Uncertainties are common in geological models and have a considerable impact on model interpretations and subsequent decision-making. This is of particular significance for high-risk, high-reward sectors. Recent advances allows us to view geological modeling as a statistical problem that we can address with probabilistic methods. Using stochastic simulations and Bayesian inference, uncertainties can be quantified and reduced by incorporating additional geological information. In this work, we propose custom loss functions as a decision-making tool that builds upon such probabilistic approaches.

As an example, we devise a case in which the decision problem is one of estimating the uncertain economic value of a potential fluid reservoir. For subsequent true value estimation, we design a case-specific loss function to reflect not only the decision-making environment, but also the preferences of differently risk-inclined decision makers. Based on this function, optimizing for expected loss returns an actor's best estimate to base decision-making on, given a probability distribution for the uncertain parameter of interest. We apply the customized loss function in the context of a case study featuring a synthetic 3-D structural geological model. A set of probability distributions for the maximum trap volume as the parameter of interest is generated via stochastic simulations. These represent different information scenarios to test the loss function approach for decision-making.

Our results show that the optimizing estimators shift according to the characteristics of the underlying distribution. While overall variation leads to separation, risk-averse and risk-friendly decisions converge in the decision space and decrease in expected loss given narrower distributions. We thus consider the degree of decision convergence to be a measure for the state of knowledge and its inherent uncertainty at the moment of decision-making. This decisive uncertainty does not change in alignment with model uncertainty but depends on alterations of critical parameters and respective interdependencies, in particular relating to seal reliability. Additionally, actors are affected differently by adding new information to the model, depending on their risk affinity. It is therefore important to identify the model parameters that are most influential for the final decision in order to optimize the decision-making process.

In studies of the subsurface, data availability is often limited and characterized by high possibilities of error due to signal noise or inaccuracies. This, together with the inherent epistemic uncertainty of the modes, leads to the inevitable presence of significant uncertainty in geological models, which in turn may affect interpretations and conclusions drawn from a model

Building on this probabilistic perspective, we propose the use of custom loss functions as a decision-making tool when dealing with uncertain geological models. In many applications, we are interested in some decisive model output value, for example reservoir volume. Given that such a parameter is the result of a deterministic function of uncertain variables in our model, the parameter of interest is likewise uncertain and can be represented by a probability distribution attained from stochastic simulations. A loss function can be applied to such a distribution to return a case-specific best estimate to base decision-making on.

We consider hydrocarbon exploration and production as an exemplary high-risk, high-reward sector, in which good decision-making is crucial. However, the described methods are potentially equally applicable to other types of fluid reservoirs (e.g., groundwater, geothermal, or

To illustrate our approach of using custom loss functions for decision-making, we first illustrate what such customization might look like step by step: starting off with a standard symmetrical loss function, incorporating scenario-specific conditions and assumptions, and lastly implementing a factor to represent the varying risk affinities of different decision makers.
As we assume a petroleum exploration and production decision-making scenario, our parameter of interest should be one that indicates the economic value of a potential hydrocarbon accumulation. In a larger context, including various geological and economic factors such as operational expenditures, this could be the net present value (NPV) of a project. In preproduction stages, original oil in place (OOIP) is commonly used for early assessments

Once we have set up a loss function customized to this decision problem, we can apply it to probability density functions that represent our knowledge about the true value of the parameter of interest. As mentioned above, such distributions can result from geological modeling in a probabilistic context. To illustrate this, we include a synthetic 3-D structural geological model as a case study. In this context, we define the structurally determined maximum trap volume

We view the statistical analysis of geological models from a probabilistic perspective, which is most importantly characterized by its preservation of uncertainty. Its principles have been presented and discussed extensively in the literature (see

In many cases, decisions are made on the basis of summary parameters such as mean or standard deviation. This approximation works for well-defined probability distributions but it may fail when the distribution does not have a defined structure, which is the usual case of distribution generated as a result of Bayesian inference. In this work, we aim to tackle decision problems associated with probabilistic inferences. By applying Bayesian decision theory concepts, we are capable of transforming an arbitrary complex set of distributions onto a more adequate dimension for decision-making.

Common point estimates, such as the mean and the median of a distribution, usually come with a measure for their accuracy

Loss is a statistical measure of how “bad” an estimate of the parameter

The presence of uncertainty during decision-making implies that the true parameter value is unknown and thus the truly incurred loss

Under uncertainty, the expected loss of choosing an estimate

By the law of large numbers, the expected loss of

Standard symmetric loss functions can easily be adapted to be asymmetric, for example by weighting errors to the negative side stronger than those to the positive side. Preference over estimates larger than the true value, i.e., overestimation, is thus incorporated in an uncomplicated way. Much more complicated designs of loss functions are possible, depending on purpose, objective, and application. We will describe potential design options in the following.

For our example of estimating the economic value of a hydrocarbon prospect, which is represented by the maximum trap volume

It can be assumed that several decision makers in one such environment or sector may have the same general loss function but different affinities concerning risks. This might be based, for example, on different psychological factors or economic philosophies followed by companies. It might also be based on the budgets and options such actors have available. An intuitive example is the comparison of a small and a large company. A false estimate and wrong decision might have a significantly stronger impact on a company that has a generally lower market share and few projects than on a larger company that might possess higher financial flexibility and for which one project is only one of many development options in a wide portfolio.

In steps I–IV we make assumptions about the significance of such deviations and how they differently contribute to expected losses in the general decision-making environment and introduce the concept of varying risk affinities in the final step V.

According to Eq. (

Illustration of different steps and aspects of our loss function customization. Functions are applied to an abstract score as the parameter of interest.

In Fig.

In Fig.

In Fig.

For a better understanding of how our finalized custom loss function determines the incurrence of loss, actual losses for three fixed true values and risk neutrality (

Loss based on the risk-neutral custom loss function (Eq.

It has to be emphasized that this is just one possible proposal for loss function customization. There is not one perfect design for such a case

Next we want to show that this loss function approach is not only applicable to simple probability distributions but is an equally useful tool to estimate the true value of a parameter of interest resulting from more complex geological models that encompass numerous uncertain input parameters. As a case study, we now consider a synthetic 3-D structural geological model that is placed in a probabilistic framework.

Computationally, we implement all of our methods in a Python programming environment, relying in particular on the combination of two open-source libraries: (1) GemPy (version 1.0) for implicit geological modeling and (2) PyMC (version 2.3.6) for conducting probabilistic simulations.

GemPy is able to generate and visualize complex 3-D structural geological models based on a potential-field interpolation method originally introduced by

PyMC was devised for conducting Bayesian inference and prediction problems in an open-source probabilistic programming environment

To visually compare the states of geological unit probabilities after conducting stochastic simulations, we consider the normalized frequency of lithologies in every single voxel and visualize the results in probability fields (see

Our geological example model is designed to represent a potential hydrocarbon trap system. Stratigraphically, it includes one main reservoir unit (sandstone), one main seal unit (shale), an underlying basement, and two overlying formations that are assumed to be permeable so that hydrocarbons could have migrated upwards. Structurally, it is constructed to feature an anticlinal fold that is displaced by a normal fault. All layers are tilted and dip in the opposite direction of the fault plane dip. A potential hydrocarbon trap is thus found in the reservoir rock enclosed by the deformed seal and the normal fault.

Using GemPy, we construct the geological model as follows: in principle, it is defined as a cubic block with an extent of 2000

We include uncertainties by assigning them to the

Input parameter uncertainties defined by distributions with respective means

Such probability distributions can also be allocated as homogeneous sets to point and feature groups that are to share a common degree of uncertainty (see Table

This model was designed for the primary purpose of testing our loss function method. All features, uncertainties, and parameter relations were implemented in a way that they result in model variability and complexity that is adequate and significant to the decision problem in this work. The model is not aimed at representing a completely plausible or realistic geological setting.

Design of the 3-D structural geological model. A 2-D cross section through the middle of the model (

Given full 3-D representation of geological structures, we can now define the trap volume

We argue that

By declaring these connections, we have given our model an economic significance. We can assume that the hydrocarbon trap volume is directly linked to project development decisions; i.e., the investment and allocation of resources is represented by bidding on a volume estimate.

In the course of this work, we developed a set of algorithms to enable the automatic recognition and calculation of trap volumes in geological models computed by GemPy. The volume is determined on a voxel-counting basis via four conditions illustrated in Fig.

Following these conditions, we can define four major mechanisms that control the maximum trap volume: (1) the anticlinal spill point of the seal cap, (2) the cross-fault leak point at a juxtaposition of the reservoir formation with itself, (3) leakage due to juxtaposition with overlying layers and cross-fault seal breach (failure related to the shale smear factor, SSF), and (4) stratigraphical breach of the seal when its voxels are not continuously connected above the trap. Due to the nature of our model, (3) and (4) will always result in complete trap failure. The occurrence of these trap control mechanisms can be tracked throughout stochastic simulations of the model.

Illustration of the process of trap recognition in 2-D, i.e., the conditions that have to be met by a model voxel to be accepted as belonging to a valid trap. A voxel has to be labeled as part of the target reservoir formation

The trap volume

Furthermore, we consider the possibility of updating our model by adding additional secondary information via Bayesian inference. We do this by introducing likelihood functions that constrain our primary parameters. We have to note that these inputs remain unchanged; however, their prior probability distributions are revalued given the additional statistical information. We achieve this by conducting Markov chain Monte Carlo (MCMC) simulations. Decision-making is then based on the resulting posterior probability. Using different likelihood functions, we can create and generate different posterior probability distributions for

For the application of Bayesian inference, we implement two types of likelihoods.

Although Bayesian inference was utilized in this case study, it served primarily for the generation of these different but comparable distributions on which to base our decision-making, i.e., the application of our custom loss function. For additional information on how implicit geological modeling can be embedded in a Bayesian framework and how this can be used to reduce uncertainty, we refer to the work by

We applied our custom loss function to various different

We present the following information scenarios.

Introducing

Likely thick seal

Likely thin seal

Introducing

Likely thick reservoir

Likely thick reservoir and thick seal

Introducing

SSF likely near its critical value

Likely reliable SSF and thick seal

For the comparison of results, we consider in particular the following measures: (1) probability field visualization, (2) occurrence of trap control mechanisms, (3) resulting trap volume distributions, and (4) consequent realization of expected losses and related decisions.

Normal distribution mean (

Probability field visualization illustrates well how the prior uncertainty is based on normal distributions (see Fig.

Maximum trap volumes were calculated for each model iteration and plotted as a probability distribution in Fig.

Consequently, applying our custom loss function to this distribution resulted in widely separated minimizing estimators for the differently risk-inclined actors (see Fig.

Trap volume distribution and resulting loss function realizations for Scenario 1 (prior) and Scenario 2a, in which we introduced the likelihood of a thick seal. Comparing both, we can observe how the additional information reduced the bimodality in the posterior distribution (2a), particularly by reducing the probability of complete failure and enhancing positive probabilities. Consequently, Bayes actions converged and expected losses were reduced.

We considered two scenarios of thickness likelihoods: the seal being (Scenario 2a) likely very thick or (Scenario 2b) likely very thin (see Table

In Scenario 2a, probability visualization illustrates that the presence of a thick seal is very probable (see Fig.

A high likelihood of a reliable seal cap (2a) significantly reduced the probability of trap failure, while enhancing the mode of highly positive outcomes (see Fig.

In both scenarios, Bayes actions shifted towards the respectively emphasized modes. This came with the overall convergence of decisions and reduction of expected losses. In Scenario 2a, all decision makers bid on a positive outcome. Risk-averse individuals experienced the strongest shift but also present the highest expected losses. In Scenario 2b, all individuals decide not to allocate resources. Even the risk-friendliest actor moved to a zero estimate, with the most risk-averse bid having already been placed in the prior Scenario 1. However, although all decisions coincide, expected losses increase from risk averse to risk friendly (see Table

Probability field visualizations for seal and reservoir units in Scenarios 1 (prior), 4a, and 4b. For Scenario 1, we used 3-D voxel visualizations and set a threshold at a probability of 0.5 (only voxels with a probability higher than 0.5 are shown). It can be recognized that the seal is disrupted across the fault in more than 50 % of the prior model realizations. For the other scenarios, we show the full probability field for both units on a section through the middle of the model (

We also tested scenarios for the likelihood of a thick reservoir formation alone (Scenario 3a) and in combination with the likelihood of a thick seal (Scenario 3b; see Table

In Scenario 3a, failure probabilities slightly increased, resulting in a decision shift towards lower values (see Fig.

Trap volume distribution and resulting loss function realizations for Scenario 4a and Scenario 4b. Adding a likelihood of the SSF being around its critical value led to increased bimodality and an elimination of low to moderate volume probabilities. Bayes actions diverged accordingly in Scenario 4a. Implementing a reliable SSF value likelihood (

We considered two SSF-related likelihood scenarios. In Scenario 4a, we implemented solely an SSF likelihood that was based on a narrow normal distribution (

Scenario 4a resulted in increased bimodality of the posterior distribution (see Fig.

The results for 4b were comparable to those of 2a but more pronounced. Entropies, particularly related to the seal thickness, were clearly reduced, also in the hanging wall. Probabilities of failure and low volumes were almost eliminated, further enhancing the highly positive mode. This consequently resulted in an even higher convergence of Bayes actions, as well as reduction of expected losses compared to Scenario 2a. Anticlinal spill is the decisive control mechanism in 79.5 % of cases; otherwise, only cross-fault leakage to the reservoir occurred (20.5 %).

Our results show that it is possible to apply Bayesian decision theory to geological models as an approach to obtain an objective basis for decisions by considering uncertainties in these models. Even though the concept itself is not new, the application to the context of probabilistic geological modeling requires some adaptation and care when constructing appropriate loss functions. Our results highlight the potential use of custom loss functions, first for a simplified 1-D case, and then for a more complex full 3-D model. Even though these models are both conceptual, they highlight in our point of view the interesting potential of the method, as the optimal decision, the Bayes action, is not always directly obvious when only considering posterior predictive distributions. The addition of subjective risk affinity and the risk of critical overestimation particularly lead to interesting changes in the optimal decision. Given these aspects, we consider the use of custom loss functions with probabilistic geological modeling to be a very suitable combination in the framework of Bayesian decision theory.

The case study considered here addressed a typical scenario of exploration for a fluid reservoir. We first discuss additional relevant points with regard to this specific case and then provide more general comments on extensions and the application in additional fields in which geological models are commonly used.

As we defined trap volume to be in essence a deterministic function of uncertain model input parameters, uncertainties propagate to this parameter of interest when conducting stochastic simulations. We consider the resulting volume probability distributions to be expressions of the respective state of knowledge (or information) on which the decision-making is to be based. As this should include all parameters and conditions relevant for decision-making, we furthermore propose that the overall uncertainty inherent in this probability distribution can be referred to as “decision uncertainty” and that this entity should be viewed separately from geological model uncertainty.

By viewing decision-making as a problem of optimizing a case-specific custom loss function applied to such a state of knowledge and decision uncertainty, we were able to observe clear differences in the respective behavior of distinctly risk-inclined actors.

The position and separation of their minimizing estimators, i.e., their decisions, manifested according to the properties of the value distributions. The general spread and the occurrence of modes relative to the overall distribution and the relevant decision space appear to be particularly significant. High spread and bimodal tendencies, i.e., high overall uncertainty, resulted in a wider separation of different actions. Reduction of the distribution to one mode conversely led to their convergence. A decrease in decision uncertainty was furthermore accompanied by a reduction in expected loss for each Bayes estimator.

Considering these observations, we derive the degree of action convergence and respective expected losses as measures for the state of knowledge and decision uncertainty at the moment of making a decision. The better these are, the more similar the decisions of differently risk-inclined actors and the lower their loss expectations are. Given perfect information all actors would bid on the same estimate (the true value) and expect no loss, since no risk would be present. It furthermore follows from this that the relevance of risk affinity decreases with greater reduction of decision uncertainty.

We used these loss-function-related indicators to assess the significance of additional information for decision-making. We observed that the impact on decision uncertainty, induced by Bayesian inference, is not simply strictly aligned with the change in uncertainty regarding model parameters but on parameter combinations that are relevant for the outcome of the value of interest. It seems to be of central importance (1) “where” in the model uncertainty is reduced, i.e., in which spatial area or regarding which model parameters, and (2) which possible outcome is enhanced in terms of probability. An increased probability of a thick or thin seal in our model equally reduced decision uncertainty significantly by raising the probability of a positive or negative outcome, respectively. Improved certainty about our reservoir thickness, however, had far lesser impact on decision-making. This shows that some areas and parameter combinations have a much greater influence on the decision uncertainty than others, depending on the way they contribute to the outcome of the value of interest.

Some types of additional information could even lead to increased decision uncertainty. We observed this in Scenario 4a. The introduced SSF likelihood practically constrained our geological model to two possible situations: (1) a trap that is sealed off from juxtaposing layers and full to spill and (2) complete failure of the trap due to a breached seal across the fault. This made the decision problem a predominantly binary one and split the outcome distribution into two narrowed but distant modes. The resulting increase in decision divergence and expected losses show that, in some cases, adding information might leave actors in greater disagreement than before.

However, we furthermore have to consider that actors weight possible outcomes of the value distribution differently. They are consequently affected differently by the same type of information. Risk-friendly actors were the most robust in their decision-making in the face of possible trap failure. Eliminating this risk proved to be far less significant for the most risk-friendly than for risk-averse actors. Accordingly, it should be of foremost importance for risk-averse actors to reduce the uncertainty regarding critical factors, such as seal integrity, which might decide between the success and complete failure of a project. This is less relevant for risk-friendly decisions makers, who might acquire a comparable benefit from knowing more about the probability of positive outcomes. They are less afraid of failure than they are of missing out on opportunity.

Crucial risks might be easily assessed if they are dependent on only one or a few parameters, such as seal thickness. In other cases, they are derived from more complex parameter interrelations, as is the case for the shale smear factor. To approach an effective mitigation of high risks, the complexities behind decisive factors need to be assessed thoroughly, and respective parent parameters, as well as their interdependencies, need to be identified. This might enable a better understanding of which type of information is missing and where in the model additional data might be of use for improved decision-making.

More of simply any type of information does not necessarily lead to better decisions. Instead, improved decision-making is achieved by attaining the right kind of information that is able to shed light on uncertainties that are relevant to an individual's own goals and preferences, as well as the general problem at hand.

In this work, we applied our loss function approach to estimate a hydrocarbon trap volume. For this, we considered stochastic geomodeling parameters, defined deterministic functions to acquire volume, layer thicknesses, and SSF values, and linked the latter two to respective likelihoods. Regarding the bigger picture, this methodology is expandable and could include other parameters and dependencies. By taking into account other reservoir parameters and recovery factors, we could, for example, base decision-making on recoverable volumes. We could also take depth information from our model and combine this with other cost parameters to calculate drilling costs. Including additional costs, but also the selling price of hydrocarbons, we could attain the NPV as our final value of interest.

While Monte Carlo simulation is by now common in the hydrocarbon sector, it does not make decisions, as

Bayesian inference and MCMC methods have been applied for OOIP estimation and forecasting of reservoir productivity by

Our continuous approach could be integrated into common discrete decision-making frameworks, such as decision trees. In real cases, normally only a limited number of options is given. In the context of hydrocarbon exploration and production, this would relate to fixed magnitudes of resource allocation, such as a certain number of required drilling wells or the size of a production platform. Based on such previously defined actual options, we could discretize our value probability distribution into sections, which represent each decision scenario accordingly. Our minimizing estimators would then indicate the best discrete option for a decision maker.

We applied the concept of decision theory here to an implicit geological modeling method

We defined risk affinity to be dependent on arbitrarily chosen risk factors that led to according reweighting.

There are still many points that could be expanded on in future research. It would be of interest to apply the same overall concept and methodology to an authentic case based on real datasets. Given a realistic economic scenario including the capital and operational expenditures of a project, a full net-present-value (NPV) analysis could possibly be conducted by applying a loss function to an NPV distribution (see Fig.

We chose hydrocarbon systems and petroleum exploration as a sector for an exemplary application, as studies on risk related to geological modeling are most prominent in this field. However, geological modeling is of central importance to decision-making in several other fields. Directly related are all other types of subsurface fluid reservoirs, for example in groundwater extraction or geothermal energy usage. Also closely related are applications of fluid storage in subsurface reservoirs, most prominently carbon capture and storage (CCS) applications. Questions regarding storage capacity and safety deal with similar conditions and geological problems as the ones presented in this work. The described concepts can similarly be applied to other types of geological features, for example ore bodies in mineral exploration or subsurface structures and materials in geotechnical applications. In all of these cases, the geological model can have significant uncertainties and, similar to the example described in this paper, further engineering and usage aspects carry high costs. We are therefore confident that a more detailed analysis of uncertainties and the definition and understanding of custom loss functions in the context of Bayesian decision theory are very interesting paths for future research with wide possible applications.

The code and model data used in this study are available in a GitHub repository found at

The volume is calculated on a voxel-count basis. To assign model voxels to the trap feature, it is necessary to check whether the following conditions (illustrated in Fig.

For our model, we define the critical SSF to be SSF

Regarding anticlinal structures and traps, it can be observed that, geometrically and mathematically, a spill point is a saddle point of the reservoir top surface in 3-D. This was described by

Regarding a surface defined by

If

If

If

If

We first look for vertices at which the surface of interest coincides with a gradient zero point.

Then, we check for the change in gradient sign at each such point in perpendicular directions. If they are opposite to one another, we can classify the vertex as a saddle point.

Lastly, we declare the highest saddle point to be our anticlinal spill point.

For the potential point of leakage to formations underlying the seal across the normal fault (including the reservoir itself), we take the highest

In the case of a juxtaposition with seal-overlying formations and a failed SSF check, the maximum contact of the trap with the fault becomes the final spill point. Due to the shape of the trap in our model, we can then expect full leakage and set the maximum trap volume to zero.

When all trap voxels have been determined via the conditions defined in Sect.

For the example of a cubic geological model with an original extent of 2000

Posterior trap volume distributions and respective loss function realization plots for Scenarios 1 (prior), 2b, 3a, and 3b.

Probability field visualizations for Scenarios 1 to 3b.

Decision results for all considered scenarios and each actor. Respective optimal estimates (decisions) are represented by

Occurrence rate of trap control mechanisms in percent for each information scenario.

Geweke plots and traces for Scenarios 2a to 3b.

Geweke plots and traces for Scenarios 4a and 4b.

FAS, MdlV, and FW contributed to the conceptualization and method development. FAS designed the geological model, as well as the custom loss function, and conducted simulations. FAS wrote and maintained the code with the help of MdlV (geological modeling with GemPy and simulations with PyMC). FAS prepared the article with contributions from both co-authors in reviewing and editing. MdlV was involved in creating some of the figures. FW conceived the original idea and provided scientific supervision and guidance throughout the project.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Understanding the unknowns: the impact of uncertainty in the geosciences”. It is a result of the EGU General Assembly 2018, Vienna, Austria, 8–13 April 2018.

We would like to thank Cameron Davidson-Pilon for his comprehensive, free introduction into Bayesian methods, which inspired parts of this research. Special thanks to Alexander Schaaf for helping with 3-D visualizations. We would also like to acknowledge the funding provided by the DFG through DFG project GSC111.

This research has been supported by the DFG (grant no. GSC111).

This paper was edited by Lucia Perez-Diaz and reviewed by two anonymous referees.