Downhole acoustic system for determining a rate of penetration of a drill string and related methods

A method of determining a rate of penetration of a drill string, the method comprising:

• operably coupling an array of acoustic transducers within a circumference of at least one of a drill bit coupled to a drill string within a wellbore, another component of a bottomhole assembly coupled to the drill string, or the drill string;
• operably coupling the array of acoustic transducers to a controller comprising a memory and a processor;
• transmitting acoustic waves exhibiting a source frequency from at least one acoustic transducer of the array of acoustic transducers toward a wall of the wellbore in a direction defining an acute angle with a longitudinal axis of the drill string;
• measuring a frequency of reflected acoustic waves with at least one acoustic transducer of the array of acoustic transducers;
• receiving, from a flow meter operating at a different frequency than the acoustic transducers, an indication of a velocity of drilling fluid between the array of acoustic transducers and the wall of the wellbore during the time between emitting the acoustic waves and measuring the reflected acoustic waves;
• receiving an indication of a rotation rate of the drill bit during the time between emitting the acoustic waves and measuring the reflected acoustic waves;
• determining, with the processor, a frequency shift between the emitted acoustic waves and the measured acoustic waves;
• determining an effect of the velocity of the drilling fluid on the frequency shift based at least partially on the received indication of the velocity of the drilling fluid;
• determining an effect of the rotation rate of the drill bit during the time between emitting the acoustic waves and detecting the reflected acoustic waves on the frequency shift;
• based at least partially on the determined frequency shift, the effect of the velocity of the drilling fluid on the frequency shift, the effect of the rotation rate of the drill bit on the frequency shift, and the acute angle, determining a velocity of the acoustic transducers of the array of acoustic transducers relative to the wall of the wellbore;
• and
• determining a rate of penetration based at least in part on a value of the determined velocity of the acoustic transducers of the array of acoustic transducers relative to the wall of the wellbore.

A downhole acoustic system for determining a rate of penetration of a drill string, the downhole acoustic system comprising:

• a member of a drill string within a wellbore, the member of the drill string including a drill bit configured to drill through a formation;
• an array of acoustic transducers operably coupled to a member of the drill string and configured transmit acoustic waves exhibiting a first frequency to a wellbore wall in a direction defining an acute angle with a longitudinal axis of the drill string, at least one acoustic transducer of the array of acoustic transducers configured to measure a frequency of reflected acoustic waves reflected from the wellbore wall, the reflected acoustic waves exhibiting a second frequency;
• and
• a downhole controller operably coupled to the array of acoustic transducers, the downhole controller comprising a memory configured to cause a processor to:
• receive, from a flow meter operating at a different frequency than the acoustic transducers, an indication of a velocity of drilling fluid between the array of acoustic transducers and the wellbore wall during the time between transmitting the acoustic waves and measuring the reflected acoustic waves;
• receive an indication of a rotation rate of the drill bit during the time between transmitting the acoustic waves and measuring the reflected acoustic waves;
• calculate a frequency shift between the first frequency and the second frequency;
• determine an effect of the velocity of the drilling fluid on the frequency shift based at least partially on the received indication of the velocity of the drilling fluid;
• determine an effect of the rotation rate of the drill bit during the time between transmitting the acoustic waves and measuring the reflected acoustic waves on the frequency shift;
• based at least partially on the determined frequency shift, the effect of the velocity of the drilling fluid on the frequency shift, the effect of the rotation rate of the drill bit on the frequency shift, and the acute angle, determine a velocity of the acoustic transducers of the array of acoustic transducers relative to the wellbore wall;
• and
• determine a rate of penetration of the drill string through the formation based at least in part on the determined velocity of the acoustic transducers of the array of acoustic transducers relative to the wall of the wellbore.