Analytical modeling of flow and transport in highly heterogeneous anisotropic porous formations

Abstract

Flow and transport through three dimensional anisotropic heterogeneous for- mations are investigated using analytical methods. The aim of the present dissertation is to extend, to the relevant case of anisotropic formations, the Self-Consistent method which was developed in the past for isotropic forma- tions (Dagan et al., 2003; Fiori et al., 2003; Fiori et al. 2006). An Eulerian and a Lagrangian approach is employed, respectively, for the analysis of the properties of the ow and the advective transport of a plume of inert solute. The results obtained by the Self-Consitent (SC) method are compared with those obtained by the well-known First-Order (FO) approach, that, in the last 30 years, has led to several useful analytical solutions on ow and trans- port statistics. However, such solutions su er from the strong limitation of weak heterogeneity, i.e. they are formally valid for 2Y 1. Accurate nu- merical simulations (NS) free from model approximations are employed to test the semi-analytical method. Uniform ow of constant mean velocity takes place in a random hydraulic conductivity eld, which is modeled by Multi-Indicator (MI) model alterna- tive to the Multi-Gaussian employed in the FO method. The mean ow di- rection is aligned with the long axis of anisotropy. The MI formation is made up of a collection of non-overlapping oblate spheroids of random hydraulic conductivity Ki, placed in a matrix of hydraulic conductivity K0. The SC argument is used to calculate approximate analytical solutions for the ve- locity eld and the breakthrough curves. The methodology is applied to a medium with inclusions of normally distributed logpermeability Y = lnK. The thesis is organized into three sections, investigating the e ective prop- erties of the medium, the ow, and the transport . The main contribution to the study of the e ective conductivity lies in the comparisons between the results of the analytical models with the numerical simulations. It is found that the SC solution is very accurate for a broad range of the values of heterogeneity and anisotropy (f). The FO approximation provides a good estimation of the Kef in its range of applicability ( 2Y 1), but strongly deviations from the numerical simulations are observed for large 2Y . The extension of the rst order results to high heterogeneous media is not possible by using the exponential Landau- Matheron conjecture. The study of the ow is conducted through the evaluations of the sta- tistical moments of the velocity. Closed-form expressions for both average velocity and its variance were obtained, while higher order moments (up to fourth), velocity autocorrelation and integral scales are computed by numer- ical quadratures. If anisotropy is absent, the solutions converge to those developed for isotropic media by Fiori et al. (2003). The main e ect of anisotropy is to increase the variance of the longitudinal velocity and to re- duce the variances of the transverse and vertical components. Surprisingly, the growth of the vertical velocity variance with heterogeneity ( 2Y ) is not always monotonous. FO solutions generally overpredict the growth trends of the velocity variance with heterogeneity. Analysis of skewness and kurtosis suggests that the velocity probability density function (pdf) is generally far from Gaussian, except for weakly heterogeneous formations. The deviation from Gaussianity increases with increased anisotropy for all velocity compo- nents. The velocity autocorrelation function is weakly dependent on 2Y , as con rmed by the numerical simulations. Transport is analyzed in terms of breakthrough curve (BTC) of the so- lute, identical to the traveltime distribution, at a control plane at distance x from the source. The global traveltime t is evaluated through the SC approximation by summing residence times in the inclusions ti. The anisotropic formulation of SC model is not able to reproduce nu- merical simulations. The SC method proves its ability to provide adequate estimation of the BTCs only in low anisotropic domains. These results are not in accordance with the ones concerning the e ective conductivity and the velocity statistics.