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Travel times and gradients

Figure 5.
Diagram showing source-receiver relative locations with respect to the reflector.
[pdf] [png]

Referring to Figure A-1, the direct arrival time, $ T_D$ , is given by

$\displaystyle T_D = \frac{\sqrt{x^2 + (z_s - z_r)^2}}{V}

and the reflected arrival time, $ T_R$ , is similarly

$\displaystyle T_R=\frac{\sqrt{x^2 + (z_s+z_r)^2}}{V}$    .

To calculate the relative effect of small shifts in source or receiver location on these arrival times, we compute the gradients $ \nabla T_D$ and $ \nabla T_R$ with respect to changes in source location:

$\displaystyle \displaystyle V \frac{\partial T_D}{\partial x}$ $\displaystyle =$ $\displaystyle \displaystyle \frac{x}{\sqrt{x^2 + (z_s - z_r)^2}}$  
$\displaystyle \displaystyle V \frac{\partial T_D}{\partial z_s}$ $\displaystyle =$ $\displaystyle \displaystyle \frac{z_s - z_r}{\sqrt{x^2 + (z_s - z_r)^2}}$  
    $\displaystyle \ $  
$\displaystyle \displaystyle V \frac{\partial T_R}{\partial x}$ $\displaystyle =$ $\displaystyle \displaystyle \frac{x}{\sqrt{x^2 + (z_s + z_r)^2}}$  
$\displaystyle \displaystyle V \frac{\partial T_R}{\partial z_s}$ $\displaystyle =$ $\displaystyle \displaystyle \frac{z_s + z_r}{\sqrt{x^2 + (z_s + z_r)^2}}$  

and with respect to changes in receiver location:
$\displaystyle \displaystyle V \frac{\partial T_D}{\partial z_r}$ $\displaystyle =$ $\displaystyle \displaystyle \frac{z_r - z_s}{\sqrt{x^2 + (z_s - z_r)^2}}$  
    $\displaystyle \ $  
$\displaystyle \displaystyle V \frac{\partial T_R}{\partial z_r}$ $\displaystyle =$ $\displaystyle \displaystyle \frac{z_s + z_r}{\sqrt{x^2 + (z_s + z_r)^2}}$  

With these gradients in hand, let $ {\bf f}$ be a unit vector aligned with the fracture and $ {\bf g}$ be a unit vector aligned with the receiver array. Then the directional derivatives $ \nabla T_D \cdot {\bf f}$ and $ \nabla T_R \cdot {\bf f}$ give the relative sensitivities of the direct and reflected arrival times to source displacement along the fault that gave rise to some set of multiplets. Similarly, the directional derivatives $ \nabla T_D \cdot {\bf g}$ and $ \nabla T_R \cdot {\bf g}$ provide the arrival slopes of the direct and reflected arrivals respectively.

With the above, not only can we estimate where to look for weak reflections behind a direct arrival (or, conversely, how far from the microseismic source a clear reflection arose), but we can also begin to understand how much or little reflections within multiplets misalign when their associated direct arrivals are aligned. For example, the limiting case of the receiver just above the reflector has the reflected and direct arrivals arriving at the same time and changing at the same rate as the source is displaced whereas if $ z_s = z_r = x/2$ the reflected arrival displaces $ \sqrt{2}$ times further than the direct arrival. Both of these cases are serendipitous in the sense that aligning the direct arrival across channels also aligns the reflected arrival.

Let us apply these formulas to analyze the shear reflection we spotted in our Bonner multiplet example.

In the example of Figure 1, the receivers are at about 12,800 ft depth and the microseismic source was computed to be at about 13,100 ft depth and an offset of about 300 ft from the monitor well. The dipole sonic log shows a compressional velocity of about 13,750 ft/sec and a shear velocity of about 8,000 ft per second in that depth range. The difference between the direct P and the direct S arrivals would be 22 msec, in good agreement with the actual record. The delay of about 220 msec to the later shear arrival corresponds to a reflector depth of about 13,800, i.e., a thousand feet below the receivers and, sigh, well below the reservoir depth.

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Next: Bibliography Up: Farghal and Levin: Aligning Previous: Appendix B