Set $x={\log }_{10}{a}_{{\mathrm{H}}_4\mathrm{Si}{\mathrm{O}}_4^0}$ and $y={\log }_{10}(a_{\mathrm{N}{\mathrm{a}}^{+}}/a_{{\mathrm{H}}^{+}})$, treating the supplied concentration notation as an ideal-dilute activity approximation. For pure solids and unit water activity, the first chemical equilibrium relation gives $\log K_a=2y+2x$ and the second gives $\log K_b=2y+6x$. Thus plotTheir slopes are $-1$ and $-3$ respectively. They intersect at $(x_{\ast },y_{\ast })=(-5.515,8.955)$. Both the relevant silica activities and the intersection lie at negative $x$, as expected for a dilute aqueous silica species.

With water fugacity coefficient ${\varphi }_f=0.8$, use $f={\varphi }_{f}P=4800$ bar. Retain the same ideal mineral-site activities, givingHenceThe lower fluid chemical fugacity stabilizes the water-bearing products of the forward dehydration relative to the ideal-gas calculation and lowers the required temperature by about $13.19$ K. Here the stated fugacity coefficient is treated as a given constant for evaluating the equilibrium.

An activity model is required in addition to the two compositions. Use ideal multisite mineral activity: independent ideal Fe–Mg mixing on the three relevant sites per formula unit givesThe garnet/biotite activity ratio is $0.008/0.512=1/64$. All other solids are pure, so the dimensionless reaction quotient isWith the same pressure-corrected numerator as in part b,Diluting the magnesian product endmember more strongly than the reactant favors the forward dehydration and lowers its equilibrium temperature. The powers of three count Fe–Mg mixing sites, not three formula units of either mineral. Nonideal site interactions or site partitioning would change these activities. If ideal mixing of whole formula units rather than sites were intended, activities would instead be $0.8$ and $0.2$; composition alone does not select that different model. The calculation explicitly adopts the conventional ideal-site interpretation.

Integrate the solids’ molar volume contribution from $P^{\circ }=1$ bar and use ideal-gas water chemical fugacity $f=P$. For pure solids,The logarithm is dimensionless, and $\Delta V_s=-2.28\mathrm{J}\mathrm{ba}{\mathrm{r}}^{-1}$ can be multiplied directly by pressure in bars. Therefore the dehydration equilibrium with ideal-gas fluid givesHence $T=1295.57$ K, or $1022.42^{\circ }$C. Do not also add a constant gas-volume pressure term: the water contribution has already been integrated through $RT\ln (P/P^{\circ })$. Ideal-gas behavior at $6$ kbar is a prescribed model approximation, not a realistic equation of state. The solid volume correction is distinct from it.

For pure phases at the $1$ bar reference pressure, the equilibrium condition is ${\Delta }_r{G}^{\circ }={\Delta }_r{H}^{\circ }-T{\Delta }_r{S}^{\circ }=0$. ThusThe calculation uses the stipulated constant reaction enthalpy and entropy, with the fluid at its reference chemical fugacity.

Choose using target depth, scale, expected contrasts, overburden, noise and required result. A seismic reflection survey uses reflected travel times and amplitudes to image interfaces; it needs useful acoustic impedance contrasts, sufficient bandwidth, suitable offsets and dense coverage for moveout correction, stacking and seismic migration. A large clay-cap/igneous contrast gives a potentially strong reflection here, but fractures, scattering and attenuation can obscure coherent events. At $100$ Hz the nominal quarter-wavelength vertical seismic resolution is about $3.75$ m in the claystone and $8.75$ m in the igneous rock, if that frequency reaches the target.

A seismic refraction survey uses first arrivals and critically refracted waves to estimate velocities and interface structure. The increase from $1500$ to $3500\mathrm{m}{\mathrm{s}}^{-1}$ is favorable for a cap-base head-wave branch, provided the spread reaches sufficient offsets. Refraction/tomography may better constrain lateral velocity variations and fractures, but simple head-wave inversion can miss a low-velocity layer or a thin hidden layer and often has poorer boundary detail.

Use expected reflector continuity, velocity inversions, required aperture, access, near-surface complexity and cost to choose or combine the surveys. A pilot gather and travel-time modelling can test which phases will be measurable; the supplied velocities alone do not specify the reservoir depth or the best acquisition geometry.

For an approximately isotropic elastic solid,where $K$ is bulk modulus, $\mu$ is shear modulus and $\rho$ is mass density. Five relevant controls are:

These controls interact: a dense rock is not automatically faster if its moduli are also different.