Inverting hydraulic heads in an alluvial aquifer constrained with ERT data through MPS and PPM: a case study Hermans T. 1, Scheidt C. 2, Caers J. 2, Nguyen F. 1 1 University of Liege, Applied Geophysics 2 Stanford University, Energy Resources Department GEOENV 2014
The construction of hydrogeological models is generally done with sparse data There are generally two types of data : borehole log description (lithology) and hydraulic head levels 2
What happens between boreholes???? Geophysical methods provide spatially distributed information Clay Sand Bed-rock Gravel 3
The integration of hydrogeological data in geostatistics is not straightforward Geostatistical models are useful tools to integrate various types of data and to impose a priori spatial constraints However, integrating hydrogeological data remains challenging Hydraulic heads and tracer tests data result from a dynamical process which requires to solve flow (and transport) equations Possible solutions : the probability perturbation method, pilot-point method, gradual deformation, MCMC, etc. 4
Context and choice of the training image Probability Perturbation Method Synthetic Benchmark Field case : Hermalle-sous-Argenteau (Belgium) 5
Context and choice of the training image Probability Perturbation Method Synthetic Benchmark Field case : Hermalle-sous-Argenteau (Belgium) 6
We investigate the site of Hermalle-sous-Argenteau (Belgium) in the alluvial plain of the Meuse River 7
From our knowledge of the deposits, we propose four different scenarios for training images 3 facies based on high, intermediate and low hydraulic conductivity 2 scenarios for the gravel facies : channels or bars with two sizes proportions : gravel = 42%, sand = 40%, clay = 18% (from borehole) 8
We would like to ensure that training images and geophysical models are somehow compatible This would ensure that conditioning with soft data will not create conflict with features of the TI. 9
For each training images, 2D sections are randomly extracted to be compared with field models For each model, facies are assigned a value of electrical resistivity ERT data are simulated and inverted to obtain geophysical inverted models that can be compared to true field data 10
All the models (field and synthetic) are mapped using multidimensional scaling Hermans et al., IAMG 2013 Park et al., 2013 If the TI-based scenarios are consistent with ERT, field models should fall in the distribution of synthetic models 11
The probability of each scenario according to field data is computed using krenel density estimation P(TI ti D d ) k All scenarios have the same prior probability P = 0.25 obs f (d obs ti k )P(TI ti k) f (d ti )P(TI ti ) k obs k k (Park et al., 2013) Training Images conditional probabilities SC MC SB BB Mean 0.2 0.18 0.27 0.35 SC = small channels SB = small bars MC = medium channels BB = big bars Scenarios with channels have lower probabilities of occurrence than scenarios with bars 12
The electrical resistivity distribution can be transformed into probability maps to constrain MPS simulations f ( A)P(A) P(A ) f ( ) = Distribution of resistivity ρ given the facies = Probability to observe a facies given the resistivity ρ 13
Context and choice of the training image Probability Perturbation Method Synthetic Benchmark Field case : Hermalle-sous-Argenteau (Belgium) 14
The probability perturbation method (PPM) is a stochastic, iterative inversion method optimizing the geometry of the deposits Generate initial model with MPS Run flow simulation with HGS 1. Initialization with prior model constraints 1 calibrated realization Yes N i 1 (h h ) obs calc 2 i i N 3. Inner loop to optimize the perturbation No Change the random seed to generate new random numbers Choose a value of r D the perturbation parameter Generate a model with MPS Run flow simulation with HGS Converge to best r D? 2. Outer loop to ensure the convergence The structure of the algorithm is divided in three steps 15
The perturbation is computed through a single parameter in the inner loop We introduce : D = the observed hydraulic heads P(A D) is the probability to have a facies distribution according to these data. It is estimated by P(A D) = (1-r D ) I 0 (u) + r D P(A) I 0 (u) = best matching model from MPS (constrained by TI, hard data and geophysical data) = realization of the joint probability P(A B,C) r D =0 : no perturbation r D =1 : new independent realization P(A B,C,D) is calculated through the tau model 16
The process converges towards the target level of the objective function and thus to a model explaining all available data At each iteration r D is optimized The model is coherent with direct geological observations (imposed), near-surface geophysical data (soft data) and hydrogeological data (dynamic data) The process is repeated to obtain several equiprobable models 17
Context and choice of the training image Probability Perturbation Method Synthetic Benchmark Field case : Hermalle-sous-Argenteau (Belgium) 18
The synthetic benchmark was designed to simulate an alluvial aquifer TI and prior probability Hard Data Data to reproduce Soft Data = ERT data Clay lenses 22% : K = 10-6 m/s Gravel channels 20% : K = 10-2 m/s Sand matrix 58% : K = 10-4 m/s The aim is to study the role of geophysical data and the choice of the training image on final results 19
We calculate the posterior distributions based on the mean of 50 realizations With geophysical soft data Mean OF after 10 iterations = 0.028 m Without geophysical soft data Mean OF after 10 iterations = 0.06 m Geophysical data (ERT) help to build plausible models by giving information on facies between boreholes Through this improvement of models, geophysical data increase the convergence of the method 20
The training image influences the geometry of the deposits and consequently modifies groundwater flows and posterior distributions Gravel posterior probability Small channels Gravel posterior probability Medium channels Bigger channels yield more continuous high probability gravel zones in the posterior distribution 21
Geophysical data reduce the «error» introduced by the wrong training image (here bigger channels) Gravel posterior probability Medium channels Without geophysics Gravel posterior probability Medium channels With geophysics Geophysical data correct the posterior distribution where the uncertainty related to the presence of gravel should be higher 22
Context and choice of the training image Probability Perturbation Method Synthetic Benchmark Field case : Hermalle-sous-Argenteau (Belgium) 23
The methodology is applied to the site of Hermalle-sous- Argenteau with a pumping followed in 8 piezometers and the pumping well 24
Electrical resistivity models are used to constrain MPS simulations 12 parallel ERT profiles Probability map for each facies (here sand) 25
PPM is applied 50 times with a different initial model, posterior distributions are used to classify each cell of the model Posterior probability of gravel Classification 26
PPM is applied 50 times with a different initial model, posterior distributions are used to classify each cell of the model Posterior probability of gravel Classification 27
The results for the 4 TI are linearly combined using their respective probability to produce final maps Posterior probability of gravel 28
The results for the 4 TI are linearly combined using their respective probability to produce final maps Classification 29
Conclusion and perspectives 30
Development of a global methodology to integrate nearsurface geophysical, geological and hydrogeological data with multiple-point geostatistics TI Uncertainty Soft data Models Hard data + = Hydrogeological data Posterior distributions Classification Application to 3 facies models with a potential geophysical method (ERT) 31
The method can now be extended to other geological contexts, other scales and other geophysical methods Fractured limestones Water catchment scale SIP (Robert, 2012) 32
THANK YOU VERY MUCH FOR YOUR ATTENTION 33
APPENDICES 34
Traditional geostatistics (variogram-based, kriging) fail to reproduce multimodal distributions and complex structures 3 different kinds of heterogeneity having similar variograms (Caers and Zhang, 2002) Multiple-point geostatistics (MPS) were developed to overcome this problem 35
MPS uses a training image (TI) to describe the expected geological heterogeneity, generally with discrete variables TI representing sinuous fluvial channels partially interconnected 500 m The training image is scanned with a given template of neighbors The reproductions of each possible combination are counted and stored in a search tree 36
Sequential simulations are used to produce equiprobable realizations (SNESIM algorithm) The neighborhood of the simulated point is extracted, and compared to the search tree to derive the conditional probability 2 examples of equiprobable realizations Training image 100 m 100 m 500 m 37
There is a lack of sedimentological data concerning the alluvial aquifer of the Meuse, to build accurate training image A lot of boreholes are available but poorly described (only basic description of the types of sediments) and far away from each other More than 30 different descriptions of facies e.g.: gravelly loam, clayey gravel, loamy sands, sandy gravel, gravelly sand, etc. Proportions vary from site to site From the sedimentological point of view, we can expect elements from meandering and braided systems 38
We will find sand/gravel bars or channels and point bars, loam/clay crevasse splays and floodplain deposits Meandering systems 1 channel Braided systems 1 main channel + secondary channels Channel lag deposits Point bar deposits Levees Crevasse splays Floodplain deposits Channel bars Longitudinal bars Cut and fill structures Floodplain deposits Old channel fill 39
The final aim is to provide hydrogeological models of the deposits The facies must be related to the hydrogeological properties 3 facies description 1) Gravel = high hydraulic conductivity = channel or bars 2) Clay and loam = low hydraulic conductivity = floodplain, filling and crevasse splays 3) Sand and sandy gravel = intermediate hydraulic conductivity = bars, levees, low energy channels 40
Consistency 41
MPS simulations use a TI to depict the expected geological heterogeneity Size of channels? Orientation? Size of lenses? Considerable uncertainty often remains concerning the geometry and architecture of facies elements. 42
Geophysical data may provide spatial information on subsurface properties clay gravel sand Classification based on electrical resistivity Electrical resistivity tomography (ERT) yields a model of subsurface resistivity, high values being linked with sand and low values with clay. 43
The use of geophysical image to extract TI is limited due to the resolution of geophysics and the philosophy of MPS 1. Geophysical methods are not able to differentiate unequivocally different facies: small scale features may be hidden by larger elements 2. Geophysical methods may be used to constraint spatially MPS simulations, TI and soft data should be independent Training image 44
They are compared to inverted field data Example of an ERT profile acquired on the site of Hermalle-sous-Argenteau 45
Synthetic inverted sections and true field sections are compared using multi-dimensional scaling (MDS) 1. The Euclidean distance between any two inverted models (electrical resistivity of each cell) is calculated d ( ) ( ) T ij i j i j 2. This defines a metric space in which the L models are defined in terms of distance to each other 3. A change of variable is done such that 1 2 1 T B HAH Size L x L with aij dij and H I 11 2 L 4. The eigenvalue decomposition of B offers a way to project the models into a d- dimension map, using the d first eigenvalues and eigenvectors. B V V T B B B MDS X V 1/2 d B,d B,d 5. X d are the coordinates of the model in the MDS map of d-dimensions 46
The probability analysis of different scenarios can be seen as a conditional probability problem P(TI ti D d ) k obs f (d obs ti k )P(TI ti k) f (d ti )P(TI ti ) k obs k k d obs is the projection of the response from each synthetic model indicating a location on the MDS map We can estimate f (d obs ti k ) using the adaptive kernel density estimation technique with a bivariate normal function applied on the MDS map 2 1 1 (x ) (x ) f (x, x ) exp( [ ]) 2 X 1 2 1 x 2 y 2 2 2 x y 2 x y (Park et al., 2013) x 1 and x 2 are the coordinates of the models, μ x and μ y correspond to the coordinates of the reference value (the field data for example), σ x and σ y are bandwith parameters depending on the cluster 47
The proposed methodology enables to verify the «consistency» of trainingimage based scenarios with geophysical data using synthetic benchmarks This method is a way to justify and validate the choice of a training image and to propose alternatives for uncertainty analysis and stochastic simulations 48
Important remarks and limitations 1. The choice of resistivity distribution plays an important role, the geophysicist must ensure that reliable values are chosen 2. The choice of the Euclidean distance is locally dependent which may be a drawback in MPS since training image are not locally dependent 3. It would be possible to use other types of distance measurements or other attributes than resistivity 4. The choice of the dimensions of the map is often 2D or 3D for visual inspection, here it represents about 60% of the total variance, higher dimensions are possible 5. 3D geophysical surveys are more costly and time consuming, but the methodology can be extended to 3D models as well, it would be useful to distinguish between channels and elongated bars 6. The methodology shows that there is some consistency between TI and geophysical data, but not that the TI is perfect or that other scenarios are impossible 49
The initial and the history matched models both respect the same prior constraints, i.e. the same geostatistical parameters Some features are quite similar (e.g. red zone), other features are completely different (e.g. blue zone) 50
ERT data are simulated and a 3D model of resistivity is created and transformed in probability maps Gravel probability map Clay probability map High probability of gravel are related to the presence of gravel channels as imaged by ERT 51