Geological uncertainty quantification is critical to subsurface modeling and prediction, such as groundwater, oil or gas, and geothermal resources, and needs to be continuously updated with new data. We provide an automated method for uncertainty quantification and the updating of geological models using borehole data for subsurface developments within a Bayesian framework. Our methodologies are developed with the Bayesian evidential learning protocol for uncertainty quantification. Under such a framework, newly acquired borehole data directly and jointly update geological models (structure, lithology, petrophysics, and fluids), globally and spatially, without time-consuming model rebuilding. To address the above matters, an ensemble of prior geological models is first constructed by Monte Carlo simulation from prior distribution. Once the prior model is tested by means of a falsification process, a sequential direct forecasting is designed to perform the joint uncertainty quantification. The direct forecasting is a statistical learning method that learns from a series of bijective operations to establish “Bayes–linear-Gauss” statistical relationships between model and data variables. Such statistical relationships, once conditioned to actual borehole measurements, allow for fast-computation posterior geological models. The proposed framework is completely automated in an open-source project. We demonstrate its application by applying it to a generic gas reservoir dataset. The posterior results show significant uncertainty reduction in both spatial geological model and gas volume prediction and cannot be falsified by new borehole observations. Furthermore, our automated framework completes the entire uncertainty quantification process efficiently for such large models.

Uncertainty quantification (UQ) is at the heart of decision making. This is
particularly true in subsurface applications such as groundwater,
geothermal resources, fossil fuels,

Representing the unknown subsurface geological reality by a single deterministic model has been commonly done (Beven, 1993; Royse, 2010), mostly by means of a single realization of the structure (layers or faults), rock, and fluid model derived from the borehole data with other supporting geological and geophysical interpretations (e.g., Fischer et al., 2015; Kaufmann and Martin, 2008). However, relying on a single model cannot reflect the inherent geological uncertainty (Neuman, 2003). Recent advances in geostatistics have shown the importance of using multiple model realizations for uncertainty quantification in many geoscience fields, including glaciology (e.g., Cullen et al., 2017), hydrogeology (e.g., Barfod et al., 2018; Zhou et al., 2014), hydrology (e.g., Goovaerts, 2000; Marko et al., 2014), hydrocarbon reservoir modeling (e.g., Caers and Zhang, 2004; Christie et al., 2002; Dutta et al., 2019; Yin et al., 2019), and geothermal (e.g., Rühaak et al., 2015; Vogt et al., 2010). Geostatistical approaches can provide multiple geological models that are conditioned or constrained to borehole data. When new boreholes are drilled, uncertainty needs to be updated. While uncertainty updating in the form of data assimilation is commonly applied to various subsurface applications, it is rarely used for updating newly drilled borehole data, often termed “hard data” in geostatistical literatures (Goovaerts, 1997). Elfeki and Dekking (2007) used a coupled Markov chain (CMC) approach to calibrate a hydrogeological lithology model by conditioning on boreholes in the central Rhine–Meuse delta in the Netherlands, and they then ran a Monte Carlo simulation to reevaluate the hydrogeological uncertainty. A similar approach was also used by Li et al. (2016) to reduce the uncertainty in near-surface geology for the risk assessment of soil slope stability and safety in Western Australia. Jiménez et al. (2016) updated 3-D hydrogeological models by adding new geological features identified from borehole tracer tests. Eidsvik and Ellefmo (2013) and Soltani-Mohammadi et al. (2016) investigated the value of information of additional boreholes for uncertainty reduction in mineral resource evaluations.

The problem of geological uncertainty, due to its interpretative nature and the presence of prior information, is often handled in a Bayesian framework (Scheidt et al., 2018). The key part often lies in the joint quantification of the prior uncertainty on all modeling parameters, whether structural, lithological, petrophysical, or fluid. A common problem is that the observed data may lie outside the defined prior model and hence are falsified. Another major issue is that most of the state-of-the-art uncertainty updating practices deal with each geological model component separately (a silo treatment of each UQ problem). However, the borehole data inform all components jointly, and hence any separate treatment ignores the likely dependency between the model components, possibly returning unrealistic uncertainty quantification. A final concern, more practically, lies around automating any uncertainty updating. Geological modeling often requires significant individual or group expertise and manual intervention to make the model adhere to geological rules, hence often requiring months of work when new data are acquired. There is to date, no method that addresses, with borehole data, the falsification, the joint uncertainty quantification, and the automation problem.

Recently, an uncertainty quantification protocol termed Bayesian evidential learning has been proposed to address decision making under uncertainty, and it has been applied to cases in oil or gas, groundwater contaminant remediation and geothermal energy (Athens and Caers, 2019; Hermans et al., 2018, 2019; Scheidt et al., 2018). It provides explicit standards that need to be reached at each stage of its UQ design with the purpose of decision making, including model falsification, global sensitivity analysis, prior elicitation, and data-science-driven uncertainty reduction under the principle of Bayesianism. Compared to the previous works on Bayesian evidential learning (BEL), model falsification, statistical learning-based uncertainty reduction approaches, and automation are what is of concern in this paper. Also, we will deal with one specific data source: borehole data, through logging or coring, for geological uncertainty quantification. First, we will introduce a scheme to address the model falsification problem involving borehole data by using robust Mahalanobis distance. We will then extend a statistical learning approach termed direct forecasting (Hermans et al., 2016; Satija et al., 2017; Satija and Caers, 2015) to reduce uncertainty of all geological model parameters jointly, using all (new) borehole data simultaneously. To achieve this, we will present a model formulation that involves updating based on the hierarchy typically found in subsurface formation: structures, then lithology, and then property and fluid distribution. Finally, we will show how the proposed framework can be completely automated in an open-source project. With a generalized field case study of uncertainty quantification of gas volume in an offshore reservoir, we will illustrate our approach and emphasize the need for automation, minimizing the need for tuning parameters that require human interpretation.

We establish the geological uncertainty quantification framework based on BEL, which is briefly reviewed in this section. BEL is not a method, but a prescriptive and normative data-scientific protocol for designing uncertainty quantification within the context of decision making (Athens and Caers, 2019; Hermans et al., 2018; Scheidt et al., 2018). It integrates four constituents in UQ – data, model, prediction, and decision under the scientific methods and philosophy of Bayesianism. In BEL, the data are used as evidence to infer model or/and prediction hypotheses via “learning” from the prior distribution, whereas decision making is ultimately informed by the model and prediction hypotheses. The BEL protocol consists of six UQ steps: (1) formulating the decision questions and prediction variables; (2) statement of model parametrization and prior uncertainty; (3) Monte Carlo and prior model falsification with data; (4) global sensitivity analysis between data and prediction variables; (5) uncertainty reduction based on statistical learning methods that reflect the principle of Bayesian philosophy; (6) posterior falsification and decision making. Bayesian methods, particularly in the Earth sciences rely on the statement of prior uncertainty. However, such a statement may be inconsistent with data in the sense that the prior cannot predict the data, hence the important falsification step. We next provide important elements of BEL within the problem of this paper: prior model definition, falsification, and inversion by direct forecasting.

In geological uncertainty quantification, any prior uncertainty statement
needs to involve all model components jointly. A geological model

Since borehole data provide information at the locations of drilling, we
define the data variables

The prediction variable

The goal of falsification is to test that the prior model is not wrong.
The prior model should be able to predict the data. Our reasoning then is
that a prior model is falsified if the observed data

If the prior model cannot be falsified, we will use direct forecasting to
reduce geological model uncertainty. Direct forecasting (DF) is a
prediction-focused data science approach for inverse modeling
(Hermans
et al., 2016; Satija et al., 2017; Satija and Caers, 2015). The aim is to
estimate/learn the conditional distribution

In detail, the specific steps of direct forecasting are

Monte Carlo: generate

Orthogonality: PCA (principal component analysis) on data variable

Linearization: maximize linear correlation between the orthogonalized data
and variables by normal score transform and CCA (canonical component
analysis), obtaining transformed

Bayes–linear-Gauss: calculate conditional mean and covariance of the transformed prediction variable.

Sampling: sample from the posterior distribution of transformed prediction
variable

Reconstruction: invert all bijective operations, obtaining

Direct forecasting can also be extended to update model variables, by
simply replacing the prediction variable

We defined our prior uncertainty model (Eq. 2) through a sequential
decomposition of hierarchical model components. Likewise, the conditioning
of such model components to borehole data will be done, using direct
forecasting in a sequential fashion:

To apply this framework to discrete variables such as lithology, we need a
different method for dimension reduction than using PCA. PCA relies on a
reconstruction by a linear combination of principal component vectors, which
becomes challenging when the target variable is discrete.
Figure 1 shows this problem that discrete lithology
model cannot be recovered from inverse PCA. To avoid this, a level set
method of signed distance function
(Osher and Fedkiw, 2003; Deutsch
and Wilde, 2013) is employed to transform rock type models into a continuous
scalar field of signed distances before applying PCA. Here, considering

PCA on discrete lithology model:

Our objective of automation is to allow for seamless uncertainty
quantification once the prior uncertainty models have been established.
Therefore, following the above-described geological UQ strategies, we design
a workflow in Fig. 3 to automate the
implementation. The workflow starts with the prior model Monte Carlo (MC)
samples and borehole observations as input. All following steps including
the extraction of borehole data variables, prior falsification, sequential
direct forecasting, posterior prediction, and falsification (if required) are
completely automated. With this workflow, we develop an open-source Python
implementation to execute the automation (named “AutoBEL”). This
open-source project can be accessed from Github (repository:

Example of transforming categorical lithology model to continuous signed distances for performing PCA.

Proposed workflow to automate the geological uncertainty quantification.

We demonstrate the application of the automated UQ framework using a synthetic
dataset inspired by a gas reservoir located offshore of Australia. This case
study is regarded as synthetic due to simplification for generic
application and because of confidentiality issues. Its spatial size is
around 50 km (E–W)

The structure of the AutoBEL python implementation project.

Initially, the reservoir geological variation is represented on a 3-D model
(Fig. 5b) with a total of 1.5 million grid cells
with dimension of

Layer view of prior Monte Carlo model samples of thickness trend and corresponding thickness, facies, porosity, permeability (logarithmic, termed log-perm), and Sw.

The reservoir geological properties responsible for reserve appraisals are spatial variations in (1) reservoir thickness, spatial distributions of (2) lithological facies belts, (3) 3-D porosity, and (4) 3-D formation water (saturation), while the spatial heterogeneity of (5) 3-D permeability is critical to the future production of gas but is not used in volume appraisal. Constructing a prior uncertainty model for these properties requires a balance between considering aspects of the data and overall interpretation based on such data. The strategy in the BEL framework is not to state too narrow an uncertainty initially but rather to explore a wide range of possibilities. Based on interpretation from data, Table 1 contains all uncertainties and their prior distribution was constructed. We will clarify how these uncertainties were obtained.

The global model parameter

First, the thickness uncertainty is mainly due to a limited resolution of the
geophysical seismic data and uncertainty in velocity modeling (not shown in
this paper). Seismic interpretations show no faults in the geological
system, but the thickness variations follow a structural trend. To model
thickness uncertainty, we decompose thickness

The lithological facies are considered to have rather simple spatial variability and are described as “belts” (see Fig. 5a). These are common in the stratigraphic progression and typical of shallow marine environments. To describe such variation, we use a 3-D Gaussian process that is truncated (Beucher et al., 1993), thereby generating discrete variables. This truncated Gaussian process has a specific advantage in reproducing simple organizations of ordered lithologies, thus making a useful model in our case. Because four facies exist, three truncations need to be made on the single Gaussian field. The truncation bounds are determined based on facies proportions. The uncertain facies proportions are obtained from lithological interpretations on borehole gamma ray logs and geophysical seismic interpretation.

For each facies belt, rock porosity and permeability (logarithmic scale, termed log-perm) are modeled, using two correlated 3-D Gaussian processes. The cross-covariances of these processes are determined via Markov models (Journel, 1999) that only require the specification of a correlation coefficient. Laboratory measurements on the borehole rock core samples show that permeability is linearly correlated to porosity with a coefficient of 0.80 and a small experimental error (around 6 % random error according to the lab scientists by repeating the experiments). The marginal distributions of porosity and log-perm are assumed to be normal but with uncertain mean and variances. The mean of porosity and log-perm is based on borehole neutron porosity logs and core sample measurements. Similar to the thickness residual modeling, the spatial covariances are modeled via a variogram, respectively, for porosity and permeability, with uncertain range and sill. Limited wellbore observations make variogram range and sill highly uncertain, and therefore large uncertainty bounds are assigned.

Rocks contain gas and water; hence the uncertain saturation of water (Sw)
will affect the uncertain gas volume calculations. The modeling of Sw is
based on a classical empirical capillary pressure model from a Leverett
J-function (Leverett et al., 1942),
formulated as

By running Monte Carlo from the given prior distribution in
Table 1, a set of 250
geological model realizations are generated. Figure 6
displays Monte Carlo realizations of the geological model: thickness trend
and corresponding thickness model, facies, porosity, permeability (log-perm), and Sw. With prior samples of the geological model, prior prediction of GIIP is calculated, using the following linear equation:

Uncertainty quantification of GIIP based on prior uncertainty and four boreholes.

Table 1 is a subjective statement of prior uncertainty. When new data are
acquired, this statement can be tested, using a statistical test (Sect. 2.1.3) that may lead to a falsified prior. To perform falsification,
borehole data variables at the seven new well locations (from w7 to w13) are
extracted by applying the data forward operator

Prior falsification using robust Mahalanobis distance (RMD). Circle dots represent the calculated RMD for data variable samples. The red square is the RMD for borehole observations. The red dashed line is the 97.5 percentile of the chi-squared distributed RMD.

After attempting to falsify the prior uncertainty model, we use the
automated framework to jointly update model uncertainty with the new
boreholes. The joint model uncertainty reduction is performed sequentially
as explained in Sect. 2.2.2. Under the AutoBEL GitHub repository
instruction (

Example of applying PCA to thickness model. One model realization

Uncertainty in facies and thickness models can be updated jointly, as they
are two independent components for this case. AutoBEL first transforms
the categorical facies to a continuous model using signed distance function.
The transformed signed distances are then combined with the thickness model to
perform orthogonalization using mixed PCA
(Abdi et al., 2013). As shown in
Fig. 9, the first eigen image (first principal
components, PC1) of thickness reflects the global variations in reservoir
thickness, while higher-order eigen images (e.g., eigen image of PC40)
represent more local variation features. To evaluate what model variables
impact thickness variation at the boreholes, DGSA
(Fenwick et al., 2014) is then performed to
analyze the sensitivity of model variables to data.
Figure 10a plots the main effects in a Pareto plot.
As shown in the plot, DGSA identifies sensitive (measure of sensitivity

Following the steps of direct forecasting (see Sect. 2.2.1), uncertainty reduction proceeds by mapping all sensitive model variables into a lower-dimensional space such that the Bayes–linear-Gauss model can be applied. This requires the application of CCA to the selected model variables and data variables and then normal score transformation. Figure 10b shows two examples of a cross plot between model and data variables of the first and tenth canonical components, where we observe a linear correlation coefficient of 0.84 even for the tenth canonical components. Once the Bayesian model is specified, one can sample from the posterior distribution and back-transform from lower-dimensional scores into actual facies and thickness models. Figure 10c shows the distribution of the posterior model realizations in comparison to the corresponding prior, showing the reduction in the model uncertainty. Figure 10d shows the comparison between the prior and posterior distributions of the scores for the first four sensitive PCs, where the reduction in uncertainty is observed (while noting that uncertainty quantification involves all the sensitive PC score variables).

Uncertainty reduction in thickness and facies:

Figure 11 plots the reconstructed posterior global
parameters in comparison to the prior. Uncertainty reduction in sensitive
global parameters is observed, while the distribution of non-sensitive
global parameters (

Uncertainty updating of

AutoBEL is now applied to update the uncertainty in porosity, permeability, and saturation under the sequentially decomposition.

The prior Monte Carlo samples have provided a full distribution of porosity for each facies. This allows the calculation of posterior porosity to fit the obtained posterior facies models. Therefore, we condition to posterior facies model and borehole porosity observations in AutoBEL to calculate the posterior porosity. Similarly, for permeability and saturation model, AutoBEL is applied by additionally conditioning to posterior models from previous model components.

Figures 14, 15, and 16 show the results. In Fig. 14, we see sensitive global and spatial model variables that are selected for uncertainty reduction. Figure 15 shows the constructed the linear correlation between data and sensitive model variables by means of CCA. Figure 16 plots the posterior model realizations (250 realizations) computed from the Bayes–linear-Gauss model, where reduced uncertainty is observed when comparing to the prior. The posterior spatial model PC scores are also plotted in Fig. 17.

Ensemble variance of the posterior and prior thickness and facies models from the first sequence.

Results from global sensitivity analysis using DGSA at

First canonical covariates of data and model variables
from

Reduction in uncertainty of the first model canonical
component:

Prior and posterior distribution of the scores of the two
sensitive PCs with highest variances:

Finally, by back-transformation, we can reconstruct all original model variables. Figure 18 compares ensemble means and variances of the reconstructed posterior porosity, log-perm, and Sw to their corresponding prior models, with actual borehole observations plotted on top. Taking w7 for example, the actual borehole observations show low values of porosity, permeability, and Sw, while the prior model initially expects those values to be large at this location. This is adjusted in the posterior. From the ensemble variance maps, we notice that spatial uncertainty is significantly reduced from prior to posterior in areas near w7. The updates of model expectations and reduction in spatial uncertainty are also observed from the other wells. It implies that the posterior models have been constrained by the borehole observations.

Ensemble mean and variance of posterior and prior
geological models:

Figure 19 shows one example realization of the spatial models. It shows that, as with the hierarchical order in the prior (Fig. 19a), the spatial distributions of posterior porosity and log-perm follow the spatial patterns of their corresponding facies belts (Fig. 19b). However, if the joint model uncertainty reduction is performed without the sequential decomposition (not conditioning to the posterior models from previous sequences), the model hierarchy from facies to porosity and permeability is lost (marked by the purple boxes in Fig. 19c). This is because they are treated as independent model variables, which violates the imposed geological order of variables. The linear correlation between porosity and log-perm is also preserved due to the sequential decomposition. We observe similar correlation coefficients from prior (Fig. 20a) to posterior (Fig. 20b). But without sequential decomposition, this important feature cannot be maintained as the results shown from Fig. 20c: (1) the four-cloud pattern (representing the four facies) of the covariate distribution between porosity and log-perm is lost; (2) the correlation coefficient has changed significantly for facies 0, 2, and 3.

Prior and posterior facies, porosity and log-perm of realization
no. 1

Bivariate distribution between porosity and log-perm model
of realization no. 1

Gas storage volume is calculated using the posterior geological models and plotted in Fig. 21. The result highlights a steep uncertainty reduction in comparison to the initial prior prediction. The posterior predicted GIIP leads to a major shift in the expected gas volumes to a more positive direction (higher than initially expected). More importantly, the forecast range is significantly narrowed. This provides critical guidance to the financial decisions on the field development. It also in return confirms the value of the information of the newly drilled wells. In total, the whole application of AutoBEL to this test case took about 45 min (not including the time on prior modeling) when run on a laptop with an Intel Core i7-7820HQ processor and 64 GB of Ram.

To test the posterior, we perform posterior falsification using data from validating boreholes (w5 and w6). Figure 22 plots the result from applying robust Mahalanobis distance outlier detection to the posterior data of the two wells. The statistical test shows that the test borehole observation falls within the main population of data variables, below the 97.5 threshold percentile. We also want to further examine if the posterior models can predict the validating boreholes (regarded as future drilling wells) with reduced uncertainty. To do so, we compare the prior and posterior predicted thickness at the two borehole locations, together with their actual measurements (Fig. 23). For 3-D models of facies, porosity, log-perm, and Sw, this comparison is performed on vertical average values across the 75 layers. We notice that these future borehole observations are predicted by posterior models with significantly reduced uncertainty.

One main purpose of this paper is to introduce automation to geological uncertainty quantification when new borehole data are acquired. We tackle this challenge by following the protocol of Bayesian evidential learning to build an automated UQ framework. BEL formulates a protocol involving falsification, global sensitivity analysis, and statistical learning uncertainty reduction. When establishing such a framework for geological UQ, three important questions have to be addressed. The first is on how to preserve the hierarchical relationships and correlations that commonly exist in geological models. We propose a sequential decomposition by following the chain rule under Bayes' theorem. This allows us to assess the joint distribution of multiple model components while honoring the geological rules. The second one is on how to falsify the geological model hypotheses, especially when data become highly dimensional. We employ multivariate outlier detection methods. They provide quantitative and robust statistical calculations when attempting to falsify the model using high-dimensional data. The last but most practical one, is to deploy data-science-driven uncertainty reduction. Uncertainty reduction in geological models is usually time-consuming because conventional inverse methods require iterative model rebuilding. When it comes to real cases, the daunting time consumption and computational efforts of conventional methods can hamper practical implementations of automation. Direct forecasting helps to avoid this, as it mitigates the uncertainty reduction to a linear problem in a much lower dimension. There are many dimension reduction methods for complex models, such as deep neural network (Laloy et al., 2017, 2018), but here we use PCA because it is simple and bijective, and the structure models are not complex (e.g., channels). However, direct forecasting of geological model is faced with two new challenges. One is to accommodate a direct forecasting algorithm to the sequential model decomposition. This is achieved by additionally conditioning to the posterior from previous sequences. The other challenge is that DF cannot be directly applied to categorical models such as lithological facies. We therefore introduce a signed distance function to convert categorical models to continuous properties before performing the DF. Field application has shown the benefits of using the proposed framework. Since the posterior in the case study cannot be falsified, its uncertainty can be further reduced by repeating the automated procedures with validating borehole observations. This suggests that the proposed framework has potentials for life-of-field uncertainty quantification for applications where new boreholes are regularly drilled.

The main challenge addressed in this paper is to apply such an uncertainty quantification within a Bayesian framework. Most methods applied in this context simply rebuild the models by repeating the same geostatistical methods that were used to construct the prior model. In such an approach, all global variables and their uncertainty need to be reassessed. The problem with such an approach is twofold. First, it does not address the issue of falsification: the original models may not be able to predict the data. Hence, using the same approach to update models with a prior that may have been falsified may lead again to falsification, thereby leading to invalid and ineffective uncertainty quantification. As a result, the uncertainty quantification of some desirable property, such as volume, exhibits a yo-yo effect (low variance in each UQ but shifting mean). Second, there is no consistent updating of global model variables. Often such uncertainties are assessed independently of previous uncertainties. The challenge addressed in this paper is to jointly update global and spatial variables and do this jointly for all properties.

The proposed method offers a Bayesian consistency to uncertainty
quantification in the geological modeling setting. However, unlike
geostatistical methods, the posterior models do not fully match local
borehole observations. The current method is only designed to globally
adjust the model, not locally at the borehole observation. This can be an
important issue if using the model for subsurface flow simulations. To
tackle this problem, one possible path we would like to explore in the future is
to combine geostatistical conditional simulation as posterior step to the
current methodology. A second limitation is that the method does not (yet) treat discrete global variables, such as a geological interpretation. In the
case study, only one interpretation of the lithology was used. The way such
variables would be treated is by assigning prior probabilities to each
interpretation (e.g., of a depositional system) and then updating them into
posterior probabilities. This has been done by treating the interpretation
independent of other model variables in some studies
(e.g.,
Aydin and Caers, 2017; Grose et al., 2018; Wellmann et al., 2010). For
example, one could first update the probabilities of geological scenarios,
then update the other variables (Lopez-Alvis
et al., 2019). Regarding the automation of BEL, its intermediate steps can
also be adjusted depending on users' specific applications. Taking the
direct forecasting step for example, here we adapt it for uncertainty
quantification using borehole data, which is a linear problem. But for more
complex nonlinear inverse problems, it may be difficult to use CCA to
derive a Bayes–linear-Gauss relationship in DF. Statistical estimation
approaches such as kernel density estimation
(Lopez-Alvis et al., 2019) can be used for
such cases, and there are also extensions of CCA to tackle nonlinear
problems (e.g., Lai and Fyfe, 1999). AutoBEL can also
be adapted if other types of parameters (other than spatial model parameters)
are used for uncertainty quantification. This can be done by simply adding
the additional parameters to the model variable

The prior and posterior prediction of GIIP.

Posterior falsification using robust Mahalanobis distance outlier detection method using the data from w5 and w6.

Prior and posterior predicted thickness, facies, porosity, log-perm, and Sw at validating boreholes. The blue and brown colored dots, respectively, represent the prior and posterior prediction, while the red squares are the actual observations.

In conclusion, we generalized a Monte Carlo-based framework for geological uncertainty quantification and updating. This framework, based on Bayesian evidential learning, was demonstrated in the context of geological model updating using borehole data. Within the framework, a sequential model decomposition was proposed, to address the geological rules when assessing the joint uncertainty distribution of multiple model components. For each component, we divided model parameters into global and spatial ones, thus facilitating the uncertainty quantification of complex spatial heterogeneity. When new borehole observations are measured, instead of directly reducing model uncertainty, we first strengthen the model hypothesis by attempting to falsify it via statistical tests. Our second contribution was to show how direct forecasting can jointly reduce model uncertainty under the sequential decomposition. This requires a posterior model from previous sequences as additional inputs to constrain the current prior. Such sequential direct forecasting was shown to maintain important geological model features of hierarchy and correlation, whilst avoiding the time-consuming conventional model rebuilding. In terms of discrete models, such as lithology, a signed distance function was employed, before applying direct forecasting to reduce uncertainty. The third contribution, but maybe a more important one, is that the proposed framework allows the automation of geological UQ. We developed an open-source Python project for this implementation. Its application to a large reservoir model showed that the automated framework ensures that the model is objectively informed by data at each step of uncertainty quantitation. It jointly quantified and updated uncertainty of all model components, including structural thickness, facies, porosity, permeability, and water saturation. The posterior model was shown to be constrained by new borehole observations globally and locally, with dependencies and correlations between the model components preserved from the prior. It predicted validating observations (future drilling boreholes) with reduced uncertainty. Since the posterior cannot be falsified, the uncertainty-reduced GIIP prediction can be used for decision makings. The whole process takes less than 1 h on a laptop workstation for this large field case, thus demonstrating the efficiency of the automation

AutoBEL is a free, open-source Python library. It is available at GitHub:

ZY contributed the concept and methodology development, wrote and maintained the code, conducted the technical application, and drafted this paper. SS prepared data for the methodology application and provided critical insights during the research initialization. JC provided overall supervision and funding to this project, contributed major and critical ideas to the research development, and revised the paper.

The authors declare that they have no conflict of interest.

We thank Chevron for sponsoring this research project. The authors would like to express special thanks to Maisha Amaru, Jairam Kamath, Lewis Li, Sarah Vitel, Lijing Wang, and Celine Scheidt for the technical discussions and support.

This paper was edited by Thomas Poulet and reviewed by Guillaume Caumon and one anonymous referee.