«10.1 Introduction The stochastic evaluation of solute transport is of interest where not only the expected behavior of contaminant movement but also ...»
Harter Dissertation - 1994 - 321
10. CONDITIONAL SIMULATION OF
UNSATURATED SOLUTE TRANSPORT
The stochastic evaluation of solute transport is of interest where not only the expected
behavior of contaminant movement but also the uncertainty associated with the mean
concentration prediction must be evaluated. To obtain the statistical parameters of the input random field variables (RFVs, see section 2.5.1) Ks and " in (4-8), measurements must be taken to determine Ks and " at the site that needs to be evaluated (unless these data are available from similar or nearby sites). In many cases, measurements are also available that are related to those two parameters, although they represent a different physical quantity, for example soil tensiometer data or concentration measurements. Data of physical variables that are different from, but related to the constitutive parameters of unsaturated flow are often referred to as "indirect" information. The unconditional stochastic method presented in the previous two chapters ignores any available indirect data and considers only the statistical properties of the "direct" data. The approach is satisfactory in applications where either the lateral extent of the contamination source or the travel distance of interest is very large with respect to the correlation scale of the soil and if the soil is of only mild heterogeneity. In such cases, the actual solute plume is "ergodic" (see chapter 2) i.e., the stochastic mean concentration plume accurately predicts the actual plume and the concentration variance is zero. But for a point source or very localized contamination, the travel distance required for the plume to reach ergodicity may be exceedingly large. Dagan (1986) suggested that the ergodicity assumption is valid only after the plume has been displaced several hundred correlation scales. In the unsaturated zone, this may correspond to several tens of meters (see field studies referenced in the introduction to chapter 8). If soil heterogeneity is found on a number of distinct scales of increasing order, ergodicity may not be achieved at all, even in the deep unsaturated zones Harter Dissertation - 1994 - 322 encountered in semi-arid and arid environments (see chapter 9).
The non-ergodic mean plume concentration has a meaning much different from the ergodic mean plume concentration. It merely is a mass conservative, best estimate of the local, time dependent concentration probability. Unlike the pdf of other RFVs, the concentration pdf is difficult to determine due to the non-stationarity in space and time. Hence, the significance of the first two unconditional concentration moments is questionable if the variability is very large. To condition the stochastic evaluation of solute transport on all of the available information - including the deterministic value of single measurement data - is therefore a desirable approach not only to reduce the uncertainty of the concentration prediction but also to fully reflect the information content of the available field measurements.
Conditional stochastic analysis has been applied to a number of groundwater problems.
Dagan (1982, 1984) derived analytical perturbation expressions for the conditional moments of the saturated hydraulic conductivity (input variable), the conditional head, and the spatial plume moments (output variables) in a Bayesian framework. The work accounted for local measurements of the hydraulic conductivity, of the head, and of the groundwater pore velocity.
Delhomme (1979) used the geostatistical method to generate conditional random input fields of the saturated hydraulic conductivity (chapter 3; Journel, 1974). By generating random fields of Ks and solving the saturated flow equation numerically, he evaluated the conditional head moments through a Monte Carlo simulation. A similar approach was taken by Smith and Schwartz (1981) who not only analyzed the conditional head, but also the conditional solute arrival time to demonstrate the principal effect of conditioning. Binsariti (1980) and Clifton and Neuman (1982) used transmissivity and water table measurements to condition the transmissivity fields. They applied the statistical inverse method introduced by Neuman and Yakowitz (1978) to condition the hydraulic conductivity data on measurements of head. Clifton and Neuman (1982) reported a large decrease in prediction uncertainty with respect to the head moments, when head measurements are included in the conditional approach. The first effort
Graham and McLaughlin (1989). They presented a first order analytical stochastic solution based on spectral perturbation analysis and Kalman filtering. This work has to date been the only rigorous approach that allows for conditioning with concentration data. Indirect and direct information is used in the Lagrangian conditional transport analysis by Rubin (1991a), who uses a cokriging approach to obtain the conditional moments of the velocity from measurements of the saturated hydraulic conductivity and/or the head. The covariances and cross-covariances necessary for the cokriging are derived from a linear first order analysis based on Dagan's work (1984). Using the conditional velocity fields, the conditional spatial moments (center of mass and moment of inertia) of the contamination plume are evaluated in a Lagrangian framework by particle tracking. Zhang and Neuman (1994a,b,c,d) develop a new approach to obtain conditional concentration moments, conditional spatial moments of the mean concentration, and conditional solute flux moments based on the Eulerian-Lagrangian transport theory by Neuman (1993). Transmissivity and hydraulic head data are used in their work to condition the concentration moments.
To date, no attempt has been made to also analyze unsaturated transport with conditional stochastic methods. Recently, an exact formalism to predict the conditional moments of transient unsaturated flow (but not transport) in heterogeneous media has been suggested (Neuman and Loeven, 1994). In principle, all of the above approaches lend themselves for an analysis of the conditional plume and concentration moments under unsaturated conditions. The main difficulty encountered in the numerical (Monte Carlo) approach on one hand is the prohibitive amount of computation time needed to obtain just one steady-state velocity field from the conditional random input fields of Ks and ". The difficulty of the analytical approaches on the other hand is the derivation of covariance and crosscovariance functions necessary to obtain the conditional velocity moments.
The work presented in the previous chapters has overcome both limitations: An efficient numerical approach to compute steady-state unsaturated velocity fields, given a
This allows the efficient implementation of Monte Carlo simulations similar to the work by Smith and Schwartz (1981a,b). The stochastic moments of the unsaturated flow variables f=logK s, a=log" (log: natural logarithm), and soil water tension (head) h have also been derived (chapter 4; Yeh et al., 1985a,b). With these theoretical moments, Rubin's (1991a) analysis of conditional plume moments can easily be extended to unsaturated flow. His semi-analytical approach, however, is limited to small perturbations. In this chapter the (nonlinear) numerical Monte Carlo technique is applied to derive various conditional stochastic transport parameters without having to linearize either the flow or the transport equation. Linearization is only used to generate conditional input random fields f and a given data of either f, a, or h. For the conditioning, a geostatistical inverse method called cokriging is applied (Myers, 1982; Kitanidis and Vomvoris, 1983).
Conditional simulation of unsaturated transport distinguishes itself from the conditional simulation of saturated transport not so much in the principle of the approach as in the interdependencies between input and output RFVs. The same measurement data play a different role depending on whether they are applied to saturated or unsaturated flow. In unsaturated flow two independent parameters (or more - depending on the choice of the constitutive relationship) define the actual local hydraulic conductivity. The unsaturated flow problem is inherently nonlinear i.e., head and conductivity are interdependent unlike in the saturated case, where the conductivity is independent of the head. It is therefore expected that the data measured in the field and used to condition the stochastic analysis have a relevant information content that is distinctly different from the saturated case. Much of the usefulness of one type of measurement will depend on the availability of other types of measurements. Measuring, for example, either the saturated hydraulic conductivity, or the soil pore size distribution parameter, or the soil water tension each by itself should result in much less conditioning than the combined effect of all three measurements.
The main objective of this chapter is to investigate the role of both indirect information
distribution (monitoring network or sampling strategy) on the uncertainty of the conditional stochastic prediction of non-reactive solute transport under variably saturated conditions in isotropic and anisotropic soils. A second objective is to discriminate the effect that conditioning has on the various measures of solute transport. Besides analyzing the local concentration moments (Rubin (1991a, Zhang and Neuman, 1994b), the conditioning effects on the spatial plume moments (Dagan, 1982; 1984), on the arrival time (Smith and Schwartz, 1981; Zhang and Neuman, 1994c), and on the integrated breakthrough (Zhang and Neuman, 1994c) at an arbitrary compliance surface are examined. The structure of this chapter is as follows: The theoretical background and the implementation of the conditional unsaturated flow and transport model is described in sections 10.2 and 10.3. The hypothetical field soil sites for the conditioning study are a subset of the example soils described in the previous chapter and are selected in section
10.4. The impact of different sampling strategies or monitoring network designs on the reduction in the spatial moments of solute transport is investigated in sections 10.5 through 10.8.
Parameter uncertainty in the context of conditional simulation is addressed in section 10.9.
Section 10.10 discusses the role of the spatial plume moments as a measure to judge the effect of conditioning. In many applications involving environmental compliance at a particular location or surface, the variable of interest is the solute arrival time or breakthrough curve and not the spatial plume distribution. In the two sections 10.11 and 10.12, the effect of conditioning on several local and integrated measures of solute travel time is studied. The conditional mean concentration prediction at a highly conditioned site is compared to the deterministic inverse modeling prediction in section 10.13. The chapter closes with a summary and conclusion.
10.2 Theory of Conditional Simulation by Cokriging In chapter 3.3 a method was introduced to generate conditional random fields of the same random field variable (RFV) of which measurement data are available. In the context of
their dependent functions) that are not only conditioned on data from the same RFV (direct data e.g., random Ks fields conditioned on Ks data) but also or even exclusively on data from other physically related RFVs (indirect data e.g., random K s fields conditioned on head data). The conditional simulation technique used in this study is based on the same principles and numerical techniques as the conditional simulation algorithm described in chapter 3.3, equation 3-14 (Journel, 1974; Delhomme, 1979). The important difference is that cokriging rather than kriging is employed because of the multivariate nature of the problem. The kriging equations are given in (2-46) through (2-48). The cokriging equations are identical to the kriging equations (2-46), (2-47) in chapter 2 (Carr and Myers, 1985). However, the array of measured data X1 in (2-46) contains data from more than one RFV e.g., from saturated hydraulic conductivity data and head data, while the array of unknown data X2 is - as in kriging - comprised of data exclusively from one RFV e.g., the saturated hydraulic conductivity Ks. 712 in (2-46) is the weight matrix of the measured data X1 with respect to the estimate X2, which is either of the same RFV as X1 (kriging) or of a different RFV (cokriging). In either case the kriging weight matrix 712 is computed by solving the covariance matrix equation (2-47). For the cokriging case, the cross-covariances between two RFVs must be known to determine the matrices C11 and C12 in (2-47). Note, that the covariance and cross-covariance functions must be positive definite,
10.3 Conditional Monte Carlo Simulation: Methods 10.3.1 Principal Elements of the Monte Carlo Algorithm The principal procedures in the conditional Monte Carlo simulation are identical to those of the unconditional Monte Carlo simulation introduced in chapters 8 and 9 (see Figure 8.2). Conditional realizations of f and a are generated and a conditional approximate solution h is computed explicitly. The realization of each of these three RFVs is passed to MMOC2, which computes the steady-state soil water tension through a finite element solution of Richards equation, the flux field through a finite element solution of Darcy's law, and the transient solute transport by using a modified method of characteristics (chapter 5). The procedure is repeated for 150 to 300 realizations (see below). Finally, the appropriate statistical sample parameters are computed from the output of the Monte Carlo simulation. The only difference between the conditional simulations in this chapter and the unconditional simulations in chapter 9 is the algorithm used to generate the random field realizations f and a and the approximate solution h, all of which must be conditioned on measurement data, which are provided as input. As in the previous chapter, the statistical parameters describing the RFVs f and a are assumed to be known. The next sections discuss the actual implementation of the conditional random field generator and the conditional extension of the ASIGN method described in chapter 7. A flow chart of conditional ASIGNing and Monte Carlo simulation is shown in Figure 10.1.