Analysis of flow and solute transport problem in porous media are affected by uncertainty inbuilt both in boundary conditions and spatial variability in system parameters. The experimental investigation reveals that the parameters may vary in various scales by several orders. These affect the solute plume characteristics in field-scale problem and cause uncertainty in the prediction of concentration. The main focus of the present thesis is to analyze the probabilistic behavior of solute concentration in three dimensional(3-D) heterogeneous porous media. The framework for the probabilistic analysis has been developed using perturbation approach for both spectral based analytical and finite element based numerical method. The results of the probabilistic analysis are presented either in terms of solute plume characteristics or prediction uncertainty of the concentration. After providing a brief introduction on the role of stochastic analysis in subsurface hydrology in chapter 1, a detailed review of the literature is presented to establish the existing state-of-art in the research on the probabilistic analysis of flow and transport in simple and complex heterogeneous porous media in chapter 2. The literature review is mainly focused on the methods of solution of the stochastic differential equation. Perturbation based spectral method is often used for probabilistic analysis of flow and solute transport problem. Using this analytical method a nonlocal equation is solved to derive the expression of the spatial plume moments. The spatial plume moments represent the solute movement, spreading in an average sense. In chapter 3 of the present thesis, local dispersivity if also assumed to be random space function along with hydraulic conductivity. For various correlation coefficients of the random parameters, the results in terms of the field scale effective dispersivity are presented to demonstrate the effect of local dispersivity variation in space. The randomness of local dispersivity is found to reduce the effective fields scale dispersivity. The transverse effective macrodispersivity is affected more than the longitudinal effective macrodispersivity due to random spatial variation of local dispersivity. The reduction in effective field scale longitudinal dispersivity is more for positive correlation coefficient. The applicability of the analytical method, which is discussed in earlier chapter, is limited to the simple boundary conditions. The solution by spectral method in terms of statistical moments of concentration as a function of space and time, require higher dimensional integration. Perturbation based stochastic finite element method(SFEM) is an alternative method for performing probabilistic analysis of concentration. The use of this numerical method for performing probabilistic analysis of concentration. The use of this numerical method is non common in the literature of stochastic subsurface hydrology. The perturbation based SFEM which uses FEM for spatial discretization of the steady state flow and Laplace transform for the solute transport equation, is developed in chapter 4. The SFEM is formulated using Taylor series of the dependent variable upto second-order term. This results in second-order accurate mean and first-order accurate standard deviation of concentration. In this study the governing medium properties viz. hydraulic Conductivity, dispersivity, molecular diffusion, porosity, sorption coefficient and decay coefficient are considered to vary randomly in space. The accuracy of results and computational efficiency of the SFEM are compared with Monte Carle Simulation method(MCSM) for both I-D and 3-D problems. The comparison of results obtained hby SFEM and MCSM indicates that SFEM is capable in providing reasonably accurate mean and standard deviation of concentration. The Laplace transform based SFEM is simpler and advantageous since it does not require any stability criteria for choosing the time step. However it is not applicable for nonlinear transport problems as well as unsteady flow conditions. In this situation, finite difference method is adopted for the time discretization. The first part of the Chapter 5, deals with the formulation of time domain SFEM for the linear solute transport problem. Later the SFEM is extended for a problem which involve uncertainty of both system parameters and boundary/source conditions. For the flow problem, the randomness in the boundary condition is attributed by the random spatial variation of recharge at the top of the domain. The random recharge is modeled using mean, standard deviation and 2-D spatial correlation function. It is observed that even for the deterministic recharge case, the behavior of prediction uncertainty of concentration in the space is affected significantly due to the variation of flow field. When the effect of randomness of recharge condition is included, the standard deviation of concentration increases further. For solute transport, the concentration input at the source is modeled as a time varying random process. Two types of random source at the source is modeled as a time varying random process. Two types of random source condition are considered, firstly the amount of solute mass released at uniform time interval is random and secondly the source is treated as a Poission process. For the case of multiple random mass releases, the stochastic response function due to stochastic system is obtained by using SFEM. Comparing the results for the two type of random sources, it sis found that the prediction uncertainty is more when it is modeled as a Poisson process. The probabilistic analysis of nonlinear solute transport problem using MCSM is often requires large computational cost. The formulation of the alternative efficient method, SFEM, for nonlinear solute transport problem is presented in chapter 6. A general Langmuir-Freundlich isotherm is considered to model the equilibrium mass transfer between aqueous and sorbed phase. In the SFEM formulation, which uses the Taylor Series expansion, the zeroth-order derivatives of concentration are obtained by solving nonlinear algebraic equation. The higher order derivatives are obtained by solving linear equation. During transport, the nonlinear sorbing solutes is characterized by sharp solute fronts with a traveling wave behavior. Due to this the prediction uncertainty is significantly higher. The comparison of accuracy and computational efficiency of SFEM with MCSM for I-D and 3-D problems, reveals that the performance of SFEM for nonlinear problem is good and similar to the linear problem. In Chapter 7, the nonlinear SFEM is extended for probabilistic analysis of biodegrading solute, which is modeled by a set of PDEs coupled with nonlinear Monod type source/sink terms. In this study the biodegradation problem involves a single solute by a single class of microorganisms coupled with dynamic microbial growth is attempted using this methods. The temporal behavior of mean and standard deviation of substrate concentration are not monotonic, they show peaks before reaching lower steady state value. A comparison between the SFEM and MCSM for the mean and standard deviation of concentration is made for various stochastic cases of the I-D problem. In most of the cases the results compare reasonably well. The analysis of probabilistic behavior of substrate concentration for different correlation coefficient between the physical parameters(hydraulic conductivity, porosity, dispersivity and diffusion coefficient) and the biological parameters(maximum substrate utilization rate and the coefficient of cell decay) is performed. It is observed that the positive correlation between the two sets of parameters results in a lower mean and significantly higher standard deviation of substrate concentration. In the previous chapters, the stochastic analysis pertaining to the prediction uncertainty of concentration has been presented for simple problem where the system parameters are modeled as statistically homogeneous random. The experimental investigations in a small watershed, point towards a complex in geological substratum. It has been observed through the 2-D electrical resistivity imaging that the interface between the layers of high conductive weathered zone and low conductive clay is very irregular and complex in nature. In chapter 8 a theoretical model based on stochastic approach is developed to stimulate the complex geological structure of the weathered zone, using the 2-D electrical image. The statistical parameters of hydraulic conductivity field are estimated using the data obtained from the Magnetic Resonance Sounding(MRS) method. Due to the large complexity in the distribution of weathered zone, the stochastic analysis of seepage flux has been carried out by using MCSM. A batter characterization of the domain based on sufficient experimental data and suitable model of the random conductivity field may help to use the efficient SFEM. The flow domain is modeled as (i) an unstructured random field consisting of a single material with spatial heterogeneity, and (ii) a structured random field using 2-D electrical imaging which is composed of two layers of different heterogeneous random hydraulic properties. The simulations show that the prediction uncertainty of seepage flux is comparatively less when structured modeling framework is used rather than the unstructured modeling. At the end, in chapter 9 the important conclusions drawn from various chapters are summarized.

L'injection de tensioactifs est une méthode très appliquée dans le domaine de la récupération améliorée des hydrocarbures. Cependant, son efficacité repose sur la capacité de ces agents chimiques à mobiliser l'huile résiduelle en diminuant la tension interfaciale entre l'huile et l'eau. Des modèles à l'échelle du réservoir calculent l'efficacité de la récupération d'huile résiduelle par injection de solutions contenant des tensioactifs. Les mécanismes physiques pris en compte dans les modélisations font intervenir la physico-chimie du système roche-fluide et une courbe globale donnant la saturation résiduelle en huile en fonction du nombre capillaire (courbe de désaturation capillaire). Cette donnée est majeure dans le calcul de l'efficacité de récupération d'huile par injection de solutions de tensioactifs. En effet la mobilisation de l'huile résiduelle laissée en place après injection d'eau n'est possible qu'en augmentant considérablement le nombre capillaire. La prédiction de l'efficacité d'un procédé chimique de récupération passe par la compréhension, à l'échelle du pore, du processus de mobilisation des ganglions d'huile suivant la structure poreuse et le nombre capillaire. L'objet de cette thèse est de caractériser la récupération d'huile tertiaire en fonction du nombre capillaire dans diverses roches mouillables à l'eau. Ces courbes permettront de quantifier l'effet de la microstructure, les hétérogénéités du milieu poreux et diverses propriétés pétrophysiques sur la récupération d'huile. Cette thèse permettra aussi de caractériser les différents mécanismes d'action de tensioactifs sur la mobilisation d'huile résiduelle dans le milieu poreux. L'expérimentation par tomographie RX est utilisée. La tomographie RX permettra de caractériser les courbes de désaturation capillaire à l'échelle de Darcy et visualiser localement le déplacement d'huile résiduelle à travers les milieux poreux. Des essais d'écoulement diphasique sous micro-CT permettront d'observer in-situ et d'étudier les interfaces eau/huile et leurs évolutions en 3D au sein du milieu poreux en fonction du nombre capillaire. / Oil recovery by surfactant injection is related to oil-water interfacial tension and rock properties through the capillary number. In the modeling of oil recovery by surfactant injection, fluid flow physical mechanisms are represented through the capillary desaturation curve (CDC). This curve is central in the evaluation of oil recovery efficiency. In order to mobilize residual oil trapped after waterflooding by capillary forces, chemical EOR rely on increasing capillary number to extremely high values. The mechanisms governing oil release can be described at the pore scale where the balance of capillary and viscous forces is achieved. This description will help to predict the efficiency of surfactant based EOR processes by taking into account the porous geometry and topology, the physico-chemical properties of the fluids and the different phase interaction. The objective of this work is to characterize capillary desaturation curves for various strongly water-wet sandstones. These curves will be used to study the relationship between tertiary oil recovery and the pore structure, porous media heterogeneity and petrophysicals properties. The other aim of this work is to map the different mechanisms of oil recovery by surfactant injection. Experiments under X-Ray tomography are proposed. X-Ray tomography will be applied to characterize capillary desaturation curve at Darcy scale and to visualise the two phase flow saturation after injection. Pore scale experiments based on X-Ray micro-tomography imaging are performed to describe the different mechanisms of oil mobilization.