Application of soft computing approaches for modeling fluid transport ratio of slim-hole wells in one of Iranian central oil fields

Abstract

In order to have a successful drilling operation program, it is essential to consider the efficiency of the fluid transport ratio (FTR). A complicated problem that is influenced by a wide range of parameters is to transporting these cuttings. Drilling engineers have used soft computing (SC) techniques for two purposes, firstly for having a better control over the drilling operation and secondly for lowering the total costs of this process, the latter is done because these techniques can allow them to estimate the drilling operation in advance. In the present research, several of different methods, including optimization algorithms, the generalized reduced gradient (GRG) method, and SC techniques like radial basis function (RBF) and multilayer perceptron (MLP), are used for modeling FTR in slim-hole wells according to an extensive databank. In order to improve the estimating capability of SC models in the MLP training phase, a variety of algorithms are applied, including bayesian regularization (BR), levenberg–marquardt (LM), resilient backpropagation (RB), scaled conjugate gradient (SCG), fletcher-reeves conjugate gradient (FRCG), broyden fletcher goldfarb shanno (BFGS), and polak-ribiere Conjugate Gradient (PRCG). Afterwards, six models were integrated into a single unique model by applying a committee machine intelligent system (CMIS). Applying the GRG method, a new correlation was made for being able to predict the FTR more easily. The CMIS model predicted FTR very satisfactorily, as the total average absolute percent relative error (AAPRE) was 0.134%; therefore, it was shown that our suggested model have a much greater performance than the existing approaches for prediction of FTR.

Appendix

Predictive techniques

Generalized reduced gradient (GRG)

In order to solve multi-variable problems, it is possible to use the GRG method. Specifically, the mentioned method can solve both linear and non-linear problems and can also provide a prediction with a high degree of accuracy by selecting the most suitable variables for the intended equations (Gill et al. 2019). The restrictions and gradient are described simultaneously: hence, it is possible to define the objective function as the restriction gradient. So the search pace can be moved in a probable direction, and as a result, the search area could be reduced. For the objective function of f(x) which is subjected to h(x), we have:

$$\mathrm{Minimum}:f(x)=x$$

(8)

$$\mathrm{Subjected to}: {h}_{k}(x)=0$$

(9)

This method can be shown given the following equation:

$$\frac{df}{d{x}_{k}}=\nabla {x}_{k}^{t}f-\nabla {x}_{i}^{t}f{(\frac{\partial h}{\partial {x}_{i}})}^{-1}\frac{\partial h}{\partial {x}_{k}}$$

(10)

It should be noted that an essential condition to minimize f(x) is that df(x) = 0, or the same condition for an infinite minimum is that $\frac{df}{d{x}_{k}}=0$ (Ameli et al. 2016). The readers are recommended to see the previous relevant studies for more information (Haji-Savameri et al. 2020; Graves and Wolfe 1963; Sharma and Glemmestad 2013; David et al. 1986; Abadie 1969).

Multilayer perceptron

It is possible to use ANNs for finding complicated relationships among system inputs and outputs. Every ANN consists of two major elements that are acting as processing elements and links. The processing elements, also called neurons or nodes, are responsible for information processing, whereas the task of interconnections or the weights is connecting the neurons (Mohaghegh 2000; Mohagheghian et al 2015; Hemmati-Sarapardeh et al 2018; Karkevandi-Talkhooncheh 2018). Several layers exist in every MLP neural network, i.e. an input layer, an output layer, and an intermediate layer acting between the input and the output layers, also known as the hidden layers (Lashkarbolooki et al 2012). In general, hidden layers generate the internal manifestation of the relationship between the model inputs and the intended output.

One should specify the number of hidden layers as well as the number of neurons in every layer through empirical observation. In most situations, it is important to have an MLP containing just one hidden layer (Hemmati-Sarapardeh et al 2016). In general, very complex systems require two hidden layers. If an MLP, containing two hidden layers and Logsig and Tansig activation functions are assumed for the two mentioned hidden layers as well as a Purlin for the output layer. However, in the present research, seven major optimization algorithms have been utilized, i.e. BR, LM, RB, SCG, BFGS, FRCG, and PRCG. The scheme proposed for the MLP network of this research can be seen in Fig. 9.

Fig. 9

The structure of MLP applied in current research

Radial basis function

RBF is among the well-known kinds of ANNs that can be utilized for both classification and regression goals. This is a three-layers feed-forward network containing, i.e. two input and output layers and one hidden layer (Panda et al 2008; Varamesh et al 2017). The input layer includes the input nodes, where the number of the input nodes and the input parameters of the model are the same (Zhao 2016).

In the current study, the Gaussian function has been used as the transfer function for the RBF. RBF neural networks consist of an input and output layer as well as a hidden layer (Tatar et al 2013). The neurons in the hidden layers consist of a radial basis function serving as a non-linear activation function, in which, the outputs of this function have an inverse relationship with the distance from the center of the neurons. According to the linear optimization approach, the RBF is capable of attaining a general optimal solution that can be adjusted to the weights in the minimum mean square error (MSE).

Figure 10 represents the schematic structure of the RBF applied in the current study. Two key factors in the structure of Gaussian RBF are the maximum number of neurons and the spread coefficient. The ability and accuracy of the RBF depend to a great extent on the values of the mentioned parameters. Therefore, the optimization of these parameters is essential to ensure the a reliable and precise function of RBF. In this research, a trial and error approach was applied find the ideal values for the mentioned parameters.

Fig. 10

The structure of RBF applied in current research

Optimization techniques

Levenberg–marquardt

In order to find the most appropriate solution for efficient problem minimization, the LM was developed by making some changes to the conventional method introduced by Newton. In this method, the equation of Newton-like weight update applies to the Hessian matrix approximation that is shown by Eq. 11 (Daliakopoulos et al 2005):

$${x}_{k+1}={x}_{k}-{({J}^{T}J-\eta I)}^{-1}{J}^{T}e$$

(11)

where, η, e, x and J represent a scalar controlling learning process, vector indicating the residual error, the weight of the neural network and Jacobian matrix, respectively. It should be noted that these values must be minimized (Daliakopoulos et al 2005; Rostami et al 2019). If Eq. 11 is run using the approximation of the Hessian matrix, it will yield results similar to those obtained by the Newton’s approach.

Bayesian regularization

In BR model, the weights and biases were updated based on LM optimization (MaCkay 1992; Foresee and Hagan 1997). Squared errors and weights that are combined in this algorithm should be minimized make the most ideal combination allowing a generalization with high accuracy (Pan et al 2013; Ameli et al 2018). The following equation is used to define the network weights for the objective function (Yue et al 2011):

$$F(w)=\beta \times {E}_{D}+\alpha \times {E}_{w}$$

(12)

where E_ω and E_D represent the sum of squared network weights and the sum of network errors, and F(ω) is the objective function. α and β indicate the parameters of the objective function which are determined according to the Bayes’ theorem.

Scaled conjugate gradient

In this technique, the weights should be updated according to the most negative gradient that is calculated as the performance function reduces with an increase in the speed, though it should not be regarded as the fastest algorithm. A more rapid convergence as well as a steeper downward trend can be achieved for the first repetition using a search algorithm similar to the conjugate gradient. P₀ is the search direction also named the conjugate direction. To optimize the present search direction in this algorithm, the next equation should be applied to show the best probable distance (Kişi, 2005; Ameli et al 2018):

$${x}_{k+1}={g}_{k}\times {\alpha }_{k}+{x}_{k}$$

(13)

Resilient backpropagation algorithm

In the MLP algorithm, different transfer functions are applied such as Sigmoid and Tansig that contribute to decreasing the infinite input domain into a finite output domain. In activation functions such as Tansig, the line slope gets close to zero if a big input is entered. This can reason some problems whenever the steepest descent is utilized to train the network. The reason for this event is the insignificant value of the gradient; hence, some minor changes take place in biases and weights (Riedmiller and Braun 1993).

Broyden fletcher goldfarb shanno

The most effective Quasi-Newton (QN) model applied for unrestricted nonlinear schedules is the BFGS method, whichis extensively utilized in nonlinear programming. BFGS is generally different from the Newton method, in which an evaluation of the Hessian matrix is considered in preference to the true Hessian Matrix H (Bazaraa et al. 2013). We should compute and reverse the Hessian matrix, but much more time is required for calculating the process. On the other hand, using an analysis of the gradient vectors, we can conduct the updating and then inverse the Hessian matrix in QN methods. This can reduce the objective function significantly (Zhang et al. 2020).

Fletcher-Reeves conjugate gradient

The FRCG algorithm is typically utilized in the training phase of neural networks to remove some drawbacks observed in the back propagation algorithm. This algorithm is preferred to the back propagation training algorithm due to its better performance and good convergence (Fletcher and Reeves 1964; Dai and Y.-X. YUAN, , 1996). To obtain the minimum of a function f(x): min f(x), x $\in$ Rⁿ: in order to obtain the minimum of the objective function, conjugate gradient methods utilize several iterations, the process of which is given below:

$${x}_{k+1}={x}_{k}+{\alpha }_{k}{d}_{k}$$

(14)

where ${\alpha }_{k}$ indicates the step size length which is specified by line search; d_k also represents the search direction satisfying: g^T(x) d_k ˂0. Conjugate gradient algorithms begin by exploring in the steepest descent direction.

Polak-Ribiere conjugate gradient

The most widely-applied algorithm is PRCG, however, given the presence of user-dependent parameters, this algorithm is not normally very efficient in large-scale problems. It is possible to express the neural network training problem in the form of a nonlinear unconstrained optimization problem. Therefore, if the error function E is minimized using the following equation, it would be possible to realize the training process:

$$E(w,b)=\frac{1}{2}\sum_{i=1}^{{n}_{1}}({d}_{i}-{{x}^{^{\prime}}}_{i}{)}^{2}$$

(15)

where ${x}^{^{\prime}}$ is a function of w (the weight vector) and d indicates the target using the forward pass equations. The cost function quantifies the square d error between the idea and the true output vectors.