Wellbore surveying using a tilted survey sensor

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 having a sensory axis that is rotationally offset from a longitudinal axis of the collar by an offset angle in a range between 15 and 60 degrees where the tilted survey sensor is sensitive to both axial and radial components of a gravitational or magnetic field measured by the tilted survey sensor;
• 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;
• fit the plurality of sensor measurements with at least one sinusoidal fitting function to obtain fitting parameters representing offset and magnitude of the at least one sinusoidal fitting function;
• and
• determine at least one wellbore parameter that characterizes orientation of the wellbore while drilling the wellbore from the fitting parameters and the offset angle between the sensory axis of the tilted survey sensor and the longitudinal axis of the downhole tool.

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;
• and
• 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;
• and
• 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.