University Geological parameters within reservoir models define the geometry of the reservoir and the distribution of reservoir properties such as porosity and permeability. These parameters are introduced into numerical reservoir models to reproduce a geological event, for example, channel geometry, fault throw, and facies proportions.
The variation of these parameters can produce significant changes in the production profile of a reservoir model. Controlling the variation is required to generate reservoir models with geological characteristics that can be considered as realistic, i.e., they mimic geological information observed in analog outcrops, modern depositional environments, and geophysical data.
The conventional method that the hydrocarbon industry uses to evaluate the reliability of a reservoir model is the history matching process, which is based on building a reservoir model consistent with its production history.
Most of the work related to history matching has focused on the variation of reservoir properties, such as fluids, porosity, and permeability. Less attention is centered on the variation of geological properties, such as geobodies’ geometry or structural and stratigraphic frameworks. Controlling the variation of geological parameters is required to generate reservoir models that can be considered as geologically realistic. Unrealistic geological models could be history matched but mislead the development plans for a particular reservoir.
As is true of many problems in earth sciences, reservoir history matching has been identified as an inverse problem. The solutions require the specification of an a priori component of a solution (prior information) to constrain the nature of the inverse solution. Here is a method that can be used in the history matching process to control the realism of sedimentological features.
Automatic History Matching
Automatic history matching (AHM) is based on determining the uncertain reservoir parameters linked to the problem of calibrating a reservoir model with observation data. This problem can be tackled by an optimization approach, which tunes the reservoir parameters and reduces the difference between observation data and simulation outputs (misfit). AHM has been implemented to reduce employee hours and efficiently obtain a large number of history matched models.
Quantifying Uncertainty
Reservoir modeling is a process with large uncertainties because of the scarcity of information and the cost of acquiring it. The most common way to quantify the statistical uncertainty of a problem is to assign a probability distribution to the uncertain parameters related to the specific problem.
In AHM, we have used a Bayesian framework to quantify the uncertainties of geological models. This involves a systematic procedure to update the current knowledge of a system based on newly obtained data. Bayesian inference is based on the Bayes’ theorem and used to perform inferences about the value of some parameters based on prior and newly observed information.
Bayes’ theorem is usually written as:
where M is the parameter space of the reservoir model, m is a vector of model parameters that compose an arbitrary reservoir model from the space M, O is a vector of the observed (production and pressure), p(m) is the prior probability distribution. p(O|m) is the likelihood of the data (a measure of the quality of the fit of model m predictions to the observed data) and p(m|O) is the posterior probability (PPD) representing the updated knowledge about the model m based on observations O.
Prior Information
In probabilistic analysis, prior information is defined as the available knowledge about the probability of parameters having a specific value, before having any further information. Prior information plays an important role in Bayesian analysis; thus, in the equation, the relevance of this information where its use will improve the posterior estimation.
Prior distribution is classified in the following three groups:
- Noninformative prior distributions. These distributions assign the same probability to occur at all the values in the proposed range.
- Highly informative priors. This type is used when information about the possible values of a parameter in the model is available.
- Moderately informative hierarchical prior distribution. This type of prior distribution is used when information about the parameter is limited. It will allow the parameter to vary by a specific factor between each model.
Sedimentological Priors
Published information related to geobody geometry was used to model the interrelationships between the parameters that describe a sedimentary body. These models were then used as prior information to generate geological interpretation of the subsurface.
Noninformative prior information is often used in modeling facies using object-based modeling techniques. Some geomodelers use linear relationships as informative priors. These linear relationships are very restrictive and lead to underestimation of uncertainty.
Using OC-SVM Model
Modeling relationships between geological parameters is a nonlinear, multivariate problem applied to data coming from multiple sources (e.g., outcrops, seismic interpretations, modern depositional environments, well data, and analogous reservoirs).
The authors modeled the relationships among sedimentological parameters using the one-class support vector machine (OC-SVM) model. This model is used as prior information to control the realism of the combination of sedimentological parameters.
As the support vector machine’s extension to one-class classification, the OC-SVM model 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 feature space.
Modeling and Matching
By modeling sedimentological prior information using the OC-SVM method, it is possible to identify combinations of parameters (e.g., channel width and thickness, meander wavelength, and amplitude) that are realistic.
Fig. 1 shows a workflow illustrating the idea of using the realistic 4D “yellow cloud” to reject combinations of sedimentological parameters that are unrealistic (purple points).
Truth Case
To apply this method, the Stanford VI synthetic reservoir was used as the reference case (truth case). The facies present in this reservoir were channel, point bars, and floodplain deposits, and the petrophysical properties (porosity and vertical and horizontal permeability) were assigned as constant for each facies for observing only the effect of varying facies geometry on the history matching results.
During the history matching process, the measured variables were field production rates and well production and pressure. The sampling algorithm was particle swarm optimization. An ensemble of 1,000 models was generated for this case with misfit calculated using a Least Squares misfit function.
As shown in Fig. 1, the workflow for history matching starts by selecting a combination of channel geomorphic parameters (channel width and thickness, and meander wavelength, and amplitude). This combination is compared with the 4D cloud of the realistic combination of parameters. If the selected combination of parameters is inside the 4D cloud, the model will be geologically realistic. Otherwise, the model will be rejected. If the combination of geomorphic parameters is considered realistic, it is used as input for static modeling.
Results
There were 126 iterations rejected as unrealistic, the lowest misfit was reached after 223 iterations, and more history matches of similar quality were generated thereafter.
Fig. 2 compares the truth case geomorphic parameters and the parameters corresponding to the models with the lowest and the highest misfits, and an example model obtained with an unrealistic combination of parameters.
The figure shows that the dimensions of the geomorphic parameters of the realistic model with the lowest misfit (c) are closer to those of the “truth case” than the other models. In the case of the unrealistic model (d), the channel thickness value was considered as unrealistic, given the corresponding channel width, meander amplitude, and wavelength.
Fig. 3 illustrates the difference of history matching the Stanford VI reservoir, considering the sedimentological prior information model generated using the OC-SVM method and the models obtained using noninformative (flat) priors.
From Fig. 3 (top), it is possible to observe that, when using intelligent priors, there is a trend for convergence of the match quality that is not seen in the case where flat priors were used. The lowest misfit was reached twice as fast in the case of intelligent priors than in the case of using flat priors. In the forecast of field oil production rate, the range of P10 to P90 values is lower in the case of realistic prior information than of a flat prior, which suggests a reduction in uncertainty when using intelligent prior information.
Conclusion
The use of intelligent prior models allowed the rejection of models that were going to be generated using an unrealistic combination of geological parameters, which reduced uncertainty and saved computing time. It was observed that when using “intelligent” geological priors, the history matching process converges faster than when using uninformative priors, because the algorithm tends to sample from the reduced parameter space controlled by the realistic combination of sedimentological parameters.