1. Introduction
Oil production prediction is very important in the oilfield development; study on oil production predicting method is a key topic of petroleum science. At present, there are four kinds of predicting methods [1-6] with physical meanings: Empirical formula such as Arps method, Hubbert model, water-flood decline curve method; Hydrodynamic model based on fluid mechanics model; Material balance equation model; Numerical reservoir simulation model. Besides the above mentioned four methods, there is a typical type of prediction model related to modern optimization, this model type is composed of GM (1, 1) model, neural network model, support vector machine model, etc. Oil field development system is a complex multi variables non-linear dynamical systems, different predicting model has different characteristics like predicting accuracy. Neural network model and support vector machine model are two effective methods to solve multi nonlinear mapping problem. At present they are used in many disciplines, even in the oil production prediction. In this paper, neural network model and support vector machine model have been used to predict the oil production of XINJIANG oil field, and good predicting results have been achieved. Meanwhile, GM (1, 1) model has also been used to predict oil production. Predicting result shows that this model is adaptable to the case which lacks data and hard to establishes model with probabilistic method.
2. Predicting Model
2.1. Neural Network Predicting Model
At present, in the application of artificial neural networks (ANN) [7,8], most of them are back propagation (BP) ANN and their variations. It has been proved that BP neural network can approximate any multivariate continuous function. Kolmogorov rule guaranteed that any continuous function or mapping can be achieved by a 3-layer ANN.
3-layer BP ANN is used to establish the ANN model with prediction function. The first layer of BP-ANN is input layer, the second layer of it is middle layer, and the third layer of it is the output layer, see Figure 1.
Network Simulation
Assume N-samples (input-output data samples) are used during the training process:
The input variables are 
, the output variables are
. For the k-sample
, let. 
be input mode vector; 
be the expectation output vector; 
be the middle layer input vector; 
be the output unit vector of middle layer; 
be the input vector of output layer; 
be the output vector of output layer; 
be the connection weights from input layer to middle layer; 
be the connection weights from middle layer to output layer; 
be the threshold value of middle layer; 
be the threshold value of output layer; 
is the learning rate.
Let the response function of ANN 
be Sigmoidtype function: 
The input and output values of each neural unit satisfy the following relationship:
Middle layer:
(1)
Output layer:
(2)
After training of N samples, the network error is:
.
Error of output layer unit:
(3)
Error of middle layer unit:
(4)
The connection weights 
and the threshold value 
can be modified by the output layer error 
and output value of middle layer unit
:
(5)
(6)
The connection weights 
and the threshold value 
can be modified by the middle layer error 
and input value of input layer 
:
(7)
(8)
Repeat the above-mentioned learning mode, until the network converges to a given error range.
2.2. Support Vector Regression(SVR) Predicting Model
SVR [9-11] is an effective method to solve the regression problem. This regression problem can be described in mathematics:
Given training set
, where
,
. The training set is composed of independent and identically distributed sample points following certain probability distribution
. After giving loss function
, the regression function 
will be found to make the expected risk 
reach its minimum value, where 
is the nonlinear mapping, it maps the data
into a high dimensional feature space; 
and 
are weight vector and bias value, separately.
If we denote the regression problem by minimum risk problem with the loss function 
, then the basic rule of SVR method is established: by introducing kernel function 
into the nonlinear mapping, the nonlinear regression problem in lowdimensional input space transformed into a linear regression problem in high dimensional feature space and the key is to solve the parameter 
and 
in the regression function. Hence, the basic rule of SVR in solving regression problem is to solve an optimal problem with the following form:
(9)
where 
are Lagrange multipliers; 
is a constant, called penalty factor; 
is a given positive value, it is a maximum error of regression.
By solving Equation (9) gives the optimal Lagrange multipliers
, and then gives regression function when the expected risk 
gets its minimum:
(10)
where the sample satisfied 
is the support vector;
.
Select 
or 
in interval
;
If 
is selected, then
If 
is selected, then
when solving regression problem, the proper kernel function can be selected to SVR training. Introducing sample factor vector 
and predictor vector 
into Equation (10) after training gives the prediction result.
2.3. GM (1,1) Predicting Model
The Let 
be initial data series, its 1-AGO data series is
where 
denote the grey derivative of 
by:
Let 
be the generated data series of
, then
Hence, denote the gray differential equation model of GM (1, 1) [12,13] by:
It is,
(11)
In Equation (11) 
is the grey derivative, 
is the developing coefficient, 
is the albino background value, 
is the grey functional variable.
Introducing 
into Equation (13) gives
Let
Then GM (1, 1) model can be expressed as
, now the question comes down to find the value of
.
Introducing least square method to solve the estimating value:
In Equation (13), if 
is continuous variables when
, then 
is a function of time
, it is
, so the derivative of 
become
and the derivative of albino background value
becomes
. Hence, the grey differential equation becomes:
(12)
Equation (12) is the albino type of GM (1, 1) model. Given initial value
, the solution of Equation (12) becomes:
.
3. Applications and Discussions
Given the initial oil production data (from 1958 to 2012) of certain oilfield block in China, then the above-mentioned three method can be used to predict the oil production of different oilfield block (A1, A2, A3). After using the above-mentioned three different predicting model gives Figures 2-4.
In Figures 2-4, the black circle means the real oil production, the green curve is the predicting curve with ANN model, the red curve is the predicting curve with SVR model, the blue curve is the predicting curve with GM (1, 1) model. Figures 2-4 show the predicting accuracy with ANN model and SVR model is higher than
Figure 2. comparison of oil production prediction value with different method in block-A1.
Figure 3. comparison of oil production prediction value with different method in block-A2.
Figure 4. comparison of oil production prediction value with different method in block-A3.
the GM (1, 1) model, the maximum relative error is less than 10%, ANN model and SVR model can be used to short term prediction. However, the GM (1, 1) model predicts the overall trend in oil production decline; it can be used to middle-long term oil production prediction.
4. Conclusions
Prediction with ANN model and SVR model can comply with the actual oilfield production dynamics, the prediction errors of them are less than 10%. However, they are learning type of model; much data is needed to complete the prediction, so they are only suitable for the short term prediction;
Although the prediction accuracy with GM (1, 1) model is low, still the prediction result fits with the overall downward trend of oil production, so it can be used as a reference for long-term prediction.
NOTES