Stochastic groundwater simulations for highly heterogeneous ...

MAMERN09: 3rd International Conference on Approximation Methods and Numerical Modelling in Environment and Natural Resources Pau (France), June 8-11, 2009

Stochastic groundwater simulations for highly heterogeneous porous media Jocelyne Erhel1 1 INRIA

Rennes Campus de Beaulieu, 35042 Rennes, France e-mail:[email protected]

Keywords: heterogeneous porous media, stochastic models, large scale computations, groundwater ressources. Abstract. The heterogeneity of natural geological formations, induced by slow geological processes during the aquifer formation, plays an important role in the contaminant migration in the groundwater. The main focus of this talk is to describe a stochastic numerical model allowing to quantify the impact of the heterogeneity on the contaminant dispersion. This model relies on uncertainty quantification methods and on parallel large scale computations.

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Introduction

Groundwater is an important part of drinking water as high as 63 % in France. It also represents 96 % of the water intake points compared to 4 % for surface water. As opposed to surface water and to highly karstic geologies, groundwater does not flow in well-identified open streams but in the pores and the fractures of the rock. Groundwater is highly dependent on the percentage of pores (porosity), their size and connectivity (permeability) and on the biochemical reactivity. These parameters (porosity, permeability, reactivity) are highly variable. Variability can be huge not only at the transition between geological units, like between sand and clay, but also within units. In sedimentary structures, the deposition modes of grains induce permeability to shift over three to four orders of magnitude [7]. The geological complexity addresses a challenge for the management of groundwater. Even tougher questions arise when dealing with groundwater quality because a more precise physical and chemical model is required. Tracking a contaminant precisely for remediation requires a fine model of the water velocity field. Heterogeneity is a key factor for groundwater models. The classical approaches used until around 1990 rely on homogenization methods. These methods require to define a homogenization scale larger than the scale of all geological structures. However, it is not the case in many geological media. Paleo-river channels were organized like today in highly structure trees that cannot be mistaken as homogeneous [5]. Underground media very often do not show any obvious homogenization scale and moreover display characteristic structures on a widely-scattered 1

Jocelyne Erhel

scale range. Therefore, our methodology is based on numerical models counting for all geological scales, thus with physical domains discretized at a very fine scale. Particle transport is highly sensitive to the detailed heterogeneity structure [9]. Because of the widely-scattered medium heterogeneity, velocity values of groundwater span several orders of magnitudes. Particles are advected by the velocity field along the flow lines and can switch from flow lines because of diffusion and hydrodynamic dispersion and their spreading creates a so-called plume. Even with simple heterogeneity models like a finitely-correlated permeability field, the development and characterization of the plume remain a much debated topic [2, 7, 8]. First the sampling of the full velocity field is highly dependent on the flow lines and second the plume is very sensitive to injection conditions [1]. Characterization of transport laws thus requires simulating the particle transport on very long times and in turn in very large domains. Thanks to our software PARADIS [6], we could for the first time simulate the full development of the plume and determine with no ambiguity asymptotic dispersion in 2D domains [3, 4]. We work now on numerical simulations in 3D domains.

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Numerical model

In this talk, we consider a 2D or 3D porous medium defined by a rectangle or parallelepiped. The porous medium, assumed isotropic, is characterized by a random hydraulic conductivity field K, which follows a stationary log-normal probability distribution Y = ln(K), defined by a mean m and a covariance function C given by C(r) = σ 2 exp(− λr ), where σ 2 is the variance of the log hydraulic conductivity, r represents the separation distance between two points and λ denotes the correlation length scale. We assume here that the medium is saturated and that the water density is constant. Classical laws governing the steady flow in a porous medium are mass conservation and Darcy law:  v = −K∇h, (1) ∇.(v) = q where V and h are respectively the Darcy velocity and the hydraulic head. Boundary conditions are homogeneous Neumann on upper and lower sides and Dirichlet h = 0 on left side, Dirichlet h = 1 on right side. The system should not be too much elongated in order to avoid border effects. In fact the velocity field close to the lateral boundaries is highly influenced by the no-flow boundary condition. In the porous medium, we inject an inert solute, which does not exchange mass with the solid matrix. The two mechanisms, governing the inert solute transport, are the advection and the diffusion. This type of solute migration is generally described by the advection - dispersion equation, given by ∂(c) + ∇.(cv) − ∇.(D∇c) = 0 ∂t

(2)

where c is the solute concentration in the aqueous phase and D is the dynamic dispersion tensor. In order to overcome the border effects, the inert solute is injected at a distance of the left side. The numerical simulations are divided into three main steps: we first generate the domain with a random permeability field; then we solve the steady flow equations and compute the velocity field; finally, we solve the transport equation. The objective is to find out the longitudinal and transversal spatial spreadings, which are the second moments of mass distribution. In the case of a random permeability field, the estimation of macro dispersions has to be made statistically because V and c are random variables. 2

Stochastic groundwater simulations for highly heterogeneous porous media

In order to achieve this goal, a usual Monte Carlo method is used. Each Monte Carlo simulation corresponds to a random draw of the permeability field K and a set of flow and transport equations. Solving flow then transport problems for each sample gives a set of samples of the random variable c used for estimating the moments. Since we use approximation methods to solve the partial differential equations, we introduce several errors in the computations. Moreover, Monte-Carlo method gives only an approximation of the moments. Also, flow computations are linear and involve a large sparse matrix. Accuracy is directly related to the condition number of the matrix which depends on the variance of the permeability field. We run numerical experiments to analyze these errors and to validate our computations. Since we must choose a small size of the discretization grid and must deal with large computational domains, the matrix size is very large. In order to cope with high memory and CPU requirements, we develop a parallel software PARADIS. We run numerical experiments to analyze performances and scalability.

REFERENCES [1] T. Le Borgne, J.-R. de Dreuzy, P. Davy, and O. Bour. Characterization of the velocity field organization in heterogeneous media by conditional correlations. Water Reosurces Research, 43, 2007. [2] G. Dagan, A. Fiori, and I. Jankovic. Flow and transport in highly heterogeneous formations: 1. conceptual framework and validity of first-order approximations. Water Resources Research, 9, 2003. [3] J.-R. de Dreuzy, A. Beaudoin, and J. Erhel. Asymptotic dispersion in 2D heterogeneous porous media determined by parallel numerical simulations. Water Resource Research, 43(W10439, doi:10.1029/2006WR005394), 2007. [4] J.-R. de Dreuzy, A. Beaudoin, and J. Erhel. Reply to comment by a. fiori et al. on ”asymptotic dispersion in 2d heterogeneous porous media determined by parallel numerical simulations”. Water Resources Research, 44, W06604:doi:10.1029/2008WR007010, 2008. [5] Ghislain de Marsily, Frdric Delay, J. Gonalvez, Philippe Renard, Vanessa Teles, and S. Violette. Dealing with spatial heterogeneity. Hydrogeology Journal, 13:161–183, 2005. [6] J. Erhel, J.-R. de Dreuzy, A. Beaudoin, E. Bresciani, and D. Tromeur-Dervout. A parallel scientific software for heterogeneous hydrogeology. In Ismail H. Tuncer, Ulgen Gulcat, David R. Emerson, and Kenichi Matsuno, editors, PARCFD’2007 conference proceedings. Springer, to appear. [7] L. Gelhar. Stochastic Subsurface Hydrology. Engelwood Cliffs, New Jersey, 1993. [8] P. Salandin and V. Fiorotto. Solute transport in highly heterogeneous aquifers. Water Resources Research, 34:949–961, 1998. [9] B. Zinn and C. F. Harvey. When good statistical models of aquifer heterogeneity go bad: A comparison of flow, dispersion, and mass transfer in connected and multivariate gaussian hydraulic conductivity fields. Water Resources Research, 39, 2003.

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