Characterisation of the transmissivity field of a fractured and karstic aquifer, Southern France

International audienceGeological and hydrological data collected at the Terrieu experimental site north of Montpellier, in a confined carbonate aquifer indicates that both fracture clusters and a major bedding plane form the main flow paths of this highly heterogeneous karst aquifer. However, characterising the geometry and spatial location of the main flow channels and estimating their flow properties remain difficult. These challenges can be addressed by solving an inverse problem using the available hydraulic head data recorded during a set of interference pumping tests.We first constructed a 2D equivalent porous medium model to represent the test site domain and then employed regular zoning parameterisation, on which the inverse modelling was performed. Because we aim to resolve the fine-scale characteristics of the transmissivity field, the problem undertaken is essentially a large-scale inverse model, i.e. the dimension of the unknown parameters is high. In order to deal with the high computational demands in such a large-scale inverse problem, a gradient-based, non-linear algorithm (SNOPT) was used to estimate the transmissivity field on the experimental site scale through the inversion of steady-state, hydraulic head measurements recorded at 22 boreholes during 8 sequential cross-hole pumping tests. We used the data from outcrops, borehole fracture measurements and interpretations of inter-well connectivities from interference test responses as initial models to trigger the inversion. Constraints for hydraulic conductivities, based on analytical interpretations of pumping tests, were also added to the inversion models. In addition, the efficiency of the adopted inverse algorithm enables us to increase dramatically the number of unknown parameters to investigate the influence of elementary discretisation on the reconstruction of the transmissivity fields in both synthetic and field studies.By following the above approach, transmissivity fields that produce similar hydrodynamic behaviours to the real head measurements were obtained. The inverted transmissivity fields show complex, spatial heterogeneities with highly conductive channels embedded in a low transmissivity matrix region. The spatial trend of the main flow channels is in a good agreement with that of the main fracture sets mapped on outcrops in the vicinity of the Terrieu site suggesting that the hydraulic anisotropy is consistent with the structural anisotropy. These results from the inverse modelling enable the main flow paths to be located and their hydrodynamic properties to be estimated

A Fast Algorithm for Parabolic PDE-based Inverse Problems Based on Laplace Transforms and Flexible Krylov Solvers

We consider the problem of estimating parameters in large-scale weakly nonlinear inverse problems for which the underlying governing equations is a linear, time-dependent, parabolic partial differential equation. A major challenge in solving these inverse problems using Newton-type methods is the computational cost associated with solving the forward problem and with repeated construction of the Jacobian, which represents the sensitivity of the measurements to the unknown parameters. Forming the Jacobian can be prohibitively expensive because it requires repeated solutions of the forward and adjoint time-dependent parabolic partial differential equations corresponding to multiple sources and receivers. We propose an efficient method based on a Laplace transform-based exponential time integrator combined with a flexible Krylov subspace approach to solve the resulting shifted systems of equations efficiently. Our proposed solver speeds up the computation of the forward and adjoint problems, thus yielding significant speedup in total inversion time. We consider an application from Transient Hydraulic Tomography (THT), which is an imaging technique to estimate hydraulic parameters related to the subsurface from pressure measurements obtained by a series of pumping tests. The algorithms discussed are applied to a synthetic example taken from THT to demonstrate the resulting computational gains of this proposed method

Inversion using a new low-dimensional representation of complex binary geological media based on a deep neural network

Efficient and high-fidelity prior sampling and inversion for complex geological media is still a largely unsolved challenge. Here, we use a deep neural network of the variational autoencoder type to construct a parametric low-dimensional base model parameterization of complex binary geological media. For inversion purposes, it has the attractive feature that random draws from an uncorrelated standard normal distribution yield model realizations with spatial characteristics that are in agreement with the training set. In comparison with the most commonly used parametric representations in probabilistic inversion, we find that our dimensionality reduction (DR) approach outperforms principle component analysis (PCA), optimization-PCA (OPCA) and discrete cosine transform (DCT) DR techniques for unconditional geostatistical simulation of a channelized prior model. For the considered examples, important compression ratios (200 - 500) are achieved. Given that the construction of our parameterization requires a training set of several tens of thousands of prior model realizations, our DR approach is more suited for probabilistic (or deterministic) inversion than for unconditional (or point-conditioned) geostatistical simulation. Probabilistic inversions of 2D steady-state and 3D transient hydraulic tomography data are used to demonstrate the DR-based inversion. For the 2D case study, the performance is superior compared to current state-of-the-art multiple-point statistics inversion by sequential geostatistical resampling (SGR). Inversion results for the 3D application are also encouraging

Bayesian estimation of the transmissivity spatial structure from pumping test data

Estimating the statistical parameters (mean, variance, and integral scale) that define the spatial structure of the transmissivity or hydraulic conductivity fields is a fundamental step for the accurate prediction of subsurface flow and contaminant transport. In practice, the determination of the spatial structure is a challenge because of spatial heterogeneity and data scarcity. In this paper, we describe a novel approach that uses time drawdown data from multiple pumping tests to determine the transmissivity statistical spatial structure. The method builds on the pumping test interpretation procedure of Copty et al. (2011) (Continuous Derivation method, CD), which uses the time-drawdown data and its time derivative to estimate apparent transmissivity values as a function of radial distance from the pumping well. A Bayesian approach is then used to infer the statistical parameters of the transmissivity field by combining prior information about the parameters and the likelihood function expressed in terms of radially-dependent apparent transmissivities determined from pumping tests. A major advantage of the proposed Bayesian approach is that the likelihood function is readily determined from randomly generated multiple realizations of the transmissivity field, without the need to solve the groundwater flow equation. Applying the method to synthetically-generated pumping test data, we demonstrate that, through a relatively simple procedure, information on the spatial structure of the transmissivity may be inferred from pumping tests data. It is also shown that the prior parameter distribution has a significant influence on the estimation procedure, given the non-uniqueness of the estimation procedure. Results also indicate that the reliability of the estimated transmissivity statistical parameters increases with the number of available pumping tests.Peer ReviewedPostprint (author's final draft

Characterizing aquifer heterogeneity using hydraulic tomography

Fine-scale heterogeneity in a geologic medium determines rates and directions of flow and of contaminant transport. Traditional methods of determining hydraulic conductivity provide only average values of hydraulic conductivity, as opposed to a distribution throughout the aquifer. Hydraulic tomography can be used to relate phase shifts and amplitude decays of a sinusoidal pressure signal to hydraulic conductivity. Hydraulic tomography can provide fine-scale interwell resolution, but processing methods require extensive time and computing power to handle the large amounts of data necessary. This tomography study comprises a combination of multiple offset gather surveys taken at the University of Kansas' Geohydrologic Experimental and Monitoring Site, a well-studied site in northeastern Kansas. A computer program has been developed to analyze the data by extending the 3D homogeneous spherical radial equation to the heterogeneous case. The analysis program is capable of efficiently resolving zones with dimensions of about one meter on each side

A field assessment of the value of steady shape hydraulic tomography for characterization of aquifer heterogeneities

This is the published version. Copyright American Geophysical Union[1] Hydraulic tomography is a promising approach for obtaining information on variations in hydraulic conductivity on the scale of relevance for contaminant transport investigations. This approach involves performing a series of pumping tests in a format similar to tomography. We present a field-scale assessment of hydraulic tomography in a porous aquifer, with an emphasis on the steady shape analysis methodology. The hydraulic conductivity (K) estimates from steady shape and transient analyses of the tomographic data compare well with those from a tracer test and direct-push permeameter tests, providing a field validation of the method. Zonations based on equal-thickness layers and cross-hole radar surveys are used to regularize the inverse problem. The results indicate that the radar surveys provide some useful information regarding the geometry of the K field. The steady shape analysis provides results similar to the transient analysis at a fraction of the computational burden. This study clearly demonstrates the advantages of hydraulic tomography over conventional pumping tests, which provide only large-scale averages, and small-scale hydraulic tests (e.g., slug tests), which cannot assess strata connectivity and may fail to sample the most important pathways or barriers to flow

Hydraulic Conductivity Imaging from 3-D Transient Hydraulic Tomography at Several Pumping/Observation Densities

[1] 3-D Hydraulic tomography (3-D HT) is a method for aquifer characterization whereby the 3-D spatial distribution of aquifer flow parameters (primarily hydraulic conductivity, K) is estimated by joint inversion of head change data from multiple partially penetrating pumping tests. While performance of 3-D HT has been studied extensively in numerical experiments, few field studies have demonstrated the real-world performance of 3-D HT. Here we report on a 3-D transient hydraulic tomography (3-D THT) field experiment at the Boise Hydrogeophysical Research Site which is different from prior approaches in that it represents a “baseline” analysis of 3-D THT performance using only a single arrangement of a central pumping well and five observation wells with nearly complete pumping and observation coverage at 1 m intervals. We jointly analyze all pumping tests using a geostatistical approach based on the quasi-linear estimator of Kitanidis (1995). We reanalyze the system after progressively removing pumping and/or observation intervals; significant progressive loss of information about heterogeneity is quantified as reduced variance of the K field overall, reduced correlation with slug test K estimates at wells, and reduced ability to accurately predict independent pumping tests. We verify that imaging accuracy is strongly improved by pumping and observational densities comparable to the aquifer heterogeneity geostatistical correlation lengths. Discrepancies between K profiles at wells, as obtained from HT and slug tests, are greatest at the tops and bottoms of wells where HT observation coverage was lacking

Estimation of heterogeneous aquifer parameters using centralized and decentralized fusion of hydraulic tomography data

Characterization of spatial variability of hydraulic properties of groundwater systems at high resolution is essential to simulate flow and transport phenomena. This paper investigates two schemes to invert transient hydraulic head data resulting from multiple pumping tests for the purpose of estimating the spatial distributions of the hydraulic conductivity, K, and the specific storage, Ss, of an aquifer. The two methods are centralized fusion and decentralized fusion. The centralized fusion of transient data is achieved when data from all pumping tests are processed concurrently using a central inversion processor, whereas the decentralized fusion inverts data from each pumping test separately to obtain optimal local estimates of hydraulic parameters, which are consequently fused using the generalized Millman formula, an algorithm for merging multiple correlated or uncorrelated local estimates. For both data fusion schemes, the basic inversion processor employed is the ensemble Kalman filter, which is employed to assimilate the temporal moments of impulse response functions obtained from the transient hydraulic head measurements resulting from multiple pumping tests. Assimilating the temporal moments instead of the hydraulic head transient data themselves is shown to provide a significant improvement in computational efficiency. Additionally, different assimilation strategies to improve the estimation of Ss are investigated. Results show that estimation of the K and Ss distributions using temporal moment analysis is fairly good, and the centralized inversion scheme consistently outperforms the decentralized inversion scheme