Applied The Finite Difference Method To Model Spontaneous Potential Logging

Index:In the practice of well logging, it was found that there is a potential difference between an electrode in borehole and that on the surface, if the electrode in borehole is lifted along the hole, the potential changes and forms a curve. This change of potential was very stable and repeatable, so it is clear that it is not an occasional phenomenon. This potential is then called spontaneous potential. The origin of spontaneous potential (SP) has been explained as dipole layers distributed along borehole wall and formation boundaries. The result of SP logging can be used to judge lithology, calculate thickness of layer, and the salinity of connate water. The latter was connected with the resistivity of the connate water (R_w). Moreover the resistivity of the connate water (R_w) is animportant parameter to calculate the oil & gas saturation of layers. Because SP logging is a simple method and it can provide important parameter, it has always been a necessary method in well logging.The SP log not only signals the sand layers in presence of shale, it also provides some useful values to estimate the resistivity of the formation water. Unfortunately, we have the qualitative of information from the SP logs in some times. Because the methods which were used to investigate SP are simple, it is very difficult to obtain accurate and quantitative interpretations from the curves of SP logs. In this thesis, we modeled the SP logs by using finite difference method. Finite difference method is an important method to solve a PDE. In part two of this thesis, we introduce the finite difference method and the different methods which are used the FDM to solve the equations.In the part three of this thesis, we give the modeling, which is used to model the SP log. And we will get the Poisson function of SP log. In this process, we will use the typical three-layers modeling, and apply the software, which was called MATLAB, to solve the linear functions. MATLAB have the strong ability to calculate the matrix. When we plus, reduce, multiply or divide the matrixes, we only use common signs and functions. Besides it also can manage the sparse-matrixes. So it's very well to solve these functions which are produced by FDM.There are two reasons to decide the range and shape of curve of SP log. Firstly, the most important of factor is the date of spontaneous electromotive force. Secondly, the range and shape is also relative to the spontaneous current. The spontaneous current is controlled by the resistance, shape and volume of the medium, for example, the resistance of layer, the thickness of layer, the resistance of mud, the radius of borehole, and the surrounding rocks and so on. In the part four of this thesis, we introduce these reasons and summarize the character of the SP curve. According the result of calculate, we know that the curves of SP are symmetry and will have great changed in the interface of layers, and if the thickness of layer is big, the milestone of SP curve will show the interface of layer. It will be determined by the SP/SSP if the value of sandstone's SP is positive. Following the thickness of layer increasing, the value of SP/SSP will increase too. And when the thickness of layer is big, the curves in middle of layer become plainness and the value of SP will equal to the value of SSP. If the value of R_t/R_m increase, the value of SP/SSP will minish.It was given more attention that the mud invaded zone influence the curve of logs, especially in SP log. Because the origin of spontaneous potential (SP) is relative to the salinity of the liquor, the salinity of the liquor in mud invaded zone has the most response in SP log. In the part five of this thesis, we use the new model to calculate this response. In the new model, we construct the functions which show the change of the salinity of liquor in invaded zone. According the result of calculate, the response of invaded zone will show as the following. Firstly, it will have no influence for the property of symmetry and the changed in the milestone of layer. Secondly, when the thickness of invaded zone and the salinity of the stratum water are uniform, the value of SP will minish following the increasing of salinity of mud. Thirdly, if salinity of mud and the stratum water are uniform, the value of SP will minish following the increasing of the thickness of invaded zone. The lastly, when the thickness of invaded zone, the salinity of the stratum water and mud are uniform, the value of SP will change following the salinity of mud near the wall. The slower the salinity of mud near the wall changed, the smaller the values of SP were.