Methods and apparatus for discrete-frequency tube-wave logging of boreholes

A method of borehole logging, comprising the steps of:

• (a) emitting sonic energy at a first location in a borehole of at least one discrete frequency which induces propagation of tube waves in said borehole;
• (b) receiving sonic energy of said tube waves at multiple receiver locations in said borehole, said multiple receiver locations being spaced apart from one another and from said first location;
• (c) for each said receiver location, detecting a complex pressure response of the received sonic energy of said tube waves;
• and
• (d) calculating tube wave phase velocity v and tube wave attenuation constant .alpha. from said detected complex pressure response at two of said receiver locations and from the expressions:
• v(f).sup.-1 =[phase(P(z.sub.1, f))-phase(P(z.sub.2,f))]/2.pi.f(z.sub.1 -z.sub.2)
• .alpha.=ln [abs(P(z.sub.2,f)/P(z.sub.1,f))]/(z.sub.1 -z.sub.2)
• where the tube wave pressure function is expressed as:
• P(z,f)=A(f)e.sup.-.alpha.z e.sup.i2.pi.(f/v)z
• and where P is pressure;
• A is amplitude as a function of frequency and is independent of spacing between said emitting location and said receiver locations;
• z is the spacing between said emitting location and a receiver location;
• f is the discrete frequency at which the tube wave energy is propagated;
• and z.sub.1 and z.sub.2 are respective said receiver locations.

A method of borehole logging, comprising the steps of:

• (a) emitting sonic energy at a first location in a borehole of at least one discrete frequency which induces propagation of tube waves in said borehole;
• (b) receiving sonic energy of said tube waves at multiple receiver locations in said borehole, said multiple receiver locations being spaced apart from one another and from said first location;
• (c) for each said receiver location, detecting a complex pressure response of the received sonic energy of said tube waves;
• and
• (d) calculating tube wave phase velocity v and tube wave attenuation constant .alpha. from said detected complex pressure response at three of said receiver locations equally spaced at intervals .DELTA.z and from the expression:
• k=2.pi.(f/v)+i.alpha.=(ln G)/(i.DELTA.z)
• where ##EQU6## and where k is the complex wave-number of the received tube wave, f is the frequency of the received tube wave, and P.sub.n is the complex pressure response of the receiver located at an n-th one of said receiving locations.

A method of borehole logging, comprising the steps of:

• (a) emitting sonic energy at a first location in a borehole of at least one discrete frequency which induces propagation of tube waves in said borehole;
• (b) receiving sonic energy of said tube waves at multiple receiver locations in said borehole, said multiple receiver locations being spaced apart from one another and from said first location;
• (c) for each said receiver location, detecting a complex pressure response of the received sonic energy of said tube waves;
• and
• (d) determining values for a variable A and a variable B and a variable k such that the expression:
• ##EQU8## is minimized and where N is a plural number of said receiver locations, k is the complex wave-number of the received tube wave, z.sub.n is an n-th one of said receiving locations, and P.sub.n is the complex pressure response of the receiver located at z.sub.n and is expressed as:
• P.sub.n =A e.sup.ikz.sbsp.n +B e.sup.-ikz.sbsp.n.

A method of borehole logging, comprising the steps of:

• (a) emitting sonic energy at a first location in a borehole of at least one discrete frequency which induces propagation of tube waves in said borehole;
• (b) receiving sonic energy of said tube waves at multiple receiver locations in said borehole, said multiple receiver locations being spaced apart from one another and from said first location;
• (c) for each said receiver location, detecting a complex pressure response of the received sonic energy of said tube waves;
• (d) defining the detected complex pressure response P(z) at a fixed frequency to be ##EQU9## (e) defining a linear predictor h(z) so as to satisfy the relationship
• P(z) z h(z)=0
• where the notation, z, means convolution in z;
• (f) defining H(k) as the Fourier transform of h(z) in the wave-number domain, where said definitions of receiver pressure P(z) and said linear predictor h(z) can be satisfied only when H(k) is zero at k=k.sub.m ;
• (g) truncating spatial variables P(z) and h(z) to finite intervals of z;
• (h) solving the linear equations of the relationship
• P(z) z h(z)=0
• for h(z), assuming h(0)=1;
• (i) transforming the function h(z) to H(k);
• and
• (j) calculating values of k for the equation H(k)=0 to obtain estimated values for k.sub.m.

A method of borehole logging, comprising the steps of:

• (a) emitting sonic energy at a first location in a borehole of at least one discrete frequency which induces propagation of tube waves in said borehole;
• (b) receiving sonic energy of said tube waves at multiple receiver locations in said borehole, said multiple receiver locations being spaced apart from one another and from said first location;
• (c) for each said receiver location, detecting a complex pressure response of the received sonic energy of said tube waves;
• (d) calculating the function L(k) as ##EQU10## and where P.sub.n is the complex pressure response of the receiver located at z.sub.n, N is the number of receivers in the array, and the superscript * denotes conjugate transpose of the matrix or the vector;
• and
• (e) determining values of k.sub.m by searching for the locations of the local minimum of L(k).

A method of logging tube waves with a tube wave logging tool in a borehole traversing an earth formation, comprising the steps of:

• emitting from a monopole source a repeating signal sequence of sonic energy at a first location to induce propagation of tube waves in said borehole, said sonic energy being concentrated at a plurality of selected discrete frequencies separated by frequency bands containing substantially no sonic energy;
• receiving at a second and a third location sonic energy of said tube waves induced in response to the repeating signal sequence;
• for each of said plurality of frequencies, determining amplitude and phase of the sonic energy received at said second and third locations by directly measuring said tube waves in the frequency domain;
• and
• transmitting from the borehole signals representing the determined amplitude and phase of the sonic energy received at said second and third locations.

Apparatus for borehole logging, comprising:

• (a) means for emitting from a monopole source a repeating signal sequence of sonic energy at a first location in a borehole which induces propagation of tube waves in the borehole, said sonic energy being concentrated at a plurality of selected discrete frequencies separated by frequency bands containing substantially no sonic energy;
• (b) means for receiving sonic energy of said tube waves induced in response to the repeating signal sequence at multiple receiver locations in said borehole, said multiple receiver locations being spaced apart from one another and from said first location;
• and
• (c) means for detecting, for each said receiver location, a complex pressure response of the received sonic energy of said tube waves by directly measuring said tube waves in the frequency domain.

A method of borehole logging, comprising the steps of:

• (a) generating a first sinusoidal signal at a first frequency suitable for inducing tube wave propagation in a borehole;
• (b) generating a first phase reference signal representing phase of the first sinusoidal signal;
• (c) driving a transducer at a transmitter location in the borehole with said first sinusoidal signal so as to emit sonic energy at said first frequency and thereby induce propagation of tube waves in the borehole;
• (d) at each of a plurality of receiver locations spaced apart from one another and from said transmitter location, receiving sonic energy of said tube waves and producing a corresponding receiver signal;
• and
• (e) for each said receiver location, operating a phase-sensitive detector responsive to the corresponding receiver signal and to the first phase reference signal to detect a complex pressure response of the received sonic energy of the first frequency.

A method of borehole logging, comprising the steps of:

• (a) generating a first sinusoidal signal at a first frequency suitable for inducing tube wave propagation in a borehole;
• (b) generating a second sinusoidal signal at a second frequency suitable for inducing tube wave propagation in the borehole;
• (c) generating a third sinusoidal signal at a third frequency suitable for inducing tube wave propagation in the borehole;
• (d) generating a first phase reference signal representing phase of the first sinusoidal signal;
• (e) generating a phase reference signal representing phase of the second sinusoidal signal;
• (f) generating a phase reference signal representing phase of the third sinusoidal signal;
• (g) mixing said first sinusoidal signal, said second sinusoidal signal and said third sinusoidal signal to produce a transmitter signal;
• (h) driving a transducer at a transmitter location in the borehole with said transmitter signal so as to emit sonic energy and thereby induce propagation of tube waves in the borehole;
• (i) at each of a plurality of receiver locations spaced apart from one another and from said transmitter location, receiving sonic energy of said tube waves and producing a corresponding receiver signal;
• and
• (j) for each said receiver location, operating a phase-sensitive detector responsive to the corresponding receiver signal and to the first phase reference signal to detect a complex pressure response of the received sonic energy of the first frequency, operating a phase-sensitive detector responsive to the corresponding receiver signal and to the second phase reference signal to detect a complex pressure response of the received sonic energy of the second frequency, and operating a phase-sensitive detector responsive to the corresponding receiver signal and to the third phase reference signal to detect a complex pressure response of the received sonic energy of the third frequency.

A method of borehole logging, comprising the steps of:

• (a) generating a transmitter signal having a repetitive signal sequence, the transmitter signal having its energy concentrated at a number of distinct frequencies f.sub.n, each of said distinct frequencies f.sub.n being suitable for inducing tube wave propagation in a borehole and being defined by the expression f.sub.n .ltoreq.n/.tau., where .tau. is the period of the repetitive signal sequence and n is an integer;
• (b) for each of said distinct frequencies of the transmitter signal, generating a respective phase reference signal representing phase of the distinct frequency within the transmitter signal;
• (c) driving a transducer at a transmitter location in the borehole with said transmitter signal so as to emit sonic energy and thereby induce propagation of tube waves in the borehole;
• (d) at each of a plurality of receiver locations spaced apart from one another and from said transmitter location, receiving sonic energy of said tube waves and producing a corresponding receiver signal;
• and
• (e) for each said receiver location and for each said distinct frequency of the transmitter signal, operating a phase-sensitive detector responsive to the corresponding receiver signal and to the respective phase reference signal to detect a complex pressure response of the received sonic energy of each of said distinct frequencies.