Keywords

1 Introduction

Permeability governs the movement of fluids, gas and liquid, through pore spaces and networks in porous media. Its applications include enhancement of oil/gas recovery, management of water resources, \(\mathrm{{CO}}_{2}\) geological storage and geothermal energy extraction [6]. Most commonly, permeability is estimated from various well logs using either an empirical relationship or some forms of statistical regression (parametric or nonparametric). The empirical models may not be applicable in regions with different depositional environments without making adjustments to constants or exponents in the model. Also, significant uncertainty exists in the determination of irreducible water saturation and cementation factor in these models [3].

In recent years, nonparametric regression techniques such as Neural Network have been introduced to overcome the limitations of conventional methods [1] . In this context, a Multilayer Perceptron model has been used with two training algorithms for permeability estimation: back propagation and Levenberg Marquardt where petrophysical measurements and cores data of two wells from Algerian Sahara have been exploited. Well-logs data of the well well-A are used for the training of the two neural machines, at this step this well is used as a pilot and weights of connection are calculated and raw well-logs data of the depth reservoir interval [2195.8 m, 2223.5 m] are investigated. In this paper we present only data and results of the generalization well. Figure 1 is the petrophysical parameters recordings of this well (well-B).

Fig. 1
figure 1

Petrophysical parameters recordings of well-B

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2 Permeability Estimation Using Timur’s Empirical Relationship

Different empirical approaches are used to describe the observed highly non-linear dependence of permeability to porosity by exponential or power-law relationships. [3] has presented the state of the art of permeability prediction by empirical models which are based on the correlation between permeability, porosity, and irreducible water saturation. In this paper, the Timur’s relation is used [5, 7]. Figure  2c presents the core rocks permeability (CPERM) and the Timur’s relation results (PERM_Timur). Comparison between PERM_Timur and CPERM clearly shows that this kind of empirical model is not able to provide good results. For this reason, we suggest the use of Artificial Neural Network (ANN) techniques to resolve this ambiguity; ANN does not require the knowledge of a permeability relationship.

3 The Multilayer Perceptron

Neural network is basically a parallel dynamic system of highly interconnected interacting parts based on neurobiological models. Neural network mimic somewhat the learning process of a human brain instead of using complex rules and mathematical routines. Here, the nervous system consists of individual but highly interconnected nerve cells called neurons. These neurons typically receive information or stimuli from the external environment. There are many types of network such as Multi Layer Perceptron (MLP) that commonly used in many practical applications. In this case, the structure of MLP is constituted with one input layer, one hidden layer and one output layer, inputs are the recordings of two wells well-A and well-B. The output is the calculated permeability using two learning algorithms: Back Propagation (Perm_BP) and Levenberg-Marquardt (Perm_LVM).

Fig. 2
figure 2

Permeability prediction of a reservoir of well-B: a by Back propagation algorithm; b by Levenberg Marquardt algorithm; c by Levenberg-Timur’s relation

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4 Back Propagation Algorithm

This training method uses Back Propagation to calculate derivatives of performance with respect to the weight and bias variables of the network. The network training function updates weight bias values according to gradient descent momentum and an adaptive learning rate. The adaptive learning starts with an initial value, then increases or decreases by multipliers in order to keep fast and stable learning. By adjusting its learning rate the network converge faster, thereby increasing the accuracy of predictions and shortening the training time. The learning process terminates when either the maximum and number of epochs is completed or the network sum-squared error drops below the min error goal set. Detailed explanation regarding this training method can be found in [2]. Obtained results (Perm_BP) using this kind of learning algorithm are shown in Fig. 2a.

5 Levenberg Marquardt Algorithm

The Levenberg Marquardt (LM) algorithm was designed to approach second-order training speed without having to compute the Hessian matrix [4]. When the performance function has the form of a sum of squares, the Hessian matrix can be approximated as follows [4]:

$$\begin{aligned} \mathrm {H}=\mathrm {J}^\mathrm{{T}}.\mathrm {J} \end{aligned}$$

The gradient can be computed as: \(\mathrm {g}= \mathrm {J}^\mathrm{{T}}\).e

Where ”J” is the Jacobian matrix, it contains the first derivatives of the network errors with respect to the weights and biases, and ”e” is a vector of network errors.

Predicted values of permeability (Perm_LVM) using the LM learning algorithm are presented in Fig. 2b.

6 Results, Interpretation and Conclusion

Figure 2 shows the core rocks permeability (CPERM), predicted permeability using the Back propagation and Levenberg–Marquard algorithms and finally the Timur’s relationship results. It is clear from this figure that the Timur’s formula has not given good results. Calculated Root Mean Squares show a value of 0.17 for the Levenberg–Marquardt and 0.05 for the Back Propagation. By consequence the Back Propagation algorithm has proven its robustness to resolve this kind of petrophysical problem. We suggest the use of this kind of learning algorithm for permeability prediction from raw well-logs data rather than the Levenberg–Marquardt. By implementing our method, we have suggested an ANN scheme that can be used for permeability prediction of wells located in the neighborhood of the pilot well well-A.