Andrés Sahuquillo

Stochastic simulation of transmissivity fields conditional to both transmissivity and piezometric data—I. Theory

Abstract The generation of equally likely realizations of transmissivity fields conditional to both transmissivity and piezometric data is achieved using an iterative technique that couples geostatistics and optimization. By conditioning to both transmissivity fields and piezometric data we mean that the transmissivity field generated honours the transmissivity data and that the solution to the flow equation reproduces the measured piezometric heads. By equally likely realizations we mean that all realizations display the same patterns of spatial variability as observed in the field, as opposed to the transmissivity fields obtained by interpolation or by inverse modelling which are oversmooth, unrealistic, representations of the real field. In this regard, any realization could be the real but unknown transmissivity field. To achieve this goal, first a transmissivity field is generated conditional only to the transmissivity data, then the transmissivity field (and possibly the boundary conditions), are modified without destroying the spatial patterns of variability of transmissivity until the piezometric data are also honoured. The methodology is presented for steady-state flow in a confined two-dimensional aquifer. The details of the implementation are given for finite-differences solution over an aquifer discretized in square blocks.

Coupled inverse modelling of groundwater flow and mass transport and the worth of concentration data

This paper presents the extension of the self-calibrating method to the coupled inverse modelling of groundwater flow and mass transport. The method generates equally likely solutions to the inverse problem that display the variability as observed in the field and are not affected by a linearisation of the state equations. Conditioning to the state variables is measured by an objective function including, among others, the mismatch between the simulated and measured concentrations. Conditioning is achieved by minimising the objective function by gradient-based methods. The gradient contains the partial derivatives of the objective function with respect to: log conductivities, log storativities, prescribed heads at boundaries, retardation coefficients and mass sources. The derivatives of the objective function with respect to log conductivity are the most cumbersome and need the most CPU-time to be evaluated. For this reason, to compute this derivative only advective transport is considered. The gradient is calculated by the adjoint-state method. The method is demonstrated in a controlled, synthetic study, in which the worth of concentration data is analysed. It is shown that concentration data are essential to improve transport predictions and also help to improve aquifer characterisation and flow predictions, especially in the upstream part of the aquifer, even in the case that a considerable amount of other experimental data like conductivities and heads are available. Besides, conditioning to concentration data reduces the ensemble variances of estimated transmissivity, hydraulic head and concentration.

Stochastic simulation of transmissivity fields conditional to both transmissivity and piezometric data 2. Demonstration on a synthetic aquifer

In the first paper of this series a methodology for the generation of transmissivity fields conditional to both transmissivity and piezometric head data was presented. This methodology, termed the self-calibrated approach, consists of two steps: first, the generation of a seed transmissivity field conditioned only to transmissivity data, and second, the perturbation of the seed field up until the piezometric head data are reproduced. The methodology is now demonstrated on a set of controlled numerical experiments carried out on synthetic aquifers. The objective of these experiments is not just to show that the methodology works, but also to explore its robustness under different situations. A total of 12 experiments have analyzed the performance of the method as a function of: (i) the log10 T transmissivity variance (from 0.2 to 2.0); (ii) the number of log10 T conditioning data (from 10 to 30); (iii) the number of piezometric head data (from 30 to 90); (iv) the number of master points (from 25 to 1000); (v) the magnitude of allowed departure of the final T field from the seed field (up to four times the kriging standard deviation). In all cases, the method was able to generate transmissivity fields conditional to both transmissivity and head measurements, at the same time preserving the spatial variability of the transmissivity field. It was found that the performance of the method increases with both the number of log10 T data and the number of master points, whereas it decreases as either the log10 T variance or the number of piezometric head data increases.

Joint simulation of transmissivity and storativity fields conditional to steady-state and transient hydraulic head data

Abstract The self-calibrated method has been extended for the generation of equally likely realizations of transmissivity and storativity conditional to transmissivity and storativity data and to steady-state and transient hydraulic head data. Conditioning to transmissivity and storativity data is achieved by means of standard geostatistical co-simulation algorithms, whereas conditioning to hydraulic head data, given its non-linear relation to transmissivity and storativity, is achieved through non-linear optimization, similar to standard inverse algorithms. The algorithm is demonstrated in a synthetic study based on data from the WIPP site in New Mexico. Seven alternative scenarios are investigated, generating 100 realizations for each of them. The differences among the scenarios range from the number of conditioning data, to their spatial configuration, to the pumping strategies at the pumping wells. In all scenarios, the self-calibrated algorithm is able to generate transmissivity–storativity realization couples conditional to all the sample data. For the specific case studied here the results are not surprising. Of the piezometric head data, the steady-state values are the most consequential for transmissivity characterization. Conditioning to transient head data only introduces local adjustments on the transmissivity fields and serves to improve the characterization of the storativity fields.

Stochastic simulation of transmissivity fields conditional to both transmissivity and piezometric head data—3. Application to the Culebra formation at the Waste Isolation Pilot Plan (WIPP), New Mexico, USA

The self-calibrated approach is applied to the stochastic analysis of groundwater flow and advective mass transport in the WIPP site. Multiple equally likely realizations of logtransmissivity fields are generated, followed by the solution of variable density groundwater flow and particle tracking. Five different cases have been analyzed. The first one regards the modeling of variable-density groundwater flow and the remaining four regard the generation of the logtransmissivity fields. Results show that (i) it is important to model variable-density flow as accurately as possible, (ii) conditioning to piezometric head data helps in reducing the uncertainty in flow and transport predictions, (iii) accounting for uncertainty in boundary conditions helps improving the match to measured heads, and (iv) the interpreted value at location P-18 is not consistent with the model of spatial variability inferred from the data.

Significance of conditioning to piezometric head data for predictions of mass transport in groundwater modeling

Transmissivity and head data are sampled from an exhaustive synthetic reference field and used to predict the arrival positions and arrival times of a number of particles transported across the field, together with an uncertainty estimate. Different combinations of number of transmissivity data and number of head data used are considered in each one of a series of 64 Monte-Carlo analyses. In each analysis, 250 realizations of transmissivity fields conditioned to both transmissivity and head data are generated using a novel geostatistically based inverse method. Pooling the solutions of the flow and transport equations in all 250 realizations allows building conditional frequency distributions for particle arrival positions and arrival times. By comparing these fresquency distributions, we can assess the incremental gain that additional head data provide. The main conclusion is that the first few head data dramatically improve the quality of transport predictions.

Groundwater intensive use and mining in south-eastern peninsular Spain: Hydrogeological, economic and social aspects

Intensive groundwater development is a common circumstance in semiarid and arid areas. Often abstraction exceeds recharge, thus continuously depleting reserves. There is groundwater mining when the recovery of aquifer reserves needs more than 50years. The MASE project has been carried out to compile what is known about Spain and specifically about the south-eastern Iberian Peninsula and the Canary Islands. The objective was the synthetic analysis of available data on the hydrological, economic, managerial, social, and ethical aspects of groundwater mining. Since the mid-20th century, intensive use of groundwater in south-eastern Spain allowed extending and securing the areas with traditional surface water irrigation of cash crops and their extension to former dry lands, taking advantage of good soils and climate. This fostered a huge economic and social development. Intensive agriculture is a main activity, although tourism plays currently an increasing economic role in the coasts. Many aquifers are relatively high yielding small carbonate units where the total groundwater level drawdown may currently exceed 300m. Groundwater storage depletion is estimated about 15km(3). This volume is close to the total contribution of the Tagus-Segura water transfer, but without large investments paid for with public funds. Seawater desalination complements urban supply and part of cash crop cultivation. Reclaimed urban waste water is used for irrigation. Groundwater mining produces benefits but associated to sometimes serious economic, administrative, legal and environmental problems. The use of an exhaustible vital resource raises ethical concerns. It cannot continue under the current legal conditions. A progressive change of water use paradigm is the way out, but this is not in the mind of most water managers and politicians. The positive and negative results observed in south-eastern Spain may help to analyse other areas under similar hydrogeological conditions in a less advanced stage of water use evolution.

Inverse Modeling of Groundwater Flow in a 3D Fractured Media

In order to quantify the uncertainty in the prediction of three-dimensional groundwater flow and mass transport in a fractured volcanic tuff, a Monte-Carlo approach is used. An ensemble of equally likely realizations of 3-D spatially variable hydraulic conductivity is generated and used as input to groundwater flow and advective transport model.To reduce the uncertainty in the model predictions, the hydraulic conductivity fields integrate different types of data. On one hand, they are conditioned to hydraulic conductivity measurements and soft information on hydraulic conductivities taken from a structural geology model. On the other hand, they are conditioned to piezometric head measurements. While conditioning to hard and soft hydraulic conductivity data can be achieved by standard geostatis-tical techniques, conditioning to hydraulic head measurements is non trivial because hydraulic conductivity and hydraulic head are non linearly related through the groundwater flow equation. Conditioning to hydraulic head measurements is accomplished by the self-calibrated method, a technique combining geostatistics and non-linear optimization. The methodology is demonstrated in a 3-D fractured site using data on conductivity and on transient heads from a pumping test. Ensembles of hydraulic conductivity realizations conditioned to different types of information are used as input to a groundwater flow and advective transport model; the resulting hydraulic head fields and particle arrival times are compared in terms of their variability to conclude that, for this case study, incorporating hydraulic head data reduces the uncertainty in the conductivity realizations, but it does not in particle arrival times and arrival locations.

Stochastic Inverse Problems In GroundwaterModeling

The inverse problem is, by its own definition, an ill-posed problem that requires assumptions about the structure of the unknown parameters. Yet, it has been traditionally addressed in a somehow deterministic way, searching for an optimum or best estimate solution that reproduces the available data. The impossibility of measuring hydrodynamic parameters and variables all over the aquifer makes groundwater-modeling results uncertain at any given conceivable situation. This fact has been the motivation to the development of stochastic hydrogeology. In this framework, hydrodynamic parameters are treated as realizations of a stochastic process conditioned to available data and characterized by a random function model inferred from the own data. In practical cases, the difficulty to solve the stochastic partial differential equations resulting from this approach leads to proceed simulating multiple equally likely conditional parameter fields, solving the classic deterministic equations, and then analyzing the ensemble of results. In order to obtain more realistic results, the stochastic conditional parameter fields have to honor the available variable measurements (usually of head and concentrations). The implementation of new methodologies to reach this goal has defined a new field generally referred to as stochastic inverse modeling. In this paper we present some of the most advance state-of-the-art methods showing their application to a real case of risk assessment. It corresponds to a low pervious dolomite formation that might constitute the fastest pathway to the biosphere for radionuclides escaped from a nuclear waste repository. Hydraulic Engineering Software VIII, C.A. Brebbia & W.R. Blain (Editors)

Influence of Hydraulic Conductivity and Wellbore Design in the Fate and Transport of Nitrate in Multi-aquifer Systems

Nitrate concentrations in multi-aquifer systems are heavily affected by the presence of wellbores (active or abandoned) that are screened in several aquifers. The spatial variability of hydraulic conductivity in the confining layers has also an important impact on the concentrations. A synthetic three-dimensional flow and transport exercise was carried in a multi-aquifer system consisting of two aquifers separated by an aquitard in which 100 vertical wellbores had been drilled. To model the wellbores and the flow and transport connection between aquifers that they may induce, we assign a high vertical hydraulic conductivity and a low effective porosity to the cell blocks including the wells. With these parameters, a solute will travel quickly from one aquifer to the other without being stored in the well itself. The wellbores will act as preferential pathways, and the solute will move quickly between aquifers according to the hydrodynamic conditions. Not considering these preferential pathways could induce erroneous interpretations of the solute distribution in an aquifer. We also noted that when there are vertical wellbores that connect aquifers in a multi-aquifer system, low conductivity in the aquitard enhances the flow of solute through the wellbores. Time-varying pumping rates induce important fluctuations in nitrate concentrations; therefore, any estimate of the water quality of the aquifer will depend on the moment when the data has been recorded. Consequently, concentration maps obtained by interpolation of point samples are seldom a good indicator of the chemical status of groundwater bodies; alternatively, we recommend complementing the usual interpolated maps with numerical models to gain a true understanding of the spatial distribution of the solute concentration.