When natural gas is flared, a combustion reaction takes place in the form stated below [2] :

Presented below, as an example, is the combustion reaction of propane.

During a combustion reaction, several intermediate products are formed, and eventually, most are converted to CO₂ and water. Some quantities of stable intermediate products such as carbon monoxide, hydrogen, and hydrocarbons will escape as emissions.

For a complete reaction, carbon (IV) oxide and water vapour are formed. However, when the reaction is incomplete, carbon (II) oxide is formed alongside carbon (IV) oxide and water vapour .

Depending on the location, impurities such as sulphur, nitrogen and hydrogen sulphide are also found with natural gas. These gases undergo a combustion reaction to form acid gases such as oxides of nitrogen, oxides of sulphur and hydrogen sulphide H₂S.

Complete combustion requires sufficient combustion air and proper mixing of air and gas. Smoke may result from combustion, depending upon gas components and the quantity and distribution of combustion air. Gases containing methane, hydrogen, CO, and ammonia usually burn without smoke. Gases containing heavy hydrocarbons such as paraffins above methane, olefins, and aromatics, cause smoke.

Artificial Neural Network (ANN) have been used to address some of the fundamental problems in petroleum engineering that conventional predictive models have been unable to solve, especially when engineering data for design, interpretations and calculations have been less adequate. Also, with recent advances in pattern recognition, classification of noisy data, nonlinear feature detection, market forecasting, sickness recognition in human blood in medicine, and process modeling, ANN technology is very well suited for solving problems in the petroleum industry. Several authors have developed ANN models to solve several problems in the petroleum industry. Juniardi and Irashagi [3] developed ANN model to predict permeability and skin factor of faulted Reservoir. Arehart [4] developed a 3-layer back propagation neural network model to determine the grade of a drill bit while it is drilling. Ashenayi et al. [5] used a hybrid 3-layer back propagation neural network model to identify beam pump malfunctioning from down hole pump cards. Erahaghi et al. [6] used a multiple ANN to train and recognize patterns (C_D, P_D, t_D, S, d_D…) for specific conceptual reservoir model. Kumoluyi [7] discussed the general application of neural networks and their potential uses in some areas of petroleum engineering. They found that one advantage of feed forward networks in pattern recognition is their ability to recognize patterns regardless of position, rotation and scaling. The application of pattern recognition is essential in well log interpretation of multiphase flow and in seismic data processing. Mohagheh et al. [8] used a 3-layered forward back propagation neural network model to estimate the heterogeneity of some reservoirs. Briones et al. [9] developed a 3-layer radial basis neural network (RBFNN) model to relate gas-oil ratio (GOR) and API gravity to the corresponding molar composition (C₁, C₂, C₃, C₄, C₅, C₆, C₇ and CO₂) . Mc Vay et al. [10] used a feed forward back propagation neural network model to train the actual refracture treatment design, basic well information, and well performance in order to determine the Sand Volume, Fluid Type, Injection Rate and Acid Volume as the majoring factors that influence the well deliverability during hydraulic fracturing. Manmath et al. [11] used ANN model to predict fluid distribution taking oil, water, and gas production as input data. Wong et al. [12] developed a back propagation neural network (BPNN) model to estimate formation permeability in the RAVVA oil and gas field offshore in India. Garrmouch and Smaoul [13] developed a 3-layered back propagation neural network model to estimate formation permeability of tight gas reservoir. Soto et al. [14] used a neural network model to predict the permeability and porosity of zone C of the Cantagallo field in Colombia. Shelley et al. [15] developed two separate neural network models for well completion analysis and optimization to identify the factors that affect production and measure their contributions to the production result. Nikravesh et al. [16] developed several neural network models for water flood management in fractured reservoir to predict the wellhead pressure and future production in quarterly basis.

Application of neural networks in time series forecasting [17] -[20] is based on the ability of neural networks to approximate nonlinear functions. The most popular treatment of input data is feeding the neural networks with either the data at each observation, or the data from several successive observations. Denote the data at instant k as y(k), where y may be a vector, then the above treatment can be described as

or

respectively, where NN() stands for the neural network forecaster and l is the number of successive observations. This treatment considers the time series as a nonlinear time series and tends to generate a nonlinear “auto- regression” model to fit the series. So far, there have been few papers describing how to choose inputs for the neural network forecaster in order to achieve better forecasting performance. It is our belief that the performance of a neural network forecaster is much affected by input data patterns.

Autocorrelation analysis has been often used in time series forecasting using statistical approaches such as ARMA models. This analysis is mainly used in detecting the autocorrelations between successive observations of time series, and used in the well-known ARIMA models with Box-Jenkins methods that are very efficient in forecasting linear time series [21] .

Autocorrelation analysis can be used to determine the correct input patterns for nonlinear time series fore- casting with a neural network. The scheme contains three phases: detection of input patterns, determination of the number of neurons in hidden layer(s), and construction of the neural network forecaster. In the detection phase, autocorrelation analysis is used to identify input patterns of time series for training. Determination of the number of neurons in hidden layer(s) is done with Baum-Haussler rules [22] . The neural network forecaster is then constructed with the determined input patterns and the number of neurons in hidden layer(s).

We recall the linear regression model (LRM ) given as:

which is made up of the predicted part and the residual part. The residual is the difference between the observed and the predicted values which is ascribed to unknown sources. n is the number of observations, y_i is the ith observation, is the predictor variable vector related to y_i, is the parameter vector, and e_i is the error associated with ith observation.

Writing (1) in time series notation, we have

Explicitly, this is written as

where y_t is the dependent variable, x_t is the independent variable (in this case, the “years” ), α is the intercept, β is the parameter associated with the independent variable, x_t , and e_t is the stochastic term or error associated with the model.

We minimize (3) with respect to α and β , and, to obtain two normal equations respec-

tively. Solving the normal equations, we obtain the estimates of the parameters and:

The predicted model becomes

and the residual is given as

The statistical neural network (SNN) model structurally is composed of two parts: the predictive and the residual, as is in classical regression, given as

where . Thus Equation (6) can be written as

is the vector of the input variable, g(.) is the transfer (or activation) function and are the weights (or parameters) associated with the input vector, hidden neuron and the transfer function respectively, while e_i is the error associated with the network. We note that when there is no hidden neuron, the SNN reduces to the ordinary regression model.

We propose a simple time series neural network model,

The terms and symbols are as explained in the SNN model, except that t refers to “time” or “period” .

The weights are estimated using Taylor’s first order approximation,

where

If , and, then we can write Equation (6) as

where

The least squares estimate of the parameter is

and the estimated model is

while the network error is given as

In this paper, we used the symmetric saturated linear transfer function,

The data used in this study are annual data on natural gas production, utilization and flared in Nigeria from 1958-2006, obtained from the Annual Abstract of Statistics of the Nigeria Bureau of Statistics (NBS) , formerly Federal Office of Statistics (FOS) -(1970-1990), Ministry of Petroleum Resources (MPR) -(1991-1994) and NAPIMS -(1995-2006). This makes a total of 49 data set. In each case (production, utilization and flared), the TSNN model formulation is:

1-2-1, 1-5-1, 1-10-1.

All input variables were standardized, that is, converting them to the range (0, 1) before feeding them into the network. This is to avoid the application of extremely small weighting factors in the case of large input values.

Similarly, the output values are “destandardized” to provide meaningful results since all values leaving the network are automatically output in a standardized format. This is done by simply reversing the standardization algorithm used on the input nodes.

We used SPSS for the TSRM part of the analysis, while a neural code was written for the analysis of the TSNN using MATLAB R2009a, and interesting results were obtained.

Here we discuss several criteria that have been used to choose between the two models. Several criteria are used for this purpose. In particular, we discuss these criteria: (i) R² ; (ii) adjusted; (iii) Akaike information criterion (AIC) ; and (iv) Schwarz Information criterion (SIC) . All these criteria aim at minimizing the residual sum of squares (SSE) . However, except for the first criterion, criteria (ii), (iii), and (iv) impose a penalty for including an increasingly large number of predictors. Thus there is a tradeoff between goodness of fit of the model and its complexity (as judged by the number of predictors).