Method and apparatus for determining the shape of an earth borehole and the motion of a tool within the borehole
First Claim
1. An apparatus for estimating the actual cross-sectional shape and orientation of an earth borehole comprising:
- (a) a rotatable tool having(1) a plurality of distance sensors for generating standoff signals representative of respective standoff distances from each of said distance sensors to respective points on the wall of a borehole at a plurality of measurement times, and(2) at least one angle sensor for generating rotational orientation signals representative of the rotational orientation angle of said tool with respect to a reference direction at said plurality of measurement times; and
(b) a signal processor for calculating an estimate of the actual cross-sectional shape and orientation of an earth borehole based on said standoff signals and said rotational orientation signals.
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Abstract
A method and apparatus are provided for estimating the cross-sectional shape and orientation of an earth borehole and the motion of a tool therein. The method and apparatus involve measuring the distance from the tool to the borehole wall at a plurality of locations around the periphery of the tool and fitting those measured distances to a predetermined shape function using a nonlinear parameter estimation technique to minimize the error between the estimated shape of the borehole and the measured distances. The method and apparatus may be used to estimate elliptical and higher order borehole shapes. Additionally, the method and apparatus may be used while drilling the borehole.
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Citations
81 Claims
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1. An apparatus for estimating the actual cross-sectional shape and orientation of an earth borehole comprising:
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(a) a rotatable tool having (1) a plurality of distance sensors for generating standoff signals representative of respective standoff distances from each of said distance sensors to respective points on the wall of a borehole at a plurality of measurement times, and (2) at least one angle sensor for generating rotational orientation signals representative of the rotational orientation angle of said tool with respect to a reference direction at said plurality of measurement times; and (b) a signal processor for calculating an estimate of the actual cross-sectional shape and orientation of an earth borehole based on said standoff signals and said rotational orientation signals. - View Dependent Claims (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32)
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33. An apparatus for estimating the actual cross-sectional shape and orientation of an earth borehole comprising:
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a rotating tool having a radius rt ; three acoustic sensors mounted to said tool at three sensor locations equally spaced around the circumference of said tool for generating standoff signals representative of standoff distances d1, d2, and d3, respectively, between said three sensor locations on said tool and corresponding measured points P1, P2, and P3 on the wall of the borehole at each of a plurality of measurement times; a gravitational sensor mounted to said tool for generating first rotational orientation signals representative of the rotational orientation angle of said tool with respect to a first reference direction of the earth'"'"'s gravity at each of said measurement times; a magnetic sensor mounted to said tool for generating second rotational orientation signals representative of the rotational orientation angle of said tool with respect to a second reference direction of the earth'"'"'s magnetic field at each of said measurement times; a signal processor mounted to said tool in communication with said acoustic sensors, said gravitational sensor, and said magnetic sensor for receiving said standoff signals, said first rotational orientation signals, and said second rotational orientation signals, generating radius signals representative of distances r1, r2, and r3, respectively, from the center of said tool to said measured points P1, P2, and P3 on the wall of the borehole using said standoff signals according to the equations
space="preserve" listing-type="equation">r.sub.1 =d.sub.1 +r.sub.t
space="preserve" listing-type="equation">r.sub.2 =d.sub.2 +r.sub.t
space="preserve" listing-type="equation">r.sub.3 =d.sub.3 +r.sub.tat each of said measurement times, comparing said first and second rotational orientation signals and selecting one of said first or second rotational orientation signals as a primary rotational orientation signal representative of an angle θ
with respect to a primary reference direction at each of said measurement times,generating parameter signals representative of a parameter vector {X} comprised of parameters Xm that defines an estimated ellipse to approximate the cross-sectional shape and orientation of the borehole and an estimated position of said tool within the borehole at each of said measurement times according to the following equation
space="preserve" listing-type="equation">{X}={r.sub.x,r.sub.y,ψ
, xc.sub.1,xc.sub.2,yc.sub.1,yc.sub.2, . . . ,xc.sub.N,yc.sub.N }wherein rx is the major radius of said ellipse, ry is the minor radius of said ellipse, ψ
is the angle between said major radius and said primary reference direction, and (xcn, ycn) are the coordinates of the center of said tool within the borehole at said estimated position at each of said measurement times,generating point error signals representative of a point error function e for each of said measured points according to the following equation ##EQU17## wherein (x, y) are the coordinates of said measured points for each of said measurement times, combining said point error signals and generating firing error signals representative of a firing error function En for each of said measurement times according to the equation ##EQU18## generating primary error signals representative of a primary error function {ET } according to the equation
space="preserve" listing-type="equation">{E.sub.T }={E.sub.1,E.sub.2,E.sub.3, . . . ,E.sub.N }wherein the terms En represent said firing error signals at each of said measurement times, using said parameter signals and said primary error signals to generate sensitivity signals representative of the Jacobian matrix [J] as a measure of the sensitivity of said primary error function {ET } to each of said parameters Xm according to the equation ##EQU19## wherein n=1,2,3, . . . ,N and m=1,2,3, . . . ,2N+3, using said sensitivity signals and said primary error signals to generate parameter adjustment signals representative of a parameter adjustment vector {p} according to the equation
space="preserve" listing-type="equation">{p}=-PseudoInverse([J].sup.T [J]+λ
[J])([J].sup.T {E.sub.T })wherein [J]T is the transpose of the Jacobian matrix [J], λ
is a Levenburg-Marquardt parameter, and [I] is the identity matrix,using said parameter adjustment signals to iteratively adjust said parameters according to the relation
space="preserve" listing-type="equation">{X}={X}+{p}to minimize said primary error function, and generating solution signals representative of optimal values for said parameters. - View Dependent Claims (34, 35, 36, 37, 38, 39, 40)
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41. An apparatus for estimating the actual motion of a tool within an earth borehole comprising:
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(b) a rotatable tool having (1) a plurality of distance sensors for generating standoff signals representative of respective standoff distances from each of said distance sensors to respective points on the wall of a borehole at a plurality of measurement times, and (2) at least one angle sensor for generating rotational orientation signals representative of the rotational orientation angle of said tool with respect to a reference direction at said plurality of measurement times; and (b) a signal processor for calculating an estimate of the lateral motion of said tool within an earth borehole based on said standoff signals and said rotational orientation signals. - View Dependent Claims (42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66)
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67. A method of estimating the actual cross-sectional shape of an earth borehole comprising the following steps:
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placing a tool having a plurality of distance sensors in the borehole; rotating said tool in the borehole; making distance measurements of the distance between said tool and the wall of the borehole at a plurality of points around the circumference of said tool and at a plurality of measurement times; making rotational orientation measurements of the rotational orientation of said tool with respect to a reference direction at said plurality of measurement times; establishing a set of parameters that defines an estimated cross-sectional shape and orientation of the borehole and an estimated position of said tool within the borehole at each of said measurement times; establishing a primary error function proportional to the difference between said estimated cross-sectional shape and the actual cross-sectional shape of the borehole as defined by said distance measurements and said rotational orientation measurements; computing the sensitivity of said primary error function to each of said parameters; and iteratively adjusting said parameters based on said sensitivity to minimize said primary error function and thereby obtain optimal values for said parameters. - View Dependent Claims (68, 69, 70, 71, 72)
- 70. The method of claim 68 wherein the terms of said supplemental error function are defined by the equation
- space="preserve" listing-type="equation">F.sub.n =W.sub.s |s.sub.n -s.sub.n '"'"'|
wherein Ws is a weighting coefficient, sn is the value of a measured displacement signal obtained from accelerometers for each of said measurement times, and sn '"'"' is the value of a derived displacement signal obtained from said distance sensors for each of said measurement times.
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- 71. The method of claim 68 wherein the terms of said supplemental error function are defined by the equation
- space="preserve" listing-type="equation">F.sub.n =W.sub.α
|α
.sub.n -α
.sub.n '"'"'|
wherein W.sub.α
is a weighting coefficient, α
n is the value of a derived acceleration signal obtained from said distance sensors for each of said measurement times, and α
n '"'"' is the value of a measured acceleration signal obtained from accelerometers for each of said measurement times. - space="preserve" listing-type="equation">F.sub.n =W.sub.α
- space="preserve" listing-type="equation">F.sub.n =W.sub.d |D.sup.m [s.sub.n ]-D.sup.m [s.sub.n '"'"']|
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73. A method for estimating the actual cross-sectional shape and orientation of an earth borehole comprising the following steps:
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placing a tool having a radius rt and a plurality of distance sensors in the borehole; rotating said tool in the borehole; measuring distances d1, d2, and d3 between each of three equally spaced locations around the circumference of said tool and three measured points P1, P2, and P3, respectively, on the wall of the borehole at each of a plurality of measurement times; calculating distances r1, r2, and r3 from the center of said tool to said measured points on the wall of the borehole according to the equations
space="preserve" listing-type="equation">r.sub.1 d.sub.1 +r.sub.t
space="preserve" listing-type="equation">r.sub.2 d.sub.2 +r.sub.t
space="preserve" listing-type="equation">r.sub.3 d.sub.3 +r.sub.tat each of said measurement times; making a first rotational orientation measurement of the rotational orientation of said tool with respect to a first reference direction of the earth'"'"'s gravity at each of said measurement times; making a second rotational orientation measurement of the rotational orientation of said tool with respect to a second reference direction of the earth'"'"'s magnetic field at each of said measurement times; comparing said first and second rotational orientation measurements and selecting one of said first or second rotational orientation measurements as a primary rotational orientation measurement θ
with respect to a primary reference direction at each of said measurement times;establishing a parameter vector {X} comprised of parameters Xm that defines an estimated ellipse to approximate the cross-sectional shape and orientation of the borehole and an estimated position of said tool within the borehole at each of said measurement times according to the following equation
space="preserve" listing-type="equation">{X}={r.sub.x,r.sub.y,ψ
,xc.sub.1,yc.sub.1,xc.sub.2,yc.sub.2, . . . ,xc.sub.N,yc.sub.N }wherein rx is the major radius of said ellipse, ry is the minor radius of said ellipse, ψ
is the angle between said major radius and said primary reference direction, and (xcn, ycn) are the coordinates of the center of said tool within the borehole at said estimated position at each of said measurement times;calculating a point error function e for each of said measured points according to the following equation ##EQU23## wherein (x, y) are the coordinates of said measured points for each of said measurement times; calculating a firing error function En for each of said measurement times according to the equation ##EQU24## establishing a primary error function {ET } according to the equation
space="preserve" listing-type="equation">{E.sub.T }=E.sub.1,E.sub.2,E.sub.3, . . . ,E.sub.N }wherein En is said firing error function at each of said measurement times; computing the Jacobian matrix [J] as a measure of the sensitivity of said primary error function {ET } to each of said parameters Xm according to the equation ##EQU25## wherein n=1,2,3, . . . ,N and m=1,2,3, . . . ,2N+3;
calculating a parameter adjustment vector {p} according to the equation
space="preserve" listing-type="equation">{p}=-PseudoInverse([J].sup.T [J]+λ
[I])([J].sup.T {E.sub.T })wherein [J]T is the transpose of [J], λ
is a Levenburg-Marquardt parameter, and [I] is the identity matrix; anditeratively adjusting said parameters according to the relation
space="preserve" listing-type="equation">{X}={X}+{p}to minimize said primary error function and thereby obtain a solution for said parameter vector. - View Dependent Claims (74)
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75. A method of estimating the actual motion of a tool within an earth borehole comprising the following steps:
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placing a tool having a plurality of distance sensors in the borehole; rotating said tool in the borehole; making distance measurements of the distance between said tool and the wall of the borehole at a plurality of points around the circumference of said tool and at a plurality of measurement times; making rotational orientation measurements of the rotational orientation of said tool with respect to a reference direction at said plurality of measurement times; establishing a set of parameters that defines an estimated cross-sectional shape and orientation of the borehole and an estimated position of said tool within the borehole at each of said measurement times; establishing a primary error function proportional to the difference between said estimated cross-sectional shape and the actual cross-sectional shape of the borehole as defined by said distance measurements and said rotational orientation measurements; computing the sensitivity of said primary error function to each of said parameters; and iteratively adjusting said parameters based on said sensitivity to minimize said primary error function and thereby obtain optimal values for said parameters. - View Dependent Claims (76, 77, 78, 79, 80, 81)
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Specification