We consider steady state unsaturated flow in bounded randomly heterogeneous soils under influence of random forcing terms. Our purpose is to predict pressure heads and fluxes and evaluate uncertainties associated with these predictions, without resorting to Monte Carlo simulation, upscaling or linearization of the constitutive relationship between unsaturated hydraulic conductivity and pressure head. Following Tartakovsky et al. [1999], by assuming that the Gardner model is valid and treating the corresponding exponent a as a random constant, the steady-state unsaturated flow equations can be linearized by means of the Kirchhoff transformation. This allows us develop exact integro-differential equations for the conditional first and second moments of transformed pressure head and flux. The conditional first moments are unbiased predictions of the transformed pressure head and flux, and the conditional second moments provide the variance and covariance associated with these predictions. The moment equations are exact, but they cannot be solved without closure approximations. We developed their recursive closure approximations through expansion in powers of σᵧ and σᵦ, the standard deviations of Y = lnK(s), and β = ln α, respectively, where K(s), is saturated hydraulic conductivity. Finally, we solve these recursive conditional moment equations to second-order in σᵧ and σᵦ, as well as second-order in standard deviations of forcing terms by finite element methods. Computational examples for unsaturated flow in a vertical plane, subject to deterministic forcing terms including a point source, show an excellent agreement between our nonlocal solutions and the Monte Carlo solution of the original stochastic equations using finite elements on the same grid, even for strongly heterogeneous soils.

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published_or_final_version / Electrical and Electronic Engineering / Master / Master of Philosophy

published_or_final_version / Civil and Structural Engineering / Master / Master of Philosophy

published_or_final_version / Civil Engineering / Doctoral / Doctor of Philosophy

The results of a study to model a circular mine shaft constructed in a time-dependent medium are presented. The construction sequence is considered as well as the time-dependent properties of the media around the shaft. The loads acting on the shaft liner are due to excavation of the shaft material and to the loads relieved from the media onto the liner. The results show the importance of considering the time-dependent behavior of media. The analysis was carried out using the Finite Element Method. Axisymmetric triangular and quadrilateral elements were used to model the medium, and axisymmetric shell elements were used to model the liner. The construction sequence was modeled by analyzing the system under small load increments where each load increment represents a construction step. The time behavior was modeled by using the initial strain method, which assigns a different strain value for each element in the medium. The strains are transferred to stresses and then to forces, and an incremental process is started to cover the time range desired. The results for a 400-foot shaft are shown, and changes in liner stresses were monitored as time passes. Different rock materials were modeled by using different constants in the creep law. Some materials showed significant changes in the results, and others did not. The liner horizontal displacement, and horizontal and vertical stresses increased when material constants for rock salt and anhydrite were used. Stresses in the elements adjacent to the liner decreased as time passed by, and some even went into a tensile stress site. A comparison between two solutions, one representing a multi-step construction sequence and another representing an instantaneous construction of the lined shaft, showed that liner stresses are much higher when the construction sequence is not modeled. This is due to the fact that when the excavation is modeled the forces representing the construction sequence are applied to the medium. In the other case, the forces are directly applied to the liner.

A computer algorithm is developed for clipping and capping N-sided polyhedra with arbitrary planes. The algorithm is then expanded to include the processing of general two and three dimensional geometric finite element model data. Data processing is included for the transformation of original model results to match the clipped and capped graphical display model. The algorithms are implemented in a FORTRAN program that may be directly substituted into the MOVIE.BYU/ARIZONA graphics system. The new SECTION program maintains all the functions of the original version while incorporating several major new features. These new features include the expansion of the geometric library to two and three dimensional elements and two new general forms for polygons and polyhedra. Another significant change in the processing is the implementation of the reentrant clipping and capping routines. This feature permits a previously clipped model to be clipped again and again by new and different clipping planes. The above features as well as enhanced input data schemes including a preliminary interface to NASTRAN are offered as a skeleton for future modifications. The major routines in the program have taken advantage of dynamic memory allocation via FORTRAN subroutine argument calls. Through this latter feature new capability can be concatenated to the end of the current processing in a prototype manner for rapid implementation and exploration.

No description available.

Spontaneous imbibition plays a very important role in the displacement mechanism of non-wetting fluid in naturally fractured reservoirs. We developed a new 2D two-phase finite element numerical model, as available commercial simulators cannot be used to model small-scale experiments with different boundary conditions as well as complex boundary conditions such as fractures and vugs. Starting with the basic equation of fluid flow, we derived the non-linear diffusion saturation equation. This equation cannot be put in weighted-integral weak variational form and hence Rayleigh-Ritz finite element method (FEM) cannot be applied. Traditionally, the way around it is to use higher order interpolation functions and use Galerkin FEM or reduce the differentiability requirement and use Mixed FEM formulation. Other FEM methods can also be used, but iterative nature of those methods makes them unsuitable for solving large-scale field problems. But if we truncate the non-linear terms and decouple the dependent variables, from the spatial as well as the temporal domains of the primary variable to solve them analytically, the non-linear FEM problem reduces to a simple weighted integral form, which can be put into its corresponding weak form. The advantage of using Rayleigh-Ritz method is that it has immediate effect on the computation time required to solve a particular problem apart from incorporating complex boundary conditions. We compared our numerical models with the analytical solution of this diffusion equation. We validated the FDM numerical model using X-Ray Tomography (CT) experimental data from the single-phase spontaneous imbibition experiment, where two simultaneously varying parameters of weight gain and CT water saturation were used and then went ahead and compared the results of FEM model to that of FDM model. A two-phase field size example was taken and results from a commercial simulator were compared to the FEM model to bring out the limitations of this approach.

No description available.