Mathematical Geology

, Volume 31, Issue 7, pp 793–840 | Cite as

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

  • Ali Fadili
  • Rachid Ababou
  • Roland Lenormand


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 σ2 x (t) on a finite domain L, and temporal moments on a finite time scale T, e.g., the “mass variance” of arrival times σ2 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.

tracer transport random media stratified media multilayered media macrodispersion Lagrangian particle tracking 


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Copyright information

© International Association for Mathematical Geology 1999

Authors and Affiliations

  • Ali Fadili
    • 1
  • Rachid Ababou
    • 1
  • Roland Lenormand
    • 2
  1. 1.Institut de Mécanique des Fluides de ToulouseToulouseFrance
  2. 2.Institut Français du PétroleRueil-MalmaisonFrance

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