Structural geomodeling is a key technology for the visualization and quantification of subsurface systems. Given the limited data and the resulting necessity for geological interpretation to construct these geomodels, uncertainty is pervasive and traditionally unquantified. Probabilistic geomodeling allows for the simulation of uncertainties by automatically constructing geomodel ensembles from perturbed input data sampled from probability distributions. But random sampling of input parameters can lead to construction of geomodels that are unrealistic, either due to modeling artifacts or by not matching known information about the regional geology of the modeled system. We present a method to incorporate geological information in the form of known geomodel topology into stochastic simulations to constrain resulting probabilistic geomodel ensembles using the open-source geomodeling software GemPy. Simulated geomodel realizations are checked against topology information using an approximate Bayesian computation approach to avoid the specification of a likelihood function. We demonstrate how we can infer the posterior distributions of the model parameters using topology information in two experiments: (1) a synthetic geomodel using a rejection sampling scheme (ABC-REJ) to demonstrate the approach and (2) a geomodel of a subset of the Gullfaks field in the North Sea comparing both rejection sampling and a sequential Monte Carlo sampler (ABC-SMC). Possible improvements to processing speed of up to 10.1 times are discussed, focusing on the use of more advanced sampling techniques to avoid the simulation of unfeasible geomodels in the first place. Results demonstrate the feasibility of using topology graphs as a summary statistic to restrict the generation of geomodel ensembles with known geological information and to obtain improved ensembles of probable geomodels which respect the known topology information and exhibit reduced uncertainty using stochastic simulation methods.

Structural geomodeling is an elemental part of visualizing and quantifying
geological systems

The mathematical nature of implicit modeling, in combination with the use of a probabilistic modeling process, often leads to geologically unsound model
realizations and modeling artifacts. Additionally, the modeling algorithms only
take a limited set of input data types, e.g., layer interface locations and
structural orientation data, which significantly limits the amount of geological
information that can be included in the modeling process.

While the overall idea has been demonstrated in some specific cases, the general question of how to define suitable likelihood functions for specific types of observations – given specific geological systems and diverse types of prior geological knowledge – still remains.

Geological expert knowledge contains much more information that is vital to
model creation, such as understanding the geological processes that result in
the thickening and thinning of sedimentary deposits and their relative spatial distribution. One key knowledge-based input into geomodeling is the
understanding of the kinematic evolution of the rock units into their present
configuration. While kinematic modeling software exists

We therefore hypothesize that topological information about a geological system can be used as a meaningful constraint for probabilistic 3-D geomodeling outputs.

This topological information is difficult to incorporate into the mathematical foundations of implicit modeling functions and is highly case-dependant.

The origin of topological information is generally qualitative. For this reason, choosing a likelihood function, or trying to connote any probabilistic meaning to the comparison of topological graphs, does not seem to enhance the inference

To test this approach we designed two distinct experiments: one synthetic and
one case study.

We construct a synthetic fault model and explore its topological uncertainty. We do this by describing our input data not as fixed parameters, but as probability distributions. We then use Monte Carlo sampling to obtain input data realizations from which geomodels are constructed. We then show how a single topology graph can be used as a summary statistic in an ABC-rejection scheme to approximate the posterior model ensemble that honors the added information.

To test the same ABC approach on a real-world dataset, we apply it to a model extracted from a seismic interpretation of the North Sea Gullfaks field. We also explore a more advanced sampling technique to demonstrate possibilities for reducing the computational costs of the method.

In the following section we will give an overview of the applied implicit geomodeling approach, the basic concept of Bayesian inference and its use in probabilistic geomodeling, as well as the theory behind approximate Bayesian computation. We further describe how we analyze model topology and use it as a summary statistic. We will then introduce, in detail, both the synthetic fault model and the case study, followed by a comprehensive discussion of our findings.

Several approaches exist for creating structural geomodels, which can be
separated into three main categories: (a) interpolation, (b) kinematic methods
and (c) process simulation. The interpolation of surfaces and volumes from
spatial data is currently the most widely used approach in geosciences,
typically performed manually by geoscientists, which requires robust knowledge
of the geological setting and extensive amounts of data in order to robustly
approximate reality. Additionally, highly complex structures such as extensive
fault networks and repeatedly folded areas are challenging to recreate using
current interpolation methods

The open-source, Python-based implicit modeling package

URL:

Idealized horst

Topology, referring to “properties of space that are maintained under
continuous deformation, such as adjacency, overlap or separation”

To compute the geomodel topology with the necessary computational efficiency to conduct a feasible stochastic simulation of realistic geomodels, we implemented a topology algorithm using

The lithology matrix

The topology labels matrix

The topology labels matrix

This method of topology calculation works on regular grids, which imposes a
strong bias on the result: if the main lithological and structural features are not aligned with the grid orientation, the resulting topology graph could thus contain (or miss) connections. For a more detailed discussion on the effects of model discretization see

Vertical

Bayesian inference is fundamentally different to the classical frequentist
approach of inference. It treats probabilities as

While constructing meaningful likelihood functions for physical properties such
as layer thickness or geobody volume from observed data is straightforward

Geoscientists often have extensive implicit knowledge of geological settings (e.g., our understanding of the tectonics of a system), but only a limited amount
of this knowledge can be incorporated into the geological interpolation function

To obtain the approximate posterior distribution we need to sample from our
prior parameter distributions, plug the sample values

A more advanced sampling scheme for ABC is sequential Monte Carlo sampling
(ABC-SMC). In its simplest form it can be seen as an extension of rejection
sampling by chaining rejection sampling simulations together (each referred to
as an

To use geomodel topology as a constraint for probabilistic geomodels in an ABC
framework, we need a consistent way of comparing geomodel topologies – i.e.,
suitable distance functions. We consider three possible comparison methods here.

In this work we demonstrate the second approach, as it allows us to directly
compare entire geomodel topologies. We have chosen to compare the simulated
results to a single topology graph – the initial geomodel topology. This
approach was selected as a base case to demonstrate how large variations in geomodel topology observed in the stochastic simulation of input data
uncertainties in geomodels

Stochastic simulations yield vast ensembles of geomodel realizations, and their variability (and thus uncertainty) needs to be analyzed and understood. The
uncertainty of a single geological entity (e.g., a layer or a fault) can be
estimated from its frequency of occurrence in each single geomodel voxel. In
order to analyze the whole geomodel uncertainty at once, more sophisticated
measures can be applied: the concept of

Distribution parameters for prior parameterization of the synthetic fault model.

As a proof of concept we show how ABC can be used to incorporate geological
knowledge and reasoning into an uncertain synthetic geomodel. This model
represents a folded layer cake stratigraphy that is cut by a N–S-striking normal fault to represent an idealized reservoir scenario frequently encountered in the energy industry (see Fig.

The prior parameterization is schematically visualized in Fig.

Two separate simulations were run for this experiment so we can see how topology
can constrain an uncertain geomodel compared to the Monte Carlo simulation of
input parameter uncertainties alone.

A Monte Carlo simulation of the prior parameters was run to evaluate the
uncertainty in the resulting geomodel ensemble consisting of

An approximate Bayesian computation was done using the initial model topology
graph (see Fig.

To demonstrate the applicability of the method to real datasets we apply it to a
model of part of the Gullfaks field, located in the northern North Sea. The
field is located in the western part of the Viking Graben and consists of the
NNE–SSW-trending 10–25 km wide Gullfaks fault block

For the experiment, we constructed a base geomodel (Fig.

Distribution parameters for prior parameterization of the Gullfaks case study.

The prior parameterization consists of two different kinds of uncertain
parameters: (i) vertical location of the layer interfaces within each fault
block and (ii) the lateral location of the fault interfaces. This parameterization
is similar to the synthetic fault model (all specifications are listed in Table

A Monte Carlo simulation was run of the prior uncertainty for

An ABC-REJ simulation was run using the initial geomodel topology graph (see
Fig.

An ABC-SMC simulation was run using the same initial geomodel topology graph.
We ran six SMC epochs using

Shannon entropy slices in the X–Z

Simulating the uncertainties encoded in the prior parameterization resulted in

X–Y section of entropy difference between the forward simulated entropy and the approximate posterior entropy. The plot highlights areas where the entropy was reduced (blue), increased (red) and kept constant (white).

Prior (grey) and posterior (color) kernel density estimations for the different stochastic model parameters for our synthetic fault model.

Figure

Forward simulation of the prior uncertainties of the Gullfaks geomodel resulted in

The initial topology graph is used as a constraining summary statistic using
ABC with rejection sampling (ABC-REJ) and a threshold of

The results shows that this approach leads to reduced uncertainty, as
exemplified by the entropy section shown in Fig.

The experiment was run on consumer-grade hardware and leveraging GPU computation: Intel Core i5-8600 K @ 3.60 GHz, Nvidia GeForce RTX 2070 8 GB GDDR6, 16 GB DDR4 RAM @ 2133 MHz.

. In contrast, using a sequential Monte Carlo sampling scheme (ABC-SMC) required onlyFigure

Figure

Prior (grey) and posterior (colored) kernel density estimations for
selected model parameters

X–Z section of entropy difference between the forward-simulated
entropy and the approximate posterior entropy

We showed how topology information, as an encoding for important aspects of
geological knowledge and reasoning, can be included in probabilistic geomodeling
methods in a Bayesian framework. The simulation experiments for our two case
studies demonstrated that we are able to approximate posterior distributions to obtain probabilistic geomodel ensembles that honor both our prior parameter knowledge and qualitative geological knowledge. If the applied topological
information is meaningful, then the constrained stochastic geomodel ensemble
will see a meaningful reduction in uncertainty and will subsequently allow for
more precise model-based estimates and decision-making

With our approach, we directly address a scientific challenge raised in recent work by

The work of

As more complex geomodels strongly increase the required parameterization to
accurately describe the model domain in a probabilistic framework, constraining them with topological information could help keep this parameterization at computationally feasible levels by reducing the parameter dimensionality, while still obtaining meaningful geomodels (e.g., free of modeling artifacts caused by random perturbations of the limited input data). This would not work using an
inefficient rejection sampling scheme (e.g., ABC-REJ) but would rather require the use of “adaptive” sampling algorithms to efficiently explore the posterior parameter space without wasting too much computing power on rejected models (e.g., ABC-SMC). In our Gullfaks case study, we have not only shown the efficacy of the method in a real-world example, but have also demonstrated the stark increase in
computational efficiency when using advanced sampling techniques. The SMC
sampler used in our work requires manual setting of the acceptance thresholds, which directly influence the algorithm's efficiency in acquiring samples of the
approximate posterior distribution. Adaptive SMC methods automatically tune
acceptance thresholds to increase sampling efficiency “on the fly” to minimize
computation time and avoid manual (subjective) selection of thresholds

Alternatively, Bayesian optimization for likelihood-free inference

The method demonstrated the effect of topology information on geomodel uncertainty – showing how well the parameterization of a probabilistic geomodel fits our geological assumptions. The acceptance rates during sampling could potentially be used as a proxy for the validity of our assumptions: low acceptance rates could reveal a bad fit between our model and our added geological knowledge and reasoning. Using entropy-difference plots, the effect of geological assumptions on geomodel uncertainty can be analyzed spatially, e.g., how it changes around faults and other structures in the geomodel ensemble.

We have shown how to use approximate Bayesian computation to constrain probabilistic geomodels so that the approximate posterior incorporates known topology information.

The method enables additional geological knowledge and reasoning to be explicitly encoded and incorporated into probabilistic geomodel ensembles, potentially increasing the transparency of the modeling assumptions.

As opposed to standard MC with rejection, the implemented SMC approach makes the use of ABC feasible in realistic settings. Further research into using more advanced sampling schemes could provide additional speed-ups in obtaining the posterior geomodel ensemble, which is especially relevant for computationally more expensive complex geomodels with large parameterizations.

Input data and scripts to run the model and produce the
plots for all the simulations presented in this paper are archived at Zenodo

AS was responsible for the data curation, investigation, validation and visualization. Conceptualization, design and development of the research's methodology were done by AS, MdlV and FW. Formal analysis and software development were done by AS and MdlV. Supervision, research resources and funding acquisition were provided by CB and FW. AS was responsible for writing the original draft, while all authors contributed to the review and editing process.

The authors declare that they have no conflict of interest.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We would like to thank Total E&P UK in Aberdeen for funding this research. We also thank Fabian Stamm for providing the wonderful synthetic geomodel used in this paper. We are grateful for the constructive reviews from Ashton Krajnovich and an anonymous reviewer for helping us improve this paper.

This research was conducted within the scope of a Total E&P UK-funded postgraduate research project.

This paper was edited by Thomas Poulet and reviewed by Ashton Krajnovich and one anonymous referee.