Gilles Porel

Interpretation of out-diffusion experiments on crystalline rocks using random walk modeling

Matrix diffusion in saturated rocks with very low permeability is one of the major mechanisms of solute transport. Laboratory out-diffusion experiments on rock samples may provide an estimate of the bulk diffusion coefficient. However, numerous results have shown that this average parameter does not really depict the complex mechanism of diffusion as a function of the internal heterogeneity of crystalline rocks. Two-dimensional images of the porosity distribution in a granite sample were obtained by impregnation with a radioactive resin and autoradiography. Some examples based on these images and synthetic images were used to perform numerical simulations of out-diffusion using two different random walk methods. The simulated shapes of the out-diffusion curves depend on the spatial distribution of the porosity and on the pore connectivity with the border of the sample. Such relations might explain the multiple nested slopes or the convex shapes often observed on real experimental curves.

Simulation of solute transport in discrete fracture networks using the time domain random walk method

The time domain random walk (TDRW) method has been developed for simulating solute transport in discrete fracture networks. The following transport processes have been considered: advective transport in fractures, hydrodynamic dispersion along the fracture axis, sorption reactions on the fracture walls and decay reactions. The TDRW method takes advantage of both random walk and particle-tracking methods. It allows for the one-step calculation of the particle residence time in each bond of the network while avoiding mass balance problems at fracture intersections with contrasted dispersion coefficients. The accuracy of the TDRW method has been addressed by means of synthetic test problems into single fractures and into a 2D discrete fracture network. In each case, simulated and theoretical results compare very well.

Predicting solute transport in heterogeneous media from results obtained in homogeneous ones: an experimental approach

Abstract The aim of this paper is to show that the hydrodispersive transport parameters of a solute in a heterogeneous medium can be predicted using parameters determined by several experiments with the different homogeneous media that constitute together the heterogeneous one. Elementary rules for predicting these parameters are defined. Three types of experiments were carried out. (1) Homogeneous medium transport experiments in 1-D columns. They were fitted with the help of a new numerical particle tracking method, including advection-dispersion and exchange between mobile and immobile water with first-order kinetics. (2) Layered heterogeneous media. The experiments were simulated by the numerical model without fitting, using the parameters of each of the layers of the homogeneous media included in the heterogeneous one. The results prove that, in such layered media, the global transport problem can be considered as the convolution of elementary transports in homogeneous blocks. We will therefore conclude that the reservoir simulation methods which represent the heterogeneous reservoir as a random set of elementary homogeneous blocks can be used to represent transport if each elementary block is given its appropriate hydrodispersive parameters. (3) Homogeneous mixtures of two pure components. The experiments produced three interesting results. The immobile water volume is the direct linear combination of its equivalent in the two components, weighted by the volumetric fraction of the component in the mixture. The dispersivity of the medium also evolves linearly, but is affected by an initial sharp rise due to a strong increase in local flow path heterogeneities whenever a “foreign body” is introduced into a pure component. The exchange rate between mobile and immobile water is also perturbed by mixing. In a binary mixture of one component with dual porosity and another one without, the exchange rate decreases with the increase of the volumetric fraction of the medium without dual porosity. It seems that, even if it does not modify the volumes of immobile and mobile water, it reduces the surface/volume ratio of the solid medium and consequently the exchange rate. Mixing rules could be adopted in order to modify the values of the hydrodispersive parameters that need to be introduced for the elementary blocks of heterogeneous reservoirs.

An Approach to Transport in Heterogeneous Porous Media Using the Truncated Temporal Moment Equations: Theory and Numerical Validation

In the last decade, the characterization of transport in porous media has benefited largely from numerical advances in applied mathematics and from the increasing power of computers. However, the resolution of a transport problem often remains cumbersome, mostly because of the time-dependence of the equations and the numerical stability constraints imposed by their discretization. To avoid these difficulties, another approach is proposed based on the calculation of the temporal moments of a curve of concentration versus time. The transformation into the Laplace domain of the transport equations makes it possible to develop partial derivative equations for the calculation of complete moments or truncated moments between two finite times, and for any point of a bounded domain. The temporal moment equations are stationary equations, independent of time, and with weaker constraints on their stability and diffusion errors compared to the classical advection–dispersion equation, even with simple discrete numerical schemes. Following the complete theoretical development of these equations, they are compared firstly with analytical solutions for simple cases of transport and secondly with a well-performing transport model for advective–dispersive transport in a heterogeneous medium with rate-limited mass transfer between the free water and an immobile phase. Temporal moment equations have a common parametrization with transport equations in terms of their parameters and their spatial distribution on a grid of discretization. Therefore, they can be used to replace the transport equations and thus accelerate the achievement of studies in which a large number of simulations must be carried out, such as the inverse problem conditioned with transport data or for forecasting pollution hazards.

Transport in a 2-D saturated porous medium : a new method for particle tracking

A new method for solving the transport equation based on the management of a large numbe of particles in a discretized 2-D domain is presented. The method uses numerical variables to represent the number of particles in a given mesh and is more complex than the 1-D problem. The first part of the paper focuses on the specific management of particles in a 2-D problem. The method also would be valid for three dimensions as long as the medium can be modeled similar to a layered system. As the particles are no longer tracked individually, the algorithm is fast and does not depend on the number of particles present. The numerical tests show that the method is nearly numerical dispersion free and permits accurate calculations even for simulations of low-concentration transport. Because each mesh is considered as a closed system between two successive time steps, it is easy to add adsorption phenomenon without any problem of numerical stability. The model is tested under conditions that are extremely demanding for its operating mode and gives a good fit to analytical solutions. The conditions in which it can be used to best advantage are discussed.

A dual flowing continuum approach to model denitrification experiments in porous media colonized by biofilms.

We present a modeling exercise of solute transport and biodegradation in a coarse porous medium widely colonized by a biofilm phase. Tracer tests in large laboratory columns using both conservative (fluorescein) and biodegradable (nitrate) solutes are simulated by means of a dual flowing continuum approach. The latter clearly distinguishes concentrations in a flowing porous phase from concentrations conveyed in the biofilm. With this conceptual setting, it becomes possible to simulate the sharp front of concentrations at early times and the flat tail of low concentrations at late times observed on the experimental breakthrough curves. Thanks to the separation of flow in two phases at different velocities, dispersion coefficients in both flowing phases keep reasonable values with some physical meaning. This is not the case with simpler models based on a single continuum (eventually concealing dead-ends), for which inferred dispersivity may reach the unphysical value of twice the size of the columns. We also show that the behavior of the dual flowing continuum is mainly controlled by the relative fractions of flow passing in each phase and the rate of mass transfer between phases. These parameters also condition the efficiency of nitrate degradation, the degradation rate in a well-seeded medium being a weakly sensitive parameter. Even though the concept of dual flowing continuum appears promising for simulating transport in complex porous media, its inversion onto experimental data really benefits from attempts with simpler models providing a rough pre-evaluation of parameters such as porosity and mean fluid velocity in the system.

Coupling of the time domain random walk method with the finite fragment method to simulate flow and transport in 1-D heterogeneous media

Abstract A new method to simulate flow and solute transport in 1-D heterogeneous media is presented. This method integrates the time domain random walk method (TDRW) and the finite fragment method (FFM). The TDRW, similar in concept to the classical random walk method, calculates the arrival time of a particle cloud at a given location and provides the solute breakthrough curve. The FFM, which can be seen as an enhancement of the finite difference scheme, allows a heterogenous media to be divided into homogeneous zones to which the TDRW can be applied. The main advantage of the resulting coupling is that the restrictions can be avoided on the space increments and the time steps which exist with the classical methods of finite differences and random walk. In a homogeneous zone of soil, the breakthrough curve can be calculated directly at a given distance with a reasonable number of particles. A 1-D heterogeneous domain to be simulated is then split into homogeneous zones in which the hydraulic heads and the velocities are calculated by the FFM and at the end of which the temporal solute distributions are calculated by the TDRW. A few hundred particles are generally sufficient to clearly define the breakthrough curve. Comparisons with analytical and numerical solutions and experimental data shows the reliability and advantages of this new method.

Continuous-time model identification of wells interaction on the Hydrogeological Experimental Site of Poitiers

In hydrogeology, estimating aquifer permeability is an important issue. This can be useful in understanding the flow of pollutants from one area of an aquifer to another. For this aquifer analysis sake, the Hydrogeological Experimental Site of Poitiers (France) covering a limestone aquifer is an appropriate instrumented test bed enabling measurement of hydraulic responses of a series of observation wells due to a step-type pumping out excitation at a given well. Given the input-output data, black-box continous-time modeling is quite a straight forward process as shown in this paper. The aim is then to be able to use the identified parameters to classify the different wells according to how sensitive they are to the one having been excited. A correlation between black-box parameters and hydrogeological ones is then established.

3D ERT Imaging Of The Fractured-Karst Aquifer Underlying The Experimental Site Of Poitiers (France): Comparing Wenner-Schlumberger, Pole-Dipole And Hybrid Arrays

Electrical Resistivity Tomography (ERT) surveys were undertaken to investigate the Dogger Limestone fractured-karst aquifer at the Hydrogeological Experimental Site (HES) of Poitiers, France. Three-dimensional resistivity imaging was obtained from full inversion of combined 2D ERT data collected along five parallel 470 m long profiles with a 50 m line spacing. A 3D block measuring 515 x 203 m in size with a maximum depth of 100 m was surveyed. Dogger Limestone occurs at a depth ranging between 30 and 120 m and is overlain by argillaceous limestone. This paper compares the imaging obtained from different array sequences. Calibration of the 3D resistivity block with well logs indicates that: the Wenner-Schlumberger (WS) array shows the tendency to enhance layering, to locate bodies at a shallower depth and to laterally extend them; the Pole-Dipole (PD) array shows larger lateral heterogeneities, more compact and vertically extended bodies and poor data fitting; the hybrid array sequence, obtained by the combination of WS and PD array sequences, despite a poor data fitting, similar to PD, shows a better correlation with respect to well log results. In this setting, the hybrid array sequence shows better imaging, due to the combination of the large vertical resolution of WS, large lateral resolution and penetration depth of PD. It allows passing through the thick, low resistivity shallow layer. Indeed, the results are affected by the occurrence of the shallow, 30 m thick, low resistivity argillaceous limestone that reduced the investigation depth as revealed by synthetic datasets modelling and sensitivity analysis. Modelling also revealed that the occurrence of the argillaceous limestone led to a severe underestimate of the Dogger Limestone resistivity values with respect to well resistivity logs; it also allowed verifying the detectability limits when investigating shallow karst limestone intervals located at depths of up to 50 m.

Modèle d'aide à la gestion des eaux souterraines (MAGES). 1. Théorie du modèle numérique de transport des contaminants

MAGES is software for forecasting pollution hazards of groundwater which is in the process of development at INRS-Eau (Canada). The main distinctive feature of the model is the use of stationary truncated temporal moment equations instead of the classical time dependent advection-dispersion equation to solve the transport of contaminants. The aim of this work is to describe the theory of truncated temporal moment equations and to show how the curves of the concentration versus time can be calculated from temporal moments. The discrete method used to solve the equations and its stability is also discussed.