Low frequency sonic logging

Methods and apparatus are described utilizing low frequency tube waves in determining characteristics of formations traversed by a borehole. Data received from three spaced receiving transducers are utilized in formulating quantities representative of complex compressibility of formations as well as loss parameters attributed to formation characteristics. When these two factors are recorded as a function of depth, interpretation is made readily available as to the permeability and rigidity of subsurface earth formations. Data is also provided by way of the present invention to enable interpretation concerning the presence of fractures or cracks in the earth formations. The methods and apparatus may also be utilized to examine the condition of casing in a borehole.

DESCRIPTION 
1. Technical Field 
This invention relates to methods and apparatus for low frequency sonic 
logging utilizing tube waves to produce indications of conditions 
surrounding a borehole. 
2. Background of Invention 
There has long been an interest in and efforts made toward establishing 
measured values for formation rigidity, of value in shear-wave reflection 
prospecting, as well as for permeability and other lithologic parameters. 
Interest has also been expressed in determining the location of cracks or 
fractures in earth formations traversed by a borehole. These efforts and 
interests have given rise to techniques such as that performed by sonic 
devices of the type described in U.S. Pat. No. 3,434,563. These techniques 
involve the use of high frequency acoustic energy for locating anomalies, 
such as fractures, in formations traversed by a borehole. 
Sonic logging information contained in compressional and shear acoustic 
amplitudes has also been described as useful in determining the presence 
of fractures. In a paper entitled "The Use of Compressional and Shear 
Acoustic Amplitudes for the Location of Fractures" by R. L. Morris et al, 
Paper No. SPE-723, presented at the Thirty-Eighth Annual Fall Meeting of 
the Society of Petroleum Engineers of the American Institute of Mining, 
Metallurgical and Petroleum Engineers in New Orleans, La., Oct. 6-9, 1963, 
the authors describe such a technique. However, experience of the authors 
has shown that the involved amplitude measurement is not universally 
applicable. The interpretation of amplitude logs is complicated by the 
many variable factors encountered in actual logging operations. 
It is well known that the speed of low frequency waves in a borehole is 
governed by the compressibility of the fluid in combination with the 
rigidity of the material surrounding the borehole. In the case of earth 
formations, if the rock around the borehole is lossy then the attenuation 
of these waves can be expressed in terms of complex shear rigidity. If the 
rock around the borehole is also permeable, there are introduced 
additional losses for waves traveling along the borehole. These effects 
can be grouped under the concept of a complex compressibility for the 
fluid wherein the speed and the attenuation of waves along the hole are 
influenced by fluid properties, shear rigidity and permeability. The 
concept of complex compressibility is described in a text entitled 
"Seismic Waves" by J. E. White, and published by McGraw-Hill in 1965. 
Reference is had to pages l48-160 of the text and particularly to 
expression 4-15 at page 158. 
That low frequency waves can be generated in a borehole is reported in a 
paper entitled "An Examination of Tube Wave Noise in Vertical Seismic 
Profiling Data" by B. A. Hardage appearing in the June 1981 issue of 
Geophysics at pages 892-903. There the author describes problems 
encountered in Vertical Seismic Profiling by the presence of tube waves 
which camouflage upgoing and downgoing body wave events thus acting as 
noise. The source of energy utilized was a vibrator whose output was swept 
over a predetermine frequency range. The effect of tube wave noise was 
reduced by increasing the offset of the source from the top of the 
wellbore. 
DISCLOSURE OF THE INVENTION 
The present invention relates to methods and apparatus utilizing low 
frequency sonic energy to provide indications of conditions of materials 
surrounding a borehole. Where the borehole is surrounded by earth 
formation, the conditions sensed include that of permeability, shear 
rigidity and, from the correlation of these latter conditions or 
parameters, an indication can be obtained of the presence of fractures in 
the earth formations traversed by the borehole. Where the material 
surrounding the borehole is casing, indications will be obtained of casing 
conditions such as weakened walls and the presence of holes penetrating 
the casing. 
More particularly, the method of the present invention comprises the steps 
of establishing continuous tube waves within a borehole with the tube 
waves having a fundamental frequency selected from the range of 20-100 
hertz. The appearance of the tube waves is detected at three or more 
spaced apart locations within the borehole. Electrical signals are 
produced representative of the characteristics of the detected tube waves 
and these signals processed to generate indications of conditions of the 
material surrounding the borehole. In one embodiment, the particle 
velocity of the tube waves is detected at two outer locations and the 
pressure of the tube waves is detected at a center location. In another 
embodiment, the pressure of the tube waves is detected at all three 
locations. 
In carrying out the processing of the electrical signals, the electrical 
signals from the two outer locations are subtracted one from the other and 
the ratio taken of the average of the absolute value of the difference 
with respect to the average of the absolute value of the electrical signal 
from the center location to produce a signal representative of the 
magnitude of the complex compressibility of the material adjacent three 
locations. 
The product of the difference signal and the signal from the center 
location is averaged and a ratio is taken of this product with respect to 
the average of the square of the signal from the center location to 
produce a signal representative of the complex compressibility, as well as 
the losses or attenuation. Now, by taking a ratio of the second derived 
signal, including a representation of compressibility and the losses or 
attenuation with respect to the earlier derived complex compressibility, 
there is obtained a function representative of the value of the losses or 
attenuation. These functions of complex compressibility and losses are 
separately plotted as a function of depth along the borehole to produce a 
log and, thereby, provide information concerning the shear rigidity and 
the permeability, as well as the presence or absence of fractures in the 
earth formations.

BEST MODES FOR CARRYING OUT THE INVENTION 
Referring now to the drawings and specifically to FIG. 1, there is depicted 
a borehole 10 traversing a formation 11 having a fracture or crack 12. A 
sonde or logging tool 13 is suspended within the bore hole 10 by way of 
cable 15. The sonde includes three transducers R1, R2 and R3. In one 
arrangement of the embodiment illustrated in FIG. 1, the transducers R1 
and R3 are particle velocity detectors such as geophones, and the 
transducer R2 is a pressure detector such as a hydrophone. The transducers 
R1 and R3 are spaced apart from 2 to 20 feet, preferably about 4 feet, and 
the transducer R2 is centrally located between them. In another 
arrangement of the embodiment of FIG. 1, the transducers R1, R2 and R3 are 
all pressure transducers. The same spacing criteria pertain. Bow-spring 
centralizers (not shown) are utilized to maintain the sonde 13 centered in 
the borehole 10. 
Tube waves are established within the borehole 10 by utilization of a 
suitable continuous wave sound source such as vibrator 20 located at the 
surface of the earth nearby the borehole 10. The vibrator 20 is programmed 
to generate a continuous wave of sonic energy having a fundamental 
frequency selected from the range of 20-100 hertz. The tube waves 
established within the borehole, appear at and are detected by the 
transducers R1, R2 and R3. 
The mode of transmission up the cable may be either analogue or digital. If 
it is digital, the amplified waveform values are sampled at a regular 
prescribed non-aliasing rate, typically 200 to 2000 times per second, then 
digitized in the electronic cartridge 14. They are then telemetered up the 
cable 15 as a sequence of binary numbers. If the transmission is analogue, 
the amplified waveforms are passed directly up the cable 15 and digitized 
in the surface equipment. The surface equipment typically includes a 
tool/cable interface unit 22, a central processing unit 23, a magnetic 
tape recording unit 24, and an optical film recording unit 25 and other 
equipment. The program executing in the central processing unit 23 is 
responsible for issuing commands to the tool or sonde 13 through the 
tool/cable interface unit 22 for the performance of whatever tasks are 
desired to take place downhole. 
The central processing unit 23 also retrieves the waveform data, either 
from a telemetry module in the tool/cable interface unit 22 if 
digitization is done downhole, or from a digitizer module in the 
tool/cable interface unit 22 if analogue transmission is used. In either 
case, these waveform data are recorded using the magnetic tape recording 
unit 24. The program may actually process the waveform data at the well 
site utilizing the technique described hereinafter and record the 
resulting complex compressibility and losses using the optical film 
recording unit 25. Otherwise, processing is performed by a central 
processsing unit located in a remote center using the tapes of the 
waveform data. 
The signals from receiving transducers R1, R2 and R3 are initially 
processed to generate an electrical signal E1. Where the receiving 
transducers are comprised of two geophones and a pressure detector, the 
signal E1 is the difference between the instantaneous values of the 
signals from the velocity detectors or geophones R1 and R3. A second 
electrical signal E2 represents the output from the pressure detector or 
transducer R2. It is obvious that the generation of the signals E1 and E2 
can take place downhole in the electronic cartridge 14. In that event, the 
signals E1 and E2 are transmitted by way of the cable 15 and the 
tool/cable interface 22 to the computer processing unit 23. It will be 
preferred, hwever, to telemeter the signals from transducers R1, R2 and R3 
to the surface where they will be recorded by the magnetic tape recording 
unit 24 and available for processing at a remote center if so desired. 
The signals E1 and E2 generated either downhole or at the surface are shown 
in FIG. 3A in an analogue representation for ease of understanding, it 
being understood that in practice, they will be digital. These signals are 
processed in accordance with the present invention to produce indications 
of conditions of the material surrounding the borehole 10. 
The first step in the processing of signals E1 and E2 is to rectify E1, as 
shown in FIG. 3B, and convolve rectified signal with a weighted window 30 
having a decreasing exponential characteristic, as shown in FIG. 4, so as 
to obtain a weighted averaged absolute value of E1 or .vertline.E.sub.1 
.vertline.. The purpose of utilizing a weighted window is to attribute the 
greatest significance to the most recent measured signal as the tool is 
moved along the borehole. With this realization, it becomes apparent that 
though it is preferred to use a weighted window having an exponential 
characteristic 
##EQU1## 
where t is the present time and To is the effective time length of the 
window, other weighted functions may be utilized as windows or moving 
average operators to achieve the desired end result. 
The next step is to rectify voltage E2 to the form shown in FIG. 3B and to 
convolve this rectified voltage with the same window or moving average 
operator having a decreasing exponential characteristic and thus produce a 
signal representative of an absolute value for E2 or .vertline.E.sub.2 
.vertline.. The ratio of the averaged absolute value of E1 to the averaged 
absolute value of E2 then yields a quantity Y1 which is proportional to 
the magnitude of the complex compressibility. This quantity Y.sub.1 is 
represented by the expression: 
EQU Y1=.omega..DELTA.zC (1) 
where 
.omega.=2.pi.f 
f=frequency of the detected tube wave 
.DELTA.z =spacing between the transducer R1 and R3; and 
C=magnitude of the complex compressibility 
Division of expression (1) by .omega..DELTA.z, the value of which is known, 
in fact, yields a quantitative value of the magnitude of complex 
compressibility C. 
Next there is determined a value for the losses L. This entails initially 
the multiplication of the instantaneous value of the signal E1 by the 
instantaneou value of the signal E2 and the convolution of this product 
with the window or moving average operator 30 of FIG. 4 to obtain E1 E2. 
The instantaneous values of E2 are squared to obtain E2.sup.2, as shown in 
FIG. 3C, convolved with the window 30 of FIG. 4 to obtain E2.sup.2 . The 
ratio 
##EQU2## 
yields a quantity Y2 proportional to the loss parameter sin.theta. or L. 
The quantity Y2 is represented by the expression: 
EQU Y2=.omega..DELTA.zC sin.theta. (3) 
where 
sin.theta. is the loss parameter. A quantitative value of the loss 
parameter is obtained by taking the ratio 
##EQU3## 
The determined values of C and L are plotted as a function of tool depth 
along the borehole to produce logs of the type illustrated in FIGS. 5 and 
6 where the ordinate represents depth along the borehole and the value of 
compressibility C and the values of the loss parameter L are plotted along 
the abscissa. 
As shown in FIG. 5, a limestone formation will exhibit values of 
compressibility C that are relatively low. The values of the Losses L will 
also be low. When the tool encounters a sandstone formation of low 
permeability, the value of compressibility C gradually increases in the 
transition from limestone to sandstone; however, the values for the loss 
parameter L remain substantially the same. Now, as there is encountered a 
zone of high permeability, the value of the compressibility C gradually 
increases to a value much higher than that previously encountered in 
limestone or sandstone and, at the same time, the value of the loss 
parameter L also increases. Accordingly, the correlation as between the 
two traces or logs L and C gives an interpreter a tool for accurate 
evaluation or interpretation of downhole conditions as they relate to 
rigidity and permeability. 
Further in FIG. 5, upon the traverse of a crack or fracture 12 (FIG. 1) in 
the formation, the value of compressibility C will suddenly change to a 
very, very high value and so will the value of the loss parameter L. The 
presence of a crack can be immediately distinguished from zones of high 
permeability by noting the nature of the transition of the traces. It is 
observed that in the zone of high permeability, the transition is gradual 
to final values; whereas, in the presence of a crack, the change is a step 
or abrupt function. The change in value of C and L, in the presence of a 
crack, occurs immediately upon one of the transducers R1 or R3 traversing 
the crack and this signal level stays high until the last transducer 
passes the crack. Hence, for all practical purposes no matter how wide the 
crack, the log will have a high value, in the area of the crack, whose 
duration will be determined by the spacing between the outer transducers 
R1 and R3. 
FIG. 6 illustrates a log produced in accordance with the present invention 
where the formations being traversed by the wellbore are of varying 
degrees of rigidiity. In this case, there is noted a gradual increase in 
compressibility C from the limestone into the sandstone and a further 
increase into a zone of low rigidity, for example, a shale. It is to be 
noted that in cases where there is low permeability, as illustrated in 
FIG. 6, the value of the loss parameter L remains fairly constant at a low 
level enabling an interpreter to distinguish between zones of high 
permeability and zones of low rigidity. 
Having now described one complete embodiment of the present invention, 
attention is again directed to FIG. 1 for a description of a second 
embodiment wherein the detectors R1, R2 and R3 are all pressure 
transducers or hydrophones. In this instance, the signals generated by the 
transducers R1, R2 and R3 are processed either downhole or at the surface 
to produce 
EQU E1=R1+R3-2R2 (5) 
As in the first embodiment, the signal E2 is equal to the output from 
transducer R2. Signals E1 and E2 are separately rectified as shown in FIG. 
3B and averaged by convolving the absolute rectified values with window of 
FIG. 4 to produce .vertline.E1.vertline. and .vertline.E2.vertline.. The 
ratio of .vertline.E1.vertline. and .vertline.E2.vertline. produces a 
function X1 which is proportional to the magnitude of the complex 
compressibility C as shown in the following expression 
##EQU4## 
where .rho. is the value of drilling fluid or mud density. 
The value of the mud density .rho. can be determined by well-known 
techniques. In typical cases, the mud density is a constant along the 
length of the borehole. That being the case, the expression 
##EQU5## 
is a constant and by dividing expression (6) by this constant, there can 
be obtained a quantitative value for the complex compressibility C. 
Having derived a value for complex compressibility, there is now undertaken 
the steps to generate values for the loss parameter sin.theta. or L. As an 
initial step, the instantaneous values of E1 are averaged by convolution 
with the window 30 of FIG. 4 to generate a signal or function E1. The 
function E1 is multiplied with the instantaneous values of E2 and the 
product convolved with the window 30 to produce the signal or function E1 
E2. 
The instantaneous values of the signal E2 are now squared and convolved 
with the window of FIG. 4 to produce the function E2.sup.2. The ratio of 
the averaged product function with the averaged squared function is 
proportional to the loss parameter sin.theta. or L in accordance with the 
following expression 
##EQU6## 
The quantitative value for the losses sin.theta. or L is obtained by taking 
the ratio of expression 5 to expression 4 and multiplying the expression 
by .omega. or stated otherwise 
##EQU7## 
The output quantities C and L generated in accordance with the second 
embodiment are like the output quantities C and L generated in accordance 
with the first embodiment. They give substantially the same information 
when recorded as a function of depth to produce logs of the type 
illustrated in FIGS. 5 and 6. This being the case, there is no need to 
repeat the description of the logs of FIGS. 5 and 6. 
The value of To, a parameter utilized in the determination of the 
exponential characteristic of the window of FIG. 4 is approximately 20 
times the period of the frequency selected for the generation of the low 
frequency tube waves. 
Referring now to FIG. 2, there is illustrated a downhole tool l3A embodying 
a modification wherein the tube waves are established in the borehole 10 
through the utilization of two spaced sound sources S1 and S2. The 
transducers R1, R2 and R3 are the same as the transducers utilized in the 
embodiment of FIG. 1 and are located between the sound sources S1 and S2 
with the distance between sound source S1 and transducer R1 being equal to 
the distance between the sound source S2 and the transducer R3. The sound 
sources have a continuous low frequency output and are operated in phase 
in establishing the tube waves in the borehole. The sound sources S1 and 
S2 may be of the electromechanical type or any type or form of low 
frequency continuous wave downhole sound source which will produce sonic 
energy in the range of from 20-100 hertz. 
In one arrangement of the tool l3A, the transducers will comprise two 
velocity detectors R1 and R3 and a pressure detector R2. In another 
arrangement, the transducers will be comprised of three pressure 
detectors. In the described arrangements of sources and detectors, the 
data produced from the tool l3A will be essentially the same data produced 
by the tool 13 of FIG. 1 and these data are processed in exactly the same 
manner as above described to produce quantitative values of the complex 
compressibility C and the loss parameter L. 
The downhole sound source arrangement assures that the receivers R1, R2 and 
R3 are always in an optimum detecting position in that the maximum 
pressure always exists in the vicinity of transducer R2 and the motion of 
the drilling mud is always near a minimum. This means that in the first 
arrangement where the signal E1 represents the difference between signals 
produced by transducers R1 and R2, the difference signal is of the same 
order of magnitude as the individual signals from the transducers R1 and 
R3. This also means that in the second arrangement where the signal E1 is 
equal to the sum of the signals from transducers R1 and R3, less twice the 
signal from transducer R2, the output signal E1 is a maximum with respect 
to the magnitude of pressure in the vicinity of the transducer R2. 
The methods and apparatus of the present invention have thus far been 
described principally with regard to logging open hole and determining 
characteristics of earth formations as well as determining the presence of 
fractures or cracks. It will be evident to those skilled in the art that 
the same techniques employed in open hole logging can be utilized in 
logging a cased hole for the purpose of examining the condition of the 
casing. The rigidity factor in the cased hole will be determined by the 
thickness of the casing and, therefore, should the casing be weakened in 
any way by corrosion or other causes, the casing walls will be less rigid 
and give rise to a characteristic signal indicative of this condition. 
Likewise, should there happen to be holes formed in the casing, a signal 
will be produced similar to that which is generated in the presence of 
cracks when logging open hole. 
Now that the invention has been described, further variations and 
modifications will occur to those skilled in the art. It is intended that 
such modifications and variations be encompassed within the scope of the 
appended claims.