Forward Modelling of Resistivity Logs using Neural Networks

In the oil industry, one uses the di erence in electrical conductivity between hydrocarbons (oil and gas) and water to determine the hydrocarbon saturation of rocks. To this end, a measuring device is lowered in a well bore to record a so-called resistivity log. Due to environmental e ects this resistivity log di ers from the resistivity of the formation, the true resistivity. To invert the measured log to the true resistivity one uses an iterative forward modelling process, involving the numerical solution of di erential equations. Although the current modelling algorithms have signi cantly improved in speed comparing to a few years ago, they are still not fast enough for inversion on a well site. Therefore, we have investigated the feasibility of using neural networks to perform the forward modelling process. Once trained, neural networks are very fast in producing output to certain input. The one-way mapping between the earth models (true resistivity model and environmental conditions) and the tool response can be learned by a \standard" fully connected net. However, problems arise from the high number of inputs that is needed to represent the earth model ( 450) and the high number of connections between the input layer and the hidden layer of the net ( 6750). The generalization performance of these nets is not su cient for the purpose of resistivity log inversion. We have studied two preprocessing methods to reduce the number of inputs: principal component analysis and the wavelet transform. This preprocessing of the input is quite successful. We have also studied methods to reduce the number of connections in the net by using locally connected nets instead of fully connected nets. We have found an architecture which proves to be quite e cient for this mapping: a convolutional-regression net. This type of net is based on the locally connected nets and shared weights. Due to these shared weights the hidden layer performs a convolution of the input layer. The convolution kernel, the set of shared weights, is learned by the net itself. The weight sharing reduces the number of connections in the net and these constraints improve the generalization ability of the net. This convolutional-regression net is approximately 100 times faster than the forward model that is used at KSEPL nowadays. The performance of the net is measured in the average relative error between the network outut and the forward model output. The neural net can be used as a fast forward model, when the average relative error lies below 5 %. The convolutional-regression net achieves an accuracy of approximately 8 % on a resistivity log coming from a \real" oil well. Further improvements in accuracy can be achieved by using a more representative training set. Even with less accuracy the neural net could be used as initial start for the forward modelling process. i Preface This report is the master thesis of Pamela Demmenie for graduating from the Rijks Universiteit Leiden (RUL) at the department of Computer Science. It describes a project that was performed at the Koninklijke Shell Exploratie en Produktie Laboratorium (KSEPL) in Rijswijk from December 1994 until August 1995. The project was supervised by Dr. Guozhong An (KSEPL) and Dr. Ida Sprinkhuizen-Kuyper (RUL). The report is divided into six chapters. The rst chapter provides a background on forward modelling and a short introduction in neural networks. The second chapter describes all the methods we have used in the experiments. These methods involve input reductions and architecture constraints. Chapter 3 and 4 describe the actual experiments for data without and with invasion. In chapter 5 we present the results of the trained neural networks on realistic logging data and in chapter 6 we summarize the conclusions of this project. All neural net simulations were run on Sparc 10 and Sparc 20 (Unix) stations and should eventually be run on an IBM R600 workstation (the approximation times were measured on this type of workstation). We used the Xerion simulator, versions 3.1 and 4.0. We have also experimented with other simulators: Stuttgart Neural Network Simulator (SNNS) and Aspirin/MIGRAINES. The advantages and disadvantages of these network simulators are outlined in appendix A. Acknowledgements First of all I would like to thank my supervisors Guozhong An and Ida Sprinkhuizen-Kuyper for their support and ideas during this project. Of course the project would not have been completed without the help of Niels van Dijk and Leon Ho man, who have provided the data that we have been working with and helped me in my understanding of the forward modelling process. I also like to thank the students from KSEPL and from the university for their support and company during the nine months I have been working on the project. And last but not least everlasting gratitude go to Pim Bussink for word 2 on page 42. ii