Technical Notes - Theoretical Note on Linear Absorption Methods of Determination of Fluid Saturation in Porous Media

Boyer, Morgan and Muskat,' with improvements by Morgan, McDowell, and Doty,l and Laird and Putnam,' have described a scheme for the determination of fluid saturatic in porous media by measuring the x-ray linear absorbing power of particular fluids in the media. This method is one of several which depend on the linear absorbing power of the fluids in the porous medium. Other examples of linear absorption methods involve measurement of dielectric constant, gamma ray absorption, and ultrasonic attenuation. It is the purpose of this note to demonstrate by theortical considerations that the absorption versus saturation curve for a linear absorption method is not necessarily a single valued function of the saturation. Such non-unique behavior must be considered and evaluated before suitable saturation determinations can be made by a linear absorption method. To illustrate the discussion, examples will be given only for the simplest cases of combinations of two different dielectrics. The dielectric constant method was chosen for illustration because it was thought to demonstrate most graphically the point in question. According to Argue and Maass,4 the resultant dielectric constant of a combination of two different dielectrics depends on the orientation of the dielectrics in the electric field of the test condenser used to measure the dielectric constant. One ease involves alternate slabs of dielectrics E1 and E2 perpendicular to the electric field of the test condenser, as shown in Fig. 1A. For this case: l/'E, = A/E, + B/E,.......(1) where E,,, is the dielectric constant of the combination, A is the ratio of the path length through dielectric E, to the distance between the plates of the condenser (assuming the dielectrics have the same cross sectional area as the condenser), and B is the ratio of the path length through dielectric E2 to the distance between the plates of the condenser. It will be noted that the constants A and B are numerically equal to the fractional "saturation" of the region between the test plates of the condenser by the two dielectrics E1 and E2, respectively. Another case involves alternate columns of the two dielectrics parallel to the electric field and forming a continuous path between the condenser plates, as shown in Fig. 1B. For this case: E., = AE1 + BE2.........(2) where E., is the dielectric constant of the combination, A is the ratio of the cross sectional area of dielectric E, to the cross sectional area of the condenser plates (assuming the lengths of the dielectrics equal the distance between the condenser plates) and B is the ratio of the cross sectional area of dielectric E2 to the cross sectional area of the condenser plates. As before the constants A and B are the effective "saturation" of the condenser by the two dielectrics. Still another case is described by Lichtenecker5 where dielectric E2 is shown in the form of small particles distributed at random in dielectric E1. This is shown in Fig. 1C. For this case: log Em = A log E1 + B log E2.....(3) where Em is the dielectric constant of the combination. A is the ratio of the volume of dielectric E1 to the volume of the condenser, i.e., the effective "saturation" of the condenser by dielectric E,. B is the ratio of the volume of dielectric E2 to the volume of the condenser, i.e., "saturation" of dielectric El. Equations (l), (2) and (3) are plotted in Fig. 2 for E1 = 2.5 (oil) and E2 = 80 (water). Equations (1) and (2) may he thought of as representing the limiting cases of the configuration of two fluids which might be expected to exist in a porous medium. Equation (3),