Fluid Substitution

One of the most common problems in rock physics is the prediction of seismic velocities in rocks saturated with one fluid from the velocities in rocks saturated with a second fluid or from dry rock velocities. The low-frequency Gassmann's equations [Gassmann 1951] relate the bulk and shear moduli of a saturated porous medium to the moduli of the same medium in a drained (dry) state. The effective bulk modulus K_sat of the saturated rock is given by

where K₀, K_dry, and K_f are the bulk moduli of the mineral material, the dry rock and the pore fluid, respectively. Gassmann's equations show that the shear modulus is mechanically independent of the properties of any fluid present in the pore space: µ _dry = µ _sat .

Gassmann's equations assume that the porous medium contains only one type of solid constituent with a homogeneous mineral modulus and that the pore space is statistically isotropic. The equation is valid for quasi-static conditions or at frequencies which are sufficiently low such that the induced pore pressures are in equilibrium throughout the pore space. These conditions coincide exactly with the conditions simulated with the finite-element approach. Fontainebleau sandstone is both clean and homogeneous and in our simulations we impose a uniform modulus in the solid phase. We would therefore expect the numerical data for Fontainebleau to obey Gassmann's equations for different pore fluids. A comparison of the numerically predicted moduli of the Fontainebleau images for dry, water- and oil-saturated conditions to Gassmann's equations provides a good test of the accuracy of the numerical results. The results of such a comparison are summarized in Figure 5. The numerical prediction for both the bulk and shear modulus are in excellent agreement with Gassmann's equations.

Figure 5: Comparison of the results of the simulations for dry, water-saturated and oil-saturated to Gassmann prediction based on the dry rock data. In [a] we give the numerical predictions for the dry rock data (squares) and show a best fit to the data points (solid line). We use this fit and Eqn. 2 to predict the values of the water- and oil-saturated curves (dotted curves). We show the numerical predictions for the water- and oil-saturated results. The fit to the Gassmann's equations is excellent, further indicating the ability of this methodology to quantitatively predict geophysical properties. In [b] we show that the shear modulus is independent of the pore fluid as predicted by Gassmann [Gassmann 1951].