# Gassmann's equation

Gassmann's equation is in fluid substitution calculations and stems from poroelasticity theory. The effective bulk modulus of the porous material composed of solid mineral fragments and pore filling fluids is:

${\displaystyle \kappa _{\mathrm {eff} }=\kappa _{\mathrm {d} }+{\frac {\left(1-\kappa _{\mathrm {d} }/\kappa _{\mathrm {s} }\right)^{2}}{{\frac {1-\kappa _{\mathrm {d} }/\kappa _{\mathrm {s} }-\phi }{\kappa _{\mathrm {s} }}}+{\frac {\phi }{\kappa _{\mathrm {f} }}}}}}$

Where, ${\displaystyle \kappa _{\mathrm {eff} }}$ is effective bulk modulus; ${\displaystyle \kappa _{\mathrm {d} }}$ is dry or drained bulk modulus, also called the frame modulus; ${\displaystyle \kappa _{\mathrm {s} }}$ is bulk modulus of the constituent solid minerals; ${\displaystyle \kappa _{\mathrm {f} }}$ is effective fluid bulk modulus; ${\displaystyle \phi }$ is effective porosity.

In practice, it can be quite difficult to determine a value of ${\displaystyle \kappa _{\mathrm {d} }}$ for a saturated rock sample. The act of drying out the rock, in an effort to remove the fluid effects on the frame, has actually been shown to alter the dry mineral modulus. Rearranging the equation above for ${\displaystyle \kappa _{\mathrm {d} }}$ yields:

${\displaystyle \kappa _{\mathrm {d} }={\frac {1+\kappa _{\mathrm {eff} }((\phi -1)/\kappa _{\mathrm {s} }-\phi /\kappa _{\mathrm {f} })}{{\frac {1-\kappa _{\mathrm {eff} }/\kappa _{s}+\phi }{\kappa _{\mathrm {s} }}}-{\frac {\phi }{\kappa _{\mathrm {f} }}}}}}$

The terms on the right hand side of this equation are more easily measured quantities. Hence this lesser quoted equation may actually be the useful starting points for fluid substitution computations.