# Uncertainty Quantification in Reservoir Prediction: Part 1—Model Realism in History Matching Using Geological Prior Definitions

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## Abstract

Bayesian uncertainty quantification of reservoir prediction is a significant area of ongoing research, with the major effort focussed on estimating the likelihood. However, the prior definition, which is equally as important in the Bayesian context and is related to the uncertainty in reservoir model description, has received less attention. This paper discusses methods for incorporating the prior definition into assisted history-matching workflows and demonstrates the impact of non-geologically plausible prior definitions on the posterior inference. This is the first of two papers to deal with the importance of an appropriate prior definition of the model parameter space, and it covers the key issue in updating the geological model—how to preserve geological realism in models that are produced by a geostatistical algorithm rather than manually by a geologist. To preserve realism, geologically consistent priors need to be included in the history-matching workflows, therefore the technical challenge lies in defining the space of all possibilities according to the current state of knowledge. This paper describes several workflows for Bayesian uncertainty quantification that build realistic prior descriptions of geological parameters for history matching using support vector regression and support vector classification. In the examples presented, it is used to build a prior description of channel dimensions, which is then used to history-match the parameters of both fluvial and deep-water reservoir geostatistical models. This paper also demonstrates how to handle modelling approaches where geological parameters and geostatistical reservoir model parameters are not the same, such as measured channel dimensions versus affinity parameter ranges of a multi-point statistics model. This can be solved using a multilayer perceptron technique to move from one parameter space to another and maintain realism. The overall workflow was implemented on three case studies, which refer to different depositional environments and geological modelling techniques, and demonstrated the ability to reduce the volume of parameter space, thereby increasing the history-matching efficiency and robustness of the quantified uncertainty.

## Keywords

Inverse problems Uncertainty Model calibration Geostatistics Reservoir modelling Prior knowledge Support vector Classification Natural analogues Fluvial geology## 1 Introduction

The main problem in forecasting reservoir behaviour is the lack of information available to populate any reservoir model of a heterogeneous geological system away from the wellbores. A good analogy to the problems faced in creating reservoir models was supplied by Christie et al. (2005), who likened forecasting reservoir production to drawing a street map and then predicting traffic flows based on what you see from several street corners in thick fog. Expanding on Christie’s analogy, geological modelling is the equivalent of knowing that roads can be classified by the number of lanes of traffic in each direction, their location, their ranges of speed limit and the common features that can exist on them. It will not tell us how long or tortuous the road is, how often the road type changes, nor which features that affect the flow of traffic are present.

Bayesian uncertainty quantification is a commonly used approach in reservoir engineering, where the model mismatch to some data (e.g. production data) is used to estimate the likelihood and, based on some assumption of the prior, infer the posterior. It is useful as it is a quantitative way of estimating uncertainty and allows reservoir engineers to define reservoir prior probability based on distributions of model parameters. For much of the research carried out in this area, the aim has been to improve the efficiency of discovering minima in parameter space to achieve good history matches. Less effort has been applied to the prior, its importance and how to encapsulate it in the uncertainty quantification workflows, in short how to encapsulate geological features into parameters that can be sampled (a process called *geological parameterisation*).

Geological priors have the added complexity of potentially being highly non-uniform/non-Gaussian, while uniform or Gaussian are the typical models used to describe prior uncertainty. Therefore, applying simple definitions of prior probability can result in production of models that are geologically unrealistic, creating history matches that potentially lead to poor forecasts. The differences between available geological modelling approaches also complicate matters, as the parameter set for each approach differs, thus each modelling method potentially requires the creation of a set of unique prior probabilities describing the geological uncertainties for each approach.

Elicitation of prior probabilistic information is also a complex issue, as discussed in Baddeley et al. (2004) and Curtis and Wood (2004). Geologists deal with the sparsity of data by using prior knowledge about what is and is not geologically possible, to reduce the number of possible models. These expert judgements are based on the experience of the geologist in inferring probabilities about the data using different but related data sources. An example of this may be to infer porosity and net/gross values to estimate hydrocarbon volumes for an undrilled exploration well, based on previously drilled wells in the region or outcrops of reservoir facies exposed at the surface. Such data are qualitative rather than quantitative, thus any estimates of uncertainty for these data/parameters are based on the judgements of the geologist.

This paper proposes a set of novel workflows that encapsulate the geological prior, then demonstrates their benefits, showing improvements to model predictions and estimates of uncertainty over the use of non-geological priors. The work is illustrated by using three example data sets and increasing complexity of workflow to deal with them. Sections 2 and 3 describe the background of the uncertainty quantification approaches and how to integrate these into geological modelling workflows, demonstrated on a simple case study. In Sect. 4, a machine learning approach to encapsulate more complex geological prior probabilities into Bayesian workflows is described and then demonstrated using two different case studies.

## 2 Assisted History Matching, Likelihood Estimation and Uncertainty Quantification

There are a plethora of uncertainty quantification techniques available to reservoir engineers, developed for the purposes of using history-match quality to predict reservoir uncertainty. This work applies a general Bayesian workflow for uncertainty quantification, updating model descriptions based on production response.

*m*are model parameters to be estimated based on some observations

*O*, then Bayes’ theorem can be described by

*p*(

*m*) is the prior probability of the model parameters and

*p*(

*m*|

*O*) is the posterior probability given the likelihood

*p*(

*O*|

*m*). The denominator term is often considered a normalisation constant on Bayesian evidence (Elsheik et al. 2015).

*M*between the production data and the model response, where the lower the value of

*M*, the higher the likelihood of the model

*M*is calculated using one of a range of misfit functions, most commonly the least-squares misfit. The choice of the misfit function and the handling of any associated error have been shown to have a significant impact on Bayesian estimates (O’Sullivan and Christie 2005). This work uses a standard least-squares definition throughout in order to demonstrate the impact of the prior in isolation, however equal care must be taken for both the likelihood and prior to produce robust uncertainty estimates.

Stochastic optimisers are used for their ability to find multiple different clusters of models (minima) that provide similar match quality with varying forecasts, and the example algorithms above are used particularly for their efficiency in locating minima from small numbers of simulation runs. This outcome may be further enhanced by the use of multi-objective versions of the algorithms (MOO) (Christie et al. 2013).

The final step is to apply a post-processor such as NAB (Sambridge 1999a, b) to calculate the Bayesian credible intervals from the entire ensemble of history-matched models, then forecast with uncertainty to produce good approximations to the results of a thorough Markov chain Monte Carlo (MCMC) analysis (Mohamed et al. 2010). NAB does this by employing Voronoi cells to tessellate the parameter space of the misfit ensemble, creating a cell for each iteration of the ensemble, where the cell size depends on the sample density.

Each cell represents a region of equal misfit in parameter space around a particular history-matched iteration, where the volume is inversely proportional to the sampling density in that area. Across the cell, the misfit is set equal to the misfit value of that iteration. A Gibbs sampler is run over the Voronoi space, and a sample is accepted/rejected based on the visited cell’s misfit. The sampling is therefore dependent on both the misfit of a particular cell as well as the chance of visiting that cell. Low-likelihood samples tend to have fewer neighbouring samples and therefore have larger cells. These cells get more visits, but the samples are accepted less frequently. Higher-likelihood cells tend to be in refined regions of space, thus the cells are typically smaller, but they will be accepted more often.

## 3 Incorporating Geological Modelling Workflows in History Matching

For ease of implementation in real-world settings, the workflow shown in Fig. 1 is typically applied to the parameters of the dynamic simulation model that are easiest to adjust algorithmically by practising engineers. Parameters of the geological model, such as statistical model parameters, object sizes or the complex set of properties that must be changed to adjust faults, surfaces or horizons are less frequently adjusted in practice. The two main reasons for this are (i) most static models are populated by some form of geostatistical approach, and commercial modelling approaches rely on complex workflows that are hard to interact with, and (ii) that geostatistical models are typically built by geoscientists, therefore multiple technical disciplines must interact in the history-matching process for these models to be included. For simplicity, most companies keep the history-matching workflow under the control of only the reservoir engineer. Parameters such as skin, productivity index, relative permeability and porosity/permeability multipliers are typically applied to tune the geological uncertainties, however these change the continuity and structure of the geological model which was built to represent the likely correlation structure of the reservoir, potentially in ways that are not geologically possible.

To fully include geological uncertainties as parameters into the history-matching process, the geostatistical model must be included in any workflow, which requires the solution of the following challenges: (i) how to create the new workflow to include the appropriate (not just easy) geostatistical model so that its parameters can be updated by the optimisation approach, and (ii) how to define the prior ranges of the geostatistical model parameters based on the real-world geological measurements.

Broadly, there are three main approaches available to geostatistical modelling that can be applied: (i) Two-point statistical approaches based on kriging and variography to define the correlation structure of the geological features. The most common of these are methods such as sequential indicator simulation (SIS) and truncated Gaussian simulation (TGS) for modelling discrete parameters and sequential Gaussian simulation (SGS) for continuous parameters (Pyrcz and Deutsch 2014); (ii) Object modelling approaches (Pyrcz and Deutsch 2014) using pre-defined objects to represent geobodies in the reservoir (e.g. a channel, lobe or sheet) rather than a statistical representation of the correlation structure. The object is parameterised by a set of physically definable dimensions to describe its shape and size plus a proportion objective that the algorithm aims to achieve; (iii) Multi-point statistics (MPS), which uses a training image instead of a variogram or object shape/dimensions to represent the correlation model. The training image is a rich representation of the reservoir correlation that describes the spatial relationships of several facies types. Like its simpler two-point counterpart, MPS can quickly produce realisations of the reservoir model, however MPS can apply multiple training images to allow different, complex representations of the correlation structure for different geological interpretations.

Several authors have updated the geostatistical model reservoir properties directly using a variety of methods such as gradual deformation (Hu 2000) or assimilation methods such as EnKF (Evensen et al. 2007; Chang et al. 2015). Changing the structural model also requires more complex geological parameterisations such as the example shown in Cherpeau et al. (2012), which demonstrated an approach to history-match a reservoir model by automatically relocating faults whose location was uncertain. In all this work, the geological prior is of critical importance to maintain the belief in model forecasts, thus the impact of geological priors on uncertainty quantification workflows is discussed next. Preserving geological realism through the model update has attracted increasing interest in several publications that address different aspects of this issue: conditioning geological models to geophysical observation (Jessell et al. 2010), uncertainty quantification using several scenarios using a metric space approach (Scheidt and Caers 2009a), scenario discovery based on different reservoir performance metrics (Jiang et al. 2016), posterior inference based on training images (Melnikova et al. 2015), inverse modelling in seismic inversion (Linde et al. 2015) and inverse modelling with different fracture scenarios (Sun et al. 2017).

## 4 Geological Priors in History Matching and Uncertainty Quantification

Defining geomodel priors and their interdependence from measured data is less commonly tackled. Wood and Curtis (2004) illustrated an example by incorporating all geological sources, including static and dynamic data, into a model of stacked marine sequences, where a good definition of the geological prior allowed good geological matches to wells. The next section describes the importance of applying geological priors to quantifying reservoir uncertainty.

### 4.1 An Example of a Geological Prior

From Fig. 2, it is clear that (i) the correlations differ from each other, such that no one correlation can predict all fluvial systems, (ii) the likely space of possible models is bounded by the spread of the correlations, and (iii) this space is much smaller and unusually shaped in comparison with the uniform prior.

### 4.2 Case Study 1: La Seretta Outcrop using Simple Geological Priors

The La Serreta outcrop was first studied in detail by Hirst (1991). Located in the Ebro Basin, Northern Spain, La Serreta is part of the Oligo–Miocene Huesca distributive fluvial system. Located in a proximal position in the system, the resulting architecture is of multilateral/multistorey sandbodies with high net/gross, connectivity and overall good reservoir quality. There are three types of channel facies identified at the outcrop, representing channelised flow in major and intermediate channels and overbank ribbon sands, here termed minor channels. The outcrop exposure was used as the basis for creating a box model with three distinct channel types and a background facies with dimensions of \(2.5\,\)km\(~\times 2\,\)km\(~\times 80\,\)m in size. A 700,000 (\(140 \times 125 \times 40\)) grid cell *truth case* model was created, and its production forecast, against which history matching would be carried out. The truth case was constructed using object models to create sinuous channels, where channel dimensions were set to values within the realistic prior range (purple) of Fig. 2. The model was constructed using the RMS™ software, then populated using object modelling for each of the three channel facies types and sequential Gaussian simulation (SGS) for the petrophysical properties.

Case study 1 geological prior

Model parameter (prior) | Prior range | Geological or uniform prior |
---|---|---|

NTG | 0.3–0.8 | Uniform |

Channel depth (m) | 2–15 | Geological |

Channel width (m) | 50–300 | Geological |

Channel azimuth (deg) | 0–180 | Uniform |

Channel amplitude (m) | 300–1200 | Uniform |

Channel sinuosity | 1–2 | Uniform |

History matching was carried out on two cases, a *Basecase* using only uniform priors and a *Geological Prior* case that applies the purple prior definition shown in Fig. 2. History matching was carried out using the neighbourhood algorithm (NA) (Sambridge 1999a, b), using 6500 iterations for each case to achieve a match. All realisations of the model were conditioned to porosity and permeability data in the producer and injector wells (produced from the truth case model).

- 1.
Sample from the uniform prior for each parameter;

- 2.
Where geological prior exists, apply the accept/reject step to see if the model is possible. The geological prior was encoded for the width/depth ratio shown in Fig. 2 then run as a pre-processor, taking the output from the sampling algorithm and checking its validity before deciding on running the reservoir simulator;

- 3.If yes:
- (a)
Change the RMS™ input parameters;

- (b)
Create a realisation of the object model based on the new parameters;

- (c)
Export the static grid into the reservoir simulator then run the model;

- (d)
Calculate the misfit and update the optimisation algorithm.

- (a)

This simple example clearly illustrates the benefit of using non-uniform (complex in shape) geological prior definitions. The prior definition can be improved by using all available channel dimensions data to capture non-linear multivariate dependences in the definition of the prior (rather than the correlations to the data), by including more measured properties/parameters and by improving the definition of the channel model. The additional complexity of these prior definitions is worth the effort by reducing compute times and by avoiding inaccurate uncertainty estimates.

## 5 Building Geological Priors Using Machine Learning

The application of a geological prior to object models of channelised reservoirs in Sect. 4.2 is relatively easy as it only deals with a two-dimensional problem and the model parameters are equivalent to the geological prior data from palaeohydrological measurements. Defining the prior for a large number of uncertainty geological parameters whose ranges must be elicited from available data is much more complex due to the difficulty of working in high dimension. Using other modelling methods, such as multi-point statistics, may provide a better characterisation of the reservoir, as it can capture complex geological structures while conditioning perfectly to well data. Using MPS in history matching however, adds complexity to the problem, as model tuning parameters no longer link directly to the geological properties (i.e. affinity and rotation are not properties of the geological system).

To work with larger numbers of dimensions and to work with a range of geostatistical methods, any new approach must solve: (i) how to define the complex shape of high-dimensional geological priors, elicited from all available data, (ii) how to apply this prior model to the inverse problem and (iii) how to implement useful relationships between geologically induced priors and model control parameters, which may not be described by direct observations, in the inverse modelling.

This paper presents three novel approaches that together aim to solve these problems. These are: (i) Explore relationships between geomorphic parameters and model multivariate dependencies using the machine learning approach support vector regression (SVR). SVR is tested as a suitable approach to describe the prior in Sect. 5.1. (ii) Build informative realistic multi-dimensional prior for geomorphic parameters by classifying the space of realistic models using one-class support vector machine (OC-SVM). This is described in Sect. 5.2. (iii) Apply a multi-layer perceptron (MLP) to link MPS model parameters with the measurable geomorphic parameters. This is described in Sect. 5.3. Sections 5.4 and 5.5 cover two case studies to demonstrate the impact of the new workflow (Fig. 15) on history-matching performance.

### 5.1 Machine Learning to Construct Geological Priors

Machine learning presents us with a set of tools that can elicit relationships between parameters directly from the data and encapsulate these relationships in a useful way. Unlike the deterministic approach presented in the previous case study, machine learning provides a way to develop an accurate representation of complex relationships between the data without embedding hard pre-determined assumptions lacking sufficient justification.

*K*is a symmetric positive-definite function (e.g. a Gaussian of some width \(\sigma \)). The support vector coefficients \(\alpha _{i}\) identify the subset of the input data (\(x_{i}\)) and are determined through a training process.

The training process is illustrated in Fig. 8 and compared with a linear regression model. In Fig. 8a, the linear regression function is fitted to data with known error intervals, showing that the ability of the function to fit the complex data is limited. In Fig. 8b, SVR creates a tube around a centre line kernel regression model (black line) with width \(\epsilon \) to create the support vectors (green lines). In this approach, the margin represents the window of acceptable fit to the data and is set as the spread of the support vectors with samples within the epsilon tube contributing zero weight. The complex non-linear kernel regression shown in Fig. 8b is first created linearly in feature space [reproducing kernel Hilbert space (RKHS)], then transformed back as a non-linear model within the input (in this case, model parameter) space. The fundamental property of kernel functions is that they actually provide a dot product between input samples in some high-dimensional space, the reproducing kernel Hilbert space (RKHS). This property is primarily used to introduce the non-linear extensions of the traditional methods (such as kernel principal component analysis).

It is necessary to tune SVR parameters by finding an optimal model that is not overfitted and has a predictive power to generalise. The parameters to tune are: the kernel width (\(\sigma \)), which controls the smoothing such that low \(\sigma \) overfits the data and generates high testing error and low training error; an \(\epsilon \)-insensitive loss function, to allow noise within the data, albeit with increased training error; and the factor *C*, which determines the balance between goodness of fit and training error.

SVR is trained in a way that minimises the complexity of the regression function with the trade-off penalty \(\xi \) (Fig. 8b) for outliers outside the support vectors (green lines in Fig. 8b). This problem is solved by standard quadratic programming for a given combination of *C*, \(\epsilon \) and kernel width (\(\sigma \)). These parameters can be tuned through cross-validation/testing (Kanevski et al. 2009) or inferred in a Bayesian way through an inverse problem (Demyanov et al. 2008). In this study, the SVR parameters were tuned in a manual way, just to demonstrate how the non-linear geomorphic relations can be captured from data, while fine-tuning of the SVR predictor lies beyond the scope of this study.

In order to apply this machine learning approach to history matching, it must be extended to deal with the problem of classifying which regions of parameter space contain realistic models (i.e. an informative high-dimensional prior). An approach to solving this is explained in the next section.

### 5.2 Applying a Machine Learning Prior to History Matching Using One-Class SVM Classification

To identify the realistic regions of parameter space from available data (e.g. Fig. 7), a one-class SVM (OC-SVM) is used, which is the SVM extension to the one-class classification problem. Scholkopf et al. (2001) defined OC-SVM as an unsupervised kernel-based method, which is used to estimate the support of probability density distributions. The main application of this technique is to detect novelty, outliers and rare events in a high-dimensional RKHS. A more thorough description of SVM is given in the part 2 paper (Demyanov et al. 2018).

*x*, the value of

*f*(

*x*) is determined by evaluating which side of the hyperplane in the RKHS it falls on. When data are mapped in the RKHS by means of a kernel function, the data are linearly separated from the origin by maximising the margin of the hyperplane. Outliers are constrained to be close to the origin, while the core of the distribution is pushed away with a maximum margin.

The results of applying a OC-SVM to construct a prior for a four-dimensional channel property data set can be seen in Fig. 11. Figure 11a shows the point cloud of data for four parameter dimensions of fluvial channels with clear correlation. Figure 11b shows the outcome of OC-SVM in defining the prior region of parameter space that is realistic, which is very similar in shape to that defined in Figs. 7 and 9. Probabilistic outcome of the classification is derived from Bayesian interpretation of SVM output through linearisation class-condition densities between the margins using a sigmoid form (Platt 1999).

The OC-SVM prior can now be used by allowing the history-matching (HM) workflow to test if a sample is inside (value = 1) or outside (value = 0) the prior space based on four supplied coordinate for the four-dimensional space. The stochastic sampling algorithm uses OC-SVM as an accept/reject step (as per the example in Sect. 4.2) to decide whether a proposed iteration should be evaluated.

### 5.3 Using Geological Priors with Other Modelling Approaches: a Multi-point Statistics Example

MPS uses a training image built by a geologist to represent the spatial distribution of facies (or other properties) in the interpreted depositional environment. While MPS is a stochastic algorithm and can generate realisations of the reservoir to account for some uncertainty, some key geometrical relations represented in the training image are not adjusted directly by the stochastic process.

It is possible to vary some of the geometrical relationships by applying the affinity transformation in one or more spatial directions. As the facies model gains extra flexibility with affinity transformation, it can be used as a parameter to account for prior uncertainty of geobody geometries and explore the range of possible models that provide history matches.

*x*,

*y*,

*z*) to provide the variations of the output facies geometry vs. with the ones in the training image, as described by Remy et al. (2009) and illustrated in Fig. 12. For an affinity parameter of [1 1 1], all the dimensions of the training image are preserved in the output model, but by increasing the affinity in the

*x*axis by 2 to [2 1 1], the output model will double the dimensions from the training image along the

*x*axis. By reducing the affinity parameter in the

*y*axis by \( \frac{1}{4} \) to [1 0.25 1], the output model will be \( \frac{1}{4} \) of the training image size in

*y*axis. Figure 12 illustrates the effect of the affinity variation into the output model compared with the training image when simulating fluvial channels and shows the useful flexibility in covering a range of geological outcomes from a limited set of training images.

A critical step in this process is to extract the geometric parameters from the model in order to validate the MLP prediction for the training set using an R script. Channel facies bodies from the centre of the model were extracted to take representative sections of the channel sandbody. Figure 14 illustrates how the channel parameters were measured. The width was estimated as the minimum length perpendicular to the known flow direction, wavelength was the maximum inter-sand distance along the flow direction, and the amplitude was measured by obtaining the difference between the distances of the closest (L1) and farthest (L2) to the west border of the grid channel facies grid block perpendicular to the flow direction and subtracting the estimated channel width. This approach is simple and will not work where channels are oriented at anything other than \(0^\circ \), as the measures will no longer be orthogonal to gridding, but was applicable to this example.

A multilayer perceptron (MLP) neural network was used to predict the spread of the geomorphic parameters based on the combination of the MPS model affinity parameters. The MLP was trained with a set of initially simulated realisations that cover a range of affinity values and for which the spread of the geomorphic parameter values was estimated. In order to obtain a robust and reliable training database of channel geometry parameters affected by affinity variation, 81 different affinity combinations were considered with 5 stochastic realisations for each, which totals to 1055 realisations.

- 1.
Measure all the geometrical parameters of the channel outputs (

*W*channel width,*wl*meander wavelength,*T*channel thickness,*A*meander amplitude) after every realisation; - 2.Standardise the measures obtained in step (1) with the channel geometrical parameters of the training image in order to make these neural networks usable for any training image,where$$\begin{aligned} \text {Fraction}(W,T,a,wl)= \frac{\text {Realisation} X(W,T,a,wl)}{\text {TI}(W,T,a,wl)}, \end{aligned}$$(4)
*X*is a realisation,*W*is channel width,*T*is channel thickness,*a*is meander amplitude,*wl*is meander wavelength, and TI is training image. - 3.
Build and train an MLP neural network using as inputs the

*x*,*y*,*z*variations of affinity to predict the range of the simulated geometries based on the following MLP outputs: the mean (m) and the first and third quartiles (Q1 and Q3) of each of the channel geometrical parameters. - 4.
The obtained results are expressed in fractions, then used as multipliers to the dimensions of the geometrical parameters of the training image in order to obtain geometrical parameters in distance units.

### 5.4 Case Study 2: Stanford VI Model Using SVM Machine Learning Priors

Channel geomorphic parameter variation in fluvial reservoir case study 2

Parameter | Truth case reservoir model distribution | Training image | Uniform prior ranges | |||||
---|---|---|---|---|---|---|---|---|

Min. (m) | Ave. (m) | Max. (m) | Min. (m) | Ave. (m) | Max. (m) | Min. (m) | Max. (m) | |

Channel width | 150 | 180 | 200 | 200 | 300 | 400 | 100 | 600 |

Channel thickness | 15 | 15 | 15 | 5 | 8 | 10 | 5 | 200 |

Meander amplitude | 900 | 1080 | 1150 | 1300 | 1600 | 2000 | 500 | 3000 |

Meander wavelength | 1100 | 1500 | 1700 | 700 | 900 | 1200 | 500 | 3000 |

Petrophysical properties for each sedimentary facies for case study 2 meandering fluvial channels

Facies | Porosity (%) | Horizontal permeability (mD) | Vertical permeability (mD) | Facies proportions (%) |
---|---|---|---|---|

Channel | 5 | 50 | 25 | 24 |

Point bar | 20 | 500 | 250 | 36 |

Floodplain | 0.1 | 0.001 | 0.0005 | 40 |

Uncertainty in facies geometry was introduced in the MPS model through the variation of affinity parameters, which modifies the geometric relations embedded in the training image. This translates into the uncertainty of the geomorphic parameters channel width and thickness, channel meander amplitude and wavelength. The translation between affinity and the channel geometry is provided by the data-driven relationship obtained by MLP. The uniform prior ranges for the geomorphic parameters are given in Table 2.

A flow simulation black oil model was set up on a \(50 \times 50 \times 40\) grid with cell size of \(75 \times 100 \times 1\) m. The production from 18 wells was established by the water flood from 11 injection wells. The flow simulation has injection rate control and bottom hole pressure control at the production wells. The history-matching task was set up to minimise the misfit between the historical data and the simulated oil and water production rates across individual wells. The misfit was defined as a standard least-squares norm to oil and water rates, and history matching was performed using PSO.

Table 2 compares HM using informative geological priors obtained with OC-SVM with HM using flat uninformative priors from the parameter intervals. The plot for the misfit evolution using HM with uniform priors (Fig. 18e) shows that models of comparable misfit quality can be obtained. However, there are many fewer low-misfit models when using HM with uniform priors, suggesting that it is far less efficient and does not produce enough low-misfit models.

Some of the low-misfit models produced with uniform priors may have unrealistic geomorphic parameter relationships. Figure 18d shows an unrealistic model realisation, which does not preserve the correct channel thickness/width ratio. Figure 19 further compares the geomorphic parameter values versus the model misfit obtained from HM with intelligent informative priors and uniform uninformative priors. The plots for meander amplitude, wavelength and channel width show that the HM models obtained with intelligent priors come closer to the values of the reference case than the model obtained from HM with uniform priors. The channel width parameter shows greater deviation of the lowest-misfit model from the true model parameter value, however there are still models with fair misfit that scatter around the width of the truth case model.

The final HM result in Fig. 20 shows the oil production rate history match and P10/P50/P90 forecast inferred from multiple HM models for wells 1, 5 and 12. It compares the forecasts obtained based on the models generated using intelligent and uniform priors. In general, the P10–P90 range of the well oil production rates (WOPR) in models generated using intelligent priors is smaller than the P10–P90 range of the models using the uniform ones. Another important aspect is that the P50 curve is closer to the history data of the model with intelligent priors than the models that used uniform prior information.

### 5.5 Case Study 3: Deep Water

Another case study was designed based on a different depositional environment, deep marine channels, to demonstrate the value of incorporating geological information in the form of informative priors in history matching. A synthetic reservoir (DM-Field) with facies geometry reflecting the description of deep marine channels of the Baliste–Crécerelle Canyon Fill (Wonham et al. 2000) was generated using unconditional object modelling with sinuous channels (Fig. 21a). DM-Field contains three facies: (i) pre-canyon deep marine pelagic and hemipelagic shales, (ii) muddy channel levees and canyon fill pelagic shales and (iii) deep marine channel sandy deposits. Table 4 gives the dimensions of the deep marine channel geomorphic parameters variation in the DM-Field. Petrophysical properties for these facies were set constant for each facies to analyse the effect of varying facies geometries on history-matching processes, and avoid a smearing effect possibly introduced by the variation of the petrophysical properties. Table 5 presents the relationships between sedimentary facies and petrophysical properties.

Channel geomorphic parameter variation for deep marine reservoir, case study 3

Parameter | Truth case reservoir model distribution | Training image | Uniform prior ranges | |||||
---|---|---|---|---|---|---|---|---|

Min (m) | Ave. (m) | Max. (m) | Min. (m) | Ave. (m) | Max. (m) | Min. (m) | Max. (m) | |

Channel width | 150 | 180 | 250 | 450 | 500 | 600 | 150 | 1000 |

Channel thickness | 20 | 30 | 50 | 25 | 30 | 35 | 20 | 280 |

Meander amplitude | 600 | 700 | 1000 | 800 | 1000 | 1200 | 200 | 5500 |

Meander wavelength | 800 | 950 | 1200 | 1500 | 2000 | 2500 | 200 | 4500 |

Petrophysical properties associated to deep marine facies in the DM field

Facies | Porosity (%) | Horizontal permeability (mD) | Vertical permeability (mD) |
---|---|---|---|

Channel | 20 | 500 | 250 |

Pelagic canyon fill | 0.01 | 0.01 | 0.01 |

Pre-canyon pelagic | 0.01 | 0.01 | 0.01 |

The history-matching (HM) problem was set up in a similar way to case study 2, with the prior intervals for the channel geomorphic parameters given in Table 4. A black oil flow simulation model was matched to oil and water production rates at individual wells. The PSO algorithm was run for 1000 iterations to obtain multiple HM solutions.

A comparison of the history-matching results from both informative intelligent geological priors and uniform uninformative priors is presented in Fig. 22. Similar to case study 2, HM with informative priors showed fewer high-misfit models. This results in a tighter forecast P10/P90 interval when using informative priors than without. The ranges are wider for uninformative priors because of the impact of unrealistic models that are excluded from the sampling by the informative prior. Figure 23 illustrates examples of HM models obtained using geological priors and uniform priors. While Fig. 23a, b shows that the best and worst matches for the intelligent prior look similar, with clearly recognisable channel shapes, Fig. 23c shows that the best match for the non-geological prior has lost the shape and continuity patterns of the TI channels, resulting in a very non-geological scenario. The produced realisation has unrealistic combinations of channel amplitude (A) and wavelength (Wl) and also an inconsistent channel width/depth ratio. Figure 24 illustrates that the model obtained with intelligent geological priors tends to have geomorphic parameters that home in closer to the values for the truth case reservoir as a result of HM.

## 6 Conclusions

This paper demonstrates a workflow to deal with geological uncertainties in a manner that ensures that only geological realistic models are used in the quantification process based on creating high-dimensional representations of the geological prior using machine learning.

The workflow develops a geologically realistic prior definition using OC-SVM, then applies it to the MPS modelling approach using MLP to convert between the parameters of geological measurements and the affinity parameter space of MPS. This resulted in the ability to history-match MPS models using a single training image that is manipulated using the affinity parameter while ensuring that all models produced are realistic.

The workflow has two key benefits: (i) by using OC-SVM, a multi-dimensional set of geological information relevant to the problem can be encapsulated into geologically realistic numerical prior probability ranges, (ii) by using MLP, the geologically realistic SVM prior ranges can be linked to an equivalent realistic range of geostatistical algorithm parameters (such as object modelling and MPS), which are then sampled from in a history-matching workflow. The MLP approach allows one to freely tune more complex geostatistical models of the reservoir, such as multi-point statistical models, to match production data without the need to recreate the training image.

The overall workflow allows the application of the most appropriate modelling method to a reservoir, even when the model parameters do not relate to real-world, measurable properties, and can ensure that only realistic models are used in the uncertainty estimates.

These benefits were demonstrated, through three examples, to further increase the efficiency of history matching by ignoring unrealistic regions of parameter space and improve the accuracy of the quantified uncertainty by ignoring local minima in unrealistic regions of parameter space. The examples demonstrated a reduction in the estimated uncertainty, an improvement in convergence performance (by ignoring unrealistic regions of parameter space) and the ability to work with several modelling approaches that use either real-world (object models) or model space (MPS) parameters.

A subsequent (part 2) paper (Demyanov et al. 2018) demonstrates how to expand this workflow to deal with multiple geological scenarios/interpretations captured with different training images using further machine learning techniques.

## Notes

### Acknowledgements

The work was funded by the industrial sponsors of the Heriot-Watt Uncertainty Project (phase IV). The authors thank Stanford University for providing Stanford VI case study data and SGeMS geostatistical modelling software (Remy et al. 2009). The authors thank Prof. M. Kanevski of University of Lausanne for providing advice and ML Office software for MLP/SVM/SVR implementation. Authors acknowledge use of the Eclipse reservoir flow simulator provided by Schlumberger, tNavigator flow simulator provided by Rock Flow Dynamics, Raven history-matching software from Epistemy Ltd., Roxar RMS modelling suite provided by Emmerson and R package e1071 for OC-SVM.

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