Logging and Log Interpretation - Departure Curves for the Self-Potential Log

An analog computer study was made of the reduction of the magnitude of the SP by formation geometry and resistivity effects. Practical field .situations were studied. The analog data were converted to the form of departure charts. These charts may he used to determine the reduction in the SP due to the above eflects pith good precision Quantitative interpretation of the SP log in terms of formation water resistivity is presently based on the premise that the observed SP is equal to the electrochemical potential developed across a shale between the mud filtrate and the formation water. This procedure inherently assumes that, first, no electro kinetic (streaming) potential is present, and, second, that the SP is not reduced by resistivity or geometry effects in the formation. When, however, such a reduction of the observed SP does occur, one may write, SP = b,E, b, E, (1) where SP is the observed self-potential opposite the center of the bed under consideration. In this case, E, is the electrochemical potential, E, is the electrokinetic potential, b, is the electrochemical reduction factor, and h, is the electrokinetic reduction factor. The first assumption above is equivalent to assuming E, = O in Eq. 1 and the second assumption is equivalent to h, = b, = 1. Examination of electric logs shows that for many formations of interest the observed SP is reduced below its maximum value. The object of this work is to aid quantitative SP interpretation in at least some of these situations by obtaining values for b, and b, in Eq. 1 as functions of realistic combinations of bed thickness, depth of invasion, and formation resistivity. The only previously published work in this field known to the authors is by H. G. Doll' in 1948. He evaluated b, under conditions permitting exact analytical solution. His work clearly established the importance and qualitative nature of formation geometry and resistivity effects upon the SP, but the necessity of obtaining analytical solutions restricted his study to situations of limited practical interest. In order to extend such studies to practical combinations of formation parameters which do not lend themselves to feasible analytical solutions, the results reported in this paper were obtained by use of a special analog computer. THE ANALOG MODEL The analog model used for this study was the Guyed Analog Computer which originally was designed and built for resistivity log departure curve studies. It consists of a two-dimensional mesh of resistors, with the vertical mesh corresponding linearly to vertical distance along the borehole, and the horizontal mesh corresponding to radial distance from the borehole axis. By proper choice of horizontal resistor values, a three-dimensional, axially symmetric, borehole-formation problem can be modeled with this two-dimensional array. These and other details of the model are contained in a paper by H. Guyod'. The geometry used for the SP studies with the analog is depicted in Fig. 1. This configuration has a horizontal plane of symmetry, and, hence, the model, and the following discussion, need only to be concerned with one quadrant such as shown in Fig. 2. A further simplification arises by considering that both the natural potential sources and those used in the model have fixed internal impedance. Therefore, the current flows produced by these sources can be separated and studied independently ( i.e., the superposition theorem can be applied). SP = SP, i SP, = h,E, + b,E, ,(7)