Verification of a real-time ensemble-based method for updating earth model based on GAN

The process of drilling wells for hydrocarbon production represents a major cost in petroleum reservoir development. However, drilling of new wells is necessary to increase the total oil recovery. To maximize the value for each drilled well it is necessary to optimize the placement of the well within the reservoir structure. An optimally placed well will mobilize more of the petroleum resources, and reduce the need for injected water – reducing the environmental impact of oil production.

To place a well in its optimal position, operators apply geosteering. Here, the well trajectory is adjusted while drilling in response to real-time measurement of the geology surrounding the drill bit. The value of geosteering has been well documented in the literature [1, 2, 3].

The main objective with geosteering is to utilize the information in the measurements to make optimal decisions. Hence, geosteering can be seen as a sequential decision process under uncertainty and should be treated in a probabilistic framework [4]. Recently, a workflow based on the Ensemble Kalman Filter (EnKF) [5] has been employed to condition the geological model on measurements acquired while drilling [6, 7]. In the EnKF, the uncertainty is represented by an ensemble of equiprobable realizations. This workflow has then been combined with a global optimization method and applied as a Decision Support System (DSS)[8].

The DSS framework provides high quality decisions on synthetic cases, and outperforms most of geosteering experts in a controlled experiment [9]. However, practical challenges should be addressed for it to be applicable to real operations [8]. This includes modeling of modern commercial tool to process real measurements as well as real-time earth model that can handle realistic geological complexity. The forward deep neural network (FDNN) trained on synthetic data for extra-deep electromagnetic measurements [10] enabled the real-time ensemble-based update of layered models in 1.5D [11], and 3D [12]. Moreover, [13] showed that the model errors present in the FDNN approximation can be alleviated during the ensemble based inversion for the layered case.

Fossum et. al [14] proposed a new modeling sequence which combines the FDNN with a generative adversarial network (GAN) to produce complex geological realizations in real-time to aid geosteering, see Figure 1. The premise of the GAN is that it allows to represent the earth model by a Gaussian distribution, where all produced realizations also maintain geological realism. This allows using EnKF-like methods to update not only continuous properties but also complex geological structures, which is required for geosteering. However, the workflow implementation in [14] converged only with a little starting uncertainty. [15] improved the results and demonstrated visually-convincing structural ahead-of-bit prediction on a selected example. It is known, however, that the approximate ensemble based methods, such as EnKF and its derivatives, can be sensitive to non-linearities present in complex modeling sequences and thus predictions may be biased.

In this paper we present a robust and improved implementation of the framework presented in [14] that is able to account for an appropriate starting uncertainty. Further, we aim to test the convergence properties of the iterative ensemble randomised maximum likelihood (EnRML) method when updating the GAN-based geomodels with FDNN approximation of measurements – denoted the GAN-FDNN modeling sequence. The EnRML probabilistic output is compared to a gold standard Markov Chain Monte-Carlo (MCMC) solution for the same problem using various metrics. The numerical examples demonstrate that the EnRML – applied to the GAN-FDNN modeling sequence – generates posterior samples with excellent predictive capabilities that are good approximations to the true posterior solution.

To construct a reference earth model we generate realizations of a fluvial geological environment using a commercial software. These realizations are then sub-sampled to form a training dataset for the offline training of a Generative Adversarial Network (GAN). The GAN is then used, online, to generate plausible geological realizations from a low-dimensional Gaussian input vector. The complete earth modeling loop is described in Section 2. For modeling the extra-deep EM measurements we use a forward deep neural network (FDNN) trained on a dataset generated using a commercial simulator (Section 3). In Section 4 we discuss the exact and the approximate data assimilation (DA) methods. The two numerical experiments, designed to test the applicability of our proposed method, are derived, and the numerical results are presented in Section 5. Finally, we summarize and conclude the paper in Section 6.

GANs are a class of unsupervised machine learning methods which can learn to generate new formatted data with the same statistics as the training set. Motivated by successful applications of GANs for modeling channelized structures for reservoir-simulation workflows [16, 17, 18, 19, 20], we use a GAN for efficient earth modeling.

The GAN consists of two deep neural networks (DNNs): a generator and a discriminator. The generator takes a random Gaussian low dimensional vector as input and generates a realization of formatted data: geological realization. The discriminator takes the formatted data and gives a probability of it being ’real’, i.e., belonging to the training set. During training the DNNs contest each other in a min-max game. They are trained simultaneously. On each training step the generator creates (fake) geological realizations from the random vectors. Fake geological realizations are combined with random samples of the real earth model and are fed to the discriminator. The loss function for the generator is proportional to number of ’fakes’ correctly identified by the discriminator. The loss function for the discriminator is proportional to the total misjudged data samples. In our study we use an adapted Wasserstein GAN [21] with hierarchical deep convolutional networks [22] for the generator and the discriminator, see [21] for implementation details.

For geosteering we want to reproduce likely geological realizations of facies and porosity distributions on a 2D vertical geological section along the well to identify the oil-bearing sands ahead of bit. For training of the GAN we use a large (compared to the area of prediction) reference earth model, which should provide a realistic test case for the present study in terms of scale and actual geological features and properties. The reference earth model is constructed using a commercial software that models a synthetic structural framework, a facies model setup derived from outcrop analogue data, and synthetic petrophysical properties of individual facies derived from published literature. The resulting model measures 4000m x 1000m x 200m (xyz) with cell dimensions set to 10x10x0.5 m, yielding a regular grid of size 400x100x400, see Figure 2.

The constructed facies model represents a low net/gross fluvial depositional system. It was chosen since it provides complex 3D architectures comprising a limited number of facies, which form contrasting geometries, see Figure 2. Input numbers for statistical generation of facies and geometries are derived from a well-documented outcrop of the Cretaceous lower Williams Fork Formation (Mesa Verde Group) at Coal Canyon, Colorado, USA [23, 24, 25].

Key parameters of the facies model setup are listed in Table 1. The model is not intended as a rendering of the outcrop itself and is consequently simplified compared to descriptions of the original outcrop [26, 27, 28, 24]. The model contains three facies: Background/shale, Channels and Crevasse splays. The probability distribution of channel width in the model is adapted to include “narrow channel bodies”, and stacking of channels accounts for multi-story channels which comprise more than 80% of the observed channel bodies. The flow direction of the channel system is set towards $45\pm 10$ degrees. No trends were used to condition the spatial distribution of channels. The details of this synthetic model are also described in [15].

The geological realization is parameterized by a vector of 60 independent parameters. For each 60-dimensional vector, the generator outputs a 64x64 grid with three values in each grid block. For a grid block (with dimensions 10.0m along-well and 0.5m thickness) the three values, ’channels’, represent the probability of the grid-block belonging to the respective facies class: Background/Channel/Crevasse. Our generator is also predicting porosity/resistivity distribution within the geo-bodies, but in this study only the facies classes are used.

For training, the original 3D earth model is sampled as 64x64 2D images with three channels. The facies index from the training set is converted into one-hot three-dimensional vector. That is, the vector represents the probability of facies: the value of the true index is set to one and other channels to zero. During evaluation, the resistivity of the facies with the highest probability is applied.

To maintain real-time performance of a data assimilation workflow the forward model should be fast and support batch, preferably parallel execution. Proprietary forward models provided by measurement instrument vendors provide the most accurate results, but they are often not sufficiently fast, and not always optimized for batch execution. In [10], the authors developed a DNN approximation of such a forward model [29], which we abbreviate FDNN.

The model approximates the output of the ultra-deep electromagnetic well-bore logging instrument. The instrument is configured to transmit four shallow and nine pairs of deep directional measurements, and has sensitivity to boundaries up to 30 meters to the side from the well bore. We emphasize that the tool provides information around, but not ahead of the drilling position. An illustration of the deep measurements depth of detection is provided in Figure 11.

The input to the FDNN model is a layered geological media with up to three boundaries above and below the measurement instrument as well as the resistivity values of all seven layers. In this study we assume that the layer resistivity is isotropic and that the well is aligned with the horizontal axis.

We produce one synthetic set of measurements for every horizontal position (one per column of cells) of the gridded model which we ’drill’ through. We choose the most probable facies for each computational cell within the considered column and use the corresponding resistivity value (same as in [15]):

1. 1.

   Background, $R=4.0$ Ohm m;

2. 2.

   Channel, $R=171.0$ Ohm m;

3. 3.

   Crevasse, $R=55.0$ Ohm m.

We find the boundaries between layers composed of pixels with equal resistivities and use the boundaries and the layers’ resistivities as the input to the forward model.

The DSS for geosteering [8] uses the data assimilation loop (see Figure 1, left) to condition the earth model to measurements made while drilling. The fundamental idea is that if a poorly known earth model can be made consistent with measurements in the statistical sense, it will contain non-biased forecasts and, hence, provide a better basis for decisions (see Figure 1, right).

In this paper, the emphasis is placed on the data assimilation part of the DSS. Specifically, we investigate real-time data assimilation with the EnRML method utilizing a modeling sequence based on the two neural networks described above: a GAN-generator for complex earth modeling and a FDNN models for the synthetic extra-deep EM logs. The EnRML method is an ensemble-based iterative ensemble smoother which has received a lot of attention for history matching subsurface multi-phase flow problem, see [30] and references therein. The method uses an ensemble approximation to the sensitivity matrix, and provides a fast and approximate solution to the Bayesian problem. The method can only be shown to converge for Gaussian posterior distributions. However, the method is known to also sample accurately from moderately non-Gaussian posterior distributions. To assess the statistical convergence of the EnRML in this context we compare the method to samples generated by a Markov Chain Monte Carlo (MCMC) algorithm. Since the MCMC, when properly converged, generates independent and identically distributed samples from the Bayesian distribution, several metrics can be used to assess the statistical error of the EnRML method. The EnRML and the MCMC method is described in detail in the rest of the section.

The numerical experiments investigate how the GAN-FDNN modeling sequence can be applied in the data assimilation part of DSS when data assimilation using EnRML is applied for real-time uncertainty reduction. We design two synthetic experiments that focus on the reduction of uncertainty ahead of measurements and the ability of the algorithm to predict the sand channels in the unexplored part of the geomodel.

To quantify the quality of the EnRML approximation, when applied to the GAN-FDNN modeling sequence, we compare the posterior ensemble of EnRML with true samples from the posterior – acquired by the MCMC algorithm. The comparison is done by evaluating several metrics, including visual comparison of standard deviation, and mean, in every point of the domain, point-wise Kolmogorov-Smirnov two-sample test, and visual inspection of kernel density estimates of the marginal distribution of GAN-input vector $m_{i}$.

We perform two numerical tests. In both tests we utilize the generative neural network, introduced in Section 2, to represent the earth model with uncertainty. Hence, our goal is to condition the poorly known 60-dimensional input vector, $\vec{m}$, to measurements. The prior realizations of the earth model are generated by applying the generative network to parameters sampled from a multivariate Gaussian distribution, $\vec{m}\sim\mathcal{N}\left(\hat{\vec{m}_{0}},C_{\vec{m}}\right)$. The distribution is slightly shifted to simulate conditioning on pre-drill information. The $\hat{\vec{m}_{0}}$ represents the shifted mean and is defined by the equation equation:

where $\vec{m}_{0}$ is the synthetic truth from [15], see Figure 6. We use uncorrelated covariance matrix with marginal variance of $C_{m_{i}}=1$ for all parameters, similar to the GAN training. Figure 3 shows six generated earth model realizations from the prior model. From the figure, we observe that this setup provides significant variation in the earth model, which is reflected in the relatively flat mean and the standard deviation derived from the full ensemble of 500 realizations, see Figure 5. At the same time, all the realizations are consistent with the chosen channelized geological setting. We emphasize, that we use the same prior for both numerical experiments.

We conduct two numerical experiments, that differ with respect to the synthetic truth. In the first experiment, the truth model from [15] is applied. In the second experiment, the synthetic truth depicted in Figure 11 is applied. Hence, for experiment 2, the prior is biased towards a wrong model (indicating erroneous pre-drill information) making the data assimilation problem harder.

The numerical study is performed in the same manner for both experiments. Firstly, we sample the true posterior with the MCMC. Here, 8 Markov chains, starting from different initial points, were run for $10^{6}$ steps. At that point, based on assessing the multivariate potential scale reduction factor, the MCMC was found to be converged. Samples from the posterior were then extracted by removing the burn-in phase, and by thinning. For each of the 8 chains, the first half of the chain was removed, and every 100th iteration from the second half of the chain was retained, leaving $4\times 10^{4}$ samples from the posterior distribution. Secondly, we estimate the posterior distribution using the EnRML method introduced in Section 4.1. Due to the fast simulation time, we utilized an ensemble size of $N=500$, and in addition, we applied the correlation-based localization technique introduced in [42]. Finally, we assess the result from the EnRML by comparison with the samples from the MCMC.

In this paper, we have demonstrated that two essential parts, the earth model and the simulated extra-deep EM logs, of an ensemble-based DSS system can be substituted with neural networks. For the earth model, we utilize the GAN trained with images from a realistic geological setting, for the simulated logs we use a forward deep neural network (FDNN) trained using a large set of simulations from a commercial tool. The setup redistributes the computational cost from online to offline calculations, enabling complex earth models and deep-sensing EM logs to be part of real-time ensemble updates.

The numerical results illustrate that the GAN-FDNN modeling sequence provides excellent probabilistic predictions ahead of drilling capturing both continuous and discrete features when conditioning to only measurements with sideways sensitivity. Moreover, the numerical results show that the computationally efficient EnRML algorithm can sample the true Bayesian posterior confirmed by the MCMC algorithm. This conclusion is valid even when the prior model is slightly biased towards a wrong solution.

The proposed approach has many beneficial factors. Firstly, a GAN provides large flexibility for defining the geological setting. Here, we consider three different facies, but one can easily imagine including features like faults and pinch-outs as well as smoothly-varying properties. Secondly, we only need to condition a few parameters with Gaussian distribution to the measurements, which is very beneficial for the ensemble-based DA approach. Thirdly, since we are utilizing a neural network model to generate the simulated log, the computational cost of simulating a single ensemble member is milliseconds. Hence, the proposed approach can utilize a large ensemble for the DA part.

The numerical experiments illustrated that the posterior has a predictive capability for both MCMC and the faster EnRML method. The future work is to integrate the DA developed in this paper with the decision framework developed in [8], allowing DSS under a much more complex geological setting. Furthermore, the method can be extended to account for model errors present in machine learning approximations in real-time [13].

This work is part of the Center for Research-based Innovation DigiWells: Digital Well Center for Value Creation, Competitiveness and Minimum Environmental Footprint (NFR SFI project no. 309589, https://DigiWells.no). The center is a cooperation of NORCE Norwegian Research Centre, the University of Stavanger, the Norwegian University of Science and Technology (NTNU), and the University of Bergen. It is funded by Aker BP, ConocoPhillips, Equinor, Lundin Energy, TotalEnergies, Vår Energi, Wintershall Dea, Kongsberg Digital, Odfjell Drilling, Sekal, and the Research Council of Norway.

Part of the work was performed within the project ’Geosteering for IOR’ (NFR-Petromaks2 project no. 268122) which is funded by the Research Council of Norway, Aker BP, Equinor, Vår Energi and Baker Hughes Norway.

We would like to thank Emerson Roxar for providing an academic licence for RMS 11.1. used for the geo-modelling in this study.