LICENCIAS DE OPERACIÓN Y COMERCIALIZACIÓN

Back propagation artificial neural networks (BPANNs), used in the present study, are the most common type of feed –forward multi-layered neural network, consisting of an input

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neurons in the next layer, and the connections between neurons are weighted. Each neuron receives a net input (net j) that is computed from weighted outputs from prior neurons

connected to this neuron:

net j =



O

W

Wij is the connection weight from neuron i to neuron j

Oi is the output from neuron i in the prior layer

The output from each neuron is dictated by its activation function, a mathematical function, which calculates the neuron’s output based on the input to this neuron. The most commonly used activation function in back-propagation neural networks is the sigmoid activation function, which produces an output in the range 0 to 1 and is a continuous function. The sigmoid activation function has the mathematical formula as follows:

---(4.2)

The desired performance of a neural network is achieved through the training process. Given input and output patterns to the neural networks, it will adjust the connection weight between neurons as mentioned earlier until the predicted output is close to the desired output. An input pattern is presented to the network. This input is then propagated forward in the network until activation reaches the output layer. This constitutes the so-called forward propagation phase. The output of the layer is then compared with the output pattern. The error, that is the difference between the output Oj and the teaching input tj of a

target output neuron j, is then used together with the output of the source neuron i to compute the necessary changes of the weight Wij. To compute the errors of inner neurons

for which no teaching input is available (neurons of hidden layers), the errors of the following layer, which are already computed, are used. In this way the errors are propagated backward, so this phase is called backward propagation. The most commonly

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used back-propagation update rule is the generalised delta rule, which is mathematically expressed as follows:

--- (4.3)

Where:

ΔWij is the weight change

η is the learning rate δj is the error of neuron j

Oj is the output from neurons

The first step is to define the network architecture, which includes the number of input, hidden and output layers and the number of neurons in each layer. This is usually done by “trial and error”. Wong et al. (1995 and 1997) used one hidden layer with 5 neurons; Huang et al. (1996) used 12 neurons in a single hidden layer; Arpat et al. (1998) used one hidden layer with 15, 18 and 30 neurons; Du et al. (2003) have indicated that the neural network can be improved by adding more hidden layers.

Determination of the appropriate number of nodes for the hidden layer is difficult, and often also done by trial and error. Le (2004) suggested a simple rule of thumb as follows:

Number of neurons (hidden layer) = 2 number of input neurons1 --- (4.4)

The important feature of the back-propagation neural network is that it learns to reproduce the outputs not by just remembering that output appropriate for every input, but by learning the patterns contained within the data. Once trained, the network can make predictions from novel sets of input data.

i j ij

W  

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back-propagation neural networks. They are: the number of input wireline logs, the number of core plugs in the training dataset, the network architecture, and the number of predictions.

The first concern is the number of wireline logs being used as input to the neural network. For instance, Helle et al. (2001) used a different combination of 4 wireline logs (GR, RHOB, DT, NPHI) to predict permeability in some North Sea reservoir wells. In this study, the number of input is 4, 5, 6, and 7 corresponding to 4, 5, 6, and 7 keys wireline logs to predict SCAL parameters (Figure 4.2). The second concern is the amount of core data in the training dataset and it’s important because it plays a crucial role in terms of time and cost. The less cores that is needed, the lower would be the costs. In all published case studies to date, the number of core plugs in the training dataset was generally substantial. For instance, the lower published number of samples in a training dataset, which was called “limited”, was 45 core plugs (Arpat et al., 1998). In this study 55 core plugs were used “limited”. The third concern is the number of hidden layers and number of hidden neurons in each hidden layer. This task is usually done by “trial and error”. Arpat et al. (1998) used one hidden layer with 15, 18, and 30 neurons. In this study, 5, 6, 7, and 8 hidden layers (Figure 4.2) with 24, 34, 46, and 60 neurons were used. The last concern is the number of predictions to be used, and single prediction was used (Figure 4.2). In order to choose the most suitable learning rate, its firstly set to 0.2 and then is gradually reduced to 0. The right learning rate was then selected as the 0.2 giving the minimal error in the training dataset. The neural network classifies new patterns and predicts on output based on the learned patterns. Neural networks often have application when relationships of parameters are too complicated or require too much time to solve via conventional methods. The most frequently used type of neural network is a feed forward neural network using a back- propagation learning algorithm, due to its popularity and simplicity

Learning backpropagation algorithm is consider as an optimization problem because before any mathematical derivation it helps to develop some intuitions about the relationship between the actual output of a neuron and the correct output for a particular training case. The advantage of network, the connection weights are adjusted automatically by using input data and gave output values which match as closely as possible the real output values in the training dataset. The neural network will converge to the correct SCAL parameter

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values by backpropagation the error between its prediction and actual parameter value. In this particular case study, the application of the GFNNs approach to predict SCAL parameters to be a worthwhile technique for improved prediction and has potential for a wider scope of application such as full field review or asset evaluation where data, costs and time are normally limited.

The previous investigations (Mohaghegh et al.1996) have revealed that neural network is a powerful tool for identifying the complex relationship among permeability, porosity, fluid saturations, depositional environments, lithology, and well log data.