Velocity-porosity regression

Laboratory and sonic log measurements show that velocities usually decrease with porosity, though the velocity-porosity relations are not always unique. P and S velocities are also often well correlated. Velocity-porosity and V_P-V_S regressions have been presented in the literature for different lithologies by various authors. We used similar regressions to define a reference V_P, V_S, and density. For the sand facies we choose a $V_P-\phi$ regression of the form $V_p (km/s) = 3.05 - 1.4\phi$ with a coefficient of variation around the regression of about 2.4%. The V_P-V_S regression was V_S (km/s) = 0.775 V_P - 0.67, with about 2% coefficient of variation. These are representative of typical water saturated sands from North Sea fields. The regressions (along with the spreads around the regressions) were used to generate the reference V_P and V_S data. We did not include the effect of pressure variations on the $V_P-\phi$ and V_P-V_S regressions, though the effect of pressure on the fluid properties was taken into account. The reference velocities were transformed, applying Gassmann's equations, to the reservoir conditions using the pressure, saturations, and Gas-Oil ratio obtained from the flow simulator. This was done at each time step of interest in the reservoir production history. The fluid incompressibilities and densities as a function of pressure and Gas-Oil ratio were calculated from published laboratory relations incorporated in PetroTools Version 2.3. For the shale facies, which was assigned a zero porosity and negligible permeability, we assigned fixed V_P, V_S, and density of 2.5 km/s, 1.3 km/s, and 2.27 gm/cc respectively. Anisotropy of seismic velocities was neglected in this initial model.