Introduction

The lithology, saturation, and type of pore fluids are characteristics that the AVO/AVA analysis and seismic inversion attempt to predict using physical relationships between seismic data and rock elastic parameters (Booth et al. 2016; Ahmed et al. 2017; Samba et al. 2017). Pre-stack inversion produces more petrophysical parameters sensitive to fluid and rock properties than post-stack inversion (Maurya & Sing, 2018) and makes AVO/AVA analysis possible. Besides, post-stack data matches zero-offset data when the stack includes only the nearest offset traces and fails when the phase changes rapidly with offset. However, post-stack inversion is very fast compared to pre-stack inversion, providing high-resolution images, and although Russell and Hampson (1999) concluded that there was no right way to do a post-stack inversion, several post-stack methods, namely model-based inversion, coloured-inversion, sparse spike inversion, and band-limited inversion are available. Despite that, post-stack methods that use post-stack seismic multiple attribute regressions with rock properties (Guerrero et al. 2010; Das et al., 2017) and others using acoustic impedance got reasonable estimates (Das and Chatterjee 2016).

On the other hand, the model-based inversion generates synthetic data from an initial model and compares it with the observed ones, and if they match, the model is the solution; if not, the model is updated, and the procedure iterates until it gets an acceptable error. This optimization process varies depending on the search method in a space-solution space that seeks an optimal model. Inversion searches for the minimum-error function and will always find the minimum closest to the starting model, so AI methods become the best alternatives for finding the global minimum.

As known, artificial intelligence methods infer information from data sets (Burkov 2019; Xu 2019) and have produced a vast number of applications to solve problems in Geosciences, such as estimating recovery volumes using geological data (Crnkovic-Friis and Erlandson, 2015; Xu 2019), gas prediction and interpretation of seismic sections integrated with well-logs (Cheng et al., 2019; Xu 2019, and others improved the operators’ performance of crossover, mutation, and selection (Salamanca et al. 2017). A comprehensive review of state-of-the-art machine learning and AI to solve problems in the oil and gas industry revealed a vast set of techniques for processing and interpreting data volumes whose promising achievements address large storage volumes with high numerical efficiency (Sircar et al. 2021). Some predict subsurface properties through deep learning (Puzyrev et al. 2019) or porosity with vectorized regression and neural networks (Gholami et al. 2022). Others use machine learning to estimate acoustic impedance, porosity, and saturation (Qiang et al. 2020) or use PNN to discern nonlinear relationships between logs and seismic profiles. (Othman et al. 2021).

This research estimated reservoir properties applying a conventional inversion method and two AI algorithms to investigate the feasibility of constructing a reservoir properties model directly from a post-stack volume of the Frontier Second Wall Creek Formation member in the Teapot-Dome field, Wyoming-USA. The Probabilistic Neural Network (PNN) and the Deep Feedforward Neural Network (DFNN) provided porosity volumes (ϕ), gamma rays (GR), and Lamé impedances (µρ, λρ) through multi-attributes that related the well-logs and the stacked data, and the resulting parameters were compared with the ones provided by the based-model inversion (MBI). The three methods modeled nonlinear functions and enhanced high-frequency resolution and prediction, with comparable PNN and DFNN prediction quality and superior to MBI but consuming much greater computer resources. On the other hand, the results provided by PNN and DFNN are encouraging, especially in their ability to predict subsoil information from seismic stacked sections, making them more valuable.

Geological framework

The Teapot Dome field is southwest of the Powder River Basin (PRB), north of Casper town in Natrona County, Wyoming, US (Fig. 1A). The Powder River Basin (PRB), found between the NE of Wyoming and SE of Montana, is a 370 km length with an 18,000 ft thick sedimentary sequence that holds rocks from the Paleozoic to the Quaternary. The Paleozoic units are approximately 2500 ft thick and comprise marine carbonate and sandstones. Mesozoic rocks are 9500 ft wide and of siltstones and sandstones from marine to non-marine environments. Cenozoic units are between 4,000 and 6,000 ft thick and are carbon-containing sandstone and shale (Beikman 1962). Figure 1B shows the wells projected at the top of the Second Wall Creek unit in time (TWT), and Fig. 1C shows the Albian to Coniacian rock units of the Powder River Basin, which include the Frontier Formation. The Frontier Formation is a typical fluvial-deltaic unit deposited in continental and occasionally marine environments, composed of intercalations of shale and homogeneous quartz composition sandstone (Zupanic, 2013). The unit target is the 2nd Wall Creek (F2WC) member, up to 80 feet thick. The Rocky Mountains were thrust into a foreland basin, generating the Frontier Formation’s accommodation space, where regressive sea-level events deposited potent beachfront and delta environment units. The low accommodation environment overlapped erosive fluvial surfaces, producing significant reservoir discontinuity, and hindering interpretation (Kirschbaum and Mercier 2013).

Fig. 1
figure 1

A Location map Campo Teapot Dome, at the southwest of the Powder River Basin (PRB), north of Casper town in Natrona County, Wyoming, USA. (Modified from Merewether et al. 1979). B Wells project on top of the Second Wall Creek unit (TWT in ms). C The Albian to Coniacian rock units of the Powder River Basin, which include the Frontier Formation. (Kirschbaum and Mercier 2013)

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Theory

In the model-based inversion, synthetic data calculated from the initial model are compared with those of the stacked volume, and when they match, the model will correspond to the true one; otherwise, the model is changed, and new synthetic data generated from the updated model are compared with the stacked volume. This procedure ends when the misfit between both data becomes acceptable. Due to stacked data being acoustic data, the inversion does not provide the shear wave velocity, so it is estimated by relationships extracted from well logs. A complete and detailed explanation of MBI is beyond this paper, and an available reference is Sen and Stoffa (1995). On the other hand, a Neural Network (NN) is a program with nonlinear parallel processing that roughly mimics the human brain processes, using a learning algorithm trained on datasets. It does not require an initial model to find the relationships between two data sets and it solves classification and prediction problems through earlier supervised or unsupervised training. Supervised learning requires enough training data but allows interpreting the output based on the known problem nature. On the contrary, unsupervised learning looks for patterns in the input data by clustering them according to their properties and relationships without supplying an expected and verifiable output.

Probabilistic Neural Network (PNN) applies an interpolation scheme to an independent variable set to predict a dependent one without massive back-propagation training; instead, it replaces the data pattern with a similarity measurement between the data pattern and the input data, and its training uses a conjugate-gradient analysis to determine the parameters that minimize the validation error. PNN uses the waveform’s time parameter to recognize the occurrence and predict its strength or classification through earlier waveform processing and attributes’ extraction. For clarity, this item only explains the two-attributes \((x,y)\). PPN uses the exponential weighting function of distances \({d}_{i}({\overrightarrow{x}}_{i},{\overrightarrow{x}}_{o})\) between data points \({\overrightarrow{x}}_{i}=({x}_{i},{y}_{i})\) and\({\overrightarrow{x}}_{o}=({x}_{o},{y}_{o})\), to identify each point’s class \({C}_{j}\), and instead of using the nearest neighbor criteria, defines a sphere of influence with ranges where points classify. Assuming a Gaussian function with variance \({\sigma }^{2}\), the probability of \({\overrightarrow{x}}_{o}\) being in the class \({C}_{j}\), is:

$$P\left({\overrightarrow{x}}_{o}\in{C}_{j}\right)={g}_{{c}_{j }}\left({\overrightarrow{x}}_{o}\right)/\sum {g}_{{c}_{i }}\left({\overrightarrow{x}}_{o}\right),$$
(1)

where the weight of the Parzen estimator is given by:

$${g}_{{c}_{j}}\left({\overrightarrow{x}}_{o}\right)=\sum exp\left({-d}_{i}^{2}({\overrightarrow{x}}_{i},{\overrightarrow{x}}_{o})/{\sigma }^{2}\right).$$
(2)

To manage the case when \({\overrightarrow{x}}_{o}\) is not in either class, PNN divides classes into subclasses. \(L\left({\overrightarrow{x}}_{i}\right)\) is the log value for point \({\overrightarrow{x}}_{i},\)hence the sets of points \(\left\{{\overrightarrow{x}}_{i}\right\}\) allow estimating as:

$$\begin{aligned} &L\left({\overrightarrow{x}}_{o}\right)\\&=\sum L\left({\overrightarrow{x}}_{i}\right)exp\left({-d}_{i}^{2}\left({\overrightarrow{x}}_{i},{\overrightarrow{x}}_{o}\right)/{\sigma }^{2}\right)/\sum exp({-d}_{i}^{2}({\overrightarrow{x}}_{i},{\overrightarrow{x}}_{o})/{\sigma }^{2}).\end{aligned}$$
(3)

The classification uses discriminant analysis to predict belonging to a class by deriving linear functions from a training data set to be used later in unclassified data sets. Due to PNN usually managing copious attributes, it requires high memory, which increases its computing time. However, PNN provides probability estimates with reliable performance in the case of suitable data sets with outliers.

Unlike PNN, the Deep Forward Neural Network (DFNN) is a multi-layer back propagation NN with an input layer, hidden layers, and an output layer. Each layer has node weight connected to adjacent layer nodes, and hidden nodes try to fit the training data better with the associated over-fitting. In contrast, some of the hidden nodes supply smother results. Therefore, the higher the number of nodes, the lower the training error and the higher the validation error. The training uses the conjugate gradient to look for the best weights for the connections between nodes that predict values as close as possible to the real one. Therefore, DFNN has better control of training parameters at a higher speed than PNN. A complete reference to fundaments and developments of PNN and DFNN includes Awad and Khanna (2015), Aggarwal (2018), and Gupta and Raza (2020). On the other hand, Brown (1996), and Chen et al. (1997) supply broad descriptions and applications of attributes.

Methodology

The data includes 25 wells with logs shown in Fig. 1B, a post-stack cube of 73 km2 surface sampled at 2 ms, 8–70 bandwidth, 32 Hz dominant-frequency with 345 Inlines, and 188 Xlines. The reprocessing sequence included noise filtering, amplitude recovery, surface consistent amplitudes, surface wave attenuation, static corrections, super gather, velocity analysis, minimum phase spiking deconvolution, radon filter, Kirchhoff pre-stack migration, NMO correction, CDP stacking to provide the stacked volume and then interpreted.

MBI used an initial (ρ, Vp) model to invert the stacked volume, obtaining as a result a final (ρ, Vp) model. Next, a Vs-Vp relationship calibrated in wells supplied Vs, bringing the final (ρ, Vp, Vs) model. The µρ and λρ volumes were estimated from the final (ρ, Vp, Vs) model, and a linear ϕ-Vp and a nonlinear GHR-Vp/Vs relationships calibrated in wells predicted the ϕ and GR logs. The ability of µρ and λρ to discriminate fluid and lithology were the reasons for choosing them, whereas GR provides shale content that, together with ϕ, could distinguish sediment distribution. On the other hand, NNs can determine nonlinear relationships between seismic attributes and well-log properties to extrapolate the well-log properties at each location in the seismic volume. Unlike MBI, PNN and DFNN do not need an initial model but do need seismic attributes.

Fifty post-stack attributes were extracted from post-stack traces nearest the boreholes and correlated with ϕ, GR, µρ, and λρ profiles estimated from well-logs, choosing those attributes with correlation coefficients (CC) above 0.4. The ability of µρ and λρ to discriminate fluid and lithology were the reasons for choosing them, whereas GR provides shale content that, together with ϕ, could distinguish sediment distribution. The multiattribute approach hierarchical clusters the attributes weighted by their CC, i.e., the first selected attribute has the highest CC, the second one has the next highest CC of the remaining, and so on, up to achieving the best parameter estimation (Guerrero et al. 2010). Table 1 shows the list of post-stack attributes that provide the best GR, ϕ, µρ, and λρ estimates. After generating volumes of these attributes, those extracted attributes from the traces close to the wells are selected to constitute the data set to train PNN and DFNN.

Table 1 List of multi attribute selected to predict the \(GR, \phi , \mu \rho , \lambda \rho\) logs
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For validation, some wells are hidden from the data set to train PNN on the others, and after training, PNN is applied to the hidden wells. Unlike PNN, a subset of the training data randomly selected is removed, and the DFNN is trained on a reduced data set and later applied to the extracted subset. The DFNN consists of a four-node input layer, three ten-node hidden layers, and a four-node output layer. The training and validation are repeated for all items, and the average error becomes the validation error. Then, PNN and DFNN applied the inferred relationships between attributes and GR,ϕ,µρ, λρ to the post-stack attribute volumes to generate the GR,ϕ,µρ, λρ volumes. Finally, the comparison between the geological interpretation of the \(GR, \phi , \mu \rho , \lambda \rho\) surface slices at the top of the F2WC top, provided by all methods, allowed deciding the reliability and feasibility of each technique. The interpretation of the surface slices of ϕ, λρ, µρ, and GR at the top of the F2WC will supply a feasible model, that agrees with the geological knowledge of the area; that is, in addition to considering the values of ϕ, λρ, µρ, and GR, the interpretation must consider the depositional environment and the stratigraphic patterns.

Result and discussions

The correlation coefficients and mean errors produced by the PNN and DFNN during the training and prediction stages achieved remarkable estimation capability with high correlation coefficients for ϕ (0.94), λρ (0.83), µρ (0.92), and GR (0.84), as Table 2 shows. Despite the above, PNN performed better than DFNN during the training, while DFNN got better-predicting. The anterior comparison indicates that PNN is more susceptible to overtraining and less robust in prediction. In turn, µρ is the best-predicted property, ϕ is the worst, and λρ estimation is better than GR.

Table 2 The MLR, PNN, and DFNN algorithms produced correlation coefficients for estimating \(GR, \phi , \mu \rho\), and \(\lambda \rho\) during the training and prediction stages
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The ϕ, λρ, µρ, and GR values must agree on the geological knowledge of the area, that is, according to the depositional environment and the stratigraphy.

Figure 2A shows the Vp, ρ, and GR logs from well 41-A-X3, in the interval 540–800 ms, where the black line denotes the top of the F2WC (660 ms). The GR record identifies the bottom of very clayey seal rock (630 ms) and below, a very sandy reservoir in its upper part and less sandy towards the base (700 ms). The ρ log says that the seal rock is denser than the top of the reservoir and that the acoustic wave travels slower in the seal rock. Figure 2B and C show the intercept and gradient sections provided by the AVO analysis, with a solid line representing the top of the F2WC and a vertical blue line marking the position of well 41-A-X3 in both sections. The reflector at the top of the F2WC has a significant negative intercept value (deep blue) and a slightly negative gradient value (light blue) that identifies the F2WC sands as type III and the reservoir as suitable.

Fig. 2
figure 2

A\({ V}_{p}\), \(\phi ,\) and GR logs of the well 41-A-X3 with the F2WC top showed by a black horizontal line. B Seismic section with Intercept distribution. C Seismic section with Gradient distribution. Vertical blue lines show the positions of well 41-A-X3

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Figure 3 shows the GR logs estimated by the PNN and DFNN algorithms in the vicinity of wells 11-AX-11, 25-LX-11, and 64-IX-15, which were not used during their training. Figure 3A indicates that the mean correlation coefficient between the GR predicted by PNN and the GR logs decreases from 0.70 during validation (in Table 2) to 0.68, and Fig. 3B that the CC between GR predicted by DFNN decreased from 0.74 during validation (in Table 2) to 0.73. However, the CC decrease is more significant compared with the training, falling from 0.84 (in Table 2) to 0.68 (in Fig. 3A) for PNN and from 0,79 (in Table 2) to 0.73 (in Fig. 3B) for DFNN. The anterior comparisons indicate a decrease in the average predictive ability of both methods, with DFNN results more dependable than those of PNN, and DFNN more robust in predicting geological properties than PNN.

Fig. 3
figure 3

GR well logs (black) and the GR profiles estimated near wells 11-AX-11, 25-LX-11, 48X-28, and 64-JX-15 (not used in training) by A PNN and B DFNN. In the boxes, the correlation coefficients

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Figure 4A and B, and 4C show the GR Inline sections predicted by MBI, PNN, and DFFN, respectively, with a line that marks the top of the F2WC and the GR log of the 25-LX-11 well. Figure 4B and C show below the line a sandy unit that becomes shalier at the base, overlying a massive shaly unit, and above the line a shaly unit overlaid by another sandy one. The sandy units look laterally discontinuous and interbedded by shaly ones. However, Fig. 4A shows a shale distribution that is more uniform across the section and different from those provided by PNN and DFNN. Although GR sections provided by AI algorithms look similar, DFNN better discerns the sandy units above the F2WC, interpreted in the GR log (Fig. 2A). Complementing the above description, Fig. 4D and E, and Fig. 4F show the ϕ distribution predicted, which matches the ϕ log of the well 25-LX11, with a high ϕ (red) layer between two low ϕ units. On the other hand, the high ϕ units in Fig. 4D show lateral continuity, contrary to that observed in the same units in Fig. 4A. The λρ predicted by MBI, PNN, and DFNN in Fig. 4G and H, and IF show the F2WC with a minimum λρ (yellow) between two laterally continued high λρ fringes (green). Below, a low λρ fringe (red-yellow) coincides with the ϕ fringes in Fig. 4E and F. The three images agree with the λρ log in the borehole and look similar λρ distribution. In Fig. 4K and L, the F2WC achieves the lowest λρ and the highest µρ. Below the target, a unit with low µρ (green) overlies a unit with high µρ (red-yellow), both predicted with more lateral continuity by DFNN than PNN. The general trend shows a good correlation with intervals where the predictions match the log, considering the lack of standardization, normalization, and low quality of logs and seismic data. The above analysis of results in Fig. 4 points out the correlation between the PNN and the DFNN outputs, guaranteeing similar performance and estimation reliability. The poor performance of the MBI is due to the low correlation of the Vs-Vp, Vp–ϕ, and GR-Vp⁄Vs relationships observed in the wells.

Fig. 4
figure 4

A, D, G, and J show the GR, \( \phi , \lambda \rho\), and \($$\mu \rho$$ \) sections predicted by MBI. B, E, H, and K show the GR, \( \phi , \lambda \rho\), and μρ sections provided by PNN. C, F, T, and L show the GR, \( \phi , \lambda \rho\), and μρ sections provided by DFNN. All sections include the GR, \( \phi , \lambda \rho\), and μρ logs of the well 25-LX-1

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Fig. 5
figure 5

Distribution maps of A GR, B\(\phi \), C\(\lambda \rho\) and D\(\mu \rho\), describing a channel at the top of the F2WC

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Figure 5 displays the provided PNN distribution maps of GR, \(\phi \) and \(\mu \rho\), at the top of the F2WC unit, where map curves (white) enclose reservoir quality zones and identify deposition environment. Figure 5A and B show sandy (GR < 90) and porous (\(\phi \) >10%) zones that define a meandering paleo-channel with adjacent flood areas. The low \(\lambda \rho\) (<30) and high \(\mu \rho\) (>28) inside the enclosed zone suggest possible saturating fluids, supporting the paleo-channel interpreted in GR and \(\phi \) distribution maps. However, \(\mu \rho\) shows two zones at the south and north, which appear unconnected. On the other hand, Fig. 6 shows the provided DFNN distribution maps of GR, \(\phi , \lambda \rho ,\) and \(\mu \rho\) at the top of the F2WC, where the interpretations of GR and \(\phi \) in Fig. 6A and B suggest a similar paleo channel. However, the enclosed low \(\lambda \rho\)in Fig. 5C discriminates the channel better than in Fig. 5C, reinforcing the interpretation of a paleo-channel. On the contrary, the high \(\mu \rho\) zones at north and south delineated in Fig. 5D appear less defined in Fig. 6D than in Fig. 5D, with a weakly insinuated connecting channel.

The \(\mu \rho\) maps of Figs. 5D and 6D discern two highly liquid-saturated zones at the north and south, connected by a not homogeneously saturated channel, supporting the interpretation of a complex of channels deposited in limited accommodation space, where the eroded channels with variable shale content that look interrupted by superposed unconformities that created discontinuous reservoirs. According to well-log data, the interpreted zones correspond to complexes of distributary channels and lobes typical of beachfront and deltaic environments, subject to the action of tides, characteristics adjusted to the geological evolution of the basin in the studied area. Finally, Fig. 7 shows the interpretation of the GR, \(\phi , \lambda \rho\) and \(\mu \rho\) maps provided by the post-stack model-based inversion together with the same multiple attributes used by PNN and DFNN, which differs too much from the two anterior interpretations of Figs. 5 and 6. Since \(\mu\) is calculated from Vs and the stacked section is acoustic, the three methods provide unreliable \(\mu \rho\) images, so these images were not considered. Low correlations of the Vs-Vp, GR-Vp/Vs, and \(\phi \) -Vs relationships yield unreliable estimates of GR, \(\phi \), and \(\mu\), affecting the MBI’s performance. The above analysis pointed out that the predictions of PNN and DFNN, better than those of MBI, increase the reliability of applying PNN and DFNN with multi-attribute to post-stack sections.

Fig. 6
figure 6

DFNN distribution maps of A GR, B\(\phi \), C\(\lambda \rho\) and D\(\mu \rho\), which allow interpreting a channel at the top of the F2WC

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Fig. 7
figure 7

MBI distribution maps of A GR, B\(\phi \), C\(\lambda \rho\) and D\(\mu \rho\), which allow describing a geomorphology image at the top of the F2WC that differs from the interpreted by PNN and DFNN

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Conclusions

PNN and DFNN with multiattribute, and MBI with an initial acoustic model and petrophysical relationships, inverted a stacked volume to obtain GR, ϕ, λρ, and µρ volumes. The PNN and DFNN performed better during training than during validation, and the XLine section analysis pointed out similar outputs, which differ from the MBI’s. The GR, ϕ, λρ, and µρ distribution at the top of the 2nd Wall Creek member of the Frontier Formation were interpreted as a meandering channel characterized by low GR, high ϕ, and low λρ, deposited in limited accommodation space, and interrupted by superposed unconformities that created discontinuous reservoirs. The interpreted distributary channel complex, with lobes related to beachfront and deltaic environments under tide action, is supported by geological information. The interpreted MBI maps differ significantly from similar PNN and DFNN maps, indicating their suitability for post-stack inversion, increasing the value of stacked data to decrease prospecting risk. Since µρ is calculated from Vs and the stacked section is acoustic, the three methods provided unreliable µρ, so these images were not considered. On the other hand, low correlations of the Vs-Vp, GR-Vp/Vs, and ϕ-Vp ratios yield unreliable estimates of GR, ϕ, and µ, affecting the MBI’s performance. The above analysis pointed out that the predictions of PNN and DFNN, better than those of MBI, increase the reliability of applying PNN and DFNN with multi-attribute to post-stack sections. Although the PNN and DFNN results were reliable, the untrustworthy prediction of µρ suggests that extracting shear information is still an unsolved limitation of the post-stack inversion methods.