Excerpts from “Advanced Pore Pressure Analysis”
Excerpts from “Advanced Pore Pressure Analysis”
A course written by Chris Fletcher
The d-Exponent Model
d-Exponent History
The d-Exponent was invented at the onset of rotary drilling to assist the determination of bearing wear on a roller cone bit, allowing the driller to pull the bit before rolling a cone off and necessitating an expensive fishing trip to retrieve the cone before drilling ahead. Soon afterward it was used to generalize a fracture gradient value. Much later the d-Exponent was applied to the analysis of pore pressure.
The d-Exponent Equation
DXC = log10(ROP/60N)/log10(12WOB/106Dia)
ROP – rate of penetration ft/hr
N – rotary RPM
WOB – weight on bit in lbs
Dia – bit (total cutting surface if using a reamer) diameter in inches
Source of data
The input data for the calculation of the d-Exponent is from the data the driller uses to control the drilling parameters. ROP, N, and Dia can be assumed to be accurate. WOB is taken from the hookload sensor, referenced after making a connection with the drill string suspended off bottom while circulating and rotating. When the bit is lowered and contacts the formation the hookload is reduced. The difference between the initial hookload while the drill string was suspended and the hookload after the bit is in contact with the formation is the WOB.
WOB versus Actual Weight on Bit
Typically, the d-Exponent value will decrease as the stand of drill pipe connected is drilled, creating a saw-tooth waveform when presented on a graph, the wavelength being the length of a stand of drill pipe. With low mud weights, high weights on bit, and large bit sizes the effect is negligible. This effect is magnified by high mud weights, low weights on bit, and small bit sizes. This effect is the source of the myth “d-Exponent doesn’t work with PDC bits.”
The cause of this phenomenon is Archimedes Principle is not being accounted for. The buoyancy force on the drill pipe increases as the hole is drilled and the stand of drill pipe is submerged into the drilling mud. The actual weight on bit is less than what is being registered as WOB referencing the hookload prior to recommencing drilling. As a result, the hookload WOB is steadily increased to maintain a desired ROP as the stand is drilled down. This causes the sawtooth effect in the d-Exponent data set. When the mud weight is increased the buoyancy force increases. When a low weight on bit is required the buoyancy force becomes a significant percentage of the initial WOB determination. As an example, when 5,000 lbs is desired the buoyancy force can be more than 3,000 lbs when the stand is fully submerged into the drilling mud. To apply the desired 5,000 lbs the WOB from the hookload must be increased, thus seemingly applying an increased WOB, thus affecting the d-Exponent calculation when the actual weight on bit has not increased.
To correct for this and calculate the actual weight on bit it is necessary to apply Archimedes Principle and subtract the buoyancy force from the WOB value. With this actual weight on bit as input into the d-Exponent equation the sawtooth waveform will diminish to the point of being negligible.