Petro-Elastic Log-Facies Classification Using the Expectation–Maximization Algorithm and Hidden Markov Models

Abstract

Log-facies classification methods aim to estimate a profile of facies at the well location based on the values of rock properties measured or computed in well-log analysis. Statistical methods generally provide the most likely classification of lithological facies along the borehole by maximizing a function that describes the likelihood of a set of rock samples belonging to a certain facies. However, most of the available methods classify each sample in the well log independently and do not account for the vertical distribution of the facies profile. In this work, a classification method based on hidden Markov models is proposed, a stochastic method that accounts for the probability of transitions from one facies to another one. Differently from other available methods where the model parameters are assessed using nearby fields or analogs, the unknown parameters are estimated using a statistical algorithm called the Expectation–Maximization algorithm. The method is applied to two different datasets: a clastic reservoir in the North Sea where four litho-fluid facies are identified and an unconventional reservoir in North America where four lithological facies are defined. The results of the applications show the added value of the introduction of a vertical continuity model in the facies classification and the ability of the proposed method of inferring model parameters such as facies transition probabilities and facies posterior distributions. The application also includes a sensitivity analysis and a comparison to other statistical methods.

Appendix: The Baum–Welch Algorithm

Let forward probabilities $\alpha$ indicate conditioning on data up to the current step, that is $\alpha _i(t)=p(X_t=i|y_1,\ldots ,y_t)$ and $\alpha _{ij}(t)=p(X_{t-1}=i,X_t=j|y_1,\ldots ,y_t)$ while backwards probabilities $\gamma$ indicate conditioning on all the data, that is $\gamma _i(t)=p(X_t=i|y_1,\ldots ,y_T)$ and $\gamma _{ij}(t)=p(X_{t-1}=i,X_t=j|y_1,\ldots ,y_T)$. The Baum–Welch algorithm according to Baum et al. (1970) follows, with multivariate Gaussian likelihood models $\phi (y;\mu ,\Sigma )$ for each state.

Algorithm: Baum–Welch algorithm

Set initial $\lambda ^*=\{A^*,B^*,\pi ^*\}$ where $A^*=\{P^*\}$ and $B^*=\{\mu _i^*,\Sigma _i^*\}_{i=1\ldots ,N}$.

Iteratively do:

Forward recursions:

• Initiate

  • $C_{1} = \left[ \sum _{i=1}^N \phi (y_1;\mu _i^*,\Sigma _i^*) \times \pi _i^* \right] ^{-1}$

  • $\alpha _i(1)= C_1 \times \phi (y_1;\mu _i^*,\Sigma _i^*)\times \pi _i^* \;, \quad i=1,\ldots ,N$

• Iterate for $t=2,\ldots ,T$

  • $C_t = \left[ \sum _{i=1}^N\sum _{j=1}^N \phi (y_t;\mu _j^*,\Sigma _j^*) \times P_{ij}^* \times \alpha _i(t-1)\right] ^{-1}$

  • $\alpha _{ij}(t)=C_t\times \phi (y_t;\mu _j^*,\Sigma _j^*) \times P_{ij}^* \times \alpha _i(t-1), \quad i,j=1,\ldots ,N$

  • $\alpha _j(t) = \sum _{i=1}^N \alpha _{ij}(t), \quad j=1,\ldots ,N$

Backward recursions:

• Initiate

  • $\gamma _j(T)= \alpha _j(T), \quad j=1,\ldots ,N$

• Iterate for $t=T,\ldots ,2$

  • $\gamma _{ij}(t)= \frac{\alpha _{ij}(t)}{\alpha _j(t)} \times \gamma _j(t), \quad i,j=1,\ldots ,N$

  • $\gamma _i(t-1) = \sum _{j=1}^N \gamma _{ij}(t), \quad i=1,\ldots ,N$

Update the parameters:

• Update the parameters $\lambda$ by

  • $\pi _i = \gamma _i(1), \quad i=1,\ldots ,N$

  • $P_{ij} = \frac{\sum _{t=2}^{T} \gamma _{ij}(t)}{\sum _{t=1}^T \gamma _i(t)}, \quad i,j=1,\ldots ,N$

  • $\mu _i = \frac{\sum _{t=1}^T y_t \times \gamma _i(t)}{\sum _{t=1}^T \gamma _i(t)}, \quad i=1,\ldots ,N$

  • $\Sigma _i = \frac{\sum _{t=1}^T (y_t-\mu _i)(y_t-\mu _i)' \times \gamma _i(t)}{\sum _{t=1}^T \gamma _i(t)}, \quad i=1,\ldots ,N$

• Set $\lambda ^*=\lambda$

After convergence is reached, the final parameter set $\lambda$ is the maximum likelihood estimate according to Eq. (6). Moreover, the maximum a posteriori prediction of the categorical sequence $\mathbf {x}$ is obtained from the backward probabilities $\{\gamma _i(1),\gamma _{ij}(2), \ldots ,\gamma _{ij}(T)\}_{i,j=1,\ldots ,N}$ through the Viterbi algorithm (Viterbi 1967).