Transient Pumping Tests Research Articles

Reduced Geostatistical Approach With a Fourier Neural Operator Surrogate Model for Inverse Modeling of Hydraulic Tomography

AbstractIn this study, we propose a new approach for inverse modeling of hydraulic tomography (HT) using the Fourier neural operator (FNO) as a surrogate forward model. FNO is a deep learning model that directly parameterizes the integral kernel in Fourier space to learn solution operators for parametric partial differential equations (PDEs). We trained a highly accurate FNO to learn the solution operators of PDEs in HT, which can efficiently generate multiple channels of hydraulic head fields corresponding to specific pumping tests in new parameter fields through a single forward pass. The trained FNO surrogate model is integrated with the reduced geostatistical approach (RGA) to provide a powerful tool, named RGA‐FNO, for inverse modeling of Gaussian random fields. RGA utilizes principal components to effectively reduce problem dimensions and encode geostatistical information. FNO benefits from deep learning automatic differentiation, enabling efficient backpropagation of gradients during inverse modeling. Our results show that RGA‐FNO can efficiently produce satisfactory estimations for random transmissivity fields with an exponential covariance function, which is non‐differentiable and visually non‐smooth. The RGA‐FNO application on steady‐state multi‐well HT is better than the single‐well transient pumping test for inverse modeling. The data required for inverse modeling applications of RGA‐FNO increases with the variance of the random transmissivity fields. FNO and other neural operator models have the potential to play an important role in groundwater modeling, especially in inverse modeling and experimental design, which often requires a large number of forward simulations.

PEST and AEM Modeling for Data Acquisition Planning.

A well-planned field data collection program should be designed to (1) collect a sufficient set of data of the right types at the right locations, and (2) collect a parsimonious set of data to avoid unnecessary costs. Combining PEST and a simple analytic element method (AEM) groundwater flow model for the site of interest provides a relatively simple, low-cost method of developing such a program. AEM models are well suited to this approach because they are quick to develop yet hydraulically accurate, reducing impacts on project budgets at early data collection planning stages; and quick to run, solving rapidly for the many iterations that PEST requires to generate good parameter estimates. This article shows two examples of this method: one for a steady state watershed model, and one for a transient pumping test project to demonstrate that PEST coupled with a simple AEM model that sketches out the key features of a site conceptual model can be an efficient tool in planning key parts of a hydrogeologic site investigation.

Improving hydrogeological understanding through well-test interpretation by diagnostic plot and modelling: a case study in an alluvial aquifer in France

The study of groundwater resources using pumping test data is usually carried out with the Theis solution, which enables the hydraulic parameters of porous aquifers such as the transmissivity and storage coefficient, to be estimated from the water-level drawdown. However, the data fitting can fail and provide only an indication that the pumped aquifer has a complex structure. Here, a diagnostic plot on log-derivative drawdown is used to identify flow regimes and thus aquifer heterogeneities, leading to plausible conceptual models. Nevertheless, the diagnostic plot is insufficient and must be accompanied by further modelling because of the nonuniqueness of the drawdown log-derivative signal. The proposed approach is applied to an alluvial plain in France, known to be complex because the deposition processes change over time, resulting in channel belts limited by low-permeability deposits in the floodplain or three-dimensional (3D) interconnected structures. Six analytical models were used to simulate drawdown and its derivatives during a three-day transient pumping test. The diagnostic performed on the pumping well showed that four conceptual models, with highly contrasted hydrodynamic behaviours, may correspond to the diagnostic. The joint use of pumping-well and observation-well data allowed the only appropriate model to be identified—a dual-permeability model characterizing a multilayer aquifer. The conceptual model matched the geological observations in boreholes and corroborates the fluvial sequence stratigraphy of the alluvial plain. The pumping test used here is a tool to explore the 3D architecture of the fluvial reservoir at the scale of the depositional sequence in the floodplain.

Determination of hydraulic conductivity and its spatial variability in the Jianghan Plain using a multi-format, multi-method approach

In recent times, the groundwater in Jianghan Plain, Central China has come under severe stress from agricultural and industrial demand. The spatial variability of hydraulic conductivity (K) in the region is important for better and sustainable groundwater exploration and management. In this study, we determined the spatial distribution of K using a multi-method approach, i.e. analytic and numerical simulation as well as intelligent optimization methods (particle swarm optimization (PSO) and differential evolution (DE)) and geostatistics. Data from 55 multi-format pumping tests situated across the middle-confined aquifer (consisting of late Pleistocene (Qp3s) and middle Pleistocene (Qp2j)) were used. Results showed that for transient pumping test with constant rate by multi-wells, PSO and DE were faster and more precise, especially DE. For transient pumping test with variable rate as well as transient pumping test with constant rate by single well, the numerical method fitted the drawdown better, while the PSO and DE had large deviations at the early stage of pumping. Overall, the values of K vary from 0.032 to 10.64 m/d within Qp3s and 0.33 to 36.24 m/d within Qp2j. The spatial variability model demonstrated that the K values in Qp3s are greater than those in Qp2j. The heterogeneity elucidates the sedimentary development of the basin. These methods and results can provide new insights for estimating the spatial distribution of K for regional groundwater resources assessment in similar sedimentary environments around the world.

Depth-discrete specific storage in fractured sedimentary rock using steady-state and transient single-hole hydraulic tests

A method is presented for obtaining depth-discrete values of specific storage (Ss) from single-hole hydraulic tests in fractured rock boreholes using straddle packers (1.5–17m test intervals). Low flow constant head (CH) step tests analyzed using the Thiem method provide transmissivity (T) values free from non-Darcian error. Short-term, constant-rate pumping tests (0.5–2h) analyzed using the Cooper-Jacob approximation of the Theis method provide S from the hydraulic diffusivity using the Darcian T value from the CH step test. This synergistic use of two types of hydraulic tests avoids the common source of error when pumping tests (injection or withdrawal) are conducted at higher flow rates and thereby induce non-Darcian flow resulting in the underestimation of T. Other errors, such as well bore storage and leakage, can also substantially influence S by causing a shift in the time axis of the Cooper-Jacob semi-log plot. In this approach, the Darcian T values from the CH step tests are used in the analysis of the transient pumping test data for calculating S throughout the pumping test using the Cooper-Jacob approximation to minimize all of the aforementioned errors, resulting in more representative S values. The effect of these non-idealities on the measured drawdown is illustrated using the Theis equation with the Darcian T and S values to calculate drawdown for comparison to measured data. The Ss values for tests in sandstone obtained from this approach are more consistent with confined aquifer conditions than values derived from the traditional Cooper-Jacob method, and are within the range of field and lab values presented from a compilation of literature values for fractured sandstone. (10−7–10−5m−1) This method for obtaining Ss values from short-interval, straddle packer tests improves the estimation of both K and Ss and provides opportunity to study their spatial distribution in fractured rock.

Extending Theis' solution: Using transient pumping tests to estimate parameters of aquifer heterogeneity

AbstractA framework for interpreting transient pumping tests in heterogeneous transmissivity fields is developed to infer the overall geostatistical parameters of the medium without reconstructing the specific heterogeneous structure point wise. The methodology of Radial Coarse Graining is applied to deduce an effective radial description of multi‐Gaussian transmissivity. It was used to derive an Effective Well Flow Solution for transient flow conditions including not only the storativity, but also the geometric mean, the variance, and the correlation length of log‐transmissivity. This solution is shown to be appropriate to characterize the pumping test drawdown behavior in heterogeneous transmissivity fields making use of ensembles of simulated pumping tests with multiple combinations of statistical parameters. Based on the Effective Well Flow Solution, a method is developed for inferring heterogeneity parameters from transient pumping test drawdown data by inverse estimation. Thereby, the impact of statistical parameters on the drawdown is analyzed, allowing to determine the dependence of reliability of parameter estimates on location and number of measurements. It is shown, that the number of measurements can be reduced compared to steady state pumping tests. Finally, a sampling strategy for single aquifer analysis is developed, which allows to estimate the statistical parameters, in particular variance and correlation length for individual heterogeneous transmissivity fields making use of transient pumping test measurements at multiple locations.

Determining the optimal pumping duration of transient pumping tests for estimating hydraulic properties of leaky aquifers using global curve-fitting method: a simulation approach

Global curve-fitting method (GCFM) is regarded as an effective approach in hydrogeological parameter estimation, as it integrates and uses pumping data and water recovery data of a transient pumping test for parameter estimation. The impacts of pumping duration on hydrogeological parameter estimation by GCFM were investigated in the present study using 2 in situ pumping tests and 24 simulated transient pumping tests. Empirical formulas for determining the optimal pumping duration were derived. The study results suggest that pumping duration will have impacts on the accuracy of hydrogeological parameter estimation. When pumping duration is longer than a certain period, relative errors of hydrogeological parameter estimation keep relatively stable within an acceptable limit. Therefore, it is unnecessary to continue the pumping for a very long time after the groundwater level has become stable. When the change rate of drawdown over time (γ) in an observation well located within a distance of 100 m to the pumping well reaches 0.134, the pumping can be stopped. If there are more than one observation wells in a pumping test, the smallest γ value should be selected to determine the optimal pumping duration. This research is meaningful in the instruction of pumping tests, and will reduce test costs greatly.

Stochastic joint inversion of hydrogeophysical data for salt tracer test monitoring and hydraulic conductivity imaging

The assessment of hydraulic conductivity of heterogeneous aquifers is a difficult task using traditional hydrogeological methods (e.g., steady state or transient pumping tests) due to their low spatial resolution. Geophysical measurements performed at the ground surface and in boreholes provide additional information for increasing the resolution and accuracy of the inverted hydraulic conductivity field. We used a stochastic joint inversion of Direct Current (DC) resistivity and self-potential (SP) data plus in situ measurement of the salinity in a downstream well during a synthetic salt tracer experiment to reconstruct the hydraulic conductivity field between two wells. The pilot point parameterization was used to avoid over-parameterization of the inverse problem. Bounds on the model parameters were used to promote a consistent Markov chain Monte Carlo sampling of the model parameters. To evaluate the effectiveness of the joint inversion process, we compared eight cases in which the geophysical data are coupled or not to the in situ sampling of the salinity to map the hydraulic conductivity. We first tested the effectiveness of the inversion of each type of data alone (concentration sampling, self-potential, and DC resistivity), and then we combined the data two by two. We finally combined all the data together to show the value of each type of geophysical data in the joint inversion process because of their different sensitivity map. We also investigated a case in which the data were contaminated with noise and the variogram unknown and inverted stochastically. The results of the inversion revealed that incorporating the self-potential data improves the estimate of hydraulic conductivity field especially when the self-potential data were combined to the salt concentration measurement in the second well or to the time-lapse cross-well electrical resistivity data. Various tests were also performed to quantify the uncertainty in the inverted hydraulic conductivity field.

Using data assimilation method to calibrate a heterogeneous conductivity field conditioning on transient flow test data

A data assimilation method is developed to calibrate a heterogeneous hydraulic conductivity field conditioning on transient pumping test data. The ensemble Kalman filter (EnKF) approach is used to update model parameters such as hydraulic conductivity and model variables such as hydraulic head using available data. A synthetical two-dimensional flow case is used to assess the capability of the EnKF method to calibrate a heterogeneous conductivity field by assimilating transient flow data from observation wells under different hydraulic boundary conditions. The study results indicate that the EnKF method will significantly improve the estimation of the hydraulic conductivity field by assimilating continuous hydraulic head measurements and the hydraulic boundary condition will significantly affect the simulation results. For our cases, after a few data assimilation steps, the assimilated conductivity field with four Neumann boundaries matches the real field well while the assimilated conductivity field with mixed Dirichlet and Neumann boundaries does not. We found in our cases that the ensemble size should be 300 or larger for the numerical simulation. The number and the locations of the observation wells will significantly affect the hydraulic conductivity field calibration.

Geostatistical inverse modeling of transient pumping tests using temporal moments of drawdown

Pumping tests belong to the most common techniques of hydrogeological site assessment. While the steady state drawdown is determined by the distribution of transmissivity alone, the transient behavior is also influenced by the storativity field. In geostatistical inverse modeling the spatial distributions of both transmissivity and storativity are inferred from the drawdown curves and prior information on the spatial correlation of the parameter fields. So far, however, transient data have hardly been analyzed by geostatistical inverse methods because the computational effort is rather high. In the present study, we characterize the drawdown by its temporal moments. We present moment‐generating equations and corresponding equations to compute the sensitivity of the temporal moments of drawdown with respect to the distributions of transmissivity and storativity. We utilize these equations to infer the transmissivity and storativity fields from transient pumping tests using the quasi‐linear geostatistical approach of inverse modeling. Considering temporal moments rather than full drawdown curves drastically reduces the computational effort of the estimation procedure. In test cases we show that the first two temporal moments are sufficient to characterize the drawdown curves. We investigate how erroneous assumptions regarding the spatial variability of storativity affect the estimate of the transmissivity field, and we analyze the effect of truncating the measured drawdown curves.