Seismic experiments complement well measurements. Seismic surveys record the continuous wavefield generated by impulsive sources and reflected or transmitted by geological subsurface layers. Traditionally, the recorded data is processed to reveal the structure of the three-dimensional subsurface geology. In the past decade, geophysicists developed additional seismic processing methods to estimate concrete subsurface rock properties, such as the ratio of compressional and shear wave velocity or bulk modulo or even density (). Recently, geophysicists employ repeated seismic surveys over an actively exploited field to estimate changes in the reservoir's fluid contents. Assuming that the reservoir rock is unchanged, such time-lapse or 4-D experiments relate subtle wavefield differences to reservoir fluid changes. Unfortunately, seismic surveys themselves are complex field experiments that are difficult to repeat exactly. Variations in recording geometry, instrumentation, and wave source may lead to wavefield variations that are falsely interpreted as changes in reservoir fluid contents.

**Wave equation.**
For this study,
I chose Gassmann's formulation of the wave equation for porous rock.
Gassmann's formulation resembles the formulation for homogeneous
rocks and is widely used in rock physics studies. Furthermore,
Gassmann's formulation shares a parameterization similiar
to the parameterization for reservoir fluid flow simulations.

The generic wave equation for an isotropic, linear, elastic medium is

(9) |

The solutions to the isotropic wave equation can be separated into a compressional mode () and a shear mode (). Substitution of these two modes into the isotropic wave equation 9 yields the same simplified wave equation

(10) |

(11) |

(12) |

The density of the porous rock

(13) |

(14) |

According to Gassmann,
the effective bulk modulo of the saturated porous rock *K*_{s}
is a weighted average of
the bulk modulo of the dry rock skeleton *K*_{d},
the bulk modulo of the constituent mineral material *K*_{m}, and
the bulk modulo of the saturating fluid *K*_{f}

(15) |

(16) |

Finally, the shear moduli of the saturated rock is unchanged by the fluid pore contents

(17) |

The expressions satisfy the limiting cases.
A solid medium implies and *K*_{f} = *K*_{s} = *K*_{m}.
A fluid medium implies and *K*_{m} = 0 and *K*_{s} = *K*_{f}.
Gassmann's expressions approximate a porous rock only at low frequencies
and is strictly valid only in the static (zero-frequency) case.
The given averages only approximate the properties of a saturated
rock, if the phases in the computational cell are mixed uniformly
at very small scale.

**Problems and hope for formulation**
The formulation of a porous rock model does not guarantee its
applicability in practice.
Traditional AVO studies have shown that seismic reflection experiments
are rather insensitive to density changes.
Furthermore,
the drastic amplitude effect of gas saturation is insensitive to
the amount of gas: large and small gas saturations cause almost the same
bright spots in seismic sections ().
In general, traditional seismic surface experiments resolved little
beyond reflectivity and compressional wave velocity.
However, the innovation of time-lapse experiments
gives seismic experiments a particular sensitivity to saturation changes.
However, the optimal formulation of the simulation and inversion problem is
still unclear.

3/9/1999