Rachid Ababou

RAINFALL–RUNOFF RELATIONS FOR KARSTIC SPRINGS. PART II: CONTINUOUS WAVELET AND DISCRETE ORTHOGONAL MULTIRESOLUTION ANALYSES

Abstract Karstic watersheds appear as highly non-linear and non-stationary systems. The main focus of this paper is a heuristic study of this non-stationarity using a time-scale localisation method called the wavelet transform. First, a mathematical overview of these analysis methods is given. The wavelet transform methods used here can be divided into two main parts: the continuous Morlet wavelet transform and the multiresolution orthogonal analysis. A statistical interpretation of the wavelet coefficients is also presented, introducing wavelet spectrum analyses (univariate and cross-wavelet analyses). These wavelet methods are applied to rainfall rates and runoffs measured at different sampling rates, from daily to half-hourly sampling rate. The karstic springs under study are located in the Pyrenees Mountains (Ariege, France) and in the Causses of Larzac (Aveyron, France). They are first applied to a pumping and a naturally intermittent runoff process, allowing the separation of different sub-processes. Wavelet analyses of rainfall rates and runoffs and wavelet rainfall–runoff cross-analyses also give meaningful information on the temporal variability of the rainfall–runoff relationship. In particular, this kind of analysis provides a simple interpretation of the distribution of energy between the different scales. Finally, it is demonstrated that wavelet transforms make possible a physical explanation of the temporal structure of the basin response to rainfall allowing discrimination between a rapid response and recharge due to the karst drainage system and a slower one corresponding to infiltration response.

Rainfall-runoff relations for karstic springs. Part I : convolution and spectral analyses

Karstic basins contain large reserves of subsurface water. In this paper, three karstic systems located in the Pyrenees Mountains (Ariege, France) are studied. Long records of rainfall and discharge rates for these karstic springs are available, sampled at different rates: daily, hourly and half-hourly. This study aims at illustrating and assessing the capabilities and limitations of linear black-box methods for analysing rainfall–runoff type relationships and reconstructing runoffs from rainfall rate data using such systems. In this study, precipitation and discharge rates are considered as two autocorrelated and cross-correlated stochastic processes. A linear and stationary rainfall–runoff model is adopted, which is used for identification and simulation purposes. Different versions are analysed, including a model based on a convolution integral between the precipitation rate P(τ) and a transfer function h(t−τ) which can be thought of as the unit impulse response of the system. It is shown that this linear stochastic model (i.e. the statistical version), although accurate in some respects, does not represent the hydraulic behaviour of the system very well during low flow episodes and floods. It is also shown that the use of Fourier analysis, alone, does not lead to a satisfactory reconstitution of observed runoff sequences. For these reasons, the use of non-linear random process input–output models based on Volterra integral series is proposed and discussed.

Rainfall–runoff relations for karstic springs: multifractal analyses

Karstic watersheds appear as highly as non-linear and non-stationary systems. The behaviour of karstic springs has been previously studied using non-linear simulation methods (Volterra expansion) and non-stationary analyses methods based on wavelet transforms. The main issue of karstic spring behaviour consists of the presence and the identification of characteristic time-scales. In order to highlight more precisely the scale-properties of the rainfall–runoff relations for karstic springs, the multifractal analysis is introduced. These methods are applied daily and half-hourly rainfall rates and runoffs measured on a three French karstic springs located in the Pyrenees Mountains (Ariege, France): Aliou, Baget and Fontestorbes. They are characterised by a variable development of the drainage systems. We have at our disposal long and uninterrupted series of data over period of several years, which constitute a high quality bank data. Multifractal analyses of both daily and half-hourly rainfall rates and runoffs give evident a scale-dependant behaviour. Effectively, it highlights the presence of different multifractal processes at each sampling rate. Using a universal class of multifractal models based on cascade multiplicative processes, the identified multifractal sub-processes are characterised by the classical parameters α and C1. All these results should lead to several improvements in karstic springflow simulation models.

Upscaling fractured heterogeneous media : Permeability and mass exchange coefficient

In order to optimize oil recuperation, to secure waste storage, CO2 sequestration and describe more precisely many environmental problems in the underground, we need to improve some homogenization methods that calculate petrophysical parameters. In this paper, we discuss the upscaling of fluid transport equations in fractured heterogeneous media consisting of the fractures themselves and a heterogeneous porous matrix. Our goal is to estimate precisely the fluid flow parameters like permeability and fracture/matrix exchange coefficient at large scale. Two approaches are possible. The first approach consists in calculating the large-scale equivalent properties in one upscaling step, starting with a single continuum flow model at the local scale. The second approach is to perform upscaling in two sequential steps: first, calculate the equivalent properties at an intermediate scale called the ”unit scale,” and, second, average the flow equations up to the large scale. We have implemented the two approaches and applied them to randomly distributed fractured systems. The results allowed us to obtain valuable information in terms of sizes of representative elementary volume associated to a given fracture distribution.

Analyse en ondelettes en hydrologie karstique. 2e partie : analyse en ondelettes croisée pluie-débit

Abstract A method of cross-analysis of rainfall-runoff time series based on wavelet transforms (continuous Morlet wavelet transform and orthogonal multiresolution analysis), is presented and applied to rainfall rates and spring outflow rates measured on karstic Pyrenean watersheds. Results are compared to classical cross-correlation and Fourier cross-spectral analyses. The multiscale wavelet method appears as powerful in the study of the joint temporal variability and non-stationarity of the rainfall-runoff relationship, and it yields more precise results than Fourier cross-spectral analysis.

Equivalent upscaled hydro-mechanical properties of a damaged and fractured claystone around a gallery (Meuse/Haute-Marne Underground Research Laboratory)

Abstract In this work, we present calculations and analyses of equivalent continuum (upscaled) coefficients describing the damaged, fissured and fractured claystone around an underground gallery. We focus here on mechanical and coupled hydro-mechanical properties of the damaged claystone (the upscaled Darcy permeability of the same claystone was studied in a previous paper focused on hydraulics without mechanical deformations). Concerning the geometric structure of the damaged clay stone around the cylindrical excavation, we use a hybrid 3D geometric model of fissuring and fracturing, comprising (a) a set of 10 000 statistical fissures with radially inhomogeneous statistics (size, thickness and density increasing near the wall), and (b) a deterministic set of large curved ‘chevrons’ fractures, periodically spaced along the axis of the drift according to a 3D chevron pattern. The hydro-mechanical coefficients calculated here are second- and fourth-rank tensors, which are displayed using ellipsoids. For simplicity, we also calculate equivalent isotropic coefficients extracted from these tensors: Youngs modulus (E), bulk modulus (K), Lamé shear modulus (μ), Poissons ratio (ν), Biot coefficient (B, stress–pressure coupling) and Biot modulus (M, pressure–fluid production coupling). All of these coefficients are affected by the degree of damage and fracturing, which increases near the wall of the gallery. Both 3D and ‘2D transverse’ distributions are analysed, on grids of 3D cubic voxels and 2D pixels, respectively. Global coefficients upscaled over the entire damaged and fractured zone are also analysed. Other types of averages are presented, for example, upscaled values over a cylindrical annular shell at various radial distances from the gallery wall. The relation to the degree of fracturing is discussed, including for instance the effect of fracturing on bulk and shear stiffnesses, and on the hydro-mechanical coupling coefficients of the damaged claystone.

Statistical Analyses of Pore Pressure Signals in Claystone During Excavation Works at the Mont Terri Underground Research Laboratory

In many countries (such as Belgium, Germany, France, Japan, Switzerland, and United Kingdom), deep argillaceous formations are considered as potential host rocks for geological disposal of high-level and intermediate-level long-lived radioactive wastes. Some of these countries are investigating the suitability of high compacted clay-rich rocks at depths down to around 500 m below the ground surface. The general disposal concept comprises a network of drifts and tunnels linked to the surface by shafts and ramps, all artificially ventilated. Research is ongoing in Underground Research Laboratories, like the Mont Terri site in the Swiss Jura, to assess and ensure the safety of the repositories for the full decay life of the radioactive waste, i.e. the capacity of the hypothetical repository toprevent the migration of radionuclides towards the biosphere.[...]

Comparisons of FV-MHMM with Other Finite Volume Multiscale Methods

Upscaling and multiscale methods in reservoir engineering remain a complicated task especially when dealing with heterogeneities. In this study, we focus on flow field problem with a Darcy’s equation considered at the fine scale. The main difficulty is then to obtain an accurate description of the flow behavior by using multiscale methods available in petroleum engineering. We cross-compare three of the main finite volume formulations: multiscale finite volume method (MsFv), multiscale restriction smoothed (MsRSB) and a new finite volume method, FV-MHMM. Comparisons are done in terms of accuracy to reproduce the fine scale behavior.

Optimal Reconstruction of 3D Fracture Networks (FEBEX Field Test, Grimsel Site, Swiss Alps)

This work deals with 3D fracture networks in granite rock masses. The main objective is to optimally reconstruct the near-field 3D fractured network around the FEBEX gallery located in the Grimsel Test Site (GTS) in Switzerland. The generated fractured medium must reproduce the non-uniform map of fracture traces left on the wall of this cylindrical gallery by using morphological and geological data such as fracture orientations and densities. The Simulated Annealing optimization technique was used to calibrate a Pareto law for the fracture size ‘distribution’ (Probability Density Function), minimizing the mean square error in trace length and trace chord length histograms from observed data. The most important fractures traversing the FEBEX gallery were treated deterministically, to better reproduce the hydraulic behavior in the near field. A satisfactory concordance was obtained between the trace maps of measured and generated 3D fractured medium.

Geometric and Statistical Modeling of Fractures in the 3D Disturbed Zone of a Claystone Around a Cylindrical Gallery (Meuse-Haute Marne Underground Research Laboratory, France)

This paper deals with the generation of rock fractures in 3D space around a cylindrical excavation (here a horizontal gallery or “drift”), based on geometric and probabilistic concepts. This research is conducted in the framework of studies on the isolation properties of a geological claystone repository for radioactive waste disposal (MeuseHaute Marne Underground Research Laboratory, France). The overall objective is to quantify equivalent “upscaled” hydro-mechanical properties of the disturbed porous rock, and to analyze the effect of fracturing on macroscale rock properties, e.g., equivalent permeability [1], mechanical stiffnesses, and hydro-mechanical couplings. The present work focuses on the mathematical and probabilistic representation of fractures, and their spatial distributions around the drift. The methodology is as follows. We use a mixed random/deterministic fracturing model, comprising: (i) a statistical set of 10 000 small planar joints with radially inhomogeneous statistics (size, aperture, and spatial density increasing near the wall), and (ii) a deterministic set of large curved “chevron” fractures, periodically spaced along the axis of the gallery according to a 3D chevron pattern (or 3D herringbone pattern). In particular, the spatial statistics of the small planar joints in 3D space were worked out using inhomogeneous Poisson process and other concepts from geometric probability. We also developed a new geometric model for the large curved chevron fractures, in terms of a deterministic parametric surface (a modified conoid). In this short paper, some of the resulting fracture patterns are shown graphically; the interested reader may refer to [1] for other mathematical and technical details.