26
We propose a workflow in three steps to solve the inverse problem of matching hydraulic 566
heads and model the posterior distribution considering jointly the uncertainty in the facies 567
model and in the geological scenarios:
568
1) Construction of a geologically informed spatial prior with multiple scenarios 569
2) Validation/falsification of the prior with geophysical data 570
3) Matching dynamical data considering scenarios probabilities and geophysical data 571
using a stochastic search method 572
The originality of the method lies in the use of geophysical data both to validate/falsify 573
geological scenarios and to constrain geostatistical realizations and to perform this in a 574
manageable computational time for practical field cases. The method is sensitive to the 575
geophysical parameters used to produce simulated models in the falsification procedure. They 576
should be chosen carefully to avoid eliminating consistent scenarios. We have successfully 577
assessed the validity of the method on 4 synthetic test cases and demonstrated its applicability 578
on a case study in an alluvial aquifer using MPS and PPM. However, one of the workflow’s 579
strength is its adaptability. The workflow is not limited to training image-based scenarios, step 580
1 can be based on variogram-based scenarios or any other geostatistical methods.
581
The falsification procedure causes an additional computational cost, but it is relatively small 582
compared to the time required by the hydrogeological inverse procedure. The additional cost 583
is related to the forward and inverse modeling of synthetic geophysical data. As an example, 584
the falsification procedure for the field case required 96 synthetic models. It was run on a 3.07 585
GHz computer with 8 GB RAM in 8.5 hours without parallelization. The MDS procedure 586
only takes a few minutes. As a comparison, the time to calibrate one model with PPM on the 587
same computer, given the size of the 3D model, takes about 3 hours. The cost increases when 588
27
more synthetic models are used and is highly dependent on the type of data used for 589
falsification. It can be easily estimated knowing the time needed for one synthetic model.
590
Similarly, the application of MDS mapping for the calculation of probability in step 2 is not 591
limited to the use of geophysical data. One may combine both geophysical and informative 592
hydrogeological data sets, to assess the probability of scenarios P(Sc|DERT,Dhydro) or even use 593
hydrogeological data alone. In this study, we used geophysical data only because there are 594
more informative on the spatial distribution of facies than head data. The fast calibration of 595
models from any scenario suggests that head data are not very useful to discriminate the 596
considered scenarios in this case. Transient data would likely be more informative. Actually, 597
any method relevant to assess the pertinence of geological scenarios could be applied.
598
Combining different data types would require the definition of a distance balancing between 599
several terms and should result in some refinement in the falsification/updating procedure to 600
identify more precisely the most probable scenarios. However, it would be at the cost of more 601
computations, hence be more time consuming. Other geophysical techniques, such as the 602
spectral induced polarization method could also be used, given its high potential for 603
discriminating facies based on their permeability. Step 2 strongly depends on the definition of 604
the distance metrics, the latter could be adapted depending on the context.
605
The application of the falsification procedure on several test cases highlights that the method 606
is suitable for the rejection of inconsistent training images because the reference or field 607
model lies outside the distribution of the prior. In most cases, the methodology correctly 608
identifies the reference training image (highest updated probability). However, 609
misclassification due to spatial uncertainty may occur resulting in an incorrect training image 610
being favored. This illustrates that the geophysical data is not always sufficient to reveal 611
geological scenarios and that a large prior (several scenarios) is needed to avoid 612
underestimating uncertainty. The number of simulated models should always be sufficient to 613
28
estimate the density distribution in the MDS map. Otherwise, updated probability could be 614
badly estimated.
615
In the last step of the workflow, any sampling technique can be considered as long as it can 616
integrate the probability of geological scenarios. It means that McMC or any other techniques 617
can be used to sample the pre-posterior distributions and then combine the results using 618
equation 1.
619
In many hydrogeological problems, the model relies on relatively sparse data generally 620
limited to observed borehole facies. Potential-based methods such as electromagnetic or DC 621
resistivity methods are commonly used to characterize aquifers but, they are rarely considered 622
directly in the modeling step. While the MPS framework proposes an established way to 623
integrate soft data, very few hydrogeological studies have considered the use of geophysics as 624
spatially distributed conditioning data in MPS simulations [Trainor, 2010]. Our work shows 625
that the use of ERT as soft data is twofold. First, constraining MPS simulations with 626
geophysical data reduces the number of possible models that are proposed to the PPM 627
algorithm. It reduces the variability in the prior and subsequently reduces the variability in the 628
posterior distribution, too. Second, the convergence of the PPM is increased. This is a non-629
negligible advantage regarding the high computational demand for solving non-linear flow 630
and transport models.
631
Acknowledgement 632
The PPM code and a corresponding data set is available at 633
https://github.com/SCRFpublic/PPM. Other material can be obtained on demand, please 634
contact the corresponding author.
635
We thank the F.R.S.-FNRS and the Fondation Roi Baudouin — Prix Ernest Dubois (grant No.
636
2013-8126501-F002) for their financial support of Thomas Hermans. We are grateful to Alain 637
29
Dassargues and the Hydrogeology and Environmental Engineering unit of the University of 638
Liege for giving us access to the experimental site of Hermalle-sous-Argenteau.
639
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38 TABLE
848
Confusion matrix (%) Mean updated probability (%)
Scenario SC/SL BL1 BL2 BC P(SC/SL|ref) P(BL1|ref) P(BL2|ref) P(BC|ref)
SC/SL 100 0 0 0 88.3 1.8 0.8 9.1
BL1 3 96 1 0 3 75.1 21.7 0.2
BL2 3 15 82 0 1.7 21.6 76.6 0.1
BC 14 0 0 86 14.5 0.1 0 85.4
Table 1. Confusion matrix and updated probability for Case 1 in 2 dimensions.
849
Confusion matrix (%) Mean updated probability (%)
Scenario SC/SL SC/BL MC/SL MC/BL P(SC/SL|ref) P(SCLBL|ref) P(MC/SL|ref) P(MC/BL|ref)
SC/SL 93 4 2 1 62.9 18.2 15.4 3.5
SC/BL 11 88 1 0 18.3 68.2 10.6 2.9
MC/SL 29 10 51 10 24.2 12 42.7 21.1
MC/BL 8 6 19 67 9.9 6.1 30 54
Table 2. Confusion matrix and updated probability for Case 2 in 5 dimensions.
850
Scenario SC/SL SC/BL MC/SL MC/BL BL1 BL2 BC SC Classification
with SC/SL 70 2 4 1 0 0 0 23
without SC/SL 4 32 1 0 0 0 63
Mean updated probability
with SC/SL 46.2 7.7 14.7 7 0.8 0.4 2.2 21 without SC/SL 17 26.7 8.3 2 1.4 4 40.6
Table 3. Classification performance and updated probability for Case 3 in 5 dimensions.
851
39 852
Scenario Gravel geometry P(Sci) P(Sci|DERT) – 2D P(Sci|DERT) – 5D
Sc1 Small channels 0.25 0.2 0.29
Sc2 Medium channels 0.25 0.18 0.14
Sc3 Small bars 0.25 0.27 0.27
Sc4 Big bars 0.25 0.35 0.30
Table 4: Scenario and updated probability 853
854
40 FIGURE CAPTIONS
855
Figure 1: The eight training images used for the synthetic study are composed of gravel 856
channels and clay lobes of different sizes, SC stands for small channels, MC for medium 857
channels, BC for big channels, SL for small lobes, BL for big lobes. BL1 and BL2 have 858