Stochastic analysis of transport in hillslopes: Travel time distribution and source zone dispersion

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1 Click Here for Full Article WATER RESOURCES RESEARCH, VOL. 45, W8435, doi:1.19/8wr7668, 9 Stochastic analysis of transport in hillslopes: Travel time distriution and source zone dispersion A. Fiori, 1 D. Russo, and M. Di Lazzaro 1 Received 18 Decemer 8; revised 3 June 9; accepted 1 June 9; pulished 6 August 9. [1] A stochastic model is developed for the analysis of the traveltime distriution f t in a hillslope. The latter is descried as made up from a surficial soil underlain y a less permeale susoil or edrock. The heterogeneous hydraulic conductivity K is descried as a stationary random space function, and the model is ased on the Lagrangian representation of transport. A first-order approach in the log conductivity variance is adopted in order to get closed form solutions for the principal statistical moments of the traveltime. Our analysis indicates that the soil is mainly responsile for the early ranch of f t, i.e., the rapid release of solute which preferentially moves through the upper soil. The early ranch of f t is a power law, with exponent variale etween 1 and.5; the ehavior is mainly determined y unsaturated transport. The susoil response is slower than that of the soil. The susoil is mainly responsile for the tail of f t, which in many cases resemles the classic linear reservoir model. The resulting shape for f t is similar to the Gamma distriution. Analysis of the f t moments indicates that the mean traveltime is weakly dependent on the hillslope size. The traveltime variance is ruled y the distriution of distances of the injected solute from the river; the effect is coined as source zone dispersion. The spreading due to the K heterogeneity is less important and oscured y source zone dispersion. The model is tested against the numerical simulation of Fiori and Russo (8) with reasonaly good agreement, with no fitting procedure. Citation: Fiori, A., D. Russo, and M. Di Lazzaro (9), Stochastic analysis of transport in hillslopes: Travel time distriution and source zone dispersion, Water Resour. Res., 45, W8435, doi:1.19/8wr Introduction [] The traveltime distriution is a very important descriptor for solute transport in river asins. Traveltime t is defined as the time spent y the solute in traveling through the system, from the source (located at the surface) to the outlet (e.g., the stream). The proaility density function (pdf, or distriution) of t physically represents the reakthrough curve of an instantaneous pulse of unit mass d(t) uniformly injected at the surface. The traveltime distriution f t (t) is a gloal descriptor of the system response to a contamination event, and it encapsulates important features like the storage of solute, its spreading and the iogeochemical processes which take place in the system. The distriution f t (t) have een intensively studied in the past, leading to its determination through tracer experiments (see, e.g., the reviews y Maloszewski and Zuer [1996] and McGuire and McDonnell [6]) or numerical and analytical modeling [e.g., Haitjema, 1995; Goode, 1996; Amin and Campana, 1996; Vaché and McDonnell, 6; Dunn et al., 7; McGuire et al., 7]. 1 Dipartimento di Scienze dell Ingegneria Civile, Università di Roma Tre, Rome, Italy. Department of Environmental Physics and Irrigation, Institute of Soils, Water and Environmental Sciences, Agricultural Research Organization, Volcani Center, Bet Dagan, Israel. Copyright 9 y the American Geophysical Union /9/8WR7668$9. W8435 [3] The present work deals with the analysis of f t (t) for upland hillslopes. Transport dynamics in hillslopes are of particular interest for the analysis of solute transport in catchments, which are composed y multiple hillslopes connected y the stream network. Thus, the analysis of hillslope systems is a fundamental prerequisite for the asinscale transport analysis. Because of the relatively fast traveltime of river networks, the traveltime distriution of hillslopes plays an important role in solute transport at the asin scale, in particular when in presence of small catchments [Botter and Rinaldo, 3]. [4] Analysis of solute transport (and consequently t) in hillslopes is a very complex prolem, where flow and transport are mostly susurface processes which are strongly influenced y the structure of the porous system. Furthermore, the traveltime of a generic solute particle injected at the surface usually depends on the entire flow history of the hillslope system, eing function of the precipitation record and the external forcing conditions. However, the impact of flow nonstationarity can e assessed y solving the transport prolem under steady flow (equivalent steady state or ESS [see Russo and Fiori, 8]), and replacing calendar time with the cumulate flow exiting the system; the approach leads to the well-known flow-corrected time approximation [Niemi, 1977; Rodhe et al., 1996]. The latter provides a considerale simplification of the prolem, converting an unsteady flow prolem into a steady one, and the time-invariant traveltime distriution ecomes the core of the system response to inputs of contaminants. The approach is widely employed [see, e.g., McGuire and McDonnell, 6] and it 1of13

2 W8435 FIORI ET AL.: TRAVEL TIME DISTRIBUTION IN HILLSLOPES W8435 Figure 1. Schematic representation of the vertical cross section of the three-dimensional flow domain. was recently tested in hillslope transport prolems through a series of detailed numerical simulations [Fiori and Russo, 8]. The ESS approach is also ale to handle external fluxes like evaporation and/or transpiration, provided that the net applied recharge is adequately captured [Russo and Fiori, 8]. [5] The complex susurface system is typically composed y a surficial and highly permeale soil, underlain y a less permeale susoil or edrock where saturated groundwater flow may take place. As a rule, oth systems are heterogeneous, with spatially variale hydraulic properties. A deterministic description of the susurface is unfeasile ecause of the erratic, unpredictale spatial distriution of the hydraulic properties; for this reasons, stochastic models have een extensively (in many cases successfully) applied in several ranches of the susurface hydrology [Zhang, 1; Ruin, 3]. [6] Among the studies dealing with traveltime analysis in heterogeneous systems driven y precipitation we cite the pioneristic paper y Destouni and Graham [1995] and susequent developments [Simic and Destouni, 1999; Foussereau et al., 1; Lindgren et al., 4], as well as the numerical analyses of Russo et al. [1]. A recent work derived a three-dimensional simplified stochastic model for the analysis of transport in the comined unsaturated/ saturated zone [Russo and Fiori, 9]. The Lagrangian formulation of transport [Dagan, 1989] was employed in most of the stochastic models, and it consists in studying the solute particles y analyzing the statistics of the time needed y the solute to reach a suitale control plane. Because of the prolem complexity the theoretical analyses are usually performed after adopting some approximations, among which we cite the first-order analysis in the log conductivity of13 variance, which is valid for low to moderate degrees of heterogeneity. [7] This work develops a simplified, physically ased Lagrangian stochastic model for the traveltime pdf in upland hillslopes. The conceptual model aims at reproducing the principal components of the transport dynamics, like, for example, the susoil/edrock contriution, heterogeneity and a fully three-dimensional description of the flow. The scope of the model is not to provide a tool for applications ut rather to gain understanding in the transport processes occurring in hillslopes, with particular reference to the role played y the spatial heterogeneity of the hydraulic properties and the separate contriutions rought y the soil and susoil systems.. Theoretical Framework [8] We consider that flow and solute transport are susurface processes, as susurface flow dominates when in presence of humid climates, vegetated soils and steep hillslopes [Dunne and Leopold, 1978]. The conceptual model for the hillslope consists of two independent regions: (1) an upper soil (hereinafter denoted y suscript s) of constant thickness and slope, connected to a river, which overlies () a permeale susoil (suscript ), less permeale than the soil, whose ottom is confined y an impermeale oundary. The susoil system may also represent a fractured edrock, provided that the continuum equivalent approach can e adopted, as discussed y Fiori and Russo [7]. Figure 1 displays a sketch of the conceptual domain with the coordinate system and the oundary conditions. The domain is three dimensional, with an undefined lateral size. The upstream oundary is impermeale, and the constant

3 W8435 FIORI ET AL.: TRAVEL TIME DISTRIBUTION IN HILLSLOPES W8435 river level is imposed downstream. The two regions (soil and susoil) are heterogeneous and characterized y different hydraulic properties. Because of its high relevance for transport, saturated hydraulic conductivity is modeled as a stationary random space function, with assigned statistical properties which are different for the two systems. In particular, the susoil is considered as statistically isotropic (this in order to model fractured media; see the discussion y Fiori and Russo [7]) while the soil is anisotropic; the latter often occurs in surficial sedimentary formations. The domain is considered as ergodic; that is, its size (e.g., the horizontal length L) is much larger than the correlation length of log conductivity (I Y ), i.e., I Y L; the latter condition is likely met in most of the hillslopes, as I Y = O(1 m) while L = O (1 m 1 m) (the issue is also discussed y Fiori and Russo [7, 8]). This way, any realization of the hillslope (i.e., any realization of the random hydraulic conductivity) will determine the same traveltime distriution. The calculations are made y assuming ESS conditions [Russo and Fiori, 8; Fiori and Russo, 8], with a constant and uniform recharge R along the soil surface. The adoption of a uniform recharge seems plausile ecause of (1) the relatively small size of the upland hillslopes compared to the precipitation correlation scale and () the ESS conditions, which imply temporally averaged quantities. [9] It is easy to check that the domain setup leads to flow characteristics which do not depend on x in their average values. Conversely, fluctuations around the mean of the relevant flow quantities will also depend on the transverse coordinate x, the flow (and the derived transport) process eing fully three dimensional. [1] We emphasize that the conceptual model adopted here is a simplification of the real flow and transport dynamics occurring in hillslopes. In particular, the soil and the susoil/edrock cannot always e considered as independent, as oth water and solute fluxes may exist at the interface, as function of the distance from the river. For example, water may completely infiltrate into the susoil upstream, or move preferentially through the soil in the vicinity of the river. Nevertheless, we elieve that the conceptual model adopted is one of the simplest and it lends itself to a useful mathematical description of the transport processes occurring at the hillslope scale. [11] If the soil and the susoil are independent systems, the traveltime (hereinafter TT) pdf for the hillslope f t (t) is otained as follows f t ðtþ ¼ ff t ðtþþð1 fþf ts ðtþ where t, t, and t s are the traveltimes of the hillslope, susoil, and the soil, respectively, and f is a weight which reflects the flow proportions etween the two systems; f can e calculated as the ratio etween the susoil contriution to the hillslope discharge and the total hillslope discharge. In (1) we implicitly neglect the time spent y the solute particle in the soil efore reaching the susoil. Thus, the hillslope TT pdf f t is calculated after the pdfs of each single susystem ( f t, f ts ). [1] Along the Lagrangian approach, we split the solute plume (which is uniformly injected over the entire surface) ð1þ in particles that move according to the local random water velocity along different streamlines; each particle is injected at the location x 1 = x (see Figure 1). Since the random velocity field is ergodic and stationary along x, the ensemle moments of the traveltime of a particle injected at x 1 = x is equivalent to the spatial moments of traveltime calculated y averaging over x. Denoting for simplicity t i as the generic TT (i.e., either t s or t ), it is a random variale which depends (1) on the spatially variale velocity field, which in turn is ruled y the conductivity variations, and () on the injection location x. Thus, for a given x the traveltime t i (x ) is a random variale, and its pdf is characterized y the distriution f ti (t i ; x ) which is conditioned on the injection location x. [13] The traveltime related to the entire plume, which is the quantity employed in (1), is easily derived y averaging the TT for the single particle over x, with uniform distriution L 1 ð f ti t i Þ ¼ 1 L Z L f ti ðt i ; x Þdx ðþ where L is the horizontal length of the hillslope. [14] The crux of the matter is the calculation of the conditional pdfs f ti (t i ; x ), i.e., the pdfs associated with the single particle injected at x. In past work the latter has een determined either after evaluation of the traveltime moments and then assuming a particular traveltime pdf (lognormal, inverse Gaussian) [e.g., Simmons, 198; Shapiro and Cvetkovic, 1988; Cvetkovic et al., 199; Fiori et al., ], or y using the common assumption that the fluid particle trajectory is Gaussian and employing the relation etween the TT and trajectory pdfs [e.g., Dagan, 1989; Ruin and Dagan, 199]. We adopt here the approach y Russo [1993], Cvetkovic et al. [199], and Russo and Fiori [9], which assumes a lognormal distriution for f ti (t i ; x ) after evaluation of the moments of t i (x ). The approach is very roust and easy to implement. We will show in the sequel that any choice for the pdf f ti (t i ; x ) has little influence on the resulting distriution (). [15] In the remainder of section we derive the first two moments of t (x ), t s (x ) and also provide explicit expressions for f t (t ) and f ts (t s ) in asence of local dispersion. Pore-scale dispersion has a negligile effect on the traveltime moments [e.g., Dagan, 1989; Cvetkovic and Dagan, 1994; Berglund and Fiori, 1997], eing overwhelmed y local- and field-scale dispersion, and therefore it will not e considered in the present study. In order to simplify the moment determination we shall adopt the well-known and widely employed first-order approach (for a comprehensive description, see the ooks y Dagan [1989], Zhang [1], and Ruin [3]). We define the traveltime from the injection plane to a suitale control plane, normal to the mean trajectories, as follows: t = R a a 1 V 1 1 (x(x)) dx, where x (x) is the vector position of the particle, a 1 and a are the starting and final locations (e.g., the injection and control planes) and V 1 the velocity component aligned with x. The firstorder result is otained after a formal perturation over the random velocity V 1, which in turn is function of the log conductivity Y fluctuations. The procedure is straightfor- 3of13

4 W8435 FIORI ET AL.: TRAVEL TIME DISTRIBUTION IN HILLSLOPES W8435 ward and leads to the following expressions for the first two moments of t i (for details, see Russo and Fiori [9]) s t i ¼ Z a dx ht i i ¼ a 1 U i ðþ x Z a Z a a 1 C ui ðx x Þ a 1 U ðþu x ð Þ dxdx i i x where U i is the mean velocity in system i, and C ui is the Eulerian velocity covariance (the velocity component is aligned with U i ). In sections.1.3 we calculate the TT moments for each system..1. Transport in the Susoil [16] Both groundwater and unsaturated flow are present in the susoil/edrock. The method for solving flow and transport is similar to that of Russo and Fiori [9] except that the general procedure is simplified in a few parts. Similar to Destouni and Graham [1995], we split the particle traveltime in the susoil in two components (see Figure 1): (1) traveltime in the unsaturated zone t v (the latter is the time spent y the particle from the injection point at x to a control plane located at the water tale (CP1)) and () traveltime in the saturated zone t g, which is the time needed to reach the river from CP1. Thus, t = t v + t g and since t v, t g are uncorrelated the pdf of t is given y the convolution of the correspondent pdfs f tv and f tg, i.e., f t ðt ; x Þ ¼ Z t ð3þ ð4þ f tg ðt t ; x Þf tv ðt ; x Þdt ¼ f tg ðx Þ* f tv ðx Þ ð5þ and the prolem reduces to the determination of the pdfs f tg, f tv, through the moments of t g and t v. [17] The moment calculation is a difficult prolem, which can e consideraly simplified y adopting a few assumptions. The uniform recharge R determines a nonuniform flow in the groundwater, with spatially variale mean horizontal head gradient J. The determination of J for the prolem at hand is not simple, and therefore an approximate solution is sought here. A major simplification is achieved y adopting the Dupuit scheme for mean flow and assuming that the spatial variations in saturated thickness are relatively small compared to the average saturated thickness. This is equivalent to the linearization of the Boussinesq equation, which is extensively employed in solving phreatic flow prolems [Bear, 1988]. Although the assumed condition may not e universally valid, it leads to a major simplification of the flow prolem, and the derived solution is anyway more realistic than the uniform flow model which is sometimes employed in hillslope studies. The assumed condition was approximately met in the numerical simulations of Fiori and Russo [8] and Russo and Fiori [8]. Details of the procedure and the moments derivation are found in Appendix A... Transport in the Upper Soil [18] We consider that flow in the soil is unsaturated. It is assumed here that the constitutive relationships for unsaturated flow, K (y) and q (y) (where y is the pressure head, defined as positive), are descried locally y the Brooks and Corey [1964] parametric expressions. Good agreement etween data of K (y) and q (y) measured in media with continuous pore size distriution and the Brooks-Corey expressions was found y Brooks and Corey [1964], Lalierte et al. [1966], Bresler et al. [1978], and Russo and Bresler [198]. Neglecting local hysteresis and local anisotropy, they read q q r n s k s K s ¼ ¼ ðy =yþ ; y y 1 ; y < y ðy =yþ h ; y y 1 ; y < y where K s and k s are the (random) saturated and unsaturated soil conductivity, respectively, n s, q, and q r are the effective porosity, water content, and residual (or the irreducile) volumetric water content, respectively; y is the air entry value of y (or the uling pressure); and and h = ( + n) + are parameters (taken as constants in the present study) which are related to the soil pore size distriution, with n a parameter of the relative conductivity models [e.g., Mualem, 1976] which accounts for the dependence of the tortuosity and the correlation factors on water content. [19] The value of n can e determined on the asis of theoretical considerations. For example, the parallel capillaries k s model of Burdine [1953] gives n = 1, which leads to h =3 +, and has een used y Brooks and Corey [1964]; the serial parallel model of Childs and Collis- George [195] gives n =, which leads to h = + and has een used y Brutsaert [1967]; the undle parallel capillaries model of Gates and Lietz [195] gives n = 1, which leads to h = + and has een used y Russo and Bresler [198]. [] The pressure and velocity fields in the soil are random and complex. For example, flow is saturated or nearly saturated in the vicinity of the river and is unsaturated upstream, and the mean pressure head displays a gradient which is not known and it is generally function of the soil-susoil interactions, which in the present simplified scheme are neglected. The prolem is greatly simplified y assuming that the mean flow lines are parallel to the soil slope (which occurs when the soil is much more conductive than the susoil) and the mean gradient is approximately constant; this is the simplest approximation possile, which is in line with the other assumptions adopted in the study. Thus, we denote as J s = dh/dx 1 (H = x 3 hyi is the mean head) the horizontal component of the head gradient, the vertical one eing J s /J a, with J a =tana s ; along the previous discussion, J s /J a 1. [1] Even with the aove assumption, the calculation of the moments of t s is a formidale task, especially for the variance. Although the procedure is similar to that for the saturated zone, the calculations are more involved, and the existing solutions ased on a unit mean gradient [Russo, 1998] cannot e applied. Flow is parallel to the stratification, and for an highly anisotropic soil the velocity fluctuation u 1 is mainly determined y the hydraulic conductivity variations [Indelman and Dagan, 1999], i.e., ð6þ 4of13

5 W8435 FIORI ET AL.: TRAVEL TIME DISTRIBUTION IN HILLSLOPES W8435 u 1 = k sj s /hqi, with k s the fluctuation of the unsaturated conductivity, which, in turn, in the present simplified analysis depends on the variaility of the saturated conductivity K s. [] Hence, we calculate the moments of interest with K s as the sole random parameter, which is spatially variale. Details of the moment calculation for the upper soil are found in Appendix B..3. Traveltime Distriution in Asence of Local Dispersion (Homogeneous Hillslope) [3] For the sake of the discussion of section 4., we analyze here the traveltime distriution for an homogeneous hillslope. After assuming a lognormal shape for the pdf, the moment analysis carried out in section. allows the calculation of the traveltime pdfs f t and f ts which are needed for the determination of the TT pdf for the hillslope (1). Useful analytical expressions can e derived in asence of local dispersion, i.e., when s Y = s Ys = (i.e., homogeneous hillslope). In that case, t i (x )=ht i (x )i, the flow dynamics ecome two dimensional and f ti (t i ; x ) degenerates into a Dirac pulse. Closed form expressions for the unconditional f ti (t i ) can e easily derived y noting that t i (x ) is a monotonous increasing function of x,and the following relation holds f ti t i ð Þ ¼ f x ðx Þ dx dt i where f x (x )=L 1. Inserting in the aove the expressions for ht i (x )i derived in sections.1 and., we otain the traveltime pdfs in asence of local dispersion. Thus, for the susoil system we introduce in (7) the inverse of ht g (x )i + ht v (x )i, from (A4) and (A6), otaining f t ðt " Þ ¼ U v 1 1 þ W D # 1 U e tu=lþdu=uv LD U v where D =tana s tan a and W(x) isthelamert function; the latter is the solution of the differential equation dw/dz = W(1 + W) 1 /z, which appears in (7) after the aove passages. Expansion of (8) in Taylor series for U v /(U D ) 1 yields the simplified expression ð Þ ¼ L e tu=l 1 þ 1 e tu=l U D U f t t U v ð7þ ð8þ U v 1 U D which leads to the classic linear reservoir model for U v / (U D )!1. [4] The same procedure ased on (7) is also applied for the soil system, leading to f ts ðt s ð f ts t s Þ ¼ U s L Þ ¼ U s L t s U s y ð1 þ Þ þ1 t s t s y ð9þ t s y ts t s ðlþ ð1þ [5] The previous expressions will e discussed in section Statistical Moments at the Hillslope Scale: Mean Traveltime and Dispersion [6] The first two statistical moments of the traveltime t are derived from the pdf definition (1), accounting for the local dispersion. The mean and variance of the hillslope traveltime t, denoted as T and S, are easily derived as follows T ¼ ft þ ð1 fþt s S ¼ fs þ ð1 f ÞS s þ fð1 fþ ð T þ T s Þ ð11þ where T i, S i are the hillslope-scale mean and variance of the generic t i (either t s or t ). In particular, T i, S i are derived from the conditional moments ht i (x )i, s ti (x ) (i.e., the moments associated to the single particle injected at x ) after application of (), as follows T i ¼ 1 L S i ¼ 1 L ¼ 1 L Z L Z L Z L ht i ðx Þidx t i ðx Þ dx Ti h i s t i ðx Þþht i ðx Þi dx T i ¼ s T i þ s t i ð1þ where s ti = L 1R L s t i (x ) dx is the variance of t i at the hillslope scale and s Ti = L 1R L ht i(x )i dx T i is the variance of the mean traveltime. The terms s ti and s Ti are coined as local and source zone dispersion, respectively; the meaning of the terms will e discussed later, in section 4.. [7] Detailed calculation of the moments T i, s Ti, s ti for each susystem is carried out from the conditional moments derived in section. Starting with the susoil, we otain T ¼ L U þ LD U v ; S ¼ s T þ s t s t s T with ¼ I Y s Y L 1 þ D ¼ L U U 1 þ D U U Uv þ D U v 1 U Uv ð13þ where the Fickian regime (i.e., for x I Y ) is considered for simplicity. Similarly, we otain for the soil T s ¼ L L=y þ ð þ Þy =L ð1 þ Þ y =L U s ð þ 3 þ ; Ss Þ ¼ s T s þ s t s with ð14þ where = h and y = y /D s. 5of13

6 W8435 FIORI ET AL.: TRAVEL TIME DISTRIBUTION IN HILLSLOPES W8435 Figure. Dimensionless traveltime pdf for the susoil (t ), for a few values of U v /(U D ) and Pe = L/(s Y I Y ) (inset); I Ys /L =.1. s t s ¼ s Y s I Ys L L=y þ 4ð1 þ Þy =L ð1 þ Þ y =L Us þ ð 3 þ Þ h s T s ¼ L Us 1ð1 þ Þ L=y þ1 ð3 þ Þ y =L þ 4ð þ Þ ð 1Þ y =L 33 ð1 þ Þð3 þ Þ y =L i 4 h 1ð þ Þ 3 þ 5 þ i 1 ð15þ [8] The results of section 3 are discussed in the following. 4. Discussion 4.1. General Shape of the Traveltime pdf and Contriutions From the Soil/Susoil Systems [9] The traveltime pdf (1) is a weighted sum of the soil and susoil pdfs, f s and f. Here we analyze and discuss them separately. We start from the susoil component f (t ), which is given after introducing (5) in (), assuming a lognormal distriution for f v, f g, with statistical moments of t v, t g given in section.1. A closed form expression in asence of local dispersion (i.e., for a homogeneous hillslope) is given y (8). In the following we work for convenience with the dimensionless traveltime t /T, where T is the mean traveltime in the susoil and is given y the first of (13). Because of continuity, U and U v depends on the recharge rate R, as follows U ¼ frl ; U v ¼ fr Dn hi q ð16þ As a consequence, the mean susoil traveltime T = L/U + LD /(U v ) is weakly dependent on the hillslope length L, in agreement with the experimental analysis of McGuire et al. [5]. If the unsaturated flow contriution is neglected, T = L/U = Dn /R and the result is fully independent of L; the same formula was otained y Haitjema [1995] y a different procedure. [3] The dimensionless pdf f t (t /T ) T depends on three parameters: U v /(U D ), I Y /L, L/(s Y I Y ) = Pe, the latter 6of13 eing the Peclet numer for the susoil system, typically Pe = O(1 1 3 ) [see, e.g., Berglund and Fiori, 1997]. The most important parameter is U v /(U D ), and we depict in Figure the dimensionless f t for Pe = 1 (8), I Y /L =.1 and a few values of U v /(U D ). The pdf displays a finite value at t = and it decays with t.foru v /(U D ) (U D ) ^ the solution (8) degenerates into the linear reservoir f t (t ) = T 1 e t /T ; that is, the pdf is dominated y groundwater transport. The linear reservoir is a good approximation of (8) in a relatively wide range of U v /(U D ). [31] The impact of heterogeneity is measured through the Peclet numer Pe. In the inset of Figure we depict f t for U v /(U D ) = 1 and three values for Pe. It is seen that the impact of local dispersion on the solution is relatively modest, and it mainly manifest at the early ranch of the pdf, for low values of Peclet. In fact, heterogeneity is mainly felt in the vicinity of the river, leading to an enhanced solute flux and its spreading, ut it has a minor impact on larger travel distances. The reasons for the scarce relevance of Peclet on results shall e further analyzed later. [3] The soil pdf f ts is otained after introduction of the lognormal f ts (t s ; x ) in (), using the moments of t s (x ) analyzed in section.. Working with the dimensionless variale t s /T s, with T s the mean TT in the soil given y the first of (14), the traveltime pdf in the soil depends on the four parameters: y /(D s L), = h, I Ys /L, L/(s Ys I Ys )=Pe s, the latter eing the Peclet numer of the soil, typically Pe s = O(1 1 3 ). The infinite Peclet solution (i.e., for an homogeneous soil) is given y (1). The latter shows that the general ehavior of f ts is an asymptotically power law f ts t þ1 s ð17þ with = h. The result is qualitatively similar to the analysis of Kirchner et al. [1], who inferred from field data a Gamma pdf, which displays a power law ehavior at early time; the exponent otained there was close to

7 W8435 FIORI ET AL.: TRAVEL TIME DISTRIBUTION IN HILLSLOPES W8435 Figure 3. Dimensionless traveltime pdf for the soil (t s ), for a few values = h and Pe s = L/(s Ys I Ys ) (inset); I Ys /L =.1, y /(LD s ) = In the following we analyze the range for the exponent in (17) according to our model. [33] With h = (n + ) + [Brutsaert, 1967; Russo and Bresler, 198], = (n + 1) +. Previous analyses ased on laoratory cores [i.e., Brooks and Corey, 1964; Lalierte et al., 1966; Bresler et al., 1978; Russo and Bresler, 198], suggest that generally , with larger values for coarse-textured soil materials. The analysis of laoratory cores y Bresler et al. [1978] provided the general result =.4, leading to an exponent of (17) equal to.69. Results may e different in the field, where macropores and cracks may present. Analyses of field data from the Bet Dagan site [Russo and Bresler, 198, 1981], suggest that for a field soil classified as a sandy clay soil (Hamra Red Mediterranean soil, Rhodoxeralf), 1 and n = 1, the latter value in agreement with the model of Gates and Lietz [195]. Assuming = 1, the range of the exponent of (17) is (.8.67). The largest value is not distant from the.5 value inferred y Kirchner et al. [1], which can erecoveredyassuming =1andn = ; the latter leads to h =. Note that the comination of the linear q(y) relationships ( = 1) and the quadratic K(y) relationships (h = ) means that the soil water diffusivity, K@q/@y, is a constant given y K s y / [Russo and Bresler, 198], independent of water content, corresponding to a linear medium [Stroosnijder and Bolt, 1974]. Summarizing, the exponent of (17) is roughly ounded etween 1and1/, with a likely average value around /3 =.67. [34] The representation of f ts in terms of t s /T s is insensitive to the parameter y /(D s L). Conversely, exerts some influence on the solution. Figure 3 represents f ts (t s /T s ) T s as function of the traveltime for Pe s = 1 and I Ys /L =.1for afewvalues. It is seen that f ts decays very fast, with the main ulk of the pdf concentrated at t s /T s 1. This is in variance with f t which was more uniformly distriuted around its mean value T. As visile in Figure 3, the impact of on the shape of the pdf is not dramatic. The inset of Figure 3 shows the effect of Pe s on the pdf (for = ). It is seen that local dispersion has a negligile impact on the solution, even for the unrealistically small value Pe s =1. Again, this point will e recalled later. 7of13 [35] The pdfs f t, f ts illustrated aove contriute to the determination of the hillslope TT pdf f t, along equation (1). The relative contriution of each of the two pdfs depends on the weight f and the other parameters, the most significant eing U s /U. Besides the particular value adopted for f, the ratio U s /U estalishes the relative importance of the soil/ susoil contriutions to f t. For the sake of illustration we depict in Figure 4 the dimensionless pdf f t T as function of t/t for a few values of the ratio U s /U ; the latter is usually greater than unit for nonflat hillslopes. It is seen that the susoil contriution f t (thick line) dominates everywhere except when t/t 1 for which f ts comes into play with its power law peak; its contriution increases with U s /U.We note that the tail of f ts is very persistent, ut it drops to zero for TT values correspondent to the maximum t s (not visile in Figure 4; see the upper limit for t s reproduced in equation (1)). [36] The traveltime pdf of the hillslope f t = ff t + (1 f)f ts is y definition a linear comination of the soil and susoil pdfs. In the inset of Figure 4 we depict f t T as function of t/t for a few values of the weight f. The soil contriution increases as we decrease f, leading to a TT pdf which has a more pronounced early time peak and a slower, exponential-like tail. [37] Summarizing, the hillslope TT pdf is made up from two distriutions which correspond to the soil and the susoil contriutions. The susoil yields an exponentiallike ehavior, especially at large time, while the soil dominates the early time contriution, with a power law function of t. The small- and large-time limits of f t are consistent with the corresponding limits of the Gamma distriution, which is sometimes adopted in the representation of the f t [Kirchner et al., 1; Fiori and Russo, 8]. The large-time, exponential ehavior is well known [e.g., Haitjema, 1995; McGuire and McDonnell, 6]. The early time power law ehavior has een justified in the past y the degree of heterogeneity to which the susurface is sujected. This study suggests that the power law f t may e determined y the unsaturated transport in the upper soil, and in particular in the vicinity of the river where the velocity variations are stronger.

8 W8435 FIORI ET AL.: TRAVEL TIME DISTRIBUTION IN HILLSLOPES W8435 Figure 4. Comparison etween the soil (f ts ) and susoil (f t ) traveltime pdf for a few values of the ratio U s /U. The inset shows the travel pdf of the hillslope (f t, equation (1)) for a few values of the weight f; the other parameters are set as U v /(U D )=1, =,y /(LD s ) =.1, and Pe = Pe s = Impact of Local Heterogeneity: Source Zone Versus Local Dispersion [38] The results of section 4.1 suggest that local heterogeneity plays a minor role in the traveltime spreading. Similar conclusions were drawn y Fiori and Russo [7] on the asis of a limited set of numerical simulations; the result is generalized here. The finding can e explained y examination of the quantity S (11), the central second moment of t. The second moment depends on the TT variances for the soil and susoil, which in turn depend on two components: (1) a local variance s ti and () a source zone variance s Ti (see section 3). In words, s Ti is the variance of the deterministic component ht i (x )i (i.e., the variance of the mean TT of the particles) which depends on the spatial distriution of the injection points x and how the solute particles are routed through the system. Conversely, s ti is the mean variance of the random TT components along each trajectory, which depends on the degree of heterogeneity, measured through the log conductivity variance. [39] The second moments S = s T + s t and S s = s Ts + s ts are otained through (13) and (14), respectively; they can e rewritten as follows S s T Ss s T s ¼ 1 þ I Y s Y L g ¼ 1 þ g =Pe ¼ 1 þ I Y s s Y s L g s ¼ 1 þ g s =Pe s ð18þ where g, g s = O(1 ) are dimensionless functions which depend on the hillslope parameters. Since Pe s, Pe 1 (the orders of magnitude are descried in section 4.1), expressions (18) clearly show that the dominant component in the traveltime variance for each system is the source zone one 8of13 (s T, s Ts ). This translates into a similar result for the hillslope TT variance S, as follows S ¼ fs T ð1 þ g =Pe Þþð1 fþs T s ð1 þ g s =Pe s Þþfð1 fþðt þ T s Þ fs T þ ð1 fþs T s þ fð1 fþðt þ T s Þ ð19þ [4] Thus, hillslope spreading is dominated y source zone dispersion, which is ruled y the distriution of distances from the river of the injected particles; local heterogeneity plays a minor role. This finding is not entirely surprising and it is similar to what found in other fields, like, for example, the analysis of macrodispersion in heterogeneous porous media under mean uniform flow [see, e.g., Dagan, 1989], the spreading of the hydrological response of a river network [Rinaldo et al., 1991], or the spreading of arrival time in groundwater flows driven y wells [Koplik, 1], to mention a few. In those cases the dominant component of transport is the large-scale one (e.g., advective macrodispersion, river network dispersion, and flow path heterogeneity in wells-driven flow), which for ergodic domains overwhelms the local dispersive component (e.g., pore-scale dispersion, spreading of individual reaches of the river network, and local heterogeneity etween the wells). Our simplified analysis suggests that the same feature may occur in hillslope systems. [41] We emphasize that the aove result does not mean that heterogeneity is completely irrelevant, as it strongly determines important flow quantities like the effective conductivity or water content. For example, the effective conductivity increases with heterogeneity ecause of the emergence of preferential flow paths [Dagan, 1989; Jankovic et al., 3], especially for unsaturated flow, leading to shorter traveltimes in the hillslope for a given head gradient. Another mechanism is solute retention y low conductive areas [Fiori et al., 6], which however is

9 W8435 FIORI ET AL.: TRAVEL TIME DISTRIBUTION IN HILLSLOPES W8435 Tale 1. Parameter Values for the Analysis of the Numerical Simulation of Fiori and Russo [8] Parameter Description f =.86 Weight function etween susoil and soil (dimensionless) L = 18 Horizontal length of the hillslope (m) R = Vertical recharge (m/d) tan a s =.33 Slope of the surface (dimensionless) tan a = Slope of the impermeale ottom (dimensionless) J s =.185 Mean head gradient in the soil (dimensionless) D = 4.5 Mean depth of groundwater, for the U calculation (m) = 1.8; h = 3.8 Brooks-Corey parameters for the soil (dimensionless) y =.178 Entry pressure head (m) K sg =.5 Geometric mean of saturated hydraulic conductivity of the soil (m/d) n =.19 Effective porosity of the susoil (dimensionless) n s =.36 Effective porosity of the soil (dimensionless) s Y = s Ys = 1 Saturated log conductivity variance, susoil and soil (dimensionless) hqi/n =.55 Mean saturation of the unsaturated susoil (dimensionless) I Y = I Ys =.8 Integral scales of saturated conductivity, susoil and soil (m) not well captured y the first-order analysis employed in the present work. Nevertheless, the numerical simulations of Fiori and Russo [7] were performed with levels of heterogeneity eyond the limitations of first-order analysis, leading to similar conclusions. Thus, we are confident that our results may hold valid for a relatively wide range of heterogeneity, provided that the system is ergodic; that is, the size of the domain is much larger than the correlation length of hydraulic conductivity. 5. Comparison With Numerical Simulation [4] Here we riefly apply and test the model developed in the present study. To this aim we shall make use of the detailed numerical simulations of flow and transport in a synthetic hillslope carried out y Fiori and Russo [8]. We refer here to the long simulation which is descried y Fiori and Russo [8, section 3.4]; it consists of a 5-year simulation in which the external forcing is a precipitation record measured in an Italian asin. The simulation is fully three dimensional and the hillslope is heterogeneous, with a mild degree of heterogeneity (the variance of saturated log conductivity is unit). Flow is transient and driven y the temporally variale precipitation; oundary conditions are similar to those employed in the present study. The hillslope is represented y a highly permeale soil underlain y a less permeale edrock. The TT pdf was derived y adopting the flow-corrected time approximation y Rodhe et al. [1996], similar to the ESS assumption of Russo and Fiori [8]. Details of the numerical simulations and their results are given y Fiori and Russo [8]. The setup is similar ut not identical to the conceptual model adopted here. In particular, (1) the soil is not directly connected to the river, () the surface slope is not constant, (3) all hydraulic parameters are heterogeneous, not only the saturated hydraulic conductivity, (4) the variance of the parameters fluctuation is not small, (5) precipitation is temporally 9of13 variale, (6) van Genuchten s expressions for K (y) and q (y) are employed, and (7) the river doesn t fully penetrate the aquifer. [43] Some of the parameters can e easily derived from those of the numerical simulation, for example, y adopting average values for some of the geometrical characteristics of the hillslope and equivalent figures for some of the hydraulic parameters. For example, the soil slope is assumed as the average surface slope; the Brooks-Corey parameters are otained y matching the areas under the saturation ln y curves, otaining the equivalent,, parameters y and. Other quantities, like, for example, the geometric mean of conductivity, effective porosity or log conductivity variance are otained from the original paper, including the recharge R, which in the present ESS approach is equal to the mean annual rate of net precipitation. Things are more difficult for U (16) as the river doesn t fully penetrate the aquifer. To reflect the concentration of water and solute fluxes toward the river, which departs from the simpler scheme of our model, the mean depth D close to the river is taken as the average etween the width of the river and the aquifer depth. [44] A few quantities had to e estimated directly from the numerical simulations, the most important eing the mean head slope of the soil J s, which is found as practically constant with x 1 (R =.994), and the weight f, which was calculated through the water discharges to the river from the susoil and the soil. A detailed list of the parameters employed is given in Tale 1. None of the parameters was estimated through a fitting procedure. [45] The comparison etween f t (t) calculated y the numerical simulation and the model is given in Figure 5. The agreement is considered as satisfactory, all the approximations notwithstanding and considered that there is no parameter fitting. The theoretical model is ale to adequately represent oth the early time and late ehavior of f t (t), for which the different contriution of the two systems is quite evident. The calculated mean traveltime is T = d, which compares well with the numerical one (T = 41 d). The infinite Peclet solution, represented in Figure 5 as a dotted line, is not much different from the finite Peclet solution (solid line); this shows again the modest role played y local heterogeneity in the f t determination. 6. Summary and Concluding Remarks [46] A stochastic model has een developed for the analysis of the traveltime distriution f t in a hillslope. The scope of the model is not to provide a tool for applications ut rather to gain understanding in the transport processes occurring in hillslopes. The model is ased on the Lagrangian representation of transport. The heterogeneous hydraulic conductivity field is descried as a stationary random space function, with assigned statistical properties. The simple conceptual model adopted descries the hillslope as made up from two independent systems and, in particular, (1) a surficial and highly permeale soil underlain y () a less permeale susoil or edrock. The resulting traveltime pdf f t is a linear comination etween the pdfs of oth systems. [47] A first-order approach in the log conductivity variance is adopted in order to get closed form solutions for the principal statistical moments of the traveltime. The main

10 W8435 FIORI ET AL.: TRAVEL TIME DISTRIBUTION IN HILLSLOPES W8435 Figure 5. Comparison etween the traveltime pdf of the hillslope predicted y the theoretical model (solid and dotted lines) and the numerical results of Fiori and Russo [8]: (a) normal and () logscale; the parameter values for the model are reproduced in Tale 1. scope of the work is the analysis of the traveltime distriution as function of the hillslope geometrical and hydraulic properties. Of particular interest is the early and late time ranches of the traveltime distriution and the first two moments of f t. The model is tested against the numerical simulation of Fiori and Russo [8] with reasonaly good agreement, without the need of parameter caliration. [48] Our analysis indicates that the soil and the susoil (or the edrock) contriute in different manners to the traveltime pdf. Starting from the soil, oth the infinite Peclet solution (no heterogeneity) for the traveltime pdf and the first two statistical moments are provided in closed form. It is seen that the traveltime pdf of the soil is generally highly skewed, with a pronounced early peak. The soil is mainly responsile to the early ranch of f t, i.e., the rapid release of solute which preferentially moves through the upper soil. The early traveltime pdf is a power law, with exponent variale etween 1 and.5. This finding is in agreement with the power law structure of f t for small t which was found in a previous work [Kirchner et al., 1]. In the 1 of 13 present work the power law ehavior is determined y the unsaturated flow and transport which occur in the upper soil, and the exponent is a function of the parameters of the soil, modeled y the Brooks-Corey constitutive relations. [49] Semianalytical solutions (or closed form solution in asence of local dispersion) for the susoil traveltime pdf are also provided. The traveltime in the susoil is generally much larger that the one pertaining to the soil. The susoil response is slower than that of the soil, with a less skewed and more regular traveltime pdf. Thus, the susoil is mainly responsile for the tail of f t, which in many cases is similar to the well-known exponential model, and it provides the major contriution to the overall mean travel time of the hillslope (this result is similar to what was found experimentally y Botter et al. [8]). These findings suggests the important role played y the susoil in the determination of f t and its moments. [5] Analysis of the first two moments of f t indicates that the mean traveltime is weakly dependent on the hillslope size L. The traveltime variance, which measures the solute

11 W8435 FIORI ET AL.: TRAVEL TIME DISTRIBUTION IN HILLSLOPES W8435 spreading, is controlled y the traveltime variaility determined y the different flow paths experienced y the solute particles. If we divide the solute (which is uniformly injected over the surface) into small parcels which move with the water, each parcel experiences a different path which depends on the location where the parcel is injected. The resulting spreading is coined here as source zone dispersion. Additional spreading (denoted here as local dispersion) is caused y the spatial variaility of the hydraulic conductivity, which determines a traveltime variaility within each flow path for a given x. Our analysis suggests that for realistic values of the hillslope Peclet numer Pe = L/(s Y I Y ) (where L, s Y, and I Y are the hillslope length and the variance and integral scale of the log conductivity, respectively) the source zone dispersion overwhelms the local one. This result confirms the numerical results of Fiori and Russo [8] and it is similar to what has een oserved in hydrological studies of random environments [e.g., Dagan, 1989; Rinaldo et al., 1991]. As a consequence, any choice for the local traveltime pdf (in this study we assumed the lognormal one) would have little or no effect on f t. [51] Before concluding, we would like to stress that the results of the present work are ased on a few simplifications of the flow and transport prolems in hillslopes. First, our conceptual model of a two-layer system is a schematic representation of a real hillslope, and the flow dynamics can e more complex in reality. The steady state assumption may not e so severe as it is often adopted in most of the hillslope transport studies (the issue is discussed in the review y McGuire and McDonnell [6]). [5] The present analysis is ased on the assumption of a stationary and ergodic log conductivity random field. Roughly speaking, Y = ln K is characterized y a finite integral scale I Y which is much smaller than the hillslope length L, i.e., L I Y. If this condition is not met, the hillslope is not ergodic and each realization of the Y field determines a different traveltime distriution. Our conclusions regarding the role of heterogeneity may change in that case. [53] The hydraulic parameters of the Brooks and Corey constitutive relationship (6) have een considered here as deterministic constants. This means that the variaility in unsaturated conductivity stems entirely from the variaility in saturated conductivity, independent of water saturation and/or the cross correlation etween the various parameters of (6). Previous experimental evidence [e.g., Russo and Bresler, 1981] suggested that the parameters of (6) are spatially variale with a weak to moderate cross correlation among the parameters. In view of the limited impact of the soil heterogeneity on the traveltime distriution, however, we elieve that neglect of the spatial variaility in the parameters of (6), has only small effect on our results. [54] Another important assumption is that local heterogeneity is mild, such that the first-order analysis in the log conductivity variance can e adopted. Although the assumption is formally valid for low/mild heterogeneity, several studies [see, e.g., Bellin et al., 199; Russo et al., 1994; Fiori et al., 3] and the recent numerical simulations y Fiori and Russo [8] indicates that the first-order solution is roust and can e applied to moderate log conductivity variaility. Appendix A: Traveltime Moments in the Susoil A1. Transport in Saturated Zone [55] Adopting the assumption give at the end of section.1, the head gradient decreases linearly downstream. A similar prolem was solved at the regional scale y Ruin and Bellin [1994] and a three-dimensional extension of their results is given y Russo and Fiori [9]. Adaptation of expression (6) of Ruin and Bellin [1994] to the present case leads to J ¼ J ð1 x 1 =I Y Þ ða1þ with J the mean head gradient at the outlet (x 1 = ) and I Y the integral scale of the susoil saturated log conductivity Y. Imposing J = (i.e., zero velocity) for x 1 = L in (A1), we get = I Y /L and the mean gradient J and the mean velocity U g are given y J ¼ J 1 x 1 L U g ¼ K GJ ¼ U 1 x 1 n L ðaþ ða3þ with K G the geometric mean of the susoil saturated conductivity, n is the effective porosity and U = K G J /n is the mean velocity at the outlet; expression (A3) derives from the application of first-order analysis of the susoil saturated log conductivity Y =lnk [see, e.g., Ruin and Bellin, 1994; Russo and Fiori, 9]. [56] Inserting (A3) into (3), with a 1 = x and a =as starting and end points, respectively, we calculate the mean traveltime as follows Z dx 1 t g ðx Þ ¼ x U g ðx 1 Þ ¼ L ln 1 x U L ða4þ [57] The variance of t g is calculated along the simplification of Ruin and Bellin [1994], also adopted y Destouni and Graham [1995] and extended later y Russo and Fiori [9] to three-dimensional fields. The procedure takes advantage of a simplified, closed form expression for the velocity covariance derived y Dagan and Cvetkovic [1993]; further details are given y Russo and Fiori [9, section.5]. The simplifications are valid for = I Y /L 1, which holds true under the ergodic assumptions discussed previously. Adaptation of Russo and Fiori [9, equation ()] to the present, isotropic case leads to the following expression for the variance of t g s t g ðx Þ ¼ s Y IY B U e B x =I Y þ B x 1 I Y ða5þ 11 of 13 with s Y the variance of the saturated susoil log conductivity and B = 8/15.

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