Uncertainty Quantification in Reservoir Prediction: Part 2—Handling Uncertainty in the Geological Scenario
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Abstract
Models used for reservoir prediction are subject to various types of uncertainty, and interpretational uncertainty is one of the most difficult to quantify due to the subjective nature of creating different scenarios of the geology and due to the difficultly of propagating these scenarios into uncertainty quantification workflows. Nonuniqueness in geological interpretation often leads to different ways to define the model. Uncertainty in the model definition is related to the equations that are used to describe the modelled reality. Therefore, it is quite challenging to quantify uncertainty between different model definitions, because they may include completely different model parameters. This paper is a continuation of work to capture geological uncertainties in history matching and presents a workflow to handle uncertainty in the geological scenario (i.e. the conceptual geological model) to quantify its impact on the reservoir forecasting and uncertainty quantification. The workflow is based on inferring uncertainty from multiple calibrated models, which are solutions of an inverse problem, using adaptive stochastic sampling and Bayesian inference. The inverse problem is solved by sampling a combined space of geological model parameters and a space of reservoir model descriptions, which represents uncertainty across different modelling concepts based on multiple geological interpretations. The workflow includes building a metric space for reservoir model descriptions using multidimensional scaling and classifying the metric space with support vector machines. The proposed workflow is applied to a synthetic reservoir model example to history match it to the known truth case reservoir response. The reservoir model was designed using a multipoint statistics algorithm with multiple training images as alternative geological interpretations. A comparison was made between predictions based on multiple reservoir descriptions and those of a single one, revealing improved performance in uncertainty quantification when using multiple training images.
Keywords
Uncertainty History matching Model calibration Geostatistics Reservoir modelling Training image Multipoint statistics Fluvial geology Metric space Support vector classification1 Introduction
Geological uncertainties can be quantified in producing assets by a Bayesian process of first history matching (HM) geological parameters of the reservoir to production data using a leastsquares misfit objective function, then using a Markov chain Monte Carlo (MCMC)based postprocessor to estimate the Bayesian credible intervals. This general workflow was described previously in Arnold et al. (2018), which covers the first part of this work on geological history matching. The basic workflow is given again in Fig. 1, where geological uncertainties are included as parameters of the static model, which are sampled from within complex nonuniform priors based on measured geological data sets.
 1.
The geological interpretation of the system is known, and there is no ambiguity in the key geological features or their correlation structure.
 2.
The stratigraphic approach to dividing up the reservoir is correct. There is no ambiguity in the use of, for instance lithostratigraphic versus sequence stratigraphic concepts implemented in the model (Larue and Legarre 2004).
 3.
The data chosen for characterising the model and for history matching are appropriate for the task, and all errors in the data are accounted for in an appropriate error model (O’Sullivan and Christie 2005).
 4.
The misfit definition is appropriate for the available data, given its errors.
 5.
The modelling approach chosen to capture the reservoir is ideal for the geological scenario.
 6.
The model parameterisation covers all key uncertainties such that the forecast uncertainties are robust.
Often, the above criteria are violated due to lack of knowledge about the reservoir, therefore there is a need to understand where the ambiguities come from and at what level. The list above demonstrates the hierarchical nature of handling geological reservoir uncertainties, which therefore should be dealt with in a hierarchical way. At the highest level, one may not have a good interpretational understanding of the reservoir, therefore all choices made on the data, modelling approaches and parameterisations cascade from the initial decision on what the geology “is”.
Uncertainty in geological interpretation has been mentioned by a number of authors, including Rankey and Mitchell (2003) and Bond et al. (2007). Both showed the ambiguity present in the geological interpretation based on seismic data. Rankey and Mitchell (2003) showed a great example of overconfidence in expert opinion. Six experts were asked to interpret seismic data from a carbonate reservoir and comment on their belief in their estimates. All believed theirs was closer to reality based on the fact that the interpretation was easy for the majority of the field. However, portions of the reservoir offramp were less well defined and added considerable variation to the volumetric estimates (up to 20 % variation in gross rock volume), even though the other parts of the field were the same for all interpretations. Bond et al. (2007) showed that the same piece of seismic, given to 412 different geologists, produced a spread of different structural interpretations, which were biased by their expertise and experience.
Part of the issue is due to the various psychological biases that come into play when data must be interpreted. Baddeley et al. (2004) provides a good overview of the sources and impacts of bias in geological interpretations. In addition, geologists must also use both quantitative and qualitative information in the construction of their interpretational models (Wood and Curtis 2004).
 Scenario

This is the particular interpretation of the reservoir geology and stratigraphic model that is chosen as a basis for the modelling effort. The number of Scenarios is equal to the number of geological interpretations that exist of the geology and stratigraphic correlation.
 Model:

This is a particular combination of data and modelling method used to capture a Scenario. For any geological interpretation, there may be a number of modelling methods that are possible [e.g. the reservoir facies could be modelled using an object modelling method, a variogrambased geostatistical method such as sequential indicator simulations (SIS), multipoint statistics or a sedimentary process model] and many different data that can be applied in different ways.
 Parameterisation

This is the collection of Model parameters, and their related prior probabilities, chosen to represent the uncertainty of a reservoir. For instance, one might, for a given geomodel, have two parameterisations that focus on different sets of model parameters.
 Misfit Definition

This is the model used to describe the likelihood model of the reservoir based on the available production data and the errors in that data.
 Reservoir Description

This is the particular setup of previous levels/parts above being applied to the uncertainty quantification workflow.
 Realisation

This is a particular combination of parameters, for a particular Parameterisation of a particular Model of a particular Scenario. If one has produced, for example, 2000 iterations of a particular Parameterisation, then 1 of those 2000 iterations is a Realisation.
2 Multiple Scenarios in Uncertainty Quantification
A complete space of all possible reservoir descriptions would encompass the true prior uncertainty for the reservoir, and sampling this space with adequate realisations would allow us to predict the Bayesian posterior probability accurately. This reality is hard to accomplish, as demonstrated by the case study in Arnold et al. (2012), where a small set of uncertain Scenarios (81) made matching all these possibilities to the production data all but impossible to achieve practically.
The space of reservoir descriptions will also change over time as new data are made available, new interpretations/ideas become available from experts interrogating the data and existing reservoir descriptions become redundant. Ideally, this should be observed as a reduction in the spread/degree of estimated uncertainty, but this rarely occurs in real examples, as demonstrated by Dromgoole and Speers (1992).
The reality is that engineers are restricted computationally in how many models they can run, typically work separately from geologists during the history matching process, and tend to calibrate the most likely initial model to data rather than attempt to calibrate several. This makes including many geological uncertainties in, for instance, the structural model all but impossible without complex structural parameterisation techniques such as those demonstrated in Hu (2000) and Cherpeau and Caumon (2015). Caers et al. (2006) identified a number of modelling scenarios based on alternative training images (TIs) and different combinations of conditioning data (wells, seismic, production). Ensemble methods, such as ensemble Kalman filters, have also been used for reservoir model history matching and uncertainty prediction in combination with multipoint statistics (MPS) in Hu et al. (2013), where they considered a single training image for simplification. A machine learning approach, namely multiple kernel learning, was recently applied in Demyanov et al. (2015) to handle multiple geological scenarios in history matching by selecting and blending the most relevant spatial features from a prior ensemble of realisations.
A distance metric approach in reservoir predictions pioneered at Stanford University (Suzuki and Caers 2008) was extended to scenario modelling and optimisation with multiple training images in Jung et al. (2013). This approach was used more recently for updating development decisions through reservoir life stages under uncertainty in geological model scenarios (depositional Scheidt and Caers (2014); production Arnold et al. (2016)). More research on uncertainty quantification across multiple scenarios was published in Jiang et al. (2016), considering different reservoir performance metrics, and in Sun et al. (2017) on inverse modelling with different fracture scenarios. A closedloop workflow through reservoir development life stages—appraisal and development, early production, history matching, infill well optimisation—was presented in Arnold et al. (2016).
In this paper, we demonstrate the importance of geological interpretational uncertainty, the resulting uncertainty in appropriate model description choice and its impact on reservoir predictions. We present a workflow to account for uncertainty in the reservoir description and the model parameters which accounts for geological realism (Arnold et al. 2018). To demonstrate the issues in assessing uncertainty using only one reservoir description, we create a simple reservoir case study for history matching and uncertainty quantification in Sect. 2.1.
2.1 Case Study 1: An Example of Solving the Inverse Problem with Different Scenarios
This study compares the production responses of three possible reservoir scenarios, described by three different object shapes, with respect to history match quality and uncertainty forecasting. These were defined in three different model descriptions created in IRAP RMS™ object modelling software, and each is used to generate reservoir simulation model grids for the purposes of history matching (an oil industry term for calibrating the model response to measured reservoir production data). The geological models are included in the history matching workflow as shown in Fig. 1 for the purposes of quantifying the uncertainty in reservoir forecasts based on the match quality. Misfit is calculated using the leastsquares misfit definition, matching on oil and water rates. The model grid is a simple Cartesian grid with 30,000 grid cells with model dimensions of 2.5 by 2 km by 80 m thick, located at depth of 2500 m. The field is produced initially from a single producer/injector pair running for 5000 days at 5500 STBD. The truth case scenario in this case was synthetic, produced using channel objects with associated level and channel fill facies. Therefore, the shapes of objects are reproducible by one of the three chosen history match scenarios. The truth model was created using the same grid as the HM cases and run with the same well settings and placements.
As each of the object shapes has different numbers and types of parameters, each must have its own geological prior description—hence, it would not be appropriate to sample from the same prior for each scenario. Priors for each of the three scenarios are created (in this simple case uniform priors are used) to describe the uncertainty in object dimensions, net reservoir to gross thickness (NTG) and petrophysical properties.
History match parameters for each of the three Scenarios along with the bestmatched model parameterisation and associated misfit
Parameter number  PacMan  Hammer  Channel  

Parameter  Value  Parameter  Value  Parameter  Value  
P1  Diameter (m)  492.62  Diameter (m)  634.02  Net/gross  0.45 
P2  Facies proportion (%)  35  Facies proportion (%)  68  Channel depth (m)  5.71 
P3  Porosity (%)  24  Porosity (%)  18  Channel width (m)  71.75 
P4  Permeability (mD)  276.53  Permeability (mD)  140.57  Channel azimuth  20.43 
P5  Channel belt width (m)  740.80  
P6  Sinuosity  1.12  
P7  Number of channels  7  
P8  Belt thickness (m)  104.36  
Best misfit  204.98  47.58  137.76 
While the production forecasts with the existing well configuration are quite similar, the addition of a new well leads to larger variations in the forecast response. The PacMan model forecasts production rates lower than reality in the forecast period, while the Hammer model shows upsides in production of several hundreds of barrels per day by the end of the forecast period. Indeed, the Hammer model, which provided the best results in terms of minimum misfit (Table 1), has a wider variation in the forecast response for the three different well locations and a much larger estimate of uncertainty (Fig. 5). The use of minimum misfit as a criterion for choosing the best scenario is, therefore, not appropriate, even in this simple case. Such discrepancy between model history match and forecast quality was noted previously by Erbas and Christie (2007).
This case study demonstrates the need for a workflow that can allow history matching of multiple reservoir descriptions, where one cannot choose between multiple geological scenarios, modelling methods or parameterisations. The rest of this paper lays out a methodology to achieve a useable workflow for history matching multiple geological scenarios with geological priors using machine learning. This is demonstrated on two field examples (cases 2 and 3) and for different levels of variability between geological scenarios (cases 2 and 2a).
3 Machine Learning Approach to Handling Multiscenario Uncertainty Analysis
Uncertainty in the interpretation of the depositional environment can be addressed by a multipoint statistics (MPS) approach which uses different training images parameterised through the rotation and scaling (affinity) properties. Tuning these model parameters allows one to introduce the body size variation in aspect ratio and orientation of a training image. This approach was tested previously in Rojas et al. (2011) to account for uncertainty in training image modifications with rotation and affinity. In that work, a machine learning technique (artificial neural network) was applied to predict the uncertainty in geomorphic parameter values in a fluvial reservoir model based on the MPS training image transformation.
In this work, multidimensional scaling (MDS) was applied to the MPS reservoir model realisations based on different training images in order to map them into a unique model space and then classify the model space to group similar“looking” models. The purpose of MDS in this workflow is to reduce the complexity of the classification problem by representing the distance between the models as a point in a reduceddimensionality metric space (where the metric is the difference between, for instance, two production measurements taken at the same time).
Classification in the model space is performed using a support vector machine (SVM) trained on the initial set of realisations. The SVM classifier contoured the regions where the model realisations are based on one of the training images used. It is then possible to search the classified space in order to obtain the model realisation that would best match the historical production data. SVM classification of the model space has already been used to improve the search for betterfitting models in a HM exercise, for instance, Demyanov et al. (2010). In this work we used the adaptive stochastic algorithm particle swarm optimisation (PSO) (Mohamed et al. 2010) to navigate the combination of the SVMclassified metric space of geological scenarios and the space of the geomorphic model parameters for each reservoir description.
3.1 Mapping Reservoir Models Based on Multiple Training Images into the Metric Space Using MDS
Multidimensional scaling (MDS) has been used previously for the purposes of model clustering and classification for reservoir modelling (Scheidt and Caers 2009a, b, 2014; Arnold et al. 2016), including an application to a turbidite reservoir (Scheidt and Caers 2009b). MDS in this study uses the Euclidean distance (though other distance metrics are available) of some chosen metric (grid cell properties, production time series data) to measure the similarity/dissimilarity between models. Similar models are located closer together in metric space (Borg and Groenen 2005). MDS represents each model as a point in Ndimensional space such that the distances between models in metric space represent those of the model in the original coordinate space, where N is user defined. MDS can be used as a form of nonlinear dimensionality reduction, for instance, if the comparison is based on grid cell data (matrix) and there will frequently be more than \(1 \times 10^6\) dimensions. The dimensionality of the MDS eigenvector space depends on the number of components being considered, which can be evaluated with a stress energy diagram that shows the stress decrease with the number of eigencomponents (Wickelmaier 2003). We show an example of this later in Fig. 16a for the second case study. The reason for applying MDS to these kinds of problems is that the reduced dimensionality of metric space has advantages in improving the performance of clustering or classification processes.
A different application of distancebased methods to the petroleum industry was demonstrated by Jung et al. (2013), who used the modified Hausdorff distance metric (instead of Euclidean) between the values of grid properties then used these measures to obtain representative training images for faster MPS modelling. Park et al. (2013) performed history matching with multiple geological interpretations using the metric space, estimating the likelihood surface with a kernel density estimator. Their approach used the production data rather than the grid properties to form a distance metric (i.e. how similar the production profiles were) which compared the production of a few models with the actual production of a reservoir (prehistory matching any model) then selected the training images of the models with the production response closest to historical data.
Classification of the highdimensional metric space to aid the search for goodfitting models is a challenging problem. Classification aims to establish the correspondence between a location in the metric space and in the training image (model concept) to be used to generate a realisation. The classification solution may not be unique and is subject to how smooth or wiggly the boundaries between different classes are in the metric space. This nonuniqueness can be controlled by the complexity of the classification solution and depends on the information provided by the data realisation that describes the metric space.
3.2 Support Vector Machine Classification in Metric Space
In this work, we applied an SVM classifier that aims to balance the complexity of the model with the goodness of data fit. The SVM applied for classification in the MDS space was good at tackling the highdimensionality problem, because it controls the smoothness of the class boundaries.
The simplest approach to the classification problem is to separate two classes with a linear decision surface (a line in two dimensions, a plane in three dimensions or a hyperplane in higher dimensions). Soft margin classifiers are algorithms that allow for training errors and find a linear decision hyperplane for data that is not linearly separable (real data). This problem is avoided by using the kernel trick (which is explained below), which generates a nonlinear classifier known as SVM.
3.3 Combined Workflow for History Matching with Multiple Reservoir Descriptions and Geological Priors
 1.
Use SVM classification to generate the geological prior description from available data for the model parameters for each training image. In part 1 of this work (Arnold et al. 2018), we applied the OCSVM workflow to generate the prior for each training image though a multiclass SVM (MCSVM) that could be applied for this classification task.
 2.
Sample a number of realisations of a set of reservoir descriptions using realistic regions of parameter space (taken from step 1) and multiple TIs to generate a reservoir description space in the form of an MDS metric space and then partition it using SVM multiclass classification.
 3.
Sample the geological prior (space of geomorphic parameters) in combination with the reservoir description prior (MDS metric space) to calibrate to production data.
 4.
Infer the uncertainty of the reservoir flow response prediction from the ensemble of calibrated (history matched) models using NAB (MCMC) of the misfit ensemble to estimate the uncertainty.
3.4 Case Study 2: History Matching Using Multiple Training Images—Similar Geological Concepts
Stanford VI reservoir properties (meandering channels)
Property  Value 

Wells  18 Producers 
11 Injectors  
Fluid density (lb/ft\(^3\))  \(\hbox {Oil} = 45.1\) 
\(\hbox {Water} = 61.8\)  
Fluid viscosity (cP)  \(\hbox {Oil} = 1.18\) 
\(\hbox {Water} = 0.325 \)  
Formation volume factor  \(\hbox {Oil} = 9.98\) 
\(\hbox {Water} = 1.0 \)  
Cell dimensions  \({X}=75\hbox {m}\) 
\({Y}= 100\,\hbox {m}\)  
\({Z}=1\,\hbox {m}\)  
Number of cells  10,000 
\(50 \times 50 \times 40\)  
Top reservoir depth, TVD (ft)  2000 
Oil water contact, OWC (ft)  6000 
Petrophysical properties for each sedimentary facies for case study 2: meandering fluvial channels
Parameter  Max. (m)  Average (m)  Min. (m) 

Channel width  150  180  200 
Channel thickness  15  15  15 
Meander amplitude  900  1080  1150 
Meander wavelength  1100  1500  1700 
Petrophysical properties for each sedimentary facies for case study 2: meandering fluvial channels
History match setup  

Geomorphic parameter and range  Channel width  100–600 m 
Channel thickness  5–200 m  
Meander amplitude  500–3000 m  
Meander wavelength  500–3000 m  
Eigenvalues (metric space coordinates)  All eight eigenvalues have the same range  \(50\) to 50 
Total number of models  1000  
Minimum misfit  2000  
Sigma  6000 (STBD)  
Sampling algorithm: particle swarm optimisation  
No. of particles  20  
Group size  4  
Initial inertia  0.9  
Initial decay  0.9  
Cognitive component  1.3333  
Group component  1.333  
Social component  1.3333  
Min. steps  5  
Energy retention  0.8  
Particle behaviour  Flexible 
The characteristics of the channel geomorphic parameters are shown in Table 3 and represent uniform priors for the model parameters chosen. These were then validated using the combined OCSVM and MLP workflow described in the previous paper (part 1) (Arnold et al. 2018). Synthetic seismic data are available for Stanford VI to generate soft conditioning for each facies.
Table 4 summarises the history match settings for this case, which are the same as those used in case study 2 in Arnold et al. (2018). Forecasting the credible intervals using NAB also used the same setup as defined in that paper.
Evolution of the misfit through HM iterations with PSO is compared in Fig. 10 for the case with a single TI (using TI3 from Fig. 9c) and with the three TIs. It clearly shows that HM with multiple TIs is able to reach lower misfit values (leastsquares norm of production history \(<2000\), versus \(\approx 6000\) with a single TI) and the misfit decreases faster, with a larger number of better models being generated.
The case study setup used here demonstrates one issue in choosing very similar training images TI1 and TI2, that provide effectively the same information. This potentially underpredicts the uncertainty and adds a bias due to the truth TI coming from the space covered by the two of the three TIs used in history matching. This issue is addressed in the next section in an adaptation to this case study (case study 2a) that uses a set of three more diverse training images to capture more variation in the geological scenarios.
3.5 Case Study 2a: History Matching Using Multiple Training Images with More Varied Geological Scenarios
Case study 2 was extended to test how the workflow performs in cases where more significant differences exist between geological concepts (training images) to be used for history matching. Three training images presented in Fig. 15 represent meandering, parallel and anastomosing fluvial channel types. The MPS realisations based on these TIs are more different and, therefore, result in a more diverse clustering in the metric space (Fig. 16b). The classification problem becomes more straightforward and low dimensional, which is illustrated in the energy diagram dropping down to 99 % stress energy at 3–4 dimensions (Fig. 16a).
Figure 17 compares the misfit evolution through the history matching process with a single and multiple training images. We can observe that a lower misfit is reached using multiple training images and that more models are generated with low misfit (\(<3000\)). Figure 17 illustrates how the inclusion of multiple training images (b) reduced the misfit when compared with history matching models with a single training image. From Fig. 17, it is clear that the lowest misfit has been reduced from 3366 in case (a) to 2794 in case (b). This can be associated with the fact that, in case (b), the models with the lowest misfit were generated with a training image that describes the facies distribution of the reservoir better than the models generated with the single training image used in case (a). This is only possible by having the opportunity to sample from different training images.
Figure 18 compares history matches and the forecasts of the oil production rate (WOPR) in three wells based on one training image and multiple training images. The use of multiple training images reduced the uncertainty in wells P12 and P5, but in well P1 the uncertainty (P10–P90 range) increased. This can be due to a local connectivity effect of the facies distribution around well P1.
4 Conclusions
The paper presents a workflow to integrate multiple geological scenarios into uncertainty quantification of reservoir forecasts. The workflow demonstrates how reservoir models based on multiple training images can be updated through history matching while also ensuring that the resulting historymatched models are geologically realistic and consistent with a priori geological knowledge. Uncertainty quantification based on multiple training images demonstrated that adaptive stochastic sampling is capable of providing a close match to the observed production whilst maintaining geologically realistic model parameter relationships for a range of geological scenarios.
Automated history matching with geologically realistic priors for MPS parameterisation was applied to a meandering fluvial section of the synthetic fullfield Stanford VI reservoir case study. Application of the proposed workflow was able to identify regions of goodfitting models, which correspond to the more than one training image used. The outcomes of history matching with multiple training images were compared with those from a single training image. This comparison showed that use of multiple training images in history matching results in an ensemble of bettermatched models as well as more robust uncertainty quantification. This methodology can be seen as a way to include the widest variety of different geological interpretations available into history matching to better capture all the subsurface uncertainties.
Notes
Acknowledgements
The work was funded by the industrial sponsors of the HeriotWatt Uncertainty Project (phase IV). Authors thank Stanford University for providing Stanford VI case study data, SGeMS geostatistical modelling software (Remy et al. 2009) and advice on MDS computation. Authors acknowledge the use of Eclipse reservoir flow simulator provided by Schlumberger, tNavigator flow simulator provided by Rock Flow Dynamics and Raven history matching software from Epistemy Ltd. LibSVM R package was used for SVM classification (Chang and Lin 2011).
References
 Arnold D, Demyanov V, Tatum D, Christie M, Rojas T, Geiger S, Corbett P (2012) Hierachical benchmark case study for history matching, uncertainty quantification and reservoir characterisation. Comput Geosci 50:4–15CrossRefGoogle Scholar
 Arnold D, Demyanov V, Christie M, Bakay A, Gopa K (2016) Optimisation of decision making under uncertainty throughout field lifetime: a fractured reservoir example. Comput Geosci 95:123–139CrossRefGoogle Scholar
 Arnold D, Demyanov V, Rojas T, Christie M (2018) Uncertainty quantification in reservoir predictions: part 1  Model realism in history matching using geological prior definitions. Comput Geosci (accepted)Google Scholar
 Baddeley MC, Curtis A, Wood R (2004) An introduction to prior information derived from probabilistic judgements: elicitation of knowledge, cognative bias and herding. In: Wood R, Curtis A (eds) Geological prior information, vol 239. Geological Society Special Publication, London, pp 15–27Google Scholar
 Bond CE, Gibbs AD, Shipton ZK, Jones S (2007) What do you think this is? Conceptual uncertainty in geoscience interpretation. GSA Today 11(17):4–10CrossRefGoogle Scholar
 Borg I, Groenen PJF (2005) Modern multidimensional scaling: theory and applications. Springer, BerlinGoogle Scholar
 Caers J, Hoffman T, Strebelle S, Wen XH (2006) Probabilistic integration of geologic scenarios, seismic, and production data a West Africa turbidite reservoir case study. Lead Edge 25:240–244CrossRefGoogle Scholar
 Castro SA, Caers J, Mukerji T (2005) The Stanford VI reservoir: 18th annual report. Stanford Centre for Reservoir Forecasting, Stanford UniversityGoogle Scholar
 Chang CC, Lin CJ (2011) LIBSVM: a library for support vector machines. ACM Trans Intell Syst Technol 2(3):1–39CrossRefGoogle Scholar
 Cherpeau N, Caumon G (2015) Stochastic structural modelling in sparse data situations. Pet Geosci 21(4):233–247CrossRefGoogle Scholar
 Demyanov V, Backhouse L, Christie M (2015) Geological feature selection in reservoir modelling and history matching with multiple kernel learning. Comput Geosci 85:16–25CrossRefGoogle Scholar
 Demyanov V, Pozdnoukhov A, Christie M, Kanevski M (2010) Detection of optimal models in parameter space with support vector machines. In: Proceedings of geoENV VII Geostatistics for Environmental Applications, Springer, pp 345–358Google Scholar
 Dromgoole P, Speers R (1992) Managing uncertainty in oilfield reserves. Middle EastWell Eval Rev 12:30–41Google Scholar
 Erbas D, Christie M (2007) Effect of sampling strategies on prediction uncertainty estimation. SPE 106229Google Scholar
 Hu L (2000) Gradual deformation and iterative calibration of Gaussianrelated stochastic models. Math Geol 32:87–108CrossRefGoogle Scholar
 Hu L, Zhao Y, Liu Y, Scheepens C, Bouchard A (2013) Updating multipoint simulations using the ensemble Kalman filter. Comput Geosci 51:7–15CrossRefGoogle Scholar
 Jiang R, Stern D, Halsey TC, Manzocchi T (2016) Scenario discovery workflow for robust petroleum reservoir development under uncertainty. Int J Uncertain Quantif 6(6):533–559CrossRefGoogle Scholar
 Jung A, Fenwick DH, Caers J (2013) Training imagebased scenario modeling of fractured reservoirs for flow uncertainty quantification. Comput Geosci 17:1015–1031CrossRefGoogle Scholar
 Kanevski M, Pozdnoukhov A, Timonin V (2009) Machine learning for spatial environmental data. Theory, applications and software. EPFL Press, LausanneCrossRefGoogle Scholar
 King PR (1990) The connectivity and conductivity of overlapping sand bodies. In: Buller AT, Berg E, Hjelmeland O, Kleppe J, Torsæter O, Aasen JO (eds) North Sea oil and gas reservoirs—II: Proceedings of the 2nd North Sea oil and gas reservoirs conference organized and hosted by the Norwegian Institute of Technology (NTH), Trondheim, Norway, May 8–11, 1989, Dordrecht: Springer, Netherland, pp 353–362Google Scholar
 Larue DK, Legarre H (2004) Flow units, connectivity and reservoir characterisation in a wavedominated deltaic reservoir: Meren reservoir, Nigeria. AAPG Bull 88:303–324CrossRefGoogle Scholar
 Mohamed L, Christie M, Demyanov V (2010) Comparison of stochastic sampling algorithms for uncertainty quantification. Comput Geosci 15:31–38Google Scholar
 O’Sullivan A, Christie M (2005) Error models for reducing history match bias. Comput Geosci 9:125–153CrossRefGoogle Scholar
 Park H, Scheidt C, Fenwick D, Boucher A, Caers J (2013) History matching and uncertainty quantification of facies models with multiple geological interpretations. Comput Geosci 17:609–621CrossRefGoogle Scholar
 Rankey EC, Mitchell JC (2003) Interpreter’s corner. That’s why it’s called interpretation: impact of horizon uncertainty on seismic attribute analysis. Lead Edge 22(9):820–828CrossRefGoogle Scholar
 Remy N, Boucher A, Wu J (2009) Applied geostatistics with SGeMS: a user’s guide. Cambridge University Press, CambridgeCrossRefGoogle Scholar
 Rojas T, Demyanov V, Christie M, Arnold D (2011) Use of geological prior information in reservoir facies modelling. In: Proceedings of IAMG Annual Conference, SaltzburgGoogle Scholar
 Sambridge M (2008) Geophysical inversion with a neighbourhood algorithmII. Appraising the ensemble. Geophys J Int 138:727–746CrossRefGoogle Scholar
 Scheidt C, Caers J (2009a) Representing spatial uncertainty using distances and kernels. Math Geosci 41(4):397–419CrossRefGoogle Scholar
 Scheidt C, Caers J (2009b) Uncertainty quantification in reservoir performance using distances and kernel methods–application to a West Africa deepwater turbidite reservoir. SPE J 14(4):680–692CrossRefGoogle Scholar
 Scheidt C, Caers J (2014) Updating joint uncertainty in trend and depositional scenario for exploration and early appraisal stage. EAGE 2nd Uncertainty Quantification Workshop, DubaiGoogle Scholar
 Strebelle S (2002) Conditional simulation of complex geological structures using multiplepoints statistics. Math Geol 34:1–21CrossRefGoogle Scholar
 Sun W, Hui MH, Durlofsky LJ (2017) Production forecasting and uncertainty quantification for naturally fractured reservoirs using a new dataspace inversion procedure. Comput Geosci 21(5):1443–1458CrossRefGoogle Scholar
 Suzuki S, Caers J (2008) A distancebased prior model parameterization for constraining solutions of spatial inverse problems. Math Geosci 40:445–469CrossRefGoogle Scholar
 Vapnik VN (1998) Statistical learning theory. Wiley, New YorkGoogle Scholar
 Wickelmaier F (2003) An introduction to MDS. Sound Quality Research Unit, Aalborg University, DenmarkGoogle Scholar
 Wood R, Curtis A (2004) Geological prior information and its applications to geoscientific problems. In: Wood R, Curtis A (eds) Geological Prior Information: informing science and engineering, vol 239. Geological Society Special Publication, London, pp 1–14Google Scholar
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