Seismic-scale dependence of the effective bulk modulus of pore fluid upon water saturation

Abstract

We apply a rock-physics model established from fine-scale data (well or laboratory) to the seismically derived elastic variables (the impedances and bulk density) to arrive at the seismic-scale total porosity, clay content, and water saturation. These three outputs are defined as the volume-averaged porosity, clay content, and porosity-weighted water saturation, respectively. To use the rock-physics model, we need to know how to relate the bulk modulus of the pore fluid to water saturation in the presence of hydrocarbons. At the wellbore-measurement scale, this relation is typically the saturation-weighted harmonic average of the bulk moduli of the water and hydrocarbon. The question posed here is what this relation is at the seismic scale. The method of solution is based on the wellbore-scale data. Specifically, we seek the seismic-scale bulk modulus of the pore fluid that, if used in the rock-physics model, will yield the Backus-upscaled elastic constants at the well from the above-defined seismic-scale petrophysical variables. The answer depends on the vertical distribution of all these variables. By using examples of synthetic and real wells and assuming the lack of hydraulic communication between adjacent rock bodies, we find that this relation trends toward the arithmetic average of the individual bulk moduli of the pore-fluid phases. In fact, it falls in between the arithmetic average and the linear combination of 0.75 arithmetic and 0.25 harmonic averages. We also develop an approximate analytical solution under the assumption of weak elastic and porosity contrasts and for medium-to-high porosity sediment that indicates that the seismic-scale bulk modulus of the pore fluid is close to the arithmetic average of those in the individual layers.