*Yoram Rubin*

- Published in print:
- 2003
- Published Online:
- November 2020
- ISBN:
- 9780195138047
- eISBN:
- 9780197561676
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780195138047.003.0013
- Subject:
- Earth Sciences and Geography, Oceanography and Hydrology

In this chapter we consider the diffusive and mixing properties of fluids moving in heterogeneous porous media by means of Eulerian velocity fields. We shall discuss here the basic principles of ...
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In this chapter we consider the diffusive and mixing properties of fluids moving in heterogeneous porous media by means of Eulerian velocity fields. We shall discuss here the basic principles of advection and dispersion in heterogeneous media from an Eulerian perspective. The common theme for the various methods we shall explore is the treatment of the concentration as an SRF. The first approach models the concentration through its statistical moments, such as the expected value and the variance, and computes them through a set of differential equations, an equation for each statistical moment. The second approach is MC based, and it computes an ensemble of physically plausible realizations of the concentration field, which can then be used for computing the statistics of the concentration, and from there the probability of events such as the concentration exceeding a threshold value at specified locations and times. Let us emphasize that stochastic modeling of contaminant transport is not just about the effects of media heterogeneity. There is also room for stochastic modeling in uniform media if there is uncertainty with regard to the media's parameters. Prediction with parameter error is discussed in chapter 13. Let us consider the case of passive solutes which are injected into a fluid body at rest.
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In this chapter we consider the diffusive and mixing properties of fluids moving in heterogeneous porous media by means of Eulerian velocity fields. We shall discuss here the basic principles of advection and dispersion in heterogeneous media from an Eulerian perspective. The common theme for the various methods we shall explore is the treatment of the concentration as an SRF. The first approach models the concentration through its statistical moments, such as the expected value and the variance, and computes them through a set of differential equations, an equation for each statistical moment. The second approach is MC based, and it computes an ensemble of physically plausible realizations of the concentration field, which can then be used for computing the statistics of the concentration, and from there the probability of events such as the concentration exceeding a threshold value at specified locations and times. Let us emphasize that stochastic modeling of contaminant transport is not just about the effects of media heterogeneity. There is also room for stochastic modeling in uniform media if there is uncertainty with regard to the media's parameters. Prediction with parameter error is discussed in chapter 13. Let us consider the case of passive solutes which are injected into a fluid body at rest.

*Yoram Rubin*

- Published in print:
- 2003
- Published Online:
- November 2020
- ISBN:
- 9780195138047
- eISBN:
- 9780197561676
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780195138047.003.0010
- Subject:
- Earth Sciences and Geography, Oceanography and Hydrology

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 ...
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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.
Less

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.

*Yoram Rubin*

- Published in print:
- 2003
- Published Online:
- November 2020
- ISBN:
- 9780195138047
- eISBN:
- 9780197561676
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780195138047.003.0006
- Subject:
- Earth Sciences and Geography, Oceanography and Hydrology

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 ...
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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.
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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.