Probability perturbation method applied to the inversion of groundwater flow models using HydroGeoSphere

Conference (2013, April 04)

Solving spatial inverse problems in Earth Sciences remains a big challenge given the high number of parameters to invert for and the complexity of non-linear forward models. Techniques were developed to ... [more ▼]

Solving spatial inverse problems in Earth Sciences remains a big challenge given the high number of parameters to invert for and the complexity of non-linear forward models. Techniques were developed to reduce the number of parameters to invert for or to produce geologically consistent simulations from an initial guess. These techniques ask for a prior model to constrain the spatial distribution of the solution. Geostatistical models contain, by nature, information to control the spatial features of the inverse solutions, but the integration of dynamic data into such models remains difficult. We adapted, the “probability perturbation algorithm” (PPM) using Matlab® to invert hydrogeological data using multiple-point geostatistics to build models of pre-defined hydrofacies. The algorithm uses HydroGeoSphere (HGS) to compute the forward response of the model and SGems to produce geostatistical realizations. The algorithm only needs the proper definition of all the parameters to be used by HydroGeoSphere (grid matching with SGems, position of the wells, pumping rate, facies properties, boundary conditions, etc.). The PPM algorithm will automatically seek solutions fitting both hydrogeological data and geostatistical constraints. Through the inversion process, the initial geostatistical realization is perturbed. Only geometrical features of the model are affected, i.e. we do not attempt to directly find the optimal value of hydrogeological parameters, but the optimal spatial distribution of facies whose prior distribution is quantified in a training image. The algorithm can be divided in three steps. In the first step, we use SGems to generate an initial facies model with the multiple-point geostatistical algorithm SNESIM (single normal equation simulation). The facies model is composed of several categories representing hydrological facies (e.g. gravel, sand and clay). It can be conditioned using hard data (borehole data) and/or soft data (e.g. geophysical data). We then run a first flow simulation with HydroGeoSphere. This requires defining hydrogeological parameters (porosity, hydraulic conductivity, etc.) for each category of the facies model to create a hydrogeological model. The response of the latter model is compared to the expected one through an objective function. In the second step, a perturbation to the facies model is computed using a single parameter called rD. This perturbation is used to generate a new facies model with SGems and calculate a new objective function value via HGS, as done in the first step. An inner loop optimizes the value of rD. In the third step, we verify if the objective function of the best fitting model is smaller than a predefined value. If it is the case, we stop the algorithm, otherwise we go back to step 2 until convergence. We illustrate the methodology with a synthetic example in an alluvial aquifer. The model is based on a training image depicting gravel channels and clay lenses in a coarse sand aquifer. We simulate a pumping test and inverse water level data recorded at 9 wells using our implementation of the PPM algorithm. Using this method, it is possible to generate multiple solutions and to derive a posterior probability of the facies distribution. [less ▲]

Inversion of multi-temporal geoelectrical data sets: insights from several case studies

in Berichte der Geologischen Bundesanstalt (2012, September), 93

Inversion of multi-temporal geoelectrical field data sets: insights on noise characterization and regularization

Poster (2012, July 11)

Inversion of geoelectrical time-lapse data sets is increasingly growing as monitoring systems are being used in more applications such as seawater intrusion, landslides, remediation of contaminated sites ... [more ▼]

Inversion of geoelectrical time-lapse data sets is increasingly growing as monitoring systems are being used in more applications such as seawater intrusion, landslides, remediation of contaminated sites, landfill operation, shallow geothermal systems, or management of water resources. To date, several inversion strategies exist for taking into account the temporal dimension of the data. The most used nowadays are the independent inversion of multi-temporal data sets, the difference inversion, the temporally-constrained inversion, and the more recent process-based inversion. However, difference inversion schemes generally assume that part of the noise contained in the data cancels out when working with temporal data differences. Temporally-constrained inversion on the other hand assumes that the changes are localized and minor. Process-based inversion requires a more advanced knowledge of the system prior the inversion. In this study we demonstrate that the resolution of the time-lapse inversion scheme is mostly dependent on the quantification of the temporal behavior of the data error, on the resolution of the model-dependent pattern of the survey, and not on the regularization strategy. Our study is based on the imaging results of different data sets with different time and spatial scales, and with different degrees of geological complexity and resistivity contrast, The considered sites are a shallow sandy aquifer and a fractured hard rock aquifer where tracer experiments were performed and monitored using surface arrays. The two studied transport processes are advection, with velocities on the order of 10 m/hour and slower advection/diffusion processes. The strongest improvements were brought by using the data difference and a quantitative estimation of the data error. We found in particular a dependence of the time-lapse data error to the measured resistance (i.e., signal-to-noise-ratio), permitting to formulate an error model to describe the data error present in time-lapse data sets. We used minimum gradient support regularization to invert for model changes with enhanced contrast and found this technique more suited to time-lapse studies than for static images. Noise characterization and error models appear therefore as essential and the most impacting for a successful inversion both for static and time-lapse data whereas different spatio-temporal regularization techniques allowed to decrease artefacts but needs to be coherent with the process. [less ▲]

Training image scenarios for the Meuse alluvial aquifer and consistency with geophysical data

Poster (2012, February 08)

Recently, multiple-point statistics (MPS) introduced the training image (TI) concept to replace the variogram within an extended sequential simulation in order to describe more accurately multimodal ... [more ▼]

Recently, multiple-point statistics (MPS) introduced the training image (TI) concept to replace the variogram within an extended sequential simulation in order to describe more accurately multimodal distributions, with interconnected and curvilinear structures, such as those of alluvial plains. The role of the TI is to depict the conceptual geological patterns and it should be representative of the geological heterogeneity. MPS consists in extracting patterns from the training image, and anchoring them to subsurface data (e.g. well-log, seismic and production data). The construction of TI is one of the most critical and important step of MPS. Sedimentological studies may not be always available in a particular area. In this work, the alluvial aquifer of the Meuse river in the area of Liege is being investigated and there is few sedimentological data to build directly 3D conceptual model of the aquifer. In this context, we used general features of the Meuse river (slope, rate of flow, type of fluvial system) to select hydrofacies (based on a lithological classification in clay, sand and gravel) and their geometrical characteristics (channels and lobes of different sizes). Then several scenarios were built using these parameters to represent the uncertainty related to different possible geological scenarios. To verify the consistency of these TIs, a comparison with 2D electrical resistivity tomography data was carried out. 2D sections were randomly selected in the TIs and several cases were analyzed including the size of channels and lobes, the influence of surface resistivity in the results and the influence of the electrical resistivity of each facies. Forward and inverse electrical resistivity modelling was conducted on these synthetic models and the results were compared to field cases. The approach followed for the comparison is based on the calculation of a Euclidean distance between models and the visualization in a 2D or 3D space using multidimensional scaling (MDS). This technique allows verifying if field cases fall in the distribution represented by synthetic cases. In a second step, a cluster analysis was achieved on the MDS-map to provide a sensitivity analysis and to highlight which parameters were the most important for building training images. Then, the probability of each scenario was evaluated for the field cases using conditional probability. Conditional probability requires the calculation of the density function corresponding to the probability of the data given a geological scenario. This density function was obtained using a kernel density estimation technique based on the observations of the 2D MDS-map. Both the cluster analysis and the calculation of conditional probabilities for uncertain geological scenarios show that some parameters are not very sensitive (size of clay lenses, surface resistivity distribution) and that we can narrow the range of variations of some parameters (facies electrical resistivity values, gravel bodies size is more likely small, etc.). It shows that the investigation of the consistency of TI is an important step in each study including MPS. The next steps of the study are to generalize the analysis of the consistency of geological scenario with 3D geophysical data instead of 2D sections and to incorporate geophysical data as soft conditioning data for MPS simulations. [less ▲]

A shallow geothermal experiment in a sandy aquifer monitored using electric resistivity tomography

in Geophysics (2012), 77(1), 11-21

Groundwater resources are increasingly used around the world for geothermal exploitation systems. To monitor such systems and to estimate their governing parameters, we rely mainly on borehole ... [more ▼]

Groundwater resources are increasingly used around the world for geothermal exploitation systems. To monitor such systems and to estimate their governing parameters, we rely mainly on borehole observations of the temperature field at a few locations. Bulk electrical resistivity variations can bring important information on temperature changes in aquifers. In this paper, we demonstrate the ability of surface electrical resistivity tomography to monitor spatially temperature variations in a sandy aquifer during a thermal injection test. Heated water (48°C) was injected for 70 hours at the rate of 87 l/h in a 10.5°C aquifer. Temperature changes derived from time-lapse electrical images are in agreement with laboratory water electrical conductivity-temperature measurements. In parallel, a coupled hydrogeological saturated flow and heat transport model was calibrated on geophysical data for the conceptual model, and on hydrogeological and temperature data for the parameters. The resistivity images showed an upper flow of heated water along the well above the injection screens and lead to a new conceptualization of the hydrogeological source term. The comparison between the temperature models derived from resistivity images and from the simulations is satisfactory. Quantitatively, resistivity changes allowed estimating temperature changes within the aquifer, and qualitatively, the heated plume evolution was successfully monitored. This work demonstrates the ability of electrical resistivity tomography to study heat and storage experiments in shallow aquifers. These results could potentially lead to a number of practical applications, such as the monitoring or the design of shallow geothermal systems. [less ▲]

Integration of near-surface geophysical, geological and hydrogeological data with multiple-point geostatistics in alluvial aquifers

Scientific conference (2011, October 20)

This work takes place in the context of the following PhD research topic. Alluvial plains constitute essential geological bodies for environmental studies such contaminated sites remediation, low-enthalpy ... [more ▼]

This work takes place in the context of the following PhD research topic. Alluvial plains constitute essential geological bodies for environmental studies such contaminated sites remediation, low-enthalpy geothermal energy or groundwater resources. The general objective of this work is to contribute to a better quantification of the heterogeneity of the parameters governing flow and transport processes by combining near-surface geophysics and geostatistics. Two-points geostatistical approaches (variogram based) have been developed to quantify the heterogeneity of one geological formation but fail to reproduce the heterogeneity of fluvial deposits with multiple facies. Multiple-points geostatistics introduced the training image concept to replace the variogram within an extended sequential simulation framework. The role of the training image is to depict the conceptual geological patterns. Previous studies have demonstrated the need for more conditioning data to generate an efficient training image. The use of geophysics in this context has been studied in the petroleum research with wave-based methods (seismic reflection data). However, little research has been done to assess the use of near-surface geophysical measurements, relying on potential and wave methods, to condition multiple-point geostatistics for environmental studies. This research project will focus on three specific objectives. The first will consist in computing conceptual training images from the combination of near-surface geophysics images, geological information and borehole data. The second objective will use these images to build models of the subsurface properties with multiple-point geostatistics conditioned to site-specific near-surface geophysical data and borehole information. Finally, groundwater model parameter estimation will be performed using a geologically consistent perturbation approach based on tracer experiments, using well sampling and time-lapse geophysical data. The approach will be demonstrated on both synthetic benchmarks and real field sites. This presentation shows preliminary research results on the use of combined geophysical techniques to improve geophysical models, the building of geological scenarios (training images) in the Meuse alluvial aquifer and the use of time-lapse ERT data to improve our understanding of hydrogeological reservoirs. [less ▲]

Shallow heat injection and storage experiment : heat transport simulation and sensitivity analysis.

in Journal of Hydrology (2011), 409(1-2), 262-272

Interest in heat transport in porous media has increased because of its many applications such use as tracer or in geotechnical engineering solutions. Understanding of the physical processes and ... [more ▼]

Interest in heat transport in porous media has increased because of its many applications such use as tracer or in geotechnical engineering solutions. Understanding of the physical processes and parameters determining heat transport is therefore important. In this paper, heat transport is studied during a shallow heat injection and storage field test. The test is simulated using SEAWAT. Sensitivity analyses and collinear diagnostics are used to derive which parameters can be derived from the test and how reliable these values are. Heat transport during the test is compared with heat transport in the surficial zone at the same field site to compare parameter values. The most sensitive parameter is the thermal conductivity of the solid followed by the porosity, heat capacity of the solid and the longitudinal dispersivity. This indicates the predominance of conductive transport during the storage phase over the convective transport during the injection phase. Whereas heat transport in the surficial zone is insensitive to the longitudinal dispersivity, this parameter must be included to simulate the field test. This indicates that dispersivity can not be ignored simulating convective heat transport in aquifers. [less ▲]

How to incorporate prior information in geophysical inverse problems: deterministic and geostatistical approaches.

in EarthDoc - Near Surface 2011 - 17th European Meeting of Environmental and Engineering Geophysics (2011, September 13)

Many geophysical inverse problems are ill-posed leading to non-uniqueness of the solution. It is thus important to reduce the amount of mathematical solutions to more geologically plausible models by ... [more ▼]

Many geophysical inverse problems are ill-posed leading to non-uniqueness of the solution. It is thus important to reduce the amount of mathematical solutions to more geologically plausible models by regularizing the inverse problem and incorporating all available prior information in the inversion process. We compare three different ways to go beyond standard Occam’s inversion for electrical resistivity tomography (ERT) using electromagnetic logging data in the context of salt water infiltration: a simple reference model, a structural constraint and a geostatistical constraint based on a vertical correlation length. Results with the traditional smoothness constraint yield small contrasts of resistivity, far from the reality revealed by borehole measurements. Incorporating prior information from boreholes clearly improves the misfit with logging data. If a good reference model can always be used, it can lead to misinterpretation if its weight is too strong. When the computation of the correlation length is possible, the geostatistical inversion gives satisfactory results everywhere in the section. In this specific case, the geostatistical approach seems to be a more robust way to incorporate prior information. The structural constraint seems to be more indicated when integrating information from other geophysical methods such as GPR or seismic. [less ▲]

Using geostatistical constraints in electrical imaging for improved reservoir characterization

Conference (2010, December 14)

Developing predictive models of reservoirs is often complicated by the spatial heterogeneities and the different scales which control flow and transport processes. In numerous studies over the past two ... [more ▼]

Developing predictive models of reservoirs is often complicated by the spatial heterogeneities and the different scales which control flow and transport processes. In numerous studies over the past two decades, geophysical imaging techniques have proved very useful for reservoir characterization. However, the loss of resolution and the non-uniqueness of standard solutions to inverse problems strongly limit the use of such deterministic imaging approaches. On the other hand, the use of common geostatistical approaches for reservoir characterization, for instance from logging information, may be a difficult task, since accurate variogram information is difficult to obtain (dense sampling in the vertical and lateral directions), and also because a high number of conditioned simulations is needed to remove statistical bias. Combining the high spatial sampling of deterministic geophysical imaging methods with geostatistical constraints, valid in the whole image plane, appears as a very promising approach to enhance reservoir characterization. To do so, we use a parameterized model covariance matrix based on standard variogram functions and a prior model as regularization operator in the inversion of electrical resistance data. This way of including additional data is not restricted to electrical data but the variogram parameters may be also inferred from for example available textural or lithological information. The benefit of the presented approach is twofold: (i) It honors the spatial statistics of the reservoir and (ii) it alters the posterior model by further reducing model ambiguity inherent to the inversion compared to classical (smooth model) regularization. The proof of concept is given by synthetic studies carried out on random fields from Gauss simulations with varying (an)isotropic scale lengths using different model (co)variogram functions. We also demonstrate the approach on electrical field data combined with borehole electromagnetic data from two artificial sea inlets in the nature reserve "The Westhoek" near the French-Belgian border. The electromagnetic logs were used to calculate an experimental vertical variogram characteristic of the study site. The results enabled to determine the extension of the salt water plume laterally, and significantly enhance its extension in depth, but also in terms of total dissolved solid content. These observations are in agreement with the hydrogeological situation at the site. A comparison with borehole data shows that the results are much more plausible than results obtained with a traditional smoothness constraint used as regularization operator. In conclusion, the incorporation of geostatistical information, vertical variograms in our case, in the inverse process improves imaging capabilities for reservoir characterization significantly. [less ▲]

The use of a priori information in electrical resistivity tomography for salt water intrusion studies at the Belgian coast

Master's dissertation (2010)

Two artificial sea inlets were built in the nature reserve "The Westhoek". Consequently, sea water infiltrated in the dune aquifer filled with fresh water and exploited by a water company. Due to the ... [more ▼]

Two artificial sea inlets were built in the nature reserve "The Westhoek". Consequently, sea water infiltrated in the dune aquifer filled with fresh water and exploited by a water company. Due to the presence of a clayey layer with a low hydraulic conductivity, the vertical flow of salt water was stopped, leading to a horizontal flow outside the infiltration ponds. Electrical resistivity tomographies were carried out, in addition to borehole electromagnetic measurements. The latter were used to calculate a vertical variogram, characteristic of the study site. Then, a geostatistical constraint was imposed as a regularization tool to solve the inverse problem, with a covariance matrix based on the variogram. The important points of this type of regularization were first highlighted by several synthetic cases. The main conclusions were that the choice of a homogeneous prior model seemed judicious in this case, whereas the lack of information to determine the horizontal variogram was not a major issue. Inversion results enabled to determine the extension of the salt water plume laterally, in depth, but also in terms of total dissolved solid content (TDS). These observations are in agreement with the hydrogeological situation of the site. A comparison with borehole data showed that the results are much more satisfying than a traditional smoothness constraint used as a regularization tool. Prior information included in the inversion process enabled to improve the resolution, even if traditional image appraisal tools failed to quantify this contribution. [less ▲]