Variation of reflectivity with angle (AVA)

From SEG Wiki
Jump to navigation Jump to search
ADVERTISEMENT

Problem 3.12a

The values in Table 3.12a illustrate the differences that the interstitial fluid can produce. Calculate the reflectivity for shale-brine sand and shale/gas sand interfaces at incident angles of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 0^{\circ} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 10^{\circ} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 20^{\circ} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 30^{\circ} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 40^{\circ} .

Background

It is difficult to tell from the Zoeppritz equations how the variation of amplitude with angle of incidence is affected by changes in the various parameters involved. Shuey (1985) simplified the equations by assuming that the changes in physical properties at an interface are small, so that the raypath bending is small, resulting in Shuey’s equation:

Table 3.12a. Values for AVA/AVO calculations.
Medium Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{p} (m/s) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{s} (m/s) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): p (g/cmFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): ^{3} ) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{p} /V_{s}
Shale 2742 1394 2.062 1.967
Brine sand 2833 1470 2.078 1.927
Gas sand 2371 1473 2.044 1.610


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} R\left(\theta \right)=R_{0} +[PR_{0} +\Delta \sigma /\left(1-\sigma )^{2} \right]\sin ^{2} \theta +\left(\Delta V_{P} /2V_{P} \right)\left(\tan ^{2} \theta -\sin ^{2} \theta \right), \end{align} (3.12a)

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R_{0} =\left(Z_{2} -Z_{-1} \right)/\left(Z_{2} +Z_{1} \right)\approx \Delta \left(\rho V\right)/2\rho V=\left(1/2\right)\left(\Delta V/V+\Delta \rho/\rho\right),


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \quad P=Q-2\left(1+Q\right)\left(1-2\sigma \right)/\left(1-\sigma \right)+\Delta \sigma /R_{0} (1-\sigma )^{2} \end{align} (3.12b)


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} Q=\left(\Delta V_{P} /V_{P} \right)/\left[\left(\Delta V_{P} /V_{P} \right)+\left(\Delta \rho/\rho\right)\right] \end{align} (3.12c)

and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \sigma is Poisson’s ratio.

Hilterman (1989) introduced additional approximations resulting in


$ {\begin{aligned}R=R_{0}\cos ^{2}\theta +2.25\Delta \sigma \sin ^{2}\theta =R_{0}\cos ^{2}\theta +2.25\Delta \sigma (1-\cos ^{2}\theta )\\=R_{0}\left(1-2.25\Delta \sigma \right)\cos ^{2}\theta +2.25\Delta \sigma .\end{aligned}} $ (3.12d)

Solution

Note that 4 significant figures are required to illustrate the effect. We first calculate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \sigma for the three beds using equation (10,2) in Table 2.2a:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \sigma_{\rm shale}=\left(1.967^{2} -2\right)/2\left(1.967^{2} -1\right)=1.869/5.738=0.326,\\ \sigma_{\rm brine} =\left(1.927^{2} -2\right)/2\left(1.927^{2} -1\right)=1.713/5.427=0.316,\\ \sigma_{\rm gas} =\left(1.610^{2} -2\right)/2\left(1.610^{2} -1\right)=0.592/3.184=0.186. \end{align}

We take the following average values and increments : Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{P} =2788 , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \sigma =0.321 , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \rho=2.070 , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta V_{P} =91, \Delta \sigma =-0.010, \Delta \rho=0.016.

Using these values for the Shuey equation for the shale/brine-sand interface, equations (3.12a,b,c) give

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} R_{0} =\left(1/2\right)\left(91/2788+0.016/2.070\right)=0.0202, \\ Q=\left(91/2788\right)/\left[\left(91/2788\right)+\left(0.016/2.07\right)\right] \\ =0.0326/\left(0.0326+0.0077\right)=0.0326/0.0403=0.809, \\ P=0.809-2\left(1.809\right)\left(0.358/0.679\right)-0.010/0.0202\times 0.679^{2} \\ =0.809-1.908-1.074=-2.173, \\ R=0.0202+\left(-2.173\times 0.0202-0.010/0.679^{2} \right)\sin ^{2} \theta \\ +\left(91/2\times 2788\right)\left(\tan ^{2} \theta -\sin ^{2} \theta \right) \\ =0.0202-\left(0.0439+0.0217\right)\sin ^{2} \theta +0.0163\left(\tan ^{2} \theta -\sin ^{2} \theta \right) \\ =0.0202-0.0656\sin ^{2} \theta +0.0163\left(\tan ^{2} \theta -\sin ^{2} \theta \right) \\ =0.0202-0.0819\sin ^{2} \theta+0.0163 \tan ^{2} \theta, \\ R_{10} =0.0202-0.0819\times 0.1736^{2} +0.0163\times 0.1763^{2} =0.0182, \\ R_{20} =0.0202-0.0819\times 0.3420^{2} +0.0163\times 0.3640^{2} =0.0128, \\ R_{30} =0.0202-0.0819\times 0.5000^{2} +0.0163\times 0.5774^{2} =0.0052, \\ R_{40} =0.0202-0.0819\times 0.6428^{2} +0.0163\times 0.8391^{2} =0.0022. \end{align}

At the shale-gas sand interface, averages and increments are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{P} =2556 {\rm m/s} , $ \sigma =0.256 $, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \rho=2.053 , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta V_{P} =-371{\rm m/s} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta \sigma =-0.140 , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta \rho=-0.018.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \text {Then}, R_{0} =-\frac{1}{2} \left(\frac{371}{2556} +\frac{0.016}{2.053} \right)=-0.0765,\\ Q=\left(-371/2556\right)/\left[\left(-371/2556\right)+\left(-0.018/2.053\right)\right] \\ =0.1451/\left(0.1451+0.0088\right)=0.943, \\ P=0.943-2\left(1.943\times 0.488/0.744\right)+(-0.140/\left(-0.0765\times 0.744^{2} \right) \\ =0.943-2.549+3.306=1.700. \\ \text {Thus}, R=-0.0765+\left(-1.700\times 0.0765-0.140/0.744^{2} \right)\sin ^{2} \theta \\ +\left(-371/2\times 2556\right)\left(\tan ^{2} \theta -\sin ^{2} \theta \right) \\ =-0.0765-0.3830\sin ^{2} \theta -0.0726\left(\tan ^{2} \theta -\sin ^{2} \theta \right) \\ =-0.0765-0.3104\sin ^{2} \theta -0.0726\tan ^{2} \theta. \\ \text {So}, R_{10} =-0.0765-0.0094-0.0023=-0.0882, \\ R_{20} =-0.0765-0.0363-0.0096=-0.1221, \\ R_{30} =-0.0765-0.0776-0.0242=-0.1783, \\ R_{40} =-0.0765-0.1282-0.0511=-0.2558. \end{align}

Substituting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R_{0} =0.0202 , $ \Delta \sigma =-0.010 $ in equation (3.12d), we get for the Hilterman equation (3.12d) for the shale/brine-interface,


Table 3.12b. Comparison of predictions by Shuey and Hilterman equations.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 0^{\circ} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 10^{\circ} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 20^{\circ} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 30^{\circ} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 40^{\circ}
For shale/brine sand
Shuey equation 0.0202 0.0182 0.0128 0.0052 –0.0022
Hilterman equation 0.0202 0.0189 0.0152 0.0095 0.0026
For the shale/gas sand
Shuey equation –0.0765 –0.0881 –0.1221 –0.1782 –0.2558
Hilterman equation –0.0765 –0.0837 –0.1044 –0.1361 –0.1750
Figure 3.12a.  Reflectivity versus angle; solid line, Shuey equation; dashed, Hilterman equation.


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} R=R_{0} \left(1-2.25\Delta \sigma \right)\cos ^{2} \theta +2.25\Delta \sigma \\ =0.0202\left(1-0225\right)\cos ^{2} \theta +0.0225=0.0427\cos ^{2} \theta -0.0225; \\ R_{0} =0.0202, \\ R_{10} =0.0427\left(0.985^{2} \right)-0.0225=0.0189, \\ R_{20} =0.0427\left(0.940^{2} \right)-0.0225=0.0152, \\ R_{30} =0.0427\left(0.866^{2} \right)-0.0225=0.0095, \\ R_{40} =0.0427\left(0.766^{2} \right)-0.0225=0.0026. \end{align}

The Hilterman equation (3.12d) for the shale/gas-sand interface is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} R=-0.2385\cos ^{2} \theta -0.315, \\ R_{0} =-0.0765, \\ R_{10} =-0.2385\left(0.985^{2} \right)-0.315=0.084, \\ R_{20} =-0.2385\left(0.940^{2} \right)-0.315=0.105, \\ R_{30} =-0.2385\left(0.866^{2} \right)-0.315=0.137, \\ R_{40} =-0.2385\left(0.766^{2} \right)-0.315=0.175. \end{align}

Table 3.12b compares the values given by the Shuey and Hilterman equations and the results are graphed in Figure 3.12a.

The two equations give essentially the same results for angles up to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 20^{\circ} . The increase of amplitude with angle (offset) is larger with the Shuey equation. An additional term that becomes important at large angles is sometimes added to these equations.

Continue reading

Previous section Next section
AVO versus AVA and effect of velocity gradient Accuracy of normal-moveout calculations
Previous chapter Next chapter
Theory of Seismic Waves Geometry of seismic waves

Table of Contents (book)

Also in this chapter

External links

find literature about
Variation of reflectivity with angle (AVA)
Retrieved from "http://wiki.seg.org/w/index.php?title=Variation_of_reflectivity_with_angle_(AVA)&oldid=120583"