## 1. Introduction

[2] At the U.S. Department of Energy Hanford Site in southeastern Washington State the subsurface burial of contaminants was once common practice [*Gephart*, 2003]. These wastes have migrated through the heterogeneous vadose zone and some have been observed at the water table. Accurate predictions of soil moisture and radionuclide transport are required to evaluate future impacts to the groundwater and to evaluate remedial options.

[3] Sophisticated numerical models have been developed to predict the transfer of water, dissolved contaminants, and energy in the subsurface. However, difficulties are often encountered when deterministic models are used because of the spatially varying nature of soil, which creates uncertainty in the values of each parameter that must be assigned throughout space. One consequence of spatial variability is that the properties of a soil are a function of the size of soil samples [e.g., *Schulze-Makuch et al.*, 1999; *Iversen et al.*, 2001]. Thus hydraulic properties measured in the laboratory using small soil cores are usually not applicable to real field situations.

[4] Numerous researchers have found that hydraulic properties of heterogeneous soils are dependent on supporting scales, i.e., sample size, sampling spacing, and site size. For example, *Iversen et al.* [2001] measured air and water permeabilities of different textures at two scales (i.e., 100 cm^{3} and 6280 cm^{3}) for different soils. They found that both air and water permeabilities were higher at the larger scale compared with the small scale, but the scale-dependent effect was less pronounced in sandy soils.

[5] A heterogeneous soil may be approximated by an equivalent homogeneous medium (EHM). However, analyses for one-dimensional (1-D) transient groundwater flow [*Freeze*, 1975] and 1-D transient unsaturated water flow [*Russo and Bresler*, 1981; *Bresler and Dagan*, 1983] have shown that such an equivalent uniform soil cannot be defined by simply averaging the parameters that describe the soil hydraulic properties. *Schulze-Makuch et al.* [1999] analyzed the relationship between saturated hydraulic conductivity (K_{s}) and scale of measurement. They found that K_{s} increased with scale of measurement in heterogeneous media up to a certain volume after which the porous medium approached the properties of an EHM. *Neuman* [1990] recognized that at a given location a porous medium may be viewed as a nested sequence of more or less distinct hydrogeologic units associated with a discrete hierarchy of scales. When the semivariograms of such discrete units are superimposed, the resulting function increases with distance in a stepwise rather than a gradual fashion. Each step in such an echelon represents a natural correlation scale at which log permeability is statistically homogeneous or nearly so; other scales are locally either inactive or suppressed. As hydrogeologic conditions vary from one setting to another, so do the dominant natural scales, giving rise to an infinite variety of possible echelon-type semivariograms. *Neuman* [1994] introduced a generalized scaling of permeability using a power law semivariogram. He based this indirectly on tracer studies and also on the observation that spatial variability often increases with support scale. However, such a generalized scaling approach may not be applicable to all sites because different sites have different degrees of heterogeneity at a given scale. Using Synthetic Miller-similar anisotropic soils and numerical experiments, *Zhang et al.* [2003b] found that (1) the parameter relevant to the inverse macroscopic capillary length (α) increases with soil heterogeneity (σ_{Y}^{2} with Y = ln(K_{s})), (2) the pore size distribution parameter (n) decreased with σ_{Y}^{2}, and (3) the pore connectivity-tortuosity coefficient (L) was a function of both soil heterogeneity and anisotropy. This implies that the unsaturated hydraulic parameters are scale-dependent because soil spatial variability varies with spatial scales.

[6] The aggregate effects of soil variability can be captured by using effective or upscaled soil parameters, which usually provide more satisfactory predictions of large-scale flow. Stochastic methods may be used to compute large-scale effective parameter values from those measured at smaller spatial scales [e.g., *Yeh et al.*, 1985a, 1985b, 1985c; *Mantoglou and Gelhar*, 1987a, 1987b, 1987c; *Polmann*, 1990]. The stochastic models usually assume the Gardner hydraulic function [*Gardner*, 1958] due to its simplicity and its ability to facilitate linearization of the Richards' flow equation. *Khaleel et al.* [2002] used the stochastic method of *Yeh et al.* [1985b] to upscale flow and transport properties of perfectly stratified soils and showed that the upscaled unsaturated K based on the analytical formulas compared well with those based on Monte Carlo simulations. However, application of the stochastic model requires the knowledge of the variance of ln(K_{s}), the correlation between hydraulic parameters, and the vertical correlation length. Typically, such information is not readily available.

[7] Another way to estimate effective parameter values is by inverse methods, which attempt to minimize the differences between field observations and the simulated values using analytical or numerical solutions that contain the constitutive relations with the set of parameters to be estimated. A number of laboratory and field applications [*van Dam et al.*, 1992; *Simunek and van Genuchten*, 1996; *Lehmann and Ackerer*, 1997; *Abbaspour et al.*, 1997; *Zhang et al.*, 2000a, 2000b; *Inoue et al.*, 2000; *Zhang et al.*, 2003a] have shown the potential of inverse techniques for improving the design and analysis of vadose zone flow and transport experiments in homogenous and heterogeneous soils.

[8] The use of inverse methods to estimate soil hydraulic properties was recently reviewed by *Hopmans and Simunek* [1999] and *Hopmans et al.* [2002]. A potential problem of using inverse techniques in heterogeneous soils is that the parameter estimates may be nonunique causing some of the parameters to be unidentifiable. The nonuniqueness problem becomes worse as the number of parameters to be estimated becomes larger. Furthermore, when large numbers of parameters are inverted, large numbers of observations are required and very long simulation times are needed because the simulation time increases quadratically with the number of parameters to be estimated.

[9] The number of parameters may be reduced by similar-media scaling [*Miller and Miller*, 1956], which scales soil water retention curves and the unsaturated hydraulic conductivity functions of multiple soils with similar pore geometry to those of a reference soil. The similar-media scaling allows a single solution of the Richards equation to suffice for numerous specific cases of flow in unsaturated soils. Since the similar-media scaling scales soil hydraulic properties and hence it will also be referred to as “property scaling” to distinguish it from the “parameter scaling” proposed by *Zhang et al.* [2002, 2004]. The latter will be introduced after a brief a review of the property scaling. Assuming the soils are vertically uniform and using property scaling, *Warrick and Hussen* [1993] showed that infiltration and drainage measurements in dramatically different soil types could be successfully described by a scaled form of Richards' equation. However, this approach neglects vertical heterogeneity. In an attempt to incorporate vertical heterogeneity, *Shouse et al.* [1992] scaled soil water content (θ) with depth-dependent scaling factors with the assumption that the scaled unsaturated hydraulic conductivity is depth-independent. Use of the scaled θ with a scaled form of the Richards equation allowed heterogeneous soil hydraulic properties to coalesce into unique functions for both θ(h) and K(θ), where h is pressure head and K is unsaturated hydraulic conductivity. The limitation of this approach is that, in order to compare observations with simulated values, the scaled depth for each observation must be determined using the depth-dependent scaling factors, which are dependent on both the observed and scaled water contents.

[10] *Vogel et al.* [1991] extended the similar-media scaling by introducing a linear variability concept that expresses the soil hydraulic properties in terms of a linear transformation and used three mutually independent scaling factors for K, h, and θ. The method assumes that linear variability is an approximation of the linear component of real soil variability. The authors claim that the generalized scaling can be incorporated into numerical models for water movement in a stratified soil profile with the assumption that hydraulic properties of the soil profile are independent of profile geometry (e.g., thickness of soil layers). *Eching et al.* [1994] applied this method to analyze drainage from a horizontally heterogeneous soil. The data were inverted once to obtain reference soil hydraulic functions, from which hydraulic functions of other horizontal locations were estimated using the scaling factors. They successfully coalesced the drainage curves of soils with similar textures to a single curve after scaling.

[11] However, the property scaling based on the similar-media concept of *Miller and Miller* [1956] requires the spatial variables, i.e., the horizontal and vertical distances, be scaled. The difficulties in scaling the spatial variables limit the use of property scaling in vertically and three-dimensionally heterogeneous soils. In inverse modeling, these problems are compounded by the difficulty in converting the positions of the observations from a regular coordinate system to a scaled coordinate system. Furthermore, the texture dependence of the pore size distribution parameter is ignored in hydraulic property scaling [*Vogel et al.*, 1991].

[12] To overcome these problems, *Zhang et al.* [2002, 2004] proposed a parameter-scaling method that reduces the number of parameters to be estimated. Parameter scaling differs from property scaling in that the former scales hydraulic parameters while the latter scales hydraulic properties h(θ) and K(θ). Parameter scaling has the following characteristics: (1) unlike property scaling scaling, parameter scaling does not require the soil materials to be geometrically similar; (2) after taking parameter scaling, the values of the hydraulic parameters of all the soil textures perfectly reduce to the reference values; (3) the spatial variability of each hydraulic parameter can be expressed by the scaling factors; (4) the flow equation can always be expressed in real time and space regardless of the soil heterogeneity, and hence the parameters may be estimated inversely; and (5) when the parameters are to be estimated by an inverse procedure, the number of unknown variables is reduced by a factor of the number of textures (M), and the simulation time is reduced approximately by a factor of M^{2}. *Zhang et al.* [2002, 2004] successfully applied the method to estimate the hydraulic parameters of layered soils. When compared to the use of local-scale parameters, parameter scaling reduced the sum of squared weighted residuals by 93 to 96%.

[13] In this study, a soil is described as a composition of multiple EHMs, whose hydraulic properties are scale-dependent. The local-scale properties are typically measured in the laboratory using small soil cores and must be upscaled before application to field-scale predictions. We extend the parameter-scaling concept of *Zhang et al.* [2002, 2004] by coupling it with inverse modeling to upscale hydraulic parameters. The combined parameter scaling and inverse technique (CPSIT) only estimates the hydraulic parameters of the reference EHM and hence has the advantages of both the scaling and inverse techniques: the number of parameters to be estimated is reduced by a factor of the number of EHMs, and the field-scale parameter values are optimized. Compared with the inverse method without using parameter scaling, the CPSIT, due to the reduction of the number of parameters, suffers much less from nonuniqueness and needs fewer observations to identify all of the hydraulic parameters. More importantly, the relationship between parameter values at the local- and field-scales of the reference EHM may be transferable to other untested sites with soils of similar variability hierarchy. The CPSIT method was tested by applying it to an injection experiment at the Hanford Site, conducted over a scale of about 16 m horizontally and 18 m vertically [*Sisson and Lu*, 1984].