Quantifying and Accounting for Uncertainty

This chapter deals with a wide range of issues with a common theme: coping with uncertainty. To this end, we look at the sources of uncertainty and the types of errors we need to deal with. We then explore methods for identifying these errors and for incorporating them into our predictions. This chapter extends our discussion on these topics in chapter 1, the discussion in chapter 3 on estimation under conditions of uncertainty, and image simulation using MC techniques. A comprehensive treatment of uncertainty needs to address two different types of errors. The first type is the model error, which arises from incorrect hypotheses and unmodeled processes (Gaganis and Smith, 2001), for example, from poor choice of governing equations, incorrect boundary conditions and zonation geometry, and inappropriate selection of forcing functions (Carrera and Neuman, 1986b). The second type of error is parameter error. The parameters of groundwater models are always in error because of measurement errors, heterogeneity, and scaling issues. Ignoring the effects of model and parameter errors is likely to lead to errors in model selection, in the estimation of prediction uncertainty, and in the assessment of risk. Parameter error is treated extensively in the literature: once a model is defined, it is common practice to quantify the errors associated with estimating its parameters (cf. Kitanidis and Vomvoris, 1983; Carrera and Neuman, 1986a, b; Rubin and Dagan, 1987a,b; McLaughlin and Townley, 1996; Poeter and Hill, 1997). Modeling error is well recognized, but is more difficult to quantify. Let us consider, for example, an aquifer which appears to be of uniform conductivity. Parameter error quantifies the error in estimating this conductivity. Modeling error, on the other hand, includes elusive factors such as missing a meandering channel somewhere in the aquifer. This, in essence, is the difficulty in determining modeling error; parameter error can be roughly quantified based on measurements if one assumes that the model is correct, but modeling error is expected to represent all that the measurements and/or the modeler fail to capture. To evaluate model error, the perfect model needs to be known, but this is never possible.

Vadose Zone, Part I : Characterization and Flow Processes

Many of the principles guiding stochastic analysis of flow and transport processes in the vadose zone are those which we also employ in the saturated zone, and which we have explored in earlier chapters. However, there are important considerations and simplifications to be made, given the nature of the flow and of the governing equations, which we explore here and in chapter 12. The governing equation for water flow in variably saturated porous media at the smallest scale where Darcy’s law is applicable (i.e., no need for upscaling of parameters) is Richards’ equation (cf. Yeh, 1998)

The Lagrangian Picture, Part II : Models and Applications of the Lagrangian Approach to Solute Transport

This chapter explores applications of the ideas which we explored in chapter 7 and developed in chapter 9. It presents tools for applications and demonstrates some applications through field and numerical studies. This section discusses the statistics of particle displacements in heterogeneous flow fields. Some applications of these statistics are discussed in sections 7.1 and 9.2. We will show later in this chapter that they can also be used for computing dispersion coefficients (sections 10.2-10.5) and travel time statistics (section 10.6), which will be used later (sections 10.7) for modeling reactive transport, and for conditional modeling of transport (section 10.8). An experimental database for the Lagrangian velocity VL (9.1) is difficult but not impossible to obtain (cf. Wilson and Linderfelt, 1994, Woodbudy and Rubin, 2000). But the next best thing for an insight of the nature of VL is to compute its statistics either numerically or analytically. The emphasis in this section will be on low-order analytical approximations. Although some results are of limited applicability, they are of a fundamental nature in terms of the insight they provide.

An Overview of Stochastic Tools for Modeling Transport of Tracers in Heterogeneous Media

Spatial variability and the uncertainty in characterizing the flow domain play an important role in the transport of contaminants in porous media: they affect the pathlines followed by solute particles, the spread of solute bodies, the shape of breakthrough curves, the spatial variability of the concentration, and the ability to quantify any of these accurately. This chapter briefly reviews some basic concepts which we shall later employ for the analysis of solute transport in heterogeneous media, and also points out some issues we shall address in the subsequent chapters. Our exposition in chapters 8-10 on contaminant transport is built around the Lagrangian and the Eulerian approaches for analyzing transport. The Eulerian approach is a statement of mass conservation in control volumes of arbitrary dimensions, in the form of the advection-dispersion equation. As such, it is well suited for numerical modeling in complex flow configurations. Its main difficulties, however, are in the assignment of parameters, both hydrogeological and geochemical, to the numerical grid blocks such that the effects of subgrid-scale heterogeneity are accounted for, and in the numerical dispersion that occurs in advection-dominated flow situations. Another difficulty is in the disparity between the scale of the numerical elements and the scale of the samples collected in the field, which makes the interpretation of field data difficult. The Lagrangian approach focuses on the displacements and travel times of solute bodies of arbitrary dimensions, using the displacements of small solute particles along streamlines as its basic building block. Tracking such displacements requires that the solute particles do not transfer across streamlines. Since such mass transfer may only occur due to pore-scale dispersion, Lagrangian approaches are ideally suited for advection-dominated situations. Let us start by considering the displacement of a small solute body, a particle, as a function of time. “Small” here implies that the solute body is much smaller than the characteristic scale of heterogeneity. At the same time, to qualify for a description of its movement using Darcy’s law, the solute body also needs to be larger than a few pores. The small dimension of the solute body ensures that it moves along a single streamline and that it does not disintegrate due to velocity shear.

Moments of the Flow Variables, Part II : The Effective Conductivity

Many applications require primary information such as average fluxes as a prelude to more complex calculations. In water balance calculations one may be interested only in the average fluxes. For both cases the concept of effective conductivity is useful. The effective hydraulic conductivity is defined by where the angled brackets denote the expected value operator. The local flux fluctuation is defined by the difference qi(x) — (qi(x)). Its statistical properties as well as those of the velocity will be investigated in chapter 6. To qualify as an effective property in the strict physical sense, Kef must be a function of the aquifer’s material properties and not be influenced by flow conditions such as the head gradient and boundary conditions (Landauer, 1978). Our goal in this chapter is to explore the concept of the effective conductivity Kef and to relate it to the medium’s properties under as general conditions as possible. Additionally, we shall explore the conditions where this concept is irrelevant and applicable, the important issue being that Kef is defined in an ensemble sense, but for applications we need spatial averages. Several methods for deriving Kef will be described below. The general approach for defining Kef includes the following steps. First, H is defined as an SRF and is expressed with the aid of the flow equation in terms of the hydro-geological SRFs (conductivity, mostly) and the boundary conditions. The H SRF is then substituted in Darcy’s law and an expression in the form equivalent to (5.1) is sought. If and only if the coefficient in front of the mean head gradient is not a function of the flow conditions will it qualify as Kef. The derivation of the effective conductivity employs the flow equation. In steady-state incompressible flow, for example, Laplace’s equation is employed. Solutions derived under Laplace’s equation are applicable, under appropriate conditions, for other physical phenomena governed by the same mathematical model. For example, the electrical field in steady state is also described by Laplace’s equation.

Estimation and Simulation

Two important applications of the SRF concept developed in chapter 2 are point estimation and image simulation. Point estimation considers the SRF Z at an unsampled location, x0, and the goal is to get an estimate for z at x0 which is physically plausible and is optimal in some sense, and to provide a measure of the quality of the estimate. The goal in image simulation is to create an image of Z over the entire domain, one that not only is in agreement with the measurements at their locations, but also captures the correlation pattern of z. We start by considering a family of linear estimators known as kriging. Its appeal is in its simplicity and computational efficiency. We then proceed to discuss Bayesian estimators and will show how to condition estimates on “hard” and “soft” data, and we shall conclude by discussing a couple of simple, easy-to-implement image simulators. One of the simulators presented can be downloaded from the Internet. Linear regression aims at estimating the attribute z at x0: z0 = z(x0), based on a linear combination of n measurements of z: zi = z(xi), i = 1 , . . . ,n. The estimator of z(x0) is z*0, and it is defined by What makes this estimator “linear” is the exclusion of powers and products of measurements. However, nonlinearity may enter the estimation process indirectly, for example, through nonlinear transformation of the attribute. The challenge posed by(3.1) is to determine optimally the n interpolation coefficients λi, and the shift coefficient λ0. The actual estimation error is z*0 - z0; it is unknown, since z0 is unknown, and so no meaningful statement can be made about it. As an alternative, we shall consider the set of all equivalent estimation problems: in this set we maintain the same spatial configuration of measurement locations, but allow for all the possible combinations, or scenarios, of z values at these locations, including x0. We have replaced a single estimation problem with many, but we have improved our situation since now we know the actual z value at x0 and this will allow a systematic approach.

Fundamentals of Stochastic Site Characterization

A few schematic representations of heterogeneous geological formations are depicted in figure 2.1. These and similar types of images, often encountered in geological site investigations, demonstrate the complexity of subsurface geology. Each image shows several blocks, all nearly homogeneous in terms of some physical or chemical property z, but with possibly strong variations in properties in between. The patterns of spatial variability shown in these images are difficult to capture in the absence of a large number of measurements adequately distributed over the domain. However, the high cost of procuring such databases renders deterministic image reconstruction an elusive goal, one which is largely abandoned in favor of approaches which try to formulate the laws which govern the pattern of spatial variability. These models are known as space random functions (or SRFs, for short). Besides deconstructing complex spatial variability patterns into simple, quantitative laws, SRFs can be used to construct images which have these spatial laws in common, and to estimate z at specific locations. Constructing a SRF for a spatially variable z is based on analysis of z measurements. The goal of that analysis is to reduce the ensemble of measurements to a few useful statistics which capture mathematically the pattern of spatial variability. A few statistics were found to be very useful for exposing the laws of variability, and will be explored in detail. The data analysis includes single (univariate), two-point (bivariate), and multipoint (multivariate) analyses. Univariate analysis focuses on the same-point statistical behavior of the variable z, regardless of the behavior of its neighbors. It answer questions such as “What is the average value of Z?” or “What are the chances that Z will exceed 1000 units?” Bivariate and multivariate analyses explore the simultaneous behavior of z at two or more locations. They provide tools which answer questions such as “What is the likelihood that a region of high permeability will stretch between the contamination source and some environmentally sensitive target?” or “What is the probability to observe z larger than 1500 units at x given that a value of 1300 was measured 5 meters away from x?”

Introduction

Stochastic hydrogeology is the study of hydrogeology using physical and probabilistic concepts. It is an applied science because it is oriented toward applications. Its goal is to develop tools for analyzing measurements and observations taken over a sample region in space, and extract information which can then be used for evaluating and modeling the properties of physical processes taking place in this domain, and make risk-qualified predictions of their outcome. By invoking probabilistic concepts to deal with problems of physics, stochastic hydrogeology joins a well-established tradition followed in mining (Matheron, 1965; David, 1977; Journel and Huijbregts, 1978), turbulence (Kolmogorov, 1941; Batchelor, 1949), acoustics (Tatarski, 1961), atmospheric science (Lumley and Panofsky, 1964), composite materials and electrical engineering (Beran, 1968; Batchelor, 1974), and of course statistical mechanics. Stochastic hydrogeology broadens the scope of the deterministic approach to hydrogeology by considering the last as an end member to a wide spectrum of states of knowledge, stretching from deterministic knowledge at one end all the way to maximum uncertainty at the other, with a continuum of states, representing varying degrees of uncertainty in the hydrogeological processes, in between. It provides a formalism for addressing this continuum of states systematically. The departure from the confines of determinism is an important and intuitively appealing paradigm shift, representing the maturing of hydrogeology from an exploratory into an applied discipline. Deterministic knowledge of a site’s hydrogeology is a state we rarely, if ever, find ourselves in, although from a fundamental point of view there is no inherent element of chance in the hydrogeological processes. For example, we know that mass conservation is a deterministic concept, and we are also confident that Darcy’s law works under conditions which are fairly well understood. However, the application of these principles involves a fair amount of conjecture and speculation, and hence when dealing with real-life applications, determinism exists only in the fact that uncertainty and ambiguity are unavoidable, and might as well be studied and understood. The other end of the spectrum is where uncertainty is the largest. Generally speaking, two types of uncertainty exist: intrinsic variability and epistemic uncertainty.

Vadose Zone, Part II : Transport

This chapter is an extension of our discussion on transport in chapters 7 to 10. Our goal here is to explore a few aspects of the transport problem which are unique to variably saturated soils. The heterogeneity of soils affects transport of solutes in the vadose zone in different ways. It leads to irregular and hard-to-predict spreading of the solutes. The solutes may be channeled through highly conductive flow channels where diffusion plays only a minor role. This may lead to concentrations which are high and travel times which are fast compared to what one may anticipate by assuming that the medium is homogeneous. Evidence for such behavior was found in field experiments (cf. Wierenga et al., 1991; Ellsworth et al., 1991; Ritsema et al., 1998; Sassner et al., 1994) and in large-scale laboratory experiments (Dagan et al., 1991). Hence, the effects of heterogeneity must be recognized and modeled. The effects of heterogeneity can be modeled by employing the stochastic concepts discussed in earlier chapters. The approach for modeling contaminant transport which is the least restrictive in terms of assumptions introduced is the MC simulation. This approach will be reviewed briefly in section 12.1. Modeling of the mean concentration along our discussion in chapter 8 is computationally less demanding compared to MC simulations, yet is less informative since the concentration in the field can hardly be expected to be equal to its expected value. Applications along that line are limited since deriving the macrodispersion coefficients needed for such an undertaking is difficult. Nonetheless, we shall discussed this approach in section 12.2, for the insight into the transport processes it provides. A few simple models are available for gravitational flow through shallow depths. These methods are of course limited in applications, yet they are less demanding in terms of data requirements and the computational efforts involved. Such methods are the focus of the last section in this chapter. The concept of MC simulation was discussed in earlier chapters.

The Lagrangian Picture, Part I : Fundamentals of the Lagrangian Approach to Solute Transport

This chapter explores the principles of the Lagrangian approach to solute transport, with an emphasis on the dispersive action of the spatial variability of the velocity field. We start by developing the tools for characterizing the displacement of a single, small solute particle that will subsequently be used for characterization of the concentration’s variability and uncertainty, and we continue with a discussion of the stochastic description of solute travel times and fluxes. The principles presented in this chapter will be employed in chapter 10 to derive tools for applications such as macrodispersion coefficients, solute travel time moments, the moments of the solute fluxes and breakthrough curves, and transport of reactive solutes. As has been observed in many field studies and numerical simulations, the motion of solute bodies in geological media is complex, making the geometry of the solute bodies hard to predict. Furthermore, the concentration varies erratically, sometimes by orders of magnitude, over very short distances. The variability of the velocity field plays a significant role in shaping this complex geometry, and makes it impossible to characterize the concentration field deterministically. The alternatives we will pursue include characterizing the concentration through its moments such as the expected value and variance, and other descriptors of transport such as solute fluxes and travel times. This line was pursued in chapter 8 using the Eulerian framework. In this chapter we pursue this line from the Lagrangian perspective. Applications of these concepts are presented in chapter 10. Let us consider the displacement of a marked solute particle over time.