A comparison of seven methods for the inverse modelling of groundwater flow. Application to the characterisation of well catchments

Inverse modelling is a key step in groundwater-related hydrological studies. Several inversion techniques were developed during the last decades, but hardly any comparison between them was presented. We compare seven modern inverse methods for groundwater flow: the Regularised Pilot Points Method (both the estimation, RPPM-CE, and the Monte Carlo (MC) simulation variants, RPPM-CS), the MC variant of the Representer Method (RM), the Sequential Self-Calibration Method (SSC), the Moment Equations Method (MEM), the Zonation Method (ZM) and a non-iterative Semi-Analytical Method (SAM). These methods are applied to a two-dimensional synthetic example, depicting steady-state groundwater flow around a pumping well. Their relative performance is assessed in terms of their ability to characterise the log-transmissivity and hydraulic head fields and to predict the extent of the well catchment, both for a mildly and a strongly heterogeneous transmissivity field. The main conclusions drawn from the comparison are: (1) MC-based methods (RPPM-CS, SSC and RM) yield very similar results, regardless the degree of heterogeneity and despite they use different parameterisation schemes and objective functions; (2) statistical moments of the target quantities provided by MEM and RPPM-CE are similar to those of MC-based methods; (3) ZM and SAM are negatively affected by strong heterogeneity; and (4) in general, observed differences between the performances of all methods are not very large. MC-based inverse methods need considerably more CPU time than the other tested approaches. An advantage of MC-based methods is that they allow computing the posterior probability distribution of the target quantities, which can be directly fed to probabilistic risk-assessment procedures.     

Conditioning mean steady state flow on hydraulic head and conductivity through geostatistical inversion

Nonlocal moment equations allow one to render deterministically optimum predictions of flow in randomly heterogeneous media and to assess predictive uncertainty conditional on measured values of medium properties. We present a geostatistical inverse algorithm for steady-state flow that makes it possible to further condition such predictions and assessments on measured values of hydraulic head (and/or flux). Our algorithm is based on recursive finite-element approximations of exact first and second conditional moment equations. Hydraulic conductivity is parameterized via universal kriging based on unknown values at pilot points and (optionally) measured values at other discrete locations. Optimum unbiased inverse estimates of natural log hydraulic conductivity, head and flux are obtained by minimizing a residual criterion using the Levenberg-Marquardt algorithm. We illustrate the method for superimposed mean uniform and convergent flows in a bounded two-dimensional domain. Our examples illustrate how conductivity and head data act separately or jointly to reduce parameter estimation errors and model predictive uncertainty.This work is supported in part by NSF/ITR Grant EAR-0110289. The first author was additionally supported by scholarships from CONACYT and Instituto de Investigaciones Electricas of Mexico. Additional support was provided by the European Commission under Contract EVK1-CT-1999-00041 (W-SAHaRA-Stochastic Analysis of Well Head Protection and Risk Assessment).     

Quantifying geological uncertainty for flow and transport modeling in multi-modal heterogeneous formations

In this work, we address the problem of characterizing the heterogeneity and uncertainty of hydraulic properties for complex geological settings. Hereby, we distinguish between two scales of heterogeneity, namely the hydrofacies structure and the intrafacies variability of the hydraulic properties. We employ multiple-point geostatistics to characterize the hydrofacies architecture. The multiple-point statistics are borrowed from a training image that is designed to reflect the prior geological conceptualization. The intrafacies variability of the hydraulic properties is represented using conventional two-point correlation methods, more precisely, spatial covariance models under a multi-Gaussian spatial law. We address the different levels and sources of uncertainty in characterizing the subsurface heterogeneity, and explore their effect on groundwater flow and transport predictions. Typically, uncertainty is assessed by way of many images, termed realizations, of a fixed statistical model. However, in many cases, sampling from a fixed stochastic model does not adequately represent the space of uncertainty. It neglects the uncertainty related to the selection of the stochastic model and the estimation of its input parameters. We acknowledge the uncertainty inherent in the definition of the prior conceptual model of aquifer architecture and in the estimation of global statistics, anisotropy, and correlation scales. Spatial bootstrap is used to assess the uncertainty of the unknown statistical parameters. As an illustrative example, we employ a synthetic field that represents a fluvial setting consisting of an interconnected network of channel sands embedded within finer-grained floodplain material. For this highly non-stationary setting we quantify the groundwater flow and transport model prediction uncertainty for various levels of hydrogeological uncertainty. Results indicate the importance of accurately describing the facies geometry, especially for transport predictions.     

Worth of head data in well-capture zone design: deterministic and stochastic analysis

In this study, we examine the effects of conditioning spatially variable transmissivity fields using head and/or transmissivity measurements on well-capture zones. In order to address the challenge posed by conditioning a flow model with spatially varying parameters, an innovative inverse algorithm, the Representers method, is employed. The method explicitly considers this spatial variability.

A number of uniform measurement grids with different densities are used to condition transmissivity fields using the Representers method. Deterministic and stochastic analysis of well-capture zones are then examined. The deterministic study focuses on comparison between reference well-capture zones and their estimated mean conditioned on head data. It shows that model performance due to head conditioning on well-capture zone estimation is related to pumping rate. At moderate pumping rates transmissivity observations are more crucial to identify effects arising from small-scale variations in pore water velocity. However, with more aggressive pumping these effects are reduced, consequently model performance, through incorporating head observations, markedly improves. In the stochastic study, the effect of conditioning using head and/or transmissivity data on well-capture zone uncertainty is examined. The Representers method is coupled with the Monte Carlo method to propagate uncertainty in transmissivity fields to well-capture zones. For the scenario studied, the results showed that a combination of 48 head and transmissivity data could reduce the area of uncertainty (95% confidence interval) in well-capture zone location by over 50%, compared to a 40% reduction using either head or transmissivity data. This performance was comparable to that obtained through calibrating on three and a half times the number of head observations alone.     

Measuring the effect of structural connectivity on the water dynamics in heterogeneous porous media using speedy neutron tomography

The temporal and spatial distribution of water within a porous medium is affected by the medium’s structure, i.e., the spatial arrangement of its constituents. To analyze structural effects on the fluid dynamics, we measured the 3D water content distribution in a heterogeneous sand column during two drainage-wetting cycles using neutron transmission tomography. The sample with a volume of 105 cm³ contained 101 cubes of fine and 49 cubes of coarse sand with particles ranging from 0.01 to 0.05 and 0.03 to 0.09 cm, respectively. The pressure at the lower boundary was determined by the water reservoir positioned between 7 and 39 cm below the top of the column. The duration of one complete 3D scanning with a spatial resolution of 127 μm was 56 s. The signal to noise ratio of the measurements was low due to the short exposure time in the neutron beam, but it was possible to quantify the water content in the individual cubes and hence the effect of structure on macroscopic water distribution. Continuous structures of coarse sand drained faster than coarse sand without connection to the upper boundary. During the initial wetting phase, cubes of coarse sand material completely embedded in the fine material remained water unsaturated due to air entrapment. The effect of the coarse sand connectivity was analyzed in two-dimensional numerical simulations based on Richards equation. In contrast to the measurements, no effect of structure connectivity was found. The coarse sand cubes embedded within the fine matrix drain as quickly as the coarse sand cubes arranged in a continuous channel due to the model assumption of a continuous air phase.     

Modeling groundwater velocity uncertainty in nonstationary composite porous media

Despite the intensive research over the past decades in the field of stochastic subsurface hydrology, our ability to analyze and model heterogeneous groundwater systems remains limited. Most existing theories are either too restrictive to handle practical complexity or too expensive to be applied to realistic problem sizes. In this paper we present approximate, closed-form equations that allow modeling 2D nonstationary flows in statistically inhomogeneous aquifers, including composite aquifers containing multiple zones characterized by different statistical models. The composite representation has the effect of decreasing the variance of deviations from the mean, relaxing the limitation of the small-perturbation assumption. The simple formulas are illustrated with a number of examples and compared with a corresponding first-order nonstationary numerical analysis and Monte Carlo simulation. The results show that, despite the gross simplifications, the closed-form equations are robust and able to capture complex variance dynamics, reproducing surprisingly well the first-order numerical solutions and the Monte Carlo simulation even in highly nonstationary, variable situations.     

Comparison of sparse‐grid geometric and random sampling methods in nonlinear inverse solution uncertainty estimation

A new uncertainty estimation method, which we recently introduced in the literature, allows for the comprehensive search of model posterior space while maintaining a high degree of computational efficiency. The method starts with an optimal solution to an inverse problem, performs a parameter reduction step and then searches the resulting feasible model space using prior parameter bounds and sparse‐grid polynomial interpolation methods. After misfit rejection, the resulting model ensemble represents the equivalent model space and can be used to estimate inverse solution uncertainty. While parameter reduction introduces a posterior bias, it also allows for scaling this method to higher dimensional problems. The use of Smolyak sparse‐grid interpolation also dramatically increases sampling efficiency for large stochastic dimensions. Unlike Bayesian inference, which treats the posterior sampling problem as a random process, this geometric sampling method exploits the structure and smoothness in posterior distributions by solving a polynomial interpolation problem and then resampling from the resulting interpolant. The two questions we address in this paper are 1) whether our results are generally compatible with established Bayesian inference methods and 2) how does our method compare in terms of posterior sampling efficiency. We accomplish this by comparing our method for two electromagnetic problems from the literature with two commonly used Bayesian sampling schemes: Gibbs’ and Metropolis‐Hastings. While both the sparse‐grid and Bayesian samplers produce compatible results, in both examples, the sparse‐grid approach has a much higher sampling efficiency, requiring an order of magnitude fewer samples, suggesting that sparse‐grid methods can significantly improve the tractability of inference solutions for problems in high dimensions or with more costly forward physics.     

Direct and indirect methods for groundwater investigations: A case-study of MRS and VES in the southern part of Sweden

Magnetic Resonance Sounding (MRS) has been successfully tested for detecting groundwater in two areas in southern Sweden. Measurements of Schlumberger VES have been conducted in the same place as the MRS and the results are generally consistent. Low resistivity layers interpreted as clay are sometimes identified close to the surface. The MRS result in site 2 is a good example of signals penetrating through the clay and deeper aquifer still being detected. The MRS data suggest aquifers that are not only hosted in soft sediment materials (moraine, sand, and mixed materials), but hosted in basement rocks. Based on the MRS and borehole pumping test data, the results agree with yield, average water content and subsurface geological data.     

Comparison of theory and experiments for dispersion in homogeneous porous media

Modeling dispersion in homogeneous porous media with the convection–dispersion equation commonly requires computing effective transport coefficients. In this work, we investigate longitudinal and transverse dispersion coefficients arising from the method of volume averaging, for a variety of periodic, homogeneous porous media over a range of particle Péclet (Pe_p) numbers. Our objective is to validate the upscaled transverse dispersion coefficients and concentration profiles by comparison to experimental data reported in the literature, and to compare the upscaling approach to the more common approach of inverse modeling, which relies on fitting the dispersion coefficients to measured data. This work is unique in that the exact microscale geometry is available; thus, no simplifying assumptions regarding the geometry are required to predict the effective dispersion coefficients directly from theory. Transport of both an inert tracer and non-chemotactic bacteria is investigated for an experimental system that was designed to promote transverse dispersion. We highlight the occurrence of transverse dispersion coefficients that (1) depart from power-law behavior at relatively low Pe_p values and (2) are greater than their longitudinal counterparts for a specific range of Pe_p values. The upscaling theory provides values for the transverse dispersion coefficient that are within the 98% confidence interval of the values obtained from inverse modeling. The mean absolute error between experimental and upscaled concentration profiles was very similar to that between the experiments and inverse modeling. In all cases the mean absolute error did not exceed 12%. Overall, this work suggests that volume averaging can potentially be used as an alternative to inverse modeling for dispersion in homogeneous porous media.     

Non-uniqueness in surface-wave inversion and consequences on seismic site response analyses

Surface-wave tests are based on the solution of an inverse problem for shear-wave velocity profile identification from the experimentally measured dispersion curve. The main criticisms for these testing methodologies are related to the inverse problem solution and arise from the possible equivalence of different shear-wave velocity profiles. In this paper, some implications of solution non-uniqueness for seismic response studies are investigated using both numerical simulations and experimental data. A Monte Carlo approach for the inversion problem has been used to obtain a set of equivalent shear-wave velocity models. This selection is based on a statistical test which takes into account both data uncertainty and model parameterization. This set of solutions (i.e., soil profiles) is then used to evaluate the seismic response with a conventional one-dimensional analysis. It is shown that equivalent profiles with respect to surface-wave testing are equivalent also with respect to site amplification, thus countering the criticism related to inversion uncertainty for the engineering use of surface-wave tests.     

An inverse problem methodology to identify flow channels in fractured media using synthetic steady-state head and geometrical data

We present a methodology for identifying highly-localized flow channels embedded in a significantly less permeable medium using steady-state head and geometrical data. This situation is typical of fractured media where flows are often strongly channeled at the scales of interest (10 m–1 km). The objective is to identify both geometrical and hydraulic characteristics of the conducting structures. Channels are identified in decreasing order of importance by successive optimizations of an objective function. The identification strategy takes advantage of the hierarchical flow organization to restrict the dimension of the solution space of each individual optimization step. The characteristics of the secondary channels are strongly determined by the main flow channels. The latter are slightly modified by the secondary channels through the addition of a regularization term to the main channel characteristics in the objective function. As the objective function is strongly non-convex with numerous local minima, inversion is performed using a stochastic algorithm (simulated annealing). We assess the possibilities of the hierarchical identification strategy on simple synthetic steady-state flow configurations where hydraulic data are made up of 25 regularly spaced heads and of the boundary conditions. Those flow structures that are dominated by at most two simple channels can be identified with these head data only. Configurations comprising up to three complex and interconnected channels can still be identified with additional geometrical information including the distances of piezometers to their closest channel. The capabilities of the hierarchical identification strategy are limited to flow structures dominated by at most three equivalent flow channels. We finally discuss the perspectives of application of the method to transient-state data obtained on a more restricted number of piezometers.     

An efficient,high-order probabilistic collocation method on sparse grids for three-dimensional flow and solute transport in randomly heterogeneous porous media

In this study, a probabilistic collocation method (PCM) on sparse grids is used to solve stochastic equations describing flow and transport in three-dimensional, saturated, randomly heterogeneous porous media. The Karhunen–Loève decomposition is used to represent log hydraulic conductivity Y=ln_Ks$Y=\ln K_s$. The hydraulic head h   and average pore-velocity v$\mathbf{v}$ are obtained by solving the continuity equation coupled with Darcy’s law with random hydraulic conductivity field. The concentration is computed by solving a stochastic advection–dispersion equation with stochastic average pore-velocity v$\mathbf{v}$ computed from Darcy’s law. The PCM approach is an extension of the generalized polynomial chaos (gPC) that couples gPC with probabilistic collocation. By using sparse grid points in sample space rather than standard grids based on full tensor products, the PCM approach becomes much more efficient when applied to random processes with a large number of random dimensions. Monte Carlo (MC) simulations have also been conducted to verify accuracy of the PCM approach and to demonstrate that the PCM approach is computationally more efficient than MC simulations. The numerical examples demonstrate that the PCM approach on sparse grids can efficiently simulate solute transport in randomly heterogeneous porous media with large variances.     

Significance of injection modes and heterogeneity on spatial and temporal dispersion of advecting particles in two-dimensional discrete fracture networks

We investigate the impact of injection mode (flux and resident injection) and heterogeneity in hydraulic properties on dispersion of advecting particles in two-dimensional discrete fracture network models, using a Monte Carlo method. We find that the injection mode has a significant effect on dispersion: The resident injection mode exhibits anomalous features of transport whereas the flux injection mode tends to Gaussian transport; this observation is easily understood by considering phase diagrams where a limited number of particles entering low velocity fractures greatly increase macrodispersion. In spite of a sizeable portion of negative longitudinal velocities, it is shown that multiple crossings are negligible when quantifying longitudinal macrodispersion. A simple probabilistic expression of particle mass balance is shown to predict well the spatial distribution of advecting particles.     

Stochastic observation error and uncertainty in water quality evaluation

When evaluating water quality, the influence of physical weight of the observed index is normally taken into account, but the influence of stochastic observation error (SOE) is not adequately considered. Using Monte Carlo simulation, combined with Shannon entropy, the Principle of Maximum Entropy (POME) and Tsallis entropy, this study investigates the influence of stochastic observation error (SOE) for two cases of the observed index: small observation error and large observation error. Randomness and fuzziness represent two types of uncertainties that are deemed significant and should be considered simultaneously when developing or evaluating water quality models. To that end, three models are employed here: two of the models, named as model I and model II, consider both the fuzziness and randomness, and another model, considers only fuzziness. The results from three representative lakes in China show that for all three models, the influence of stochastic observation error (SOE) on water quality evaluation can be significant irrespective of whether the water quality index has a small observation error or a large observation error. Furthermore, when there is a significant difference in the accuracy of observations, the influence of stochastic observation error (SOE) on water quality evaluation increases. The water quality index whose SOE is minimum determines the results of evaluation.