The Use of Differential Sonic- Resistivity Plots to Find Movable Oil in Permian Formations
Authors: Lindley R.H. 
Published: December 2007

Abstract
In recent years, the plotting of sonic transit time vs the resistivity recorded by a deep-investigation device has become one of the most popular methods of interpreting electrical logs run in wells drilled with salt mud.
This paper reviews the reasons why this method is valuable. It also discusses pitfalls in the method, which may be overlooked because of its very convenience. A possible source of error emphasized is the common assumption that all zones analyzed are affected in the same degree by mud-filtrate invasion.
The use of a second "sonic-resistivity plot", this time with a shallow-investigation resistivity device, is presented as a way of avoiding entrapment by these pitfalls. The difference between the water saturations found from this second plot and from the standard plot can be used, according to rules-of-thumb to be stated, t9 estimate D, for the purpose of correcting the Laterolog reading for invasion. A sonic-resistivity plot with the corrected Laterolog readings will give better values of saturation, and the differences between these and the corresponding saturation values from the second plot are indicative of oil mobility.
Several examples of Permian formations of West Texas, combined into a composite log, are computed by both the standard sonic-resistivity plot and by the more comprehensive "differential sonic-resistivity plots". The results of these computations are compared with the production of the formations to show how the additional information can improve the log interpretation.

Introduction
In attempting to decide where and whether his well will produce oil or gas, the oil operator needs to know several things, the most important of which is the water saturation Sw. Since Sw cannot be measured directly in situ, it is usually inferred from measurements of the porosity f and the true formation resistivity Rt.
If the formation water resistivity Rw is known, it is possible to determine Sw from these indirect measurements by the use of Archie's fundamental equation.1
Equation 1
F usually is computed from the relationship F = f2, and Archie's saturation equation, therefore, may be written.
Equation 2
This equation may be readily solved arithmetically, but a more adroit method consists of a graphical solution in which the readings from a porosity log are plotted vs the corresponding resistivity readings from a deep-investigation device.3 The possibility of such a graphical solution becomes apparent by rewriting Eq. 2 in the form
Equation 3
According to Eq. 3, a plot of f in abscissa vs 1/Rt½ in ordinate will be a straight line, for constant values of Rw and Sw.
Shortly after the introduction of the Sonic log, Wyllie, et al, proposed the time-average relationship.3
Equation 4
which can also be written as
Equation 5
where ?t = the sonic transit time,
Vm = the sonic velocity in the matrix material, and
Vf = the sonic velocity in the formation fluid.
This equation states that sonic transit time in excess of 1/Vm is proportional to porosity.
Combining Eqs. 3 and 5,
Equation 6
When values of ?t are plotted in abscissa vs values of 1/Rt½ in ordinate, Eq. 6 yields a straight line provided the term Sw·Vm·Vf/Rw½ (Vm-Vt) is held constant.