Wellbore surveying using a tilted survey sensor

1 . A method for surveying a wellbore while drilling, the method comprising:

rotating a downhole tool in the wellbore while drilling the wellbore, the downhole tool including at least one tilted survey sensor, wherein the at least one tilted survey sensor includes a tilted accelerometer and a tilted magnetometer, wherein the tilted accelerometer has a sensory axis that is rotationally offset from a longitudinal axis of the downhole tool by a first offset angle in the range between 15 and 60 degrees where the tilted accelerometer is sensitive to both axial and radial components of a gravitational field measured by the tilted accelerometer, and wherein the tilted magnetometer has a sensory axis that is rotationally offset from the longitudinal axis of the downhole tool by a second offset angle in the range between 15 and 60 degrees where the tilted magnetometer is sensitive to both axial and radial components of a magnetic field measured by the tilted magnetometer,

configuring the at least one tilted survey sensor to make a plurality of sensor measurements while rotating the downhole tool in the wellbore and drilling the wellbore, wherein the plurality of sensor measurements comprises a first plurality of sensor measurements and a second plurality of sensor measurements, the first plurality of sensor measurements being made using the tilted accelerometer, and the second plurality of sensor measurements being made using the tilted magnetometer;

configuring a controller to fit the first plurality of sensor measurements with a first sinusoidal fitting function to obtain first and second fitting parameters representing offset and magnitude of the first sinusoidal fitting function;

configuring the controller to fit the second plurality of sensor measurements with a second sinusoidal fitting function to obtain third and fourth fitting parameters representing offset and magnitude of the second sinusoidal fitting function;

configuring the controller to determine a first survey parameter that characterizes inclination of the wellbore and a second survey parameter that characterizes azimuth of the wellbore, wherein the first survey parameter is determined from the first and second fitting parameters and the first offset angle between the sensory axis of the tilted accelerometer and the longitudinal axis of the downhole tool; and

configuring the controller to compute a phase difference between the first sinusoidal fitting function and the second sinusoidal fitting function to determine a difference between a gravity toolface and a magnetic toolface of the downhole tool,

wherein an angle between the longitudinal axis of the downhole tool and a local magnetic field direction is computed from the third and fourth fitting parameters and the second offset angle between the sensory axis of the tilted magnetometer and the longitudinal axis of the downhole tool, and

wherein the second survey parameter is computed from the first survey parameter, the angle between the longitudinal axis of the downhole tool and the local magnetic field direction, and the difference between the gravity toolface and the magnetic toolface.

2 . A downhole tool comprising:

a collar configured for coupling with a drill string;

at least one tilted survey sensor deployed in the collar, the at least one tilted survey sensor comprising a tilted accelerometer and a tilted magnetometer, wherein the tilted accelerometer has a sensory axis that is rotationally offset from a longitudinal axis of the downhole tool by a first offset angle in the range between 15 and 60 degrees where the tilted accelerometer is sensitive to both axial and radial components of a gravitational field measured by the tilted accelerometer, and wherein the tilted magnetometer has a sensory axis that is rotationally offset from the longitudinal axis of the downhole tool by a second offset angle in the range between 15 and 60 degrees where the tilted magnetometer is sensitive to both axial and radial components of a magnetic field measured by the tilted magnetometer; and

a controller configured to:

cause the tilted survey sensor to make a plurality of sensor measurements while the downhole tool rotates in a wellbore and the drill string drills the wellbore, wherein the plurality of sensor measurements comprises a first plurality of sensor measurements and a second plurality of sensor measurements, the first plurality of sensor measurements being made using the tilted accelerometer and the second plurality of sensor measurements being made using the tilted magnetometer;

fit the first plurality of sensor measurements with a first sinusoidal fitting function to obtain first and second fitting parameters representing offset and magnitude of the first sinusoidal fitting function;

fit the second plurality of sensor measurements with a second sinusoidal fitting function to obtain third and fourth fitting parameters representing offset and magnitude of the second sinusoidal fitting function;

determine a first survey parameter that characterizes inclination of the wellbore and a second survey parameter that characterizes azimuth of the wellbore, wherein the first survey parameter is determined from the first and second fitting parameters and the first offset angle between the sensory axis of the tilted accelerometer and the longitudinal axis of the downhole tool; and

compute a phase difference between the first sinusoidal fitting function and the second sinusoidal fitting function to determine a difference between a gravity toolface and a magnetic toolface of the downhole tool,

wherein an angle between the longitudinal axis of the downhole tool and a local magnetic field direction is computed from the third and fourth fitting parameters and the second offset angle between the sensory axis of the tilted magnetometer and the longitudinal axis of the downhole tool, and

wherein the second survey parameter is computed from the first survey parameter, the angle between the longitudinal axis of the downhole tool and the local magnetic field direction, and the difference between the gravity toolface and the magnetic toolface.

3 . A method for surveying a wellbore while drilling, the method comprising:

rotating a downhole tool in the wellbore while drilling the wellbore, the downhole tool including first, second, and third tilted accelerometers and first, second, and third tilted magnetometers, wherein the first, second, and third tilted accelerometers have sensory axes that are rotationally offset from a longitudinal axis of the downhole tool by respective offset angles in the range between 15 and 60 degrees where the first, second, and third tilted accelerometers are sensitive to both axial and radial components of a gravitational field measured by the first, second, and third tilted accelerometers, wherein the first, second, and third tilted magnetometers have sensory axes that are rotationally offset from the longitudinal axis of the downhole tool by respective offset angles in the range between 15 and 60 degrees where the first, second, and third tilted magnetometers are sensitive to both axial and radial components of a magnetic field measured by the first, second, and third tilted magnetometers, wherein the first, second, and third tilted accelerometers are rotationally offset from one another with respect to a radial reference direction and the first, second, and third tilted magnetometers are rotationally offset from one another with respect to the radial reference direction;

configuring the first, second, and third tilted accelerometers to make corresponding first, second, and third pluralities of accelerometer measurements while rotating the downhole tool in the wellbore and drilling the wellbore, and configuring the first, second, and third tilted magnetometers to make corresponding first, second, and third pluralities of magnetometer measurements while rotating the downhole tool in the wellbore and drilling the wellbore;

configuring a controller to fit the first, second, and third pluralities of accelerometer measurements with first, second, and third sinusoidal fitting functions to obtain corresponding accelerometer fitting parameters;

configuring the controller to fit the first, second, and third pluralities of magnetometer measurements with fourth, fifth, and sixth sinusoidal fitting functions to obtain corresponding magnetometer fitting parameters;

configuring the controller to determine first, second, and third wellbore inclination values that characterize inclination of the wellbore from the accelerometer fitting parameters and the respective offset angles between the sensory axes of the first, second, and third tilted accelerometers and the longitudinal axis of the downhole tool;

configuring the controller to determine first, second, and third wellbore azimuth values that characterize azimuth of the wellbore from the magnetometer fitting parameters and the respective offset angles between the sensory axes of the first, second, and third tilted magnetometers and the longitudinal axis of the downhole tool;

configuring the controller to compute first, second, and third phase differences between the first, second, and third sinusoidal fitting functions and the fourth, fifth, and sixth sinusoidal fitting functions;

configuring the controller to compute an angle between the longitudinal axis of the downhole tool and a local magnetic field direction from the corresponding magnetometer fitting parameters and the respective offset angles between the sensory axes of the first, second, and third tilted magnetometers and the longitudinal axis of the downhole tool; and

configuring the controller to compute the first, second, and third wellbore azimuth values from the first, second, and third wellbore inclination values, the angle between the longitudinal axis of the downhole tool and the local magnetic field direction, and the first, second, and third phase differences.

4 . The method of claim 3 , further comprising:

configuring the first, second, and third tilted accelerometers to make first, second, and third non-rotating accelerometer measurements while the downhole tool is rotationally stationary in the wellbore, and configuring the first, second, and third tilted magnetometers to make first, second, and third non-rotating magnetometer measurements while the downhole tool is rotationally stationary in the wellbore; and

configuring the controller to determine a static wellbore inclination value and a static wellbore azimuth value from the first, second, and third non-rotating accelerometer measurements and the first, second, and third non-rotating magnetometer measurements.

5 . The method of claim 3 , wherein:

the first, second, and third tilted accelerometers and the first, second, and third tilted magnetometers are axially spaced apart in the downhole tool; and

the controller is further configured to determine a dogleg severity of the wellbore from selected ones of the first, second, and third wellbore inclination values and the first, second, and third wellbore azimuth values.

6 . The method of claim 3 , further comprising:

configuring the controller to compute an average wellbore inclination from the first, second, and third wellbore inclination values; and

configuring the controller to compute an average wellbore azimuth from the first, second, and third wellbore azimuth values.