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1 SPE Reservoir Modelling With Neural Networks And Geostatistics: A Case Study From The Lower Tertiary Of The Shengli Oilfield, East China L. Wang, S. Tyson, Geo Visual Systems Australia Pty Ltd, X. Song, H. Cao, RIPED, PetroChina, and P.M. Wong, SPE, Univ. of New South Wales. Copyright 22, Society of Petroleum Engineers Inc. This paper was prepared for presentation at the SPE Asia Pacific Oil and Gas Conference and Exhibition held in Melbourne, Australia, 8 1 October 22. This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subect to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subect to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 3 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O. Box , Richardson, TX , U.S.A., fax Abstract This paper demonstrates a case study of a hybrid methodology based on the combination of radial basis function neural network and sequential Gaussian simulation. The methodology is demonstrated with an application to modelling the porosity distribution in an oil reservoir of the Lower Tertiary in the north of Dongying depression, Shengli Oilfield, East China. The methodology first uses radial basis function neural networks to estimate the porosity trends (fluvial directions) from high-dimensional data system with both well and seismic data. Gaussian simulation helps to do the local uncertainty analysis for the reservoir model. The final results from the hybrid methodology assure our confidence on the reservoir model both horizontally and vertically. They are realistic and honour the geological rules of the oilfield. The technique is fast and straightforward, and provides an effective computational framework for conditional simulation. Introduction Spatial descriptions of reservoir properties such as porosity and permeability are key components for performance evaluation and field development planning. When wells are sparse and limited, the statistics of the well data become unrepresentative. It is critical to integrate well data with all the available soft data to construct reliable reservoir models. In this paper, we will make use of the extensive seismic attributes and seismically interpreted results. Although some purely geostatistical techniques are capable of providing some of these functionalities 1,2, cross-correlation modelling is often difficult and tedious in practice. Also the information such as two-point statistics and linear relationships extracted from conventional reservoir data (well and seismic) may not be sufficient for describling the complexities of the reservoir model. Wang et al. 3,4, Wong et al. 5 and Caers and Journel 6 show that the neural network approach is a promising tool to deal with non-linear and multidimensional data system, hence to achieve the reliable and realistic approximation showing also the basic geological rules for reservoir model. On the other hand, Gaussian simulation provides an opportunity to do local uncertainty analysis. This paper will employ a hybrid methodology, called neural network residual simulation (NNRS) 3,4, to construct a porosity model for an oil reservoir of the Lower Tertiary in the north of Dongying depression, Shengli Oilfield, East China. The obective of this paper is to provide a case study of NNRS methodology. The reservoir in the case area is characterised by a fluvial channel with well data, seismic attributes and seismically interpreted results. The next section will first present a description of NNRS methodology, followed by the demonstrations of the case study. Methodology The integrated technique used in this paper is developed based on a combined use of neural networks and geostatistics. The technique assumes that any spatial prediction is composed of a predictable (trend) component and an error (noise or residual) component 3,4,7-9. Firstly neural networks (inexact estimators) are used to model the former component and residual kriging and simulation (exact estimators) to model the latter component. Hence the name neural network residual kriging (NNRK) and neural network residual simulation (NNRS) are used. The final estimate is simply the sum of the two components, and hence the estimator honours all the conditioning data. The methodology can be briefly described as four steps: 1) use of radial basis function neural networks to model the regional trends by integrating all the suitable seismic attributes and well data in the low resolution level; 2) the residuals at the wells are calculated in the high resolution level; 3) Gaussian simulation is used to generate multiple realisations for the residuals; and 4) adding the results of 2) and 3) together as the final predictions.

2 2 L. WANG, S. TYSON, X. SONG, H. CAO, P.M. WONG SPE Radial Basis Function Neural Networks. A radial basis function (RBF) 1 is a symmetrical transfer function, such as the Gaussian function. In spatial interpolation, it transforms the Euclidean distance between two multidimensional vectors into a function value. A RBF neural network can learn from a given set of input-output patterns and is a universal function approximator 7. The prediction is calculated based on the distance between the prediction location and the reference data location. The technique is robust and can be applied to model non-linear and non-stationary events in a multivariate environment. Setup of RBFNN. A typical RBFNN contains three layers of processing elements or neurons: input, hidden and output layers (Fig. 1). Each neuron is connected to every neuron in the preceding layer by a simple weighted link. The number of input neurons and output neurons depend on the application domain. For the sake of mathematical simplicity, we will limit our discussion to only single output networks. Each hidden neuron represents a RBF centre, which is parameterised by a position or reference vector c located in the m -dimensional covariate input space x. The Euclidean distance between the training vector and reference vector (RBF centre) can be written as: 1 2 m 2 d ( ) = R xi ci = 1,,n (1) i= 1 where R is the reliability factor for each dimension of the sampled vector. This is a confidence number, ranging between (no support) to 1 (full support) 5. The most popular RBF is the Gaussian function 2 φ ( d ) = exp( d 2σ), where σ is a constant controlling the radius of influence of the basis function. In this study, we consider the multiquadratic function φ( d ) = ( d 2 + σ 2 ) 1 2 and inverse multiquadratic function ( ) The estimator ( x) φ( d ) = d + σ as well. z at the output neuron is a weighted sum of the basis function values from the hidden layer: n z (x) = w φ( d ) (2) = 1 where { w } n 1 are the weights, n is the number of RBF centres, and φ (). represents a RBF. RBFNN Modelling. RBFNN modelling includes two stages. The first stage is to optimise the RBFNN framework including the number of RBF centres, control constant, RBF type and the reliability factor. Here we determine these parameters using trial and error technique. The details are in Wang et al. 3, 4. The weights between hidden and output layer are optimised in the second stage. From Equation (2), it is obvious that RBFNN is posed as a general linear least-squares problem: the regression of a target variable ( x ) z on an input set of covariates x given the training data pairings x, z, x, z,...,,. The unknowns are the weights {( 1 1 ) ( 2 2 ) ( x N z N )} { w } n 1 and can be obtained by setting: N k = 1 2 ( z ( ) z ( x ) w = k x k (3) Residual Simulation. Once the weights are obtained (Equation (3)), predictions can be simply made via the use of Equation (2). The predictions are compared with the target data and residuals (errors) are calculated as: Rk = zk ( x ) zk ( x ) k=1,,n (4) where N is the number of training patterns, z k ( x ) is the target of the k th training pattern, and z k ( x ) is the RBFNN prediction corresponding to the k th training pattern. Sequential Gaussian simulation (SGS) is then performed on the residuals, R k, using the corresponding residual variogram. NNRS Modelling. The SGS results can then be added to the RBFNN estimates. The NNRS results are: ( x) = z ( x) R ( x) NNRS k k SGS k z + (5) where k ( x) component (large-scale non-stationary features). ( x) z is the RBFNN estimate indicating the non-linear R are SGS k the residuals from SGS representing the stochastic error component (small-scale stationary features) which is practically impossible to be modelled by a neural network. In the following, we will apply the hybrid methodology, NNRS, to model the porosity distribution of the Lower Tertiary in the north of Dongying depression, Shengli Oilfield, East China. Shengli Oilfield Reservoir Characteristics. The case area is located in the north part of Dongying depression, Bohai Bay basin, East China. The target petroleum reservoir member cited in this paper is the sand group ES23 in the Lower Tertiary. The reservoir rocks were formed in a fluvial-flooding environment. The most favourable microfacies for reservoir is the fluvial channel bar. In the case area, we have 23 wells in 15.7km 2 area. The well logs available are sonic response (DT), spontaneous potential (SP), resistivity logs (Rt and Rxo) and the interpreted properties (porosity, permeability, V-shale and water saturation). The 3D seismic data and the interpreted reservoir properties are also available. The seismic attributes include amplitude, main frequency, averaged velocity, and the interpreted reservoir properties include porosity, and the ratio

3 RESERVOIR MODELLING WITH NEURAL NETWORKS AND GEOSTATISTICS: SPE A CASE STUDY FROM THE LOWER TERTIARY OF THE SHENGLI OILFIELD, EAST CHINA 3 of sand and mudstone. Three layers can be identified in the ES23 group. Data Preparation. In order to use NNRS to model the distribution of reservoir porosity, we first constructed two grid models with different sizes. Fig. 2 is the top structure of the grid model. The fine grid model has grids with the increment of 5 metres horizontally and 3.42 metres vertically, and the coarse grid model has grids with the same increment horizontally. The vertical resolution of the coarse grids is corresponding to the vertical resolution of the seismic recognition. Table 1 shows the correspondence between the coarse and fine grid layers. Porosity Trend (Coarse Grid Model). We first imported the well and seismic data into the coarse grids. The average porosity value from the porosity log is assigned to the grid when a well passes that grid. Table 2 presents the correlation coefficients between log porosity and the seismic attributes. The ratio of sand and mudstone, interpreted porosity from seismic and average velocity have relatively good correlation with the mean log porosity from the conditioning wells. These coefficients can be used as the reliability factors for the RBFNN model. During the RBFNN training process, we have 66 training patterns with eight input parameters of X-, Y- and Z- coordinates, average velocity, amplitude, main frequency, seismic porosity, ratio of sand and mudstone and one output of mean log porosity at the grids passing the well locations. According to the experiments with trial and error, the optimum radial basis function for this case is the inverse multiquadratic function with the global control constant of.15 (Figs. 3 and 4). In order to balance the requirements from generalisation and exactness, we empirically selected 3 radial basis centres using k-means clustering for the RBFNN training 3,4. With the use of inverse multiquadratic function and the control constant of.15, the optimum termination epoch was found to be 281 according to the root mean square error (RMSE), average error and error skewness analysis (Fig. 5) 3, 4. The resulting porosity trend is shown in Fig. 6. Residual Simulation of Log Porosity (Fine Grid Model). After the log porosity distribution trend was obtained, we exported the trend from coarse grid model to fine grid model. Then the residuals can be calculated in high resolution level using Equation (4). Fig. 7 is the variogram of the residual log porosity and the original log porosity. The range of the residual variogram is smaller, because most of the information has been extracted by the RBFNN. Sequential Gaussian simulation for the residual log porosity was then performed based on the residual variogram. Fig. 8 is a 3D view of the residual log porosity model. NNRS Results. NNRS modelling results were then obtained from Equation (5) based on the work in the above sections. Fig. 9 presents the final porosity modelling results. We can see the general distribution pattern and the fluvial direction implied in the seismic data (Fig. 6). The model follows both the geological and geophysical expectations. CONCLUSION This paper demonstrates a case study of the hybrid methodology based on the combination of radial basis function neural network and sequential Gaussian simulation. Radial basis function neural networks can learn well the basic fluvial direction, and Gaussian simulation is able to do the local uncertainty analysis for the reservoir. The results of the case study in teh Shengli Oilfield assure our confidence on the reservoir model both horizontally and vertically. They are realistic and honour the geological rules of the oilfield. The technique is fast and straightforward, and provides an effective computational framework for conditional simulation. REFERENCES 1. Deutsch, C. V. and Journel, A. G.: GSLIB: Geostatistical Software Library and User s Guide, Oxford University Press, Inc., NY (1992). 2. Holden, L., Hauge, R., Skare,. and Skorstad, A.: Modeling of Fluvial Reservoirs with Obect Models, Mathematical Geology, 3, (1998). 3. Wang, L., Wong, P.M. and Shibli, S.A.R.: Modelling porosity distribution in the Anan Oilfield: Use of geological quantification, neural networks and geostatistics, SPE Reservoir Evaluation and Engineering, 2(6), (1999). 4. Wang, L., Wong, P. M., Kanevski, M. and Gedeon, T.D.: Combining neural networks with kriging for stochastic reservoir modelling, In Situ, 23(2), (1999). 5. Wong, P.M., Tamhane, D. and Wang, L.: A neural network approach to knowledge-based well interpolation: A case study of a fluvial sandstone reservoir, Journal of Petroleum Geology, 2(3), (1997). 6. Caers, J. and Journel, A. G.: Stochastic reservoir simulation using neural networks trained on outcrop data, SPE 4926, presented at the 1998 SPE Annual Technical Conference and Exhibition, New Orleans, USA, 27-3 September, (1998). 7. Kanevski, M., Arutyunyan, R., Bolshov, L., Demyanov, V. and Maignan, M.: Artificial neural networks and spatial estimations of Chernobyl Fallout, Geoinformatics, 7, 5-11 (1995). 8. Kanevski, M., Demyanov, V. and Maignan, M.: Mapping of soil contamination by using artificial neural networks and multivariate geostatistics, International Conference on Artificial Neural Networks, LNCS 1327, (1997). 9. Kanevski, M., Demyanov, V. and Maignan, M.: Spatial estimations and simulations of environmental data by using geostatistics and artificial neural networks, Proceedings of IAMG, Part 2, (1998). 1. Bishop, C. M.: Neural Networks for Pattern Recognition, Oxford University Press (1995).

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6 6 L. WANG, S. TYSON, X. SONG, H. CAO, P.M. WONG SPE Fig. 9: NNRS modelling results for log porosity in Shengli Oilfield. The images in the left column are 3D and the images in the right column are the corresponding plan views. In both columns, the images are respectively the log porosity distributions in the second, sixth and the ninth layer in fine grids from top to bottom.