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References: replace mr template with mr parameter in CS1 templates; using AWB

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	~~The Gassmann equation~~ is ~~used in geophysics~~ and ~~its relations are receiving more attention~~ as ~~seismic data are increasingly used for~~		Like all things important, a involving care and precise detail should be investigated when it to be able to selecting your dental plan. Why, you ask? Because your personal dental plan is often a necessity and can be just as important as associated with life care policies, so take care to remember the important features when making your final decision. The three things to look out for are as below in no particular order.<br><br>An affordable, effective methods technique would brush with baking soft drinks and. Brushing your teeth with baking soda are often a great strategy naturally whiten your dental. Brushing your teeth using baking soda lead to irritation pertaining to your gums, so make positive that you make use of a gentle finger.<br><br>Many manufactured stone and natural stone veneers can be installed by homeowners which willing study and follow instructions. Some manufacturers post installation guides on their websites and recommend an individual review them carefully before commencing. "Follow need to be followed method of installation and also cheat," says Terence Meck at Rolling Rock Building Stone, a quarrier in Pikeville, Pa. "It's critical to try to to it right and to do it right the occasion." Stone veneer are minimal maintenance and simply need to be sealed if exposed to extreme temperatures or a superb deal of snow.<br><br>If ahead of time don't such shape of one's teeth or if you've stood a root canal or a filling that keeps a tooth from bleaching right color, aim veneers. Veneers look great, and almost literally change a person's entire good looks. Sometimes the dentist actually recommends veneers over braces to be able to misaligned smiles. Veneers are usually made via porcelain, and they are incredibly tiny. They are actually bonded straight away to the front of your tooth. porcelain veneers can fix discolored teeth, misshaped teeth, and chipped your. They look incredibly natural that are very durable and everlasting. Another benefit of veneers is that possibly they are stain challenging.<br><br>(4) "dentist": Can you beat a bad guy to be a sociopathic dentist? Steve Martin chewed the scenery that role, plus it is one of one of the most memorable numbers in the show. It's disturbing and hilarious and the actor free reign to ham it up all approach through. The doo-wop girls make this a complete comedic package.<br><br>Porcelain Veneers: The teeth are engrossed in veneers, which are bound for the teeth at the front end. A mold of your teeth is used to produce the veneers. In order to first shaved off to suit the veneers comfortably. They not only become straighter but also appear white.<br><br>Matt didn't get a rose and called his mother relating to this. This examiner don't know that's more embarrassing for Matt: Getting dumped in favor of anyone wearing a mask in addition to jerk like Bentley or calling your mother about getting dumped on national TV. Steven and Ryan M. also didn't obtain a rose If you have any concerns pertaining to in which and how to use [https://www.youtube.com/watch?v=O4bpBwPHoWY Veneers tampa], you can make contact with us at the web page. .
	~~reservoir monitoring~~. The ~~Gassmann equation is the most common way of performing a fluid substitution model from one known parameter.~~

	~~==Procedure==~~
	~~These formulations~~ are ~~from Avseth ''et al.'' (2006)~~.<~~ref~~>~~Avseth~~, P, ~~T Mukerji & G Mavko (2006), Quantitative seismic interpretation, Cambridge University Press, 2006~~.<~~/ref~~>

	~~Given~~ an ~~initial set~~ of ~~velocities~~ and ~~densities~~, ~~<math>V_{P}^{(1)}</math>~~, ~~<math>V_{S}^{(1)}</math>~~, and ~~<math>\rho ^{(1)}</math> corresponding~~ to ~~a rock with an initial set of fluids, you can compute the velocities and densities of~~ the ~~rock with another set of fluid~~. ~~Often these velocities~~ are ~~measured from well logs, but might also come from~~ a ~~theoretical model~~.

	'''~~Step 1:''' Extract the dynamic bulk and shear moduli~~ from ~~<math>V_\mathrm{P}^{(1)}</math>~~, ~~<math>V_\mathrm{S}^{(1)}</math>~~, and ~~<math>\rho ^{(1)}</math>:~~

	~~:<math>K_\mathrm{sat}^{(1)} = \rho \left ((V_\mathrm{P}^{(1)})^{2}-\frac{4}{3}(V_\mathrm{S}^{(1)})^{2} \right) </math>~~

	~~:<math>\mu_\mathrm{sat}^{(1)} = \rho (V_\mathrm{S}^{(1)})^{2}</math>~~

	~~'''Step 2:''' Apply Gassmann~~'s ~~relation, of~~ the ~~following form~~, to ~~transform~~ the ~~saturated bulk modulus:~~

	:<math>\frac{K_\mathrm{sat}^{(2)}}{K_\mathrm{mineral}-K_\mathrm{sat}^{(2)}}-\frac{K_\mathrm{fluid}^{(2)}}{\phi (K_\mathrm{mineral}-K_\mathrm{fluid}^{(2)})}=\frac{K_\mathrm{sat}^{(1)}}{K_\mathrm{mineral}-K_\mathrm{sat}^{(1)}}-\frac{K_\mathrm{fluid}^{(1)}}{\phi (K_\mathrm{mineral}-K_\mathrm{fluid}^{(1)})}</math>

	~~where <math>K_\mathrm{sat}^{(1)}</math>~~ and ~~<math>K_\mathrm{sat}^{(2)}</math>~~ are ~~the rock bulk moduli saturated with fluid 1~~ and ~~fluid 2, and <math>K_\mathrm{fluid}^{(1)}~~<~~/math~~> ~~and~~ <~~math~~>~~K_\mathrm{fluid}^{~~(2)~~}</math> are the bulk moduli of the fluids themselves.~~

	~~'''Step 3~~:~~''' Leave~~ the ~~shear modulus unchanged (rigidity~~ is ~~independent~~ of ~~fluid type):~~

	~~:<math>\mu_\mathrm{sat}^{(2)}=\mu_\mathrm{sat}^{(1)}</math>~~

	~~'''Step 4:''' Correct~~ the ~~bulk density for the change~~ in ~~fluid:~~

	~~:<math> \rho^{(2)}= \rho^{(1)}+\phi (\rho_\mathrm{fluid}^{(2)} -\rho_\mathrm{fluid}^{(1)})</math>~~

	~~'''Step 5:''' recompute~~ the ~~fluid substituted velocities~~

	~~:<math>V_\mathrm{P}^{(2)}=\sqrt \frac{K_\mathrm{sat}^{(2)}+\frac{4}{3} \mu_\mathrm{sat}^{(2)}}{\rho^{(2)}} </math>~~

	~~:<math>V_\mathrm{S}^{(2)} = \sqrt \frac{\mu_\mathrm{sat}^{(2)}}{\rho^{(2)}}</math>~~

	~~==Rearranging for~~ '~~'K''<sub>sat</sub>==~~

	~~Given~~

	:<math>\frac{K_\mathrm{sat}^{(2)}}{K_\mathrm{mineral}-K_\mathrm{sat}^{(2)}}-\frac{K_\mathrm{fluid}^{(2)}}{\phi (K_\mathrm{mineral}-K_\mathrm{fluid}^{(2)})}=\frac{K_\mathrm{sat}^{(1)}}{K_\mathrm{mineral}-K_\mathrm{sat}^{(1)}}-\frac{K_\mathrm{fluid}^{(1)}}{\phi (K_\mathrm{mineral}-K_\mathrm{fluid}^{(1)})}</math>

	~~Let~~

	~~:<math> S = \frac{K_\mathrm{sat}^{(1)}}{K_\mathrm{mineral}-K_\mathrm{sat}^{(1)}} </math>~~

	and

	~~:<math> F_1 = \frac{K_\mathrm{fluid}^{(1)}}{\phi (K_\mathrm{mineral}~~-~~K_\mathrm{fluid}^{(1)})}\ \ \ \ F_2 = \frac{K_\mathrm{fluid}^{(2)}}{\phi (K_\mathrm{mineral}-K_\mathrm{fluid}^{(2)})}~~ <~~/math~~>

	~~then~~

	~~:<math> K_\mathrm{sat}^{(2)} = \frac{K_\mathrm{mineral}}{\frac{1}{S-F_1+F_2} + 1}~~ <~~/math~~>

	~~Or, expanded~~

	:<math> K_\mathrm{sat}^{(2)} = \frac{K_\mathrm{mineral}}{ \left[ {\frac{K_\mathrm{sat}^{(1)}}{K_\mathrm{mineral}-K_\mathrm{sat}^{(1)}}-\frac{K_\mathrm{fluid}^{(1)}}{\phi (K_\mathrm{mineral}-K_\mathrm{fluid}^{(1)})}+\frac{K_\mathrm{fluid}^{(2)}}{\phi (K_\mathrm{mineral}-K_\mathrm{fluid}^{(2)})}} \right]^{-1} + 1} </math>

	~~==Assumptions==~~

	~~===Dry bulk modulus does not change with different saturating fluids===~~

	~~===Saturated shear modulus does not change with different saturating fluid===~~
	~~Gassmann's equations assume that the shear modulus of the saturated rock is the same as the shear modulus of the dry rock, <math>\mu_\mathrm{sat}= \mu_\mathrm{dry}</math>~~, for ~~all saturating fluids. This assumes that~~ the ~~chemical interactions between~~ the ~~fluids and the solids are always the same (or have the same consequences for rigidity)~~. ~~For example, certain clay minerals may swell only when exposed to water causing the rigidity~~ of ~~the rock~~ to ~~increase or decrease depending on~~ the ~~geometry of the clay grains~~.

	~~===Porosity does not change with different saturating fluids===~~
	~~Gassmann fluid substitution requires that~~ the ~~porosity remain constant~~. ~~The assumption being that, all other things being equal, different saturating fluids should~~ not ~~affect the porosity of the rock~~. This does not take into account [[diagenesis\|diagenetic]] processes, such as cementation or dissolution, that vary with changing geochemical conditions in the pores. For example, quartz cement is more likely to precipitate in water-filled pores than it is in hydrocarbon-filled ones (Worden and ~~Morad, 2000). So the same rock may have different porosity in different locations due~~ to ~~the local water saturation~~.

	~~===Frequency effects are negligible in the measurements===~~
	~~Gassmann~~'~~s equations are essentially the lower frequency limit of [[Biot theory\|Biot~~'s]] more ~~general equations~~ of ~~motion for poroelastic materials. At [[seismic]] frequencies (10–100 Hz), the error~~ in ~~using Gassmann's equation may be negligible~~. ~~However, when constraining the necessary parameters with [[acoustics\|sonic]] measurements at logging frequencies (~20 kHz), this assumption may be violated~~. ~~A better option, yet more computationally intense, would be~~ to use Biot's frequency-dependent equation to calculate the fluid substitution effects. If the output from this process will be integrated with seismic data, the obtained elastic parameters must also be corrected for [[Acoustic dispersion\|dispersion]] effects.

	~~==References==~~

	~~<references />~~[~~http://geoconvention.org/archives/2013abstracts/316_GC2013_A_look_into_Gassmanns_Equation.pdf Al-Khateb, N.,"A Look Into Gassmann's Equation", Geoconvention Abstracts, 2013]~~
	~~[http~~://www.~~softpedia~~.com/~~get/Science-CAD/Gassmann-s-Equation~~.~~shtml Gassmann's Equation Calculator 0~~.~~1 Beta]~~
	~~[[Category:Geophysics]]~~

en>R.e.b.: Expanding article

2012-04-23T18:02:41Z

Expanding article

New page

The Gassmann equation is used in geophysics and its relations are receiving more attention as seismic data are increasingly used for
reservoir monitoring. The Gassmann equation is the most common way of performing a fluid substitution model from one known parameter.

==Procedure==
These formulations are from Avseth ''et al.'' (2006).<ref>Avseth, P, T Mukerji & G Mavko (2006), Quantitative seismic interpretation, Cambridge University Press, 2006.</ref>

Given an initial set of velocities and densities, <math>V_{P}^{(1)}</math>, <math>V_{S}^{(1)}</math>, and <math>\rho ^{(1)}</math> corresponding to a rock with an initial set of fluids, you can compute the velocities and densities of the rock with another set of fluid. Often these velocities are measured from well logs, but might also come from a theoretical model.

'''Step 1:''' Extract the dynamic bulk and shear moduli from <math>V_\mathrm{P}^{(1)}</math>, <math>V_\mathrm{S}^{(1)}</math>, and <math>\rho ^{(1)}</math>:

:<math>K_\mathrm{sat}^{(1)} = \rho \left ((V_\mathrm{P}^{(1)})^{2}-\frac{4}{3}(V_\mathrm{S}^{(1)})^{2} \right) </math>

:<math>\mu_\mathrm{sat}^{(1)} = \rho (V_\mathrm{S}^{(1)})^{2}</math>

'''Step 2:''' Apply Gassmann's relation, of the following form, to transform the saturated bulk modulus:

:<math>\frac{K_\mathrm{sat}^{(2)}}{K_\mathrm{mineral}-K_\mathrm{sat}^{(2)}}-\frac{K_\mathrm{fluid}^{(2)}}{\phi (K_\mathrm{mineral}-K_\mathrm{fluid}^{(2)})}=\frac{K_\mathrm{sat}^{(1)}}{K_\mathrm{mineral}-K_\mathrm{sat}^{(1)}}-\frac{K_\mathrm{fluid}^{(1)}}{\phi (K_\mathrm{mineral}-K_\mathrm{fluid}^{(1)})}</math>

where <math>K_\mathrm{sat}^{(1)}</math> and <math>K_\mathrm{sat}^{(2)}</math> are the rock bulk moduli saturated with fluid 1 and fluid 2, and <math>K_\mathrm{fluid}^{(1)}</math> and <math>K_\mathrm{fluid}^{(2)}</math> are the bulk moduli of the fluids themselves.

'''Step 3:''' Leave the shear modulus unchanged (rigidity is independent of fluid type):

:<math>\mu_\mathrm{sat}^{(2)}=\mu_\mathrm{sat}^{(1)}</math>

'''Step 4:''' Correct the bulk density for the change in fluid:

:<math> \rho^{(2)}= \rho^{(1)}+\phi (\rho_\mathrm{fluid}^{(2)} -\rho_\mathrm{fluid}^{(1)})</math>

'''Step 5:''' recompute the fluid substituted velocities

:<math>V_\mathrm{P}^{(2)}=\sqrt \frac{K_\mathrm{sat}^{(2)}+\frac{4}{3} \mu_\mathrm{sat}^{(2)}}{\rho^{(2)}} </math>

:<math>V_\mathrm{S}^{(2)} = \sqrt \frac{\mu_\mathrm{sat}^{(2)}}{\rho^{(2)}}</math>

==Rearranging for ''K''<sub>sat</sub>==

Given

:<math>\frac{K_\mathrm{sat}^{(2)}}{K_\mathrm{mineral}-K_\mathrm{sat}^{(2)}}-\frac{K_\mathrm{fluid}^{(2)}}{\phi (K_\mathrm{mineral}-K_\mathrm{fluid}^{(2)})}=\frac{K_\mathrm{sat}^{(1)}}{K_\mathrm{mineral}-K_\mathrm{sat}^{(1)}}-\frac{K_\mathrm{fluid}^{(1)}}{\phi (K_\mathrm{mineral}-K_\mathrm{fluid}^{(1)})}</math>

Let

:<math> S = \frac{K_\mathrm{sat}^{(1)}}{K_\mathrm{mineral}-K_\mathrm{sat}^{(1)}} </math>

and

:<math> F_1 = \frac{K_\mathrm{fluid}^{(1)}}{\phi (K_\mathrm{mineral}-K_\mathrm{fluid}^{(1)})}\ \ \ \ F_2 = \frac{K_\mathrm{fluid}^{(2)}}{\phi (K_\mathrm{mineral}-K_\mathrm{fluid}^{(2)})} </math>

then

:<math> K_\mathrm{sat}^{(2)} = \frac{K_\mathrm{mineral}}{\frac{1}{S-F_1+F_2} + 1} </math>

Or, expanded

:<math> K_\mathrm{sat}^{(2)} = \frac{K_\mathrm{mineral}}{ \left[ {\frac{K_\mathrm{sat}^{(1)}}{K_\mathrm{mineral}-K_\mathrm{sat}^{(1)}}-\frac{K_\mathrm{fluid}^{(1)}}{\phi (K_\mathrm{mineral}-K_\mathrm{fluid}^{(1)})}+\frac{K_\mathrm{fluid}^{(2)}}{\phi (K_\mathrm{mineral}-K_\mathrm{fluid}^{(2)})}} \right]^{-1} + 1} </math>

==Assumptions==

===Dry bulk modulus does not change with different saturating fluids===

===Saturated shear modulus does not change with different saturating fluid===
Gassmann's equations assume that the shear modulus of the saturated rock is the same as the shear modulus of the dry rock, <math>\mu_\mathrm{sat}= \mu_\mathrm{dry}</math>, for all saturating fluids. This assumes that the chemical interactions between the fluids and the solids are always the same (or have the same consequences for rigidity). For example, certain clay minerals may swell only when exposed to water causing the rigidity of the rock to increase or decrease depending on the geometry of the clay grains.

===Porosity does not change with different saturating fluids===
Gassmann fluid substitution requires that the porosity remain constant. The assumption being that, all other things being equal, different saturating fluids should not affect the porosity of the rock. This does not take into account [[diagenesis|diagenetic]] processes, such as cementation or dissolution, that vary with changing geochemical conditions in the pores. For example, quartz cement is more likely to precipitate in water-filled pores than it is in hydrocarbon-filled ones (Worden and Morad, 2000). So the same rock may have different porosity in different locations due to the local water saturation.

===Frequency effects are negligible in the measurements===
Gassmann's equations are essentially the lower frequency limit of [[Biot theory|Biot's]] more general equations of motion for poroelastic materials. At [[seismic]] frequencies (10–100 Hz), the error in using Gassmann's equation may be negligible. However, when constraining the necessary parameters with [[acoustics|sonic]] measurements at logging frequencies (~20 kHz), this assumption may be violated. A better option, yet more computationally intense, would be to use Biot's frequency-dependent equation to calculate the fluid substitution effects. If the output from this process will be integrated with seismic data, the obtained elastic parameters must also be corrected for [[Acoustic dispersion|dispersion]] effects.

==References==

<references />[http://geoconvention.org/archives/2013abstracts/316_GC2013_A_look_into_Gassmanns_Equation.pdf Al-Khateb, N.,"A Look Into Gassmann's Equation", Geoconvention Abstracts, 2013]
[http://www.softpedia.com/get/Science-CAD/Gassmann-s-Equation.shtml Gassmann's Equation Calculator 0.1 Beta]
[[Category:Geophysics]]