Factor de Impacto:
4.911
Factor de Impacto de 5 años:
3.179
SJR:
1.008
SNIP:
0.983
CiteScore™:
5.2
ISSN Imprimir: 21525080
Acceso abierto
Volumes:

International Journal for Uncertainty Quantification
DOI: 10.1615/Int.J.UncertaintyQuantification.v1.i1.30
pages 3547 PROBABILISTIC PREDICTIONS OF INFILTRATION INTO HETEROGENEOUS MEDIA WITH UNCERTAIN HYDRAULIC PARAMETERSAbstract Soil heterogeneity and the lack of detailed site characterization are two ubiquitous factors that render predictions of flow and transport in the vadose zone inherently uncertain. We employ the Green{Ampt model of infiltration and the Dagan{Bresler statistical parameterization of soil properties to compute probability density functions (PDFs) of infiltration rate and infiltration depth. By going beyond uncertainty quantification approaches based on mean and variance of system states, these PDF solutions enable one to evaluate probabilities of rare events that are required for probabilistic risk assessment. We investigate the temporal evolution of the PDFs of infiltration depth and corresponding infiltration rate, the relative importance of uncertainty in various hydraulic parameters and their crosscorrelation, and the impact of the choice of a functional form of the hydraulic function. KEYWORDS: Uncertainty quantification, stochastic, infiltration rate, GreenAmpt model
1. IntroductionSoil heterogeneity and the lack of detailed site characterization are two ubiquitous factors that hamper one’s ability to predict flow and transport in the vadose zone. The continuing progress in data acquisition notwithstanding, measurements of hydraulic properties of partially saturated media remain scarce and prone to measurement and interpretive errors. Consequently, spatial distributions of hydraulic parameters (saturated and relative hydraulic conductivities, and parameters in retention curves) are typically uncertain and their statistical properties are subject to considerable debate. Despite some reservations, e.g., [1, 2], it has become common to treat saturated hydraulic conductivity K_{s}(x) as a multivariate lognormal random field whose ensemble statistics (e.g., mean, variance, and correlation length) can be inferred from spatially distributed data by means of geostatistics. No such consensus exists about statistical distributions of various hydraulic parameters entering relative hydraulic conductivity and retention curves. For example, various data analyses concluded that spatial variability of a soil parameter α_{G}(x) in the Gardner model of relative conductivity, which is often referred to as the reciprocal of the macroscopic capillary length, exhibits either a normal [3] or lognormal [4] distribution and is either correlated [5] or uncorrelated [3] with K_{s}. We defer a more detailed review of the statistical properties of both α_{G}(x) and parameters in the van Genuchten model of relative conductivity until Section 2. Here, it suffices to say that any approach to uncertainty quantification for flow and transport in the vadose zone must be flexible enough to accommodate arbitrary statistical distributions of soil properties. Statistical treatment of hydraulic parameters renders corresponding flow equation stochastic. Solutions of these equations are probability density functions (PDFs) of system states (water content, pressure, and macroscopic flow velocity) and can be used not only to predict flow in heterogeneous partially saturated porous media but also to quantify predictive uncertainty. Rather than computing PDFs of system states, standard practice in subsurface hydrology is to compute (analytically or numerically) the first two moments of system states, and to use their ensemble means as predictors of a system’s behavior and variances (or standard deviations) as a measure of predictive uncertainty. A large body of literature employing this approach to solve the stochastic Richards equation includes [611], to name just a few. With the exception of solutions based on the Kirchhoff transformation [1214], such analyses require one to linearize constitutive relations in the Richards equation, introducing errors that are hard to quantify a priori. More important, none of these solutions can be used to estimate the probability of rare events, which is of crucial importance for uncertainty quantification and risk assessment [15]. The GreenAmpt model described in some detail in Section 2 (see also [16, Section 5.2]) provides an alternative description of flow in partially saturated porous media. The relative simplicity of the GreenAmpt formulation makes it easier to solve than the Richards equation, which explains its prevalence in large numerical codes—e.g., SCS developed by U.S. Environmental Protection Agency (USEPA), DR3M developed by U.S. Geological Survey (USGS), and HIRO2 developed by U.S. Department of Agriculture (USDA)—that are routinely used to predict overland and channel flows. The first analysis of the impact of soil heterogeneity and parametric uncertainty on solutions of the GreenAmpt equations was carried out by Dagan and Bresler [17]. Saturated hydraulic conductivity—the sole source of uncertainty in their analysis—was treated as a twodimensional random field, K_{s}(x_{1},x_{2}). This enables one to model vertical infiltration with a collection of onedimensional (in the x_{3} direction) solutions each of which corresponds to a different random variable K_{s}. The DaganBresler statistical model [17], whose precise formulation is provided in Section 2, was found to yield accurate predictions of infiltration into heterogeneous soils [18, 19] and has been adopted in a number of subsequent investigations, e.g., [1924]. These and other similar analyses aimed to derive effective (ensemble averaged) infiltration equations, and some of them quantified predictive uncertainty by computing variances of system states. Driven by the needs of probabilistic risk assessment, we focus on the derivation of singlepoint PDFs (rather than the first two moments) of system states describing infiltration into heterogeneous soils with uncertain hydraulic parameters. Our analysis employs the GreenAmpt model of infiltration with the DaganBresler parameterization, both of which are formulated in Section 2. This Section also contains an overview of experimentally observed statistical properties of the coefficients entering the Gardner and van Genuchten expressions of relative hydraulic conductivity K_{r}. A general framework for derivation of PDF solutions of the GreenAmpt model is presented in Section 3. In Section 4 we investigate the temporal evolution of the PDFs of a wetting front (Setion 4.1) and corresponding infiltration rate (Section 4.2), the relative importance of uncertainty in various hydraulic parameters (Section 4.3) and their crosscorrelation (Section 4.4), and the impact of the choice of a functional form of K_{r} (Section 4.5). Concluding remarks are presented in Section 5. 2. Problem FormulationConsider infiltration into a heterogeneous soil with saturated hydraulic conductivity K_{s}, porosity ϕ, residual water content θ_{r}, and relative hydraulic conductivity K_{r}(ψ;α) that varies with pressure head ψ in accordance with a constitutive model and model parameters α. While the subsequent analysis can be applied to any constitutive relation, we focus on the Gardner model [16, Table 2.1]
and the van Genuchten model (ibid)
The model parameters α (α ≡ α_{G} and {α_{vG},n,m = 1  1/n} for the Gardner and van Genuchten models, respectively) and the rest of the hydraulic properties mentioned above vary in space and are sparsely sampled. To quantify uncertainty about values of these properties at points x = (x_{1},x_{2},x_{3})^{T} where measurements are unavailable, we treat them as random fields. Thus, a soil parameter (x,ω) varies not only in the physical domain, x ∈, but also in the probability space ω ∈ Ω. A probability density function p_{}, which describes the latter variability, is inferred from measurements of by invoking ergodicity. Experimental evidence for the selection of PDFs p_{} for various soil parameters is reviewed in Section 2.1, and the DaganBresler statistical model used in our analysis is formulated in Section 2.2. The overreaching aim of the present analysis is to quantify the impact of this parametric uncertainty on predictions of both the dynamics of wetting fronts and infiltration rates. Uncertainty in the former may significantly affect the accuracy and reliability of fieldscale measurements of soil saturation [25], while uncertainty in the latter is of fundamental importance to flood forecasting [23]. 2.1 Statistics of Soil ParametersSaturated hydraulic conductivity. In addition to the experimental studies reviewed in [12], the data analyses reported in [4, 24], etc., support our treatment of saturated hydraulic conductivity K_{s} as a lognormal random field. Gardner's constitutive parameter. The (scarce) experimental evidence reviewed in [12] suggests that α_{G}, the reciprocal of the macroscopic capillary length, can be treated alternatively either as a Gaussian (normal) or as a lognormal random field. While the approach described below is capable of handling both distributions, in the subsequent computational examples we will treat α_{G} as a lognormal field, which is a model adopted in more recent computational investigations (e.g., [4, 10]). Van Genuchten's constitutive parameters. The van Genuchten hydraulic function (2) is a twoparameter model obtained from its more general form by setting m = 1  1/n and l = 1/2 (hence, the power m/2 in the denominator). We employ this form because of its widespread use [16, Table 2.1], but the approach described below can be readily applied to quantify uncertainty in more general formulations with arbitrary m and l. The experimental evidence presented in [4, 26, 27] shows that the coefficient of variation of α_{vG} is much larger than that of n. These data suggest that α_{vG} can be treated as a lognormal field and the shape factor n as a deterministic constant. Correlations between hydraulic parameters. Experimental evidence presented in [4, 12] suggests that the coefficient of variation (CV) of K_{s} is generally much larger than that of either α_{G} or α_{vG}. These parameters were found to be either perfectly correlated or uncorrelated or anticorrelated (see also [28]). Our analysis allows for an arbitrary degree of correlation between K_{s} and either α_{G} or α_{vG}. Finally, since the difference between the full and residual saturations Δθ = ϕ  θ_{r} typically exhibits lower spatial variability than both K_{s} and α_{G} (or α_{vG}), we treat it as a deterministic constant to simplify the presentation. Our approach can be adopted to quantify uncertainty in Δθ and the shape factor n in the van Genuchten hydraulic function, as discussed in Section 3. 2.2 Statistical Model for Soil ParametersFollowing [17], we restrict our analysis to infiltration depths that do not exceed vertical correlation lengths l_{v} of (random) soil parameters (x,ω). Then = (x_{1},x_{2},ω), so that a heterogeneous soil can be represented by a collection of onedimensional (in the vertical direction x_{3}) homogeneous columns of length L_{3}, whose uncertain hydraulic properties are modeled as random variables (rather than random fields). The restriction l_{v} > L_{3} formally renders the DaganBresler parameterization [17] suitable for heterogeneous topsoils, and thus can be used to model surface response to rainfall events [23, 24] and transport phenomena in topsoil [21]. Yet it was also used to derive effective properties of the whole vadose zone [4, 28]. Rubin and Or [19] provide an additional justification for the DaganBresler parameterization by noting that “the determination of soil hydraulic properties through field methods...homogenize the properties vertically, thus eliminating the variability in the vertical direction in a practical sense.” Consider a threedimensional flow domain Ω = Ω_{h} × [0,L_{3}], where Ω_{h} represents its horizontal extent. A discretization of Ω_{h} into N elements represents Ω by an assemblage of N columns of length L_{3} and facilitates the complete description of a random field (x_{1},x_{2},ω)—in the analysis below, stands for K_{s}, α_{G}, and α_{vG} but can also include other hydraulic properties and the ponding pressure head ψ_{0} at the soil surface x_{3} = 0—with a joint probability function p_{}(A_{1},...,A_{N}). Probability density functions (PDFs) of hydraulic properties of the ith column are defined as marginal distributions,
Since statistical properties of soil parameters are inferred from spatially distributed data by invoking ergodicity, the corresponding random fields (or their fluctuations obtained by data detrending) must be stationary so that
Furthermore, if such soil parameters (e.g., K_{s} and α_{G}) are correlated, their statistical description requires the knowledge of a joint distribution. For multivariate Gaussian Y _{1} = lnK_{s} and Y _{2} = lnα_{G} (or Y _{2} = lnα_{vG}), their joint PDF is given by
where
Y_{i}and σ_{Y i} denote the mean and standard deviation of Y _{i} (i = 1,2), respectively; and 1 ≤ ρ ≤ 1 is the linear correlation coefficient between Y _{1} and Y _{2}. The lack of correlation between Y _{1} and Y _{2} corresponds to setting ρ = 0 in (5). 2.3 GreenAmpt Model of InfiltrationDuring infiltration into topsoils, the DaganBresler parameterization of soil heterogeneity can be supplemented with an assumption of vertical flow. The rationale for, and implications of, neglecting the horizontal component of flow velocity can be found in [17, 19, 20] and other studies reviewed in the Introduction. This assumption obviates the need to solve a threedimensional flow problem, replacing the latter with a collection of N onedimensional flow problems to be solved in homogeneous soil columns with random but constant hydraulic parameters. Such a framework was used to predict mean (ensemble averaged) flow with either the GreenAmpt model [17, 20] or the steadystate Richards equation with the Gardner hydraulic function [19]. We employ the GreenAmpt description because it enables one to handle transient flow and to employ arbitrary hydraulic functions, without resorting to linearizing approximations [29]. Let I(t) denote (uncertain) cumulative infiltration due to ponding water of height ψ_{0} at the soil surface x_{3} = 0. The GreenAmpt model of infiltration approximates an Sshaped wetting front with a sharp interface x_{f}(t) that separates fully saturated soil (saturation ϕ) from dry soil (saturation θ_{r}). The latter is also known as infiltration depth. If the x_{3} coordinate is positive downward, Darcy’s law defines macroscopic (Darcy’s) flux q as (e.g., [16, Eq. (5.1)])
Pressure head ψ_{f} at the infiltration depth x_{f}(t) is empirically set to a “capillary drive”,
where ψ_{in} is the initial pressure head in the dry soil. Mass conservation requires that I(t) = (ω  θ_{r})x_{f}(t) and the infiltration rate i ≡ dI/dt equals q. The first condition yields
which, combined with the second condition and (6), leads to a (stochastic) ordinary differential equation for the position of the wetting front,
Our goal is to relate uncertainty in hydraulic parameters K_{s} and α_{G} (or α_{vG}) to predictive uncertainty about the infiltration depth x_{f}(t) and the infiltration rate i(t), i.e., to express the PDFs of the latter, p_{f}(x_{f};t) and p_{i}(i;t), in terms of the PDF of the former (5). 3. PDF SolutionsTo simplify the presentation, we assume that the height of ponding water, ψ_{0}, does not change with t during the simulation time T . Then an implicit solution of (9) takes the form
For small t, (10) can be approximated by an explicit relation [16, Eq. (5.12)]
For large t, flow becomes gravity dominated, i ~ K_{s}, and [16, p. 170]
For intermediate t, various approximations, e.g., [30] and [16, p. 170], can be used to replace the implicit solution (10) with its explicit counterparts. We will use the implicit solution (10) to avoid unnecessary approximation errors. Several of the simplifying assumptions made above can be easily relaxed. First, since K_{s} and Δθ enter the stochastic Eq. (9) and its implicit solution (10) as the ratio K_{s}^{*} = K_{s}/Δθ, one can easily incorporate uncertainty in (randomness of) Δθ by replacing the PDF of K_{s} with the PDF of K_{s}^{*}. Second, the implicit relation F(x_{f},K_{s}/Δθ,α;t) = 0 given by (10) and (7) allows one to express the PDF of x_{f} in terms of the PDFs of any number of hydraulic parameters by following the procedure described below. Third, uncertainty in, and temporal variability of, the height of ponding water ψ_{0}(t) can be dealt with by replacing (10) with an appropriate solution of (9). 3.1 PDF of Infiltration DepthLet G_{f}(x_{f}^{*}) = P(x_{f} ≤ x_{f}^{*}) denote the cumulative distribution function of x_{f}, i.e., the probability that the random position of the wetting front x_{f} takes on a value not larger than x_{f}^{*}. Since (10) provides an explicit dependence of random K_{s} on random x_{f} and α (where α stands for either α_{G} or α_{vG}), i.e.,
it follows from the definition of a cumulative distribution function that
The denominator in (14) reflects the transition from (5), the joint Gaussian PDF for Y _{1} and Y _{2}, to the lognormal variables K_{s} = exp(Y _{1}) and α = exp(Y _{2}). The PDF of the random (uncertain) infiltration depth p_{f}(x_{f}^{*};t) can now be obtained as
Using Leibnitz’s rule to compute the derivative of the integral in (14) and (15), we obtain
Equation (16) holds for an arbitrary implicit solution of the GreenAmpt equation, F(x_{f},K_{s}/Δθ,α;t) = 0, and hence, the PDF solution (16) is applicable to a large class of infiltration regimes that are amenable to the GreenAmpt description. For the flow regime considered in the present analysis, K_{s}(x_{f}^{*},α) is given by (13), and (16) takes the form
3.2 PDF of Infiltration RateLet G_{i}(i^{*}) = P(i ≤ i^{*}) denote the cumulative distribution function of i, i.e., the probability that the random infiltration rate i takes on a value not larger than i^{*}. Since q = i, Eqs. (6) and (7) define a mapping K_{s} = K_{s}(i,α). This enables one to compute the cumulative distribution function G_{i}(i^{*}) as
and the PDF of infiltration rate, p_{i} = dG_{i}/di^{*}, as
The derivative ∂K_{s}/∂i^{*} is computed from (6) as the inverse of
3.3 Dimensionless Form of PDFsTo facilitate an analysis of the effects of various sources of parametric uncertainty on the PDF p_{f}(x_{f}^{*};t) of the uncertain (random) infiltration depth x_{f}(t), given by the analytical solution (17), we introduce the following dimensionless quantities. Let the averaged quantities (α)^{1} and K_{s} represent a characteristic length scale and a characteristic value of saturated hydraulic conductivity, respectively. Then a characteristic time scale τ can be defined as
and the following dimensionless quantities can be introduced,
This leads to a PDF solution for the dimensionless infiltration depth x_{f}' = αx_{f},
Likewise, the PDF of the dimensionless infiltration rate i' = i/K_{s} takes the form
In the following, we drop the primes to simplify the notation. 4. Results and DiscussionIn this Section, we explore the impact of various aspects of parametric uncertainty on the uncertainty in predictions of infiltration rate i(t) and infiltration depth x_{f}(t). Specifically, we investigate the temporal evolution of the PDFs of the wetting front (Section 4.1) and the infiltration rate (Section 4.2), the relative importance of uncertainty in K_{s} and α_{i} (Section 4.3), and the effects of crosscorrelation between them (Section 4.4). This is done for the Gardner hydraulic function (1), in which case (7) results in the interfacial pressure head ψ_{f} = α_{G}^{1}. In Section 4.5, we explore how the choice of a functional form of the hydraulic function, i.e., the use of the van Genuchten model (2) instead of the Gardner relation (1), affects the predictive uncertainty. Unless explicitly noted otherwise, the simulations reported below correspond to the dimensionless initial pressure head ψ_{in} = 9999.9, the dimensionless height of ponding water ψ_{0} = 0.1, Δθ = 0.45, the coefficients of variation CV _{ln K} ≡ σ_{Y 1}/Y _{1} = 3.0 and CV _{ln α} ≡ σ_{Y 2}/Y _{2} = 0.5 with the means Y _{1} = 0.25 and Y _{2} = 0.1, and the crosscorrelation coefficient ρ = 0. (The use of the soil data in Table 1 of [26] in conjunction with these dimensionless parameters would result in the height of ponding water ψ_{0} = 0.6 cm.) 4.1 PDF of Wetting FrontSince the initial position of the wetting front is assumed to be known, x_{f}(t = 0) = 0, the PDF p_{f}(x_{f};0) = δ(x_{f}), where δ(·) denotes the Dirac delta function. As the dimensionless time becomes large (t →∞), p_{f} ~ p_{Ks} in accordance with (12). The PDF p_{f}(x_{f};t) in (23) describes the temporal evolution of predictive uncertainty between these two asymptotes, with Fig. 1 providing snapshots at dimensionless times t = 0.01, 0.1, and 0.5. (For the soil parameters reported in Table 1 of [26], this corresponds to dimensional times 1.5, 15, and 75 min, respectively). The uncertainty in predictions of infiltration depth increases rapidly, as witnessed by wider distributions with longer tails. FIG. 1: Temporal evolution of the PDF of infiltration depth p_{f}(x_{f};t). 4.2 PDF of Infiltration RateFigure 2 provides snapshots, at dimensionless times t = 0.01, 0.1, and 0.5, of the temporal evolution of the PDF of infiltration rate p_{i}(i;t) given by (24). Both the mean infiltration rate and the corresponding predictive uncertainty decrease with time. At later times (the dimensionless time t = 5.0, for the parameters used in these simulations), the PDF appears to become time invariant. This is to be expected on theoretical grounds, see (12), according to which p_{i}(i';t') → p_{K}(K_{s}') as t'→∞. The reduced ^{2} test confirmed this asymptotic behavior at dimensionless time t = 100.0. FIG. 2: Temporal evolution of the PDF of the infiltration rate p_{i}(i;t). 4.3 Effects of Parametric UncertaintyThe degree of uncertainty in hydraulic parameters lnK_{s} and lnα_{G} is encapsulated in their coefficients of variation CV _{ln K} and CV _{α}, respectively. Figure 3 demonstrates the relative effects of these two sources of uncertainty upon the predictive uncertainty, as quantified by the infiltration depth PDF p_{f}(x_{f};t), computed at t = 0.1. Uncertainty in saturated hydraulic conductivity K_{s} affects predictive uncertainty more than uncertainty in the Gardner parameter α_{G} does. Although not shown in Fig. 3, we found similar behavior at later times t = 0.5 and 1.0. These findings are in agreement with those reported in [17, 31], wherein variances of state variables were used to conclude that uncertain saturated hydraulic conductivity K_{s} is the dominant factor affecting predictive uncertainty. FIG. 3: The infiltration depth PDF p_{f}(x_{f};t = 0.1) for different levels of uncertainty in (a) saturated hydraulic conductivity K_{s} and (b) the Gardner parameter α_{G}. 4.4 Effects of CrossCorrelationThe question of whether various hydraulic parameters are correlated with each other remains open, with different data sets supporting opposite conclusions (see Section 2.1). This suggests that the presence or absence of such crosscorrelations is likely to be sitespecific rather than universal. The general PDF solution (23) enables us to investigate the impact of crosscorrelations between saturated hydraulic conductivity K_{s} and the Gardner parameter α_{G} on predictive uncertainty. This is done by exploring the dependence of the PDF of the wetting front p_{f}(x_{f};t) on the correlation coefficient ρ. Figure 4 presents p_{f}(x_{f};t = 0.1) for ρ = 0.99, 0.0, and 0.99, which represent perfect anticorrelation, independence, and perfect correlation between K_{s} and α_{G}, respectively. The perfect correlation between K_{s} and α_{G} (ρ = 0.99) results in the minimum predictive uncertainty (the width of the distribution), while the perfect anticorrelation (ρ = 0.99) leads to the maximum predictive uncertainty. Predictive uncertainty resulting from the lack of correlation between K_{s} and α_{G} (ρ = 0.0) falls amid these two limits. The impact of crosscorrelation between soil hydraulic parameters (a value of ρ) decreases with time, falling from the maximum difference of about 21% at t = 0.01 to about 3% at t = 0.1. FIG. 4: The infiltration depth PDF p_{f}(x_{f};t = 0.1) for different levels of correlation ρ between hydraulic parameters K_{s} and α_{G}. 4.5 Effects of Selection of Hydraulic FunctionFinally, we examine how the choice of a hydraulic function K_{r}(ψ;α) affects predictive uncertainty. Guided by the data analyses presented in Section 2.1, we treat α_{vG} as the only uncertain parameter in the van Genuchten hydraulic function with n = 1.5. To make a meaningful comparison between predictions based on the Gardner (1) and van Genuchten (2) relations, we select statistics of their respective parameters α in a way that preserves the mean effective capillary drive defined by (7) [29, 32]. Specifically, we use the equivalence criteria to select the mean of lnα_{vG} (1.40, for the parameters used in these simulations) that maintains the same mean capillary drive as the Gardner model with lnα_{G} = 0.1, and choose the variance of lnα_{vG} as to maintain the original values of the coefficients of variation CV _{ln αvG} = CV _{ln αG} = 0.5. Figure 5 reveals that the choice between the van Genuchten and Gardner models has a significant effect on predictive uncertainty of the wetting front dynamics, although this influence diminishes with time. For example, the difference between the variances is 40% at t = 0.01 and 23% at t = 0.1. FIG. 5: The infiltration depth PDF p_{f}(x_{f};t = 0.1) resulting from use of the Gardner and van Genuchten hydraulic functions. 5. ConclusionWe presented an approach for computing probability density functions (PDFs) of both infiltration rates and wetting fronts propagating through heterogeneous soils with uncertain (random) hydraulic parameters. Our analysis employs the GreenAmpt model of infiltration and the DaganBresler statistical parameterization of soil properties. Our analysis leads to the following major conclusions.
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