Dispersive Particle Transport: Identification of Macroscale Behavior in Heterogeneous Stratified Subsurface Flows

Dispersive mass transport processes in naturally heterogeneous geological formations (porous media) are investigated based on a particle approach to mass transport and on its numerical implementation using LPT3D, a Lagrangian Particle Tracking 3D code. We are currently using this approach for studying microscale and macroscale space–time behavior (advection, diffusion, dispersion) of tracer plumes, solutes, or miscible fluids, in 1,2,3-dimensional heterogeneous and anisotropic subsurface formations (aquifers, petroleum reservoirs). Our analyses are based on a general advection-diffusion model and numerical scheme where concentrations and fluxes are discretized in terms of particles. The advection-diffusion theory is presented in a probabilistic framework, and in particular, a numerical analysis is developed for the case of advective transport and rotational flows (numerical stability of the explicit Euler scheme). The remainder of the paper is devoted to the behavior of concentration, mass flux density, and statistical moments of the transported tracer plume in the case of heterogeneous steady flow fields, where macroscale dispersion occurs due to geologic heterogeneity and stratification. We focus on the case of perfectly stratified or multilayered media, obtained by generating many horizontal layers with a purely random transverse distribution of permeability and horizontal velocity. In this case, we calculate explicitly the exact mass concentration field C(x, t), mass flux density field f(x, t), and moments. This includes spatial moments and dispersion variance ² _x (t) on a finite domain L, and temporal moments on a finite time scale T, e.g., the mass variance of arrival times ² _T (x). The moments are related to flux concentrations in a way that takes explicitly into account finite space–time scales of analysis (time-dependent tracer mass; spatially variable flow through mass). The multilayered model problem is then used in numerical experiments for testing different ways of recovering information on tracer plume migration, dispersion, concentration and flux fields. Our analyses rely on a probabilistic interpretation that emerges naturally from the particle approach; it is based on spatial moments (particle positions), temporal moments (mass weighted arrival times), and probability densities (both concentrations and fluxes). Finally, as an alternative to direct estimations of the flux and concentration fields, we formulate and study the Moment Inverse Problem. Solving the MIP yields an indirect method for estimating the space–time distribution of flux concentrations based on observed or estimated moments of the plume. The moments may be estimated from field measurements, or numerically computed by particle tracking as we do here.     

On the condition number of covariance matrices in kriging, estimation, and simulation of random fields

The numerical stability of linear systems arising in kriging, estimation, and simulation of random fields, is studied analytically and numerically. In the state-space formulation of kriging, as developed here, the stability of the kriging system depends on the condition number of the prior, stationary covariance matrix. The same is true for conditional random field generation by the superposition method, which is based on kriging, and the multivariate Gaussian method, which requires factoring a covariance matrix. A large condition number corresponds to an ill-conditioned, numerically unstable system. In the case of stationary covariance matrices and uniform grids, as occurs in kriging of uniformly sampled data, the degree of ill-conditioning generally increases indefinitely with sampling density and, to a limit, with domain size. The precise behavior is, however, highly sensitive to the underlying covariance model. Detailed analytical and numerical results are given for five one-dimensional covariance models: (1) hole-exponential, (2) exponential, (3) linear-exponential, (4) hole-Gaussian, and (5) Gaussian. This list reflects an approximate ranking of the models, from best to worst conditioned. The methods developed in this work can be used to analyze other covariance models. Examples of such representative analyses, conducted in this work, include the spherical and periodic hole-effect (hole-sinusoidal) covariance models. The effect of small-scale variability (nugget) is addressed and extensions to irregular sampling schemes and higher dimensional spaces are discussed.     

Solution of stochastic groundwater flow by infinite series,and convergence of the one-dimensional expansion

This paper investigates analytical solutions of stochastic Darcy flow in randomly heterogeneous porous media. We focus on infinite series solutions of the steady-state equations in the case of continuous porous media whose saturated log-conductivity (lnK) is a gaussian random field. The standard deviation of lnK is denoted . The solution method is based on a Taylor series expansion in terms of parameter , around the value =0, of the hydraulic head (H) and gradient (J). The head solution H is expressed, for any spatial dimension, as an infinite hierarchy of Green's function integrals, and the hydraulic gradient J is given by a linear first-order recursion involving a stochastic integral operator. The convergence of the -expansion solution is not guaranteed a priori. In one dimension, however, we prove convergence by solving explicitly the hierarchical sequence of equations to all orders. An infinite-order stochastic solution is obtained in the form of a -power series that converges for any finite value of . It is pointed out that other expansion methods based on K rather than lnK yield divergent series. The infinite-order solution depends on the integration method and the boundary conditions imposed on individual order equations. The most flexible and general method is that based on Laplacian Green's functions and boundary integrals. Imposing zero head conditions for all orders greater than one yields meaningful far-field gradient conditions. The whole approach can serve as a basis for treatment of higher-dimensional problems.     

Linear and nonlinear input/output models for karstic springflow and flood prediction at different time scales

Karstic formations function as three-dimensional (3D) hydrological basins, with both surface and subsurface flows through fissures, natural conduits, underground streams and reservoirs. The main characteristic of karstic formations is their significant 3D physical heterogeneity at all scales, from fine fissuration to large holes and conduits. This leads to dynamic and temporal variability, e.g. highly variable flow rates, due to several concurrent flow regimes with several distinct response times. The temporal hydrologic response of karstic basins is studied here from an input/output, systems analysis viewpoint. The hydraulic behaviour of the basins is approached via the relationship between hydrometeorological inputs and outputs. These processes are represented and modeled as random, self-correlated and cross-correlated, stationary time processes. More precisely, for each site-specific case presented here, the input process is the total rainfall on the basin and the output process is the discharge rate at the outlet of the basin (karstic spring). In the absence of other data, these time processes embody all the available information concerning a given karstic basin. In this paper, we first present a brief discussion of the physical structure of karstic systems. Then, we formulate linear and nonlinear models, i.e. functional relations between rainfall and runoff, and methods for identifying the kernel and coefficients of the functionals (deterministic vs. statistical; error minimisation vs. polynomial projection). These are based mostly on Volterra first order (linear) or second order (nonlinear) convolution. In addition, a new nonlinear threshold model is developed, based on the frequency distribution of interannual mean daily runoff. Finally, the different models and identification methods are applied to two karstic watersheds in the french Pyrénées mountains, using long sequences of rainfall and spring outflow data at two different sampling rates (daily and semi-hourly). The accuracy of nonlinear and linear rainfall-runoff models is tested at three time scales: long interannual scale (20 years of daily data), medium or seasonal scale (3 months of semi-hourly data), and short scale or “flood scale” (2 days of semi-hourly data). The model predictions are analysed in terms of global statistical accuracy and in terms of accuracy with respect to high flow events (floods).     

Fast Upscaling of the Hydraulic Conductivity of Three-Dimensional Fractured Porous Rock for Reservoir Modeling

Mathematical Geosciences - A fast upscaling procedure for determining the equivalent hydraulic conductivity of a three-dimensional fractured rock is presented in this paper. A modified...     

Analyse multirésolution croisée de pluies et débits de sources karstiquesMultiresolution cross-analysis of rainfall rates and karstic spring runoffs

In order to quantify the quality of the rainfall/discharge relationship across time-scales, we propose the use of both orthogonal wavelet multiresolution analysis and cross-correlation analysis. By using the two techniques together, it is possible to show, scale-by-scale, the influence of the input to the system (rainfall) on the response (discharge) of the aquifer and also to relate these results to the internal structure of the aquifer and to the degree of organisation of the karst drainage. An application of this method to three Pyrenean karsts is also shown. To cite this article: D. Labat et al., C. R. Geoscience 334 (2002) 551–556.