Woolheiser DA, Goodrich DC (1988) Effect of storm rainfall in-tensity patterns on surface runoff. J Hydrol 102:335–354

Woolheiser DA, Liggett JA (1967) Unsteady, one-dimensionalflow over a plane – the rising hydrograph. Water Resour Res3(3):753

Woolheiser DA, Osborn HB (1985) A stochastic model ofdimensionless thunderstorm rainfall. Water Resour Res21(14):511–522

Woolheiser DA, Smith RE, Giraldez JV (1996) Effects of spatialvariability of saturated hydraulic conductivity on Hortonianoverland flow. Water Resources Res 32(3):671–678

Wu YH, Yevjevich V, Woolheiser DA (1983) Effects of surfaceroughness and its spatial distribution on runoff hydrographs.Colorado State University, Fort Collins, Hydrol Pap 96

Yu YS, McNown JS (1964) Runoff from impervious surfaces.J Hydraul Res 2(1):2

Original article

Environmental Geology (2002) 41:797–805 805