A new approach to mechanical brittleness index modeling based on conventional well logs using hybrid algorithms

Abstract

Mechanical brittleness index ($B{I}_{mech}$) of the rock is a necessary parameter for selecting appropriate drilling bits and proper depth intervals for hydraulic fracturing. The $B{I}_{mech}$ measurement is possible through continuous coring followed by laboratory tests, which are extremely cost- and time-intensive. Although petrophysical logs provide us with some continuous pieces of information, the complex nonlinear relationship between the logs and the $B{I}_{mech}$ calls for implementing intelligent approaches before one can predict $B{I}_{mech}$ from the petrophysical logs. Therefore, the present research is an attempt to develop $B{I}_{mech}$ prediction models using least-squares support-vector machine (LSSVM) and multilayer perceptron (MLP) neural network (NN) as well as their hybrid forms with cuckoo optimization algorithm (COA), particle swarm optimization (PSO), and genetic algorithm (GA) on data from wells penetrating Maroon Oilfield. For this purpose, we began by approximating the $B{I}_{mech}$ from the Poisson’s ratio and static Young’s modulus – with the later obtained from laboratory test results-calibrated petrophysical logs. Next, the entire set of available data was split into two subsets, namely modeling and validation subsets. Application of the second version of the nondominated sorting genetic algorithm (NSGA-II) combined with the MLP-NN for feature selection showed that, among the eight features considered, five made the best set for developing estimator models, including P- and S-wave velocities, depth, density (RHOB), and gamma-ray (GR) readings. Accordingly, intelligent algorithms were developed by means of these five features on the basis of the modeling data. Results of the training and testing phases showed that the hybrid algorithms were more accurate than the simple forms of either MLP or LSSVM. Among the hybrid algorithms, the highest levels of accuracy and generalizability were achieved with the LSSVM-COA. Application of the developed models on the validation data and a well in the Azadegan oil field further confirmed the high accuracy and generalizability of this model for predicting the $B{I}_{mech}$. Therefore, at a high level of confidence, the proposed model can be recommended for predicting $B{I}_{mech}$ at wells penetrating similar formations.

Appendices

Appendices

k-fold cross validation

This technique is used to cluster a dataset into k sets of equal size. Out of this k sets, (k – 1) sets are then used to train the model and the remaining set is used to have it tested. The training–testing process is repeated for k runs so that each set is once used for testing and (k – 1) times used for training the model. At each run, the value of error is recorded and an average error is calculated at the end. In this approach, since all data points contribute to the training process, the dependence of final results on the training data alone is reduced and undesirable effect of random initializations (for hyperparameters of the estimator model or initial population of optimization algorithms) on the model development processes is attenuated, ending up with an error value which closely captures the reality.

Model evaluation criteria

Percent deviation (PD) or relative error (RE) for each data point (i) in the dataset (including n data points) are calculated based on the measured brittleness index ($B{I}_{measured}$) and predicted brittleness index ($B{I}_{predicted}$) by means of the following relationship:

$$P{D}_{i}=\frac{B{I}_{measured}-B{I}_{predicted}}{B{I}_{measured}}\times 100$$

(A1)

Given the value of PD for each data point in the dataset, average percent deviation (APD) can be evaluated through Equation (A2).

$$APD=\frac{\sum_{i=1}^{n}P{D}_{i}}{n}$$

(A2)

Absolute average percent deviation (AAPD) can be computed from Equation (A3).

$$AAPD=\frac{\sum_{i=1}^{n}\left|P{D}_{i}\right|}{n}$$

(A3)

Standard deviation (SD) of error is also given from Equation (A4) based on mean error values ($E{r}_{mean}$) and error value at each data point ($Er$).

$$SD= \sqrt{\frac{\sum_{i=1}^{n}{\left(E{r}_{i}-E{r}_{mean}\right)}^{2}}{n-1}}$$

(A4)

Root-mean-square error (RMSE) for model can be calculated from Equation (A5).

$$RMSE= \sqrt{\frac{1}{n}\sum_{i=1}^{n}{\left(P{{P}_{measured}}_{i}-P{{P}_{predicted}}_{i}\right)}^{2}}$$

(A5)

Finally, coefficient of determination (COD, ${R}^{2}$) is expressed as follows:

$${R}^{2}=1-\frac{\sum_{i=1}^{n}{\left(P{{P}_{measured}}_{i}-P{{P}_{predicted}}_{i}\right)}^{2}}{\sum_{i=1}^{n}{\left(P{{P}_{predicted}}_{i}-\frac{\sum_{i=1}^{n}P{P}_{measured}}{n}\right)}^{2}}$$

(A6)