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PART II, PAGE 2
The interaction of acoustic waves in a rock matrix affect velocity and attenuation of the acoustic signal. Rock density, porosity, saturation and fracturing have different affects that contribute to a very complex acoustic signal which contains an abundance of information.
Acoustic tools are designed to generate an acoustic signal (sound wave) having significant energy that will travel through the borehole fluid, rock formation and back through the borehole fluid to an acoustic receiver. Acoustic transmitters may be piezoelectric crystal or magnetostrictive. Both types of transmitters convert electrical energy into mechanical energy. The mechanical energy in the form of an acoustic wave travels into the surrounding formation and is received by an acoustic receiver usually of the piezoelectric type.
Acoustic waves traveling in an elastic media are subject to stress and strain. Most of the interaction of acoustic waves in a rock formation can be described mathematically using the theory of elastic waves in an elastic media. In an elastic medium with small strains, the strain is directly related to the stress that caused it. This relation is referred to as “Hooke’s Law”. The total strain is the sum of the strains produced by the individual stresses.
An acoustic waveform is subject to reflection and refraction. Because the waveform moves in all directions and encounters varying interactions, when waveforms rejoin, they may have constructive or destructive interference with each-other. Among the interactions, Plane waves referred to as “P waves” and shear waves referred to as “S waves” a formed. These waves, called “head waves” are among the first to arrive at the receiver. Various other waves are produced; conical waves, pseudo-Rayleigh waves, and Stoneley or tube waves.
It should be noted that P waves travel faster than S waves.
Example: Sandstone (WY) having density of 2.28 gm/cc ;
Young’s modulus: Poisson’s ratio: Vp: Vs: Vp/Vs: Vs as % of Vp:
0.014 0.060 7470 5130 1.46 68.42%
Matrix velocity – If a rock were truly elastic it would have no porosity. But, rock contains pore spaces, fractures and other discontinuities. The velocity of propagation in a rock matrix is therefore affected by pressure, porosity, pore fluid, fraction of the pore space containing a given fluid (saturation), fluid type and temperature. Pressure has the effect of closing micro-fractures and increasing velocity. Porosity has the effect of reducing velocity of S waves. Temperature affects fluid velocity and therefore has an effect on overall velocity.
Attenuation - Attenuation is greatly increased by increasing porosity. Attenuation decreases with increasing pressure.
Attenuation of both P and S waves are minimum in dry rock. Attenuation for both modes increases as partial saturation occurs. P waves are attenuated twice as much as S waves in partially saturated rock. As total saturation is reached, P wave attenuation is REDUCED and S wave attenuation reaches MAXIMUM.
Velocity of an acoustic wave (Vp) is measured in terms of feet per second. The reciprocal (Dt) is measured in seconds per foot. In the case of rock matrices units are micro-seconds (usec) per foot. Dt is also referred to as “transit time”.
Typical transit times:
Shale 130 – 175 usec per ft. Example: Shale (AZ) Vp = 6372 ft/sec., delta-t = 157 usec per ft.
Sandstone 52.5 – 55.5 usec per ft.
Limestone 47.5 usec per ft. Example: Limestone (PA) = Vp = 10899 ft. per sec., delta-t = 91.7
Dolomite 42.5 usec per ft.
Acoustic logging tools use either one transmitter and two receivers. Two transmitters and two receivers are used to compensate for adverse borehole conditions and is called Borehole Compensation (BHC). A process of subtracting travel times from one transmitter to one receiver then the other, has the effect of giving the resulting formation travel time only.
Using the convention T12 = time from transmitter 1 to receiver 2;
For change in hole diameter (Dd):
T12 – T11 = l/v + Dd/vf , T21 – T22 = l/v + Dd/vf
l = distance between receivers
vf = velocity of sound in fluid
Dd = change in hole diameter
(T21 – T22 + T12 – T11)/2 = l/v
If the tool is tilted:
(T21 – T22 + T12 – T11)/2 = l cos(a)/v
Where: a = tilt angle
Note: tilt angle can usually be ignored due to the length of the tool and the small diameter of hole, cos(a) = 1.
In practice, acoustic logging must be performed in a fluid filled well and the tool must be centralized in the hole.
Dry hole acoustic sondes are held stationary and pressed against the side of the hole for each measurement.
POROSITY FROM ACOUSTIC TOOL:
Porosity (f) is calculated as follows:
f = (Dt(log) - Dt(ma)) / (Dt(f) – D(ma))
Where: Dt(log) is delta-t obtained from the log.
Dt(ma) is the delta-t of matrix rock having no porosity.
Dt(f) is the delta-t of the fluid contained in the pore space.
Typical Dt(f) for water or mud is 188 to 200 microseconds (usec) per ft.
Note: vf is less in mud.
CORRECTION FOR SHALE:
Correction for shale is performed when Dt(sh) exceeds 100 usec per ft.
Multiply the calculated porosity by 100/ Dt(sh)
CORRECTION FOR GAS:
When pore space contains gas, Dt(fluid) increases. The result is optimistic porosity calculations.
In many cases acoustic information is unreliable due to cycle skipping caused by long Dt intervals.