Project Completion Report OWRT Project A- 076 -ARIZ The work upon which this publication is based was supported in part by the United States Department of the Interior as authorized under the Water Research Act of 1964, as amended. / Constant head borehole infiltration tests are widely used for the in situ evaluation of saturated hydraulic conductivities of unsaturated soils above the water table. The formulae employed in analyzing the results of such tests disregard the fact that some of the infiltrating water may flow under unsaturated conditions. Instead, these formulae are based on various approximations of the classical free surface theory which treats the flow region as if it were fully saturated and enclosed within a distinct envelope, the so- called "free surface." A finite element model capable of solving free surface problems is used to examine the mathematical accuracy of the borehole infiltration formulae. The results show that in the hypothetical case where unsaturated flow does not exist, the approximate formulae are reasonably accurate within a practical range of borehole conditions. To see what happens under conditions closer to those actually encountered in the field, the effect of unsaturated flow on borehole infiltration is investigated by means of two different numerical models: A mixed explicit - implicit finite element model, and a mixed explicit -implicit integrated finite difference model. Both of these models give nearly identical results; however, the integrated finite difference model is considerably faster than the finite element model. The relatively low computational efficiency of the finite element scheme is attributed to the large humber of operations required in order to reevaluate the conductivity (stiffness) matrix at each iteration in this highly nonlinear saturated -unsaturated flow problem. The saturated -unsaturated analysis demonstrates that the classical free surface approach provides a distorted picture of the flow pattern in the soil. Contrary to what one would expect on the basis of this theory, only a finite region of the soil in the immediate vicinity of the borehole is saturated, whereas a significant percentage of the flow takes place under unsaturated conditions. As a consequence of disregarding unsaturated flow, the available formulae may underestimate the saturated hydraulic conductivity of fine grained soils by a factor of two, three, or more. Our saturated -unsaturated analysis leads to an improved design of borehole infiltration tests and a more accurate method for interpreting the results of such tests. The analysis also shows how one can predict the steady state rate of infiltration as well as the saturated hydraulic conductivity from data collected during the early transient period of the test.

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This dissertation develops an analytical-numerical approach to deterministically predict the space-time evolution of concentrations in heterogeneous geologic media conditioned on measurements of hydraulic conductivities (transmissivities) and/or hydraulic heads. Based on the new conditional Eulerian-Lagrangian transport theory by Neuman, we solve the conditional transport problem analytically at early time, and express it in pseudo-Fickian form at late time. The stochastically derived deterministic pseudo-Fickian mean concentration equation involves a conditional, space-time dependent dispersion tensor. The latter not only depends on properties of the medium and the velocity but also on the available information, and can be evaluated numerically along mean "particle" trajectories. The transport equation lends itself to accurate solution by standard Galerkin finite elements on a relatively coarse grid. This approach allows computing without using Monte Carlo simulation and explicitly the following: Concentration variance/covariance (uncertainty), origin of detected contaminant and associated uncertainty, mass flow rate across a "compliance surface", cumulative mass release and travel time probability distribution across this surface, uncertainty associated with the latter, second spatial moment of conditional mean plume about its center of mass, conditional mean second spatial moment of actual plume about its center of mass, conditional co-variance of plume center of mass, and effect of non-Gaussian velocity distribution. This approach can also account for uncertainty in initial mass and/or concentration when predicting the future evolution of a plume, whereas almost all existing stochastic models of solute transport assume the initial state to be known with certainty. We illustrate this approach by considering deterministic and uncertain instantaneous point and nonpoint sources in a two-dimensional domain with a mildly fluctuating, statistically homogeneous, lognormal transmissivity field. We take the unconditional mean velocity to be uniform, but allow conditioning on log transmissivity and hydraulic head data. Conditioning renders the velocity field statistically nonhomogeneous with reduced variances and correlation scales, renders the predicted plume irregular and non-Gaussian, and generally reduces both predictive dispersion and uncertainty.

The main aim of this study is to develop a suitable method for the calibration and validation of mathematical models of large and complex aquifer systems. Since the calibration procedure depends on the nature of the model to be calibrated and since many kinds of models are used for groundwater, the question of model choice is broached first. Various aquifer models are critically reviewed and a table to compare them as to their capabilities and limitations is set up. The need for a general calibration method for models in which the flow is represented by partial differential equations is identified from this table. The calibration problem is formulated in the general mathematical framework as the inverse problem. Five types of inverse problems that exist in modeling aquifers by partial differential equations are identified. These are, to determine (1) parameters, (2) initial conditions, (3) boundary conditions, (4) inputs, and (5) a mixture of the above. Various methods to solve these inverse problems are reviewed, including those from fields other than hydrology. A new direct method to solve the inverse problem (DIMSIP) is then developed. Basically, this method consists of transforming the partial differential equations of flow to algebraic equations by substituting in them the values of the various derivatives of the dependent variable (which may be hydraulic pressure, chemical concentration or temperature). The parameters are then obtained by formulating the problem in a nonlinear optimization framework. The method of sequential unconstrained minimization is used. Spline functions are used to evaluate the derivatives of the dependent variable. Splines are functions defined by piecewise polynomial arcs in such a way that derivatives up to and including the order one less than the degree of polynomials used are continuous everywhere. The natural cubic splines used in this study have the additional property of minimum curvature which is analogous to minimum energy surface. These and the derivative preserving properties of splines make them an excellent tool for approximating the dependent variable surfaces in groundwater flow problems. Applications of the method to both a test situation as well as to real-world data are given. It is shown that the method evaluates the parameters, boundary conditions and inputs; that is, solves inverse problem type V. General conditions of heterogeneity and anisotropy can be evaluated. However, the method is not applicable to steady flows and has the limitation that flow models in which the parameters are functions of the dependent variable cannot be calibrated. In addition, at least one of the parameters has to be preassigned a value. A discussion of uncertainties in calibration procedures is given. The related problems of model validation and sampling of aquifers are also discussed.

Multiple aquifer systems similar to those that exist at coal gasification sites are complicated groundwater situations. In these types of systems, the aquifers are separated by aquitards through which interaction between aquifers can occur. The movement of the products of combustion into the coal seam and adjacent aquifers is a serious problem of interest. This dissertation presents two-dimensional finite element models for water flow and water quality in multiple aquifer systems. These models can be applied for general problems as well as the problems associated with the burned cavities in coal gasification sites. The Galerkin weightedresidual method is used in both models. Eight-noded isoparametric elements are used. Spatial numerical integration is performed using Gaussian quadrature. A weighted finite difference scheme is used, in both of them, for time integration. The two models are written in FORTRAN V for the CDC CYBER 175. They are applicable to one- or two-dimensional problems involving steady-state or transient flow. Each aquifer can have different initial conditions and boundary conditions. Boundary conditions, pumping rates, and the recharge can be specified as a function of time. The output of the flow program-nodal heads and velocity components is used as an input to the quality program. The numerical models were validated for simple problems that have available analytical solutions.

The water level in an open well tapping a confined formation is influenced by natural forces including the solid Earth tide (SET) and atmospheric pressure variation (APV). The spectral method is used to derive an analytical solution for well response to both the random and the periodic components of the combined SET and APV (CSA) forcings. Previously posed models for the individual SET and APV forcings are subsets of this more general model. An inverse theory and an algorithm are developed in order to provide improved results when using such models to estimate the hydraulic parameters associated with a given formation. A complex vector estimation criterion is used in developing a nonlinear, Gauss-Marquardt estimation algorithm. When compared to previous methods of fitting modulus and phase, the complex vector estimation methodology has less bias and variance, and is more robust. An examination of the response surface of the estimation criterion reveals that storativity (S) is relatively non-unique, and thus is not considered in the context of the parameter estimation problem. However, since there is little correlation between transmissivity (T) and S estimators, a good estimate for T is still possible independent of having knowledge of S. An estimate of T is possible only if the data contain sufficient information so that the analysis occurs within an identifiability window, which is defined with respect to the dimensionless transmissivity of the system. The CSA estimation methodology is compared to individual SET and APV schemes. The CSA scheme gives the greatest probability that sufficient information is contained in a data record so that T is identifiable. The results of applications to synthetic data indicate that the OEA scheme gives a T estimate with the most precision, and also that it requires collecting fewer observations.

The mechanisms responsible for overland transport of faecal pathogens, particularly Cryptosporidium oocysts, from animal sources to water bodies are not fully understood. Surface properties of microbes, such as electrostatic charge and hydrophobicity, are thought to contribute to their aggregation and attachment to solid surfaces. There is conflicting evidence that methods used to purify Cryptosporidium oocysts from faecal material may affect the oocyst surface, leading to biased conclusions from transport studies. By studying oocyst surface properties, aggregation and soil attachment, this thesis addressed whether oocyst purification methods influence overland transport studies, and whether oocysts are likely to be associated with particles during transport. When using the microbial adhesion to hydrocarbon (MATH) assay with octane, oocyst hydrophobicity was shown to be method and isolate dependent, with oocysts displaying moderate to high hydrophobicity in 0.01 M KNO3. There was no observed attachment, however, to the hydrophobic octyl-SepharoseTM bead ligands when using the same suspension solution. Oocyst age did not appear to influence their hydrophobicity. A small but statistically significant proportion of oocysts displayed a net negative surface charge as observed by their attachment to an anion exchange ligand (DEAE). There was no difference in hydrophobicity or surface charge observed between purified oocysts and oocysts that had been extracted without the use of harsh chemicals and solutions with dehydrating properties. Purified oocysts did not aggregate at pH values between 3.3 and 9.0, nor in solutions lower than 0.59 M in ionic strength at a pH 2.7 which is approaching the reported isoelectric point of oocysts. This finding suggests that oocysts may not form aggregates under general environmental conditions. The association of purified oocysts with soil particles was observed in settling columns. Attachment to soil particles was not conclusive since the settling of the soil particles may have entrained single oocysts. Nonetheless, approximately 27% of oocysts were estimated to be unbound to soil or associated with small soil particles. Hence models for oocyst overland transport should consider a significant fraction as single entities or associated with soil particles less than about 3 μm in size.

This dissertation considers the effect of measuring randomly varying local hydraulic conductivities K(x) on one's ability to predict steady state flow within a bounded domain, driven by random source and boundary functions. That is, the work concerns the prediction of local hydraulic head h(x) and Darcy flux q(x) by means of their unbiased ensemble moments (h(x))(κ) and (q(x))(κ) conditioned on measurements of K(x). These predictors satisfy a deterministic flow equation in which (q(x))(κ) = -(κ)(x)∇(h(x))(κ) + r(κ)(x) where κ(x) is a relatively smooth unbiased estimate of K(x) and r(κ)(x) is a "residual flux." A compact integral expression is derived for r(κ)(x) which is rigorously valid for a broad class of K(x) fields, including fractals. It demonstrates that (q(x))(κ) is nonlocal and non-Darcian so that an effective hydraulic conductivity does not generally exist. It is shown analytically that under uniform mean flow the effective conductivity may be a scalar, a symmetric or a nonsymmetric tensor, or a set of directional scalars which do not form a tensor. For cases where r(κ)(x) can neither be expressed nor approximated by a local expression, a weak (integral) approximation (closure) is proposed, which appears to work well in media with pronounced heterogeneity and improves as the quantity and quality of K(x) measurements increase. The nonlocal deterministic flow equation can be solved numerically by standard methods; the theory here shows clearly how the scale of grid discretization should relate to the scale, quantity and quality of available data. After providing explicit approximations for the prediction error moments of head and flux, some practical methods are discussed to compute κ(x) from noisy measurements of K(x) and to calculate required second moments of the associated estimation errors when K(x) is log normal. Nonuniform mean flow is studied by conducting high resolution Monte Carlo simulations of two dimensional seepage to a point sink in statistically homogeneous and isotropic log normal K(x) fields. These reveal the existence of radial effective hydraulic conductivity which increases from the harmonic mean of K(x) near interior and boundary sources to geometric mean far from such sources for σ^2/Υ (the variance of ln K) at least as large as 4. They suggest the possibility of replacing r(κ)(x) by a local expression at distances of few conditional integral scales from the interior and boundary sources. Special attention is paid to the "art" of random field generation, and comparisons are made between four alternative methods with five different random number generators.

The mechanisms responsible for overland transport of faecal pathogens, particularly Cryptosporidium oocysts, from animal sources to water bodies are not fully understood. Surface properties of microbes, such as electrostatic charge and hydrophobicity, are thought to contribute to their aggregation and attachment to solid surfaces. There is conflicting evidence that methods used to purify Cryptosporidium oocysts from faecal material may affect the oocyst surface, leading to biased conclusions from transport studies. By studying oocyst surface properties, aggregation and soil attachment, this thesis addressed whether oocyst purification methods influence overland transport studies, and whether oocysts are likely to be associated with particles during transport. When using the microbial adhesion to hydrocarbon (MATH) assay with octane, oocyst hydrophobicity was shown to be method and isolate dependent, with oocysts displaying moderate to high hydrophobicity in 0.01 M KNO3. There was no observed attachment, however, to the hydrophobic octyl-SepharoseTM bead ligands when using the same suspension solution. Oocyst age did not appear to influence their hydrophobicity. A small but statistically significant proportion of oocysts displayed a net negative surface charge as observed by their attachment to an anion exchange ligand (DEAE). There was no difference in hydrophobicity or surface charge observed between purified oocysts and oocysts that had been extracted without the use of harsh chemicals and solutions with dehydrating properties. Purified oocysts did not aggregate at pH values between 3.3 and 9.0, nor in solutions lower than 0.59 M in ionic strength at a pH 2.7 which is approaching the reported isoelectric point of oocysts. This finding suggests that oocysts may not form aggregates under general environmental conditions. The association of purified oocysts with soil particles was observed in settling columns. Attachment to soil particles was not conclusive since the settling of the soil particles may have entrained single oocysts. Nonetheless, approximately 27% of oocysts were estimated to be unbound to soil or associated with small soil particles. Hence models for oocyst overland transport should consider a significant fraction as single entities or associated with soil particles less than about 3 μm in size.