Introduction:

Definition: According to the Oilfield Glossary [3], the Rate of Penetration is the speed at which the drill bit can break the rock under it and thus deepen the wellbore. This speed is usually reported in units of feets per hour or meters per hour.

Measurment: ROP logs are recorded as the well is being drilled. In those logs, we can find ROP expressed as either distance per unit of time or time per unit of distance. Typically, ROP is calculated by measuring the length of time required to drill 1 ft of depth. This was usually done by reading the chart on the geolograph. But now we use a digital encoder or transducer that is attached to a part of the rig that moves in proportion to the movement of the drill string. Common attachment points are the drill line, drawworks drum, or crown sheaves. 

Applications of ROP Measurments: ROP logs serve as historical records of drilling perfomance and can help optimizing drilling performance and formation evaluation.

• Bit Selection: ROP logs will help to choose the best bit type with appropriate mechanic and hydaulic parameters.
• Mud weight adjustment: The drilling rate of shale is very sensitive to the pressure balance between the shale and wellbore. Shales drilled underbalanced are characterized by fast drilling, high gas readings, and large cuttings. Shales drilled overbalanced are characterized by slow drilling, low gas readings, and small cuttings. Recognizing this relation can aid in the selection of a mud weight to optimize drilling speed.
• Correlation: The ROP log is the first source of data that can be used to correlate to nearby wells. This is done by comparing ROP to gamma ray or SP curves. This correlation can help determine structural and stratigraphic position and is used to predict when the well will reach a zone of interest.
• Lithology: Shales generally drill slower than sandstones or carbonates, and as a result, ROP logs tend to reflect indirectly the lithology of the formation.

Factors Affecting ROP:

 In order to develop an ROP model, we should first look at the predominant factors that are affecting the rate of penetration. Experience and research suggest these variables:

• Bit type
• Formation characteristics
• Drilling Fluid properties
• Bit operating conditions
• Bit hydraulics
• Bit tooth wear
• Personal efficiency

Bit Type: The bit type selection has a predominant effect on the ROP. For roller-cone bits, the initial penetration rates are often the highest specially when using bits with long teeth and a large offset. However, this is followed by a rapid decline in penetration rates specially in hard formations.

Fixed-cutter bits give a wedging-type rock destruction mechanism in which the bit penetration per revolution depends on the number of blades and the bottom-cutting angle. Diamond and PDC bits are designed for a given penetration per revolution by the selection of the size and number of diamond or PDC cutters. Developments in PDC bits that have helped in achieving higher ROPs and longer bit life, but also involve a compromise between open, light-set bits for speed and heavy-set bits for durability. Hydraulic design improvements prevent bit balling, while mechanical-design enhancements increase the ROP.

Formation Characteristics: several characteristics of the formation affect the ROP, the most important are certainly the elastic limit and the ultimate strength of the formation. Mohr failure criterion is often used to characterize the strength of a formation (Now new models do exist providing more accurate values, but a majority are a modification of Mohr's equations under specific hypothesis). 

The other factors such as permeability of the formation has a significant effect on the ROP. In permeable rocks, the drilling fluid filtrate can move into the rock ahead of the bit and equalize the pressure differential acting on the chips formed beneath each tooth. The nature of fluids contained in pore spaces of the rock also affects the mechanism, more filtrate volume will be required to equalize the pressure in a rock containing a lighter fluid (or even gas) than in a rock containing a heavy fluid.

The mineral composition of the rock also has some effect on ROP. Rocks containing hard, abrasive minerals can cause rapid dulling of the bit teeth. Rocks containing gummy clay minerals can cause the bit to ball up and then drilling in an inefficient manner.

Drilling Fluid Properties: the properties of the drilling fluid that affect the penetration rate are essentially density, rheological properties, filtration characteristics, solids content and the chemical composition. 

Penetration rate tends to decrease with increasing fluid density, viscosity, and solids content, and tends to increase with increasing filtration rate. The density, solids content, and filtration characteristics of the mud control the pressure differential across the zone of crushed rock beneath the bit. The fluid viscosity controls the parasitic frictional losses in the drill string and, thus, the hydraulic energy available at the bit jets for cleaning. There is also experimental evidence that increasing viscosity reduces penetration rate even when the bit is perfectly clean. The chemical composition of the fluid has an effect on penetration rate by the influence of the hydration rate and bit-ballling tendency of some clays that are affected by the chemical composition of the fluid.

An increase in drilling fluid density causes an increase in the bottomhole pressure beneath the bit, and then, an increase in the pressure differential between the borehole pressure and the formation fluid pressure, so when this differential is positive, it's called an overbalance state. The Mohr failure criterion predicts similar effects of overbalance either we use a roller-cone bit or fixed-cutter bit. The relation between overbalance and penetration rate could be represented approximately by a straight line on semilog paper for the range of overbalance commonly used in field practice [1].

*Exponential relation between penetration rate and overbalance for roller-cone bits [1]

Where R is the penetration rate and R0 is the penetration rate at zero overbalance.

So, if  the pressure differential between the borehole pressure and the formation fluid pressure is negative, this is an underbalance state. And if we look at the curve for ''negative'' values of overbalance, we observe that we achieve high penetration rates, and this is effectively the case and also one of the main advantages of the Underbalanced Drilling technique.

Bit Operating Condition: here we are talking about the rotation speed and the weight on bit. First we'll evaluate the effect of each of them separately and finally expose a combined model for roller-cone bits that also takes in consideration bit size and rock strength.

          1.  Rotary Speed: penetration rate (R) usually increases linearly with increasing value of rotary speed (N) for low values of rotary speed. At higher values, the rate of increase diminishes. The point where we loose linearity is called the Foundering point (point b in the figure below). The phenomenon is essentially due to less efficient bottomhole cleaning and is also dependent to drilling fluid parameters (density, buoyancy factor).

*Relation between ROP and rotation speed

          2. Weight-On-Bit (WOB): a plot of penetration rate(R) vs. WOB (W) is illustrated in the figure below (assuming no tooth wear). No significant penetration rate is obtained until the threshold formation stress is exceeded (Point a). Penetration rate increases gradually and linearly with increasing values of bit weight for low values of bit weight (Segment ab). A linear curve is again observed at higher bit weights (Segment bc), segment bc has a much steeper slope, representing increased drilling efficiency. Point b is the transition point where the rock failure mode changes from scraping or grinding to shearing. Beyond Point c, subsequent increases in bit weight cause only slight improvements in penetration rate (Segment cd). In some cases, a decrease in penetration rate is observed at extremely high values of bit weight (Segment de, point d is the Foundering point). The poor response of penetration rate at high WOB values is usually attributed to less-efficient hole cleaning because of a higher rate of cuttings generation, or because of a complete penetration of a bit’s cutting elements into the formation being drilled, without clearance for fluid bypass.

*Relation between ROP and the weight-on-bit

Maurer developed a theoretical equation for rolling cutter bits relating ROP to bit weight, rotary speed, bit size, and rock strength. The equation can be written as:

Where R is the penetration rate, K is the constant of proportionality; S is the compressive strength of the rock; W is the bit weight; db is the
bit diameter; (W/db)t is the threshold bit weight per inch of bit diameter; and N is the rotary speed.

This theoretical relation assumes perfect bottomhole cleaning and incomplete bit tooth/insert penetration. It was verified for low values of bit weight and rotary speed.

Bit Hydraulics: The introduction of the nozzle-type roller-cone bits showed that significant improvements in penetration rate could be achieved through an improved jetting action at the bit. The improved jetting action promoted better cleaning of the bit teeth as well as improved bottomhole jetting action.

Historically, there have been many positions taken as to the best hydraulics parameter to use in characterizing the effect of hydraulics on bit performance. Bit hydraulic horsepower, jet-impact force, flow rate, and nozzle velocity are all parameters commonly discussed. The level of hydraulics achieved at the bit is also thought by many to affect the foundering point of the bit.

Eckel proposed the following model based on Reynolds number [1]:

where K is a scaling constant, р is the drilling-fluid density, v is the flow rate, d is the nozzle diameter, and μa is
the apparent viscosity of drilling fluid.

The results are shown in the graphs below. We notice that penetration rate is increased by increasing the Reynolds-number function for the full range of Reynolds numbers studied. When the bit weight was increased, the correlation curve simply was shifted upward as shown in the figure below. The behavior at the founder point was not studied by Eckel [1].

*Effect of Reynolds number and WOB on ROP [1]

Bit Tooth Wear: Most bits tend to drill slower as the drilling time elapses because of tooth wear. The tooth length of milled tooth rolling cutter bits is reduced continually by abrasion and chipping. The teeth are altered by hard facing to promote a self-sharpening type of tooth wear. However, while this tends to keep the tooth pointed, it does not compensate for the reduced tooth length. The teeth of tungsten carbide insert-type rolling cutter bits and PDC bits fail by breaking rather than by abrasion. Often, the entire tooth is lost when breakage occurs. Reductions in penetration rate due to bit wear usually are not as severe for insert bits as for milled tooth bits unless a large number of teeth are broken during the bit run.

Personal Efficiency: Manpower skill and experience –  Personnel are the key to the success or failure of those operations and ROP is one of them. Overall well costs as a result of any drilling problem can be extremely high; therefore, continuous training for personnel directly or indirectly involved is essential in order to achieve desired ROP.

References:

[1] R. F. Mitchell, S. Z. Miska ''Fundamentals of Drilling Engineering'', 2011.

[2] E. Hossain ''Factors Affecting ROP and optimization techniques''

[3] ''ROP'' Oilfield glossary Link

[4] ''Rate of Penetration'' AAPG wiki Link