Drilling optimization models provide a technique for predicting and controlling drilling processes. Rate of penetration (ROP) prediction before drilling provides a means of improving overall drilling efficiency, minimization of drilling time and by extension reduction of drilling cost. Several ROP models are available in the oil and gas industry today, however most of these models are inadequate leading to inaccurate predictions. This work employs a different technique for ROP prediction by modifying original Burgoyne and Young’s (B&Y) model using data from Niger Delta. The ROP prediction and percentage error were determined at each of the selected well depth intervals The results obtained showed that the Burgoyne &Young model performed well by error of about 0.03%. The model was further optimized to minimize the error and the overall result shows that the error was minimized from 0.03% to 0.003%. Results obtained from the optimized Burgoyne and Young model imply that the model is suitable for ROP prediction in the Niger Delta.

Keywords: drilling operation, optimization, rate of penetration, burgoyne, young model

The major goal of every drilling operation is to make holes in the ground from surface to a reference depth mainly for oil or gas exploitation. Petroleum industry is a capital intensive industry. There is a need to save time, cost and increases efficiency. One of the most costly aspects of the industry is exploration and drilling and therefore has lot of potential for optimization and reducing cost. Planning and predicting future drilling operation base on controllable variables will be essential in order to realize these efficiency gains.^1–3 This may aid ROP prediction using mathematical models.^4–12 The study area is located in the onshore part of Niger Delta sedimentary basin. The subarea covered by the Niger Delta basin is about 7500km² a total area of 300,000km² and sediment fill has a depth between 9–12km.¹³ It is composed of several different geological formations that indicate how this basin was formed, as well as the regional and large scale tectonics of the area. In addition this basin is an extensional basin surrounded by many other basin in the area that all formed from similar process. The Niger Delta basin is bounded by the Cameroon volcanic line and the transform passive continental margin.¹³

The stratigraphic structure of the Niger Delta basin is divided into three (3) unit Benin formation, Agbada formation and Akata formation. Benin formation is the topmost formation followed by Agbada formation at the middle then the Akata formation which is lowest. The Benin formation is made up continental sand deposit with shale intercalation covered with topmost low velocity layer, which in most cases is weathered within which surface wave are excited and generated. The Agbada formation is below the Benin formation. It contains reservoir sand which traps the hydrocarbon resources of the Niger Delta Basin. The Akata formation is dominated by shale it serves as the main source of hydrocarbon in the Niger Delta Basin. Economically Niger Delta basin has a very high economic value. It contains a very predictive petroleum system, it produce more than 2 million barrels of oil per day. The entire system is predicted to contain 34.5 billion barrels of oil and 95 trillion cubic feet of natural gas.¹⁴ These make it one of the largest oil production provinces in the world. 

Rate of penetration models

Mathematical drilling models provide method to predict and control drilling process and minimize drilling cost. Drilling models also provide a means of recognizing unusual effect when the observed but performance deviate from prediction rate, some of this models are Burgoyne and Young (B&Y), Mechanical Specific Energy (M.S.E), D-exponent, modified D-exponent, Cunningham, Maurer, Bingham, Moore, Warren Motahari etc. drilling models. Drilling parameters obtained from two (2) wells (well A & B) drilled within the Niger.¹⁵

Burgoyne and young (B&Y) drilling models is the most complete mathematical that has been used for rolling cutter bit. In 1973 B&Y suggested a drilling model considering the effect of several drilling variables on the rate of penetration. In this model the effect of the parameters such as WOB, RPM, Bit tooth wear and other assumed to be independent of one another.

$R=(f1)(f2)(f3)(f4)------(fn)$ (1)

$f1=e^{2.303a1}=k$ (2)

$f2=e^{2.303a2(10000-D)}$ (3)

$f3=e^{2.303a3D^{0.69(gp-9.0)}}$ (4)

$f4=e^{2.303a4D(gp-pc)}$ (5)

$f5={\left[\frac{(\frac{w}{db})-{(\frac{w}{db})}_t}{4-{(\frac{w}{db})}_t}\right]}^{a5}$ (6)

$f6={\left(\frac{N}{60}\right)}^{a6}$ (7)

$f7=e^{-a7h}$ (8)

$f8={\left(\frac{F_j}{1000}\right)}^{a8}$ (9)

D= true vertical depth (ft)

gp= Pore pressure gradient (lbm/ft)

$\rho c$ =equivalent circulating density

$\left(\frac{w}{db}\right)t=$ Threshold bit weight per inch of bit diameter at which the bit begin to drill 1000lbf/inch

h=fractional hook dullness

Fj=hydraulic impact force beneath the bit force lbf

a1 to a8 = constant that must be chosen based on local drilling conditions

f1=function represent the effect of formation strength and bit type on penetration rate.

f₂=Function account for the rock strength increase due to normal compaction with depth.

f₃=function model effect of under compaction experienced in abnormal pressure formation.

f₄=function model the effect of bit weight and rotary speed on penetration rate.

f₅&f₆ function account for the rock strength increase due to normal compaction with depth.

f₇=Function models the effect of tooth wear.

f₈==function model the effect of bit hydraulic on rate of penetration.

h=fractional tooth dullness

F_j=hydraulic impact force beneath the bit (lbf)

The constant a₁ through a₈ can be computed using past drilling data obtained in the area when drilling data is available. The above drilling model can be use for drilling optimization calculation and for detection of changes in formation pore pressure.¹⁶

Rate of penetration prediction was done using Burgoyne and Young drilling model. Well data in Niger Delta Basin was collected from Nigeria Petroleum Development Company a subsidiary of Nigeria National Petroleum Company (N.N.P.C). The name of the well was deleted from the given data for confidential purpose, the name of the well was renamed as Well A with a total depth of 9664ft. The Well data consist of drilling parameters such as well depth, rate of penetration, weight on bit, flow rate, rotation per minute, torque, bit diameter, stand pipe pressure, etc. These parameters were analyzed using B&Y models. Well depth from 1000ft to 9000ft at 200ft interval was selected for this analysis. In this model there are some unknown parameters co-efficient which must be determined based on past drilling data obtained from a field in order to determine the unknown parameters, a linear regression technique will be applied which as follows.

Y = α₀+α₁β1+α₂β2+α₃β+α₄β+α₅β+----αnβn (10)

Where Y is the dependent variable,α₀ is the intercept term and the regression co-efficient α₁,α_2,α_3,------ α_n are the analogues of the shape of linear regression. From the above equation Y is the ROP; relevant drilling parameters will make up the regression variable [β_1&β_n]. α₀ to α_n Co-efficient will be determined by using a software called statistical package for social science (SPSS Software). This SPSS software will perform the regression analysis after all the relevant drilling parameters has been uploaded into it and then run. The analysis will then provide an output computed data. The generated output data are now co-efficient of interest. The first value generated will be α₀ while the values after this are the co-efficient i.e. (α₁ to α_n). These values are to be multiplying with regression variable according to their order which is given as follows:

$ROP={\alpha }_o+{\alpha }_{1}WOB+{\alpha }_{2}FR+{\alpha }_{3}RPM+{\alpha }_{4}TRQ+{\alpha }_{5}Bd+{\alpha }_{6}SPP$ (11)

After substituting the values of the generated co-efficient (i.e.α₀ to α₆) and drilling factors (β₁ to β₆) in the above equation using Microsoft excel software a new predicted ROP is now obtained in ft. /hrs at every selected depth. 

Error calculation

The percentage error at each selected depth was calculated using the formula below

$PercentageError=\frac{\mathrm{Pr}edictedROP-ActualROP}{ActualROP}*100\%$ (12)

While the average percentage error was also calculated using the formula below.

$AveragePercentageError=\frac{{\displaystyle \sum \left(\frac{\mathrm{Pr}edictedROP-ActualROP}{ActualROP}\right)}}{N}*100\%$ (13)

The above error calculation was carried out using Microsoft excel.

Error minimization

In order to minimize the error the B&Y linear regression model was further developed as follows

$ROP={\alpha }_o+{\alpha }_{1}WOB+{\alpha }_{2}FR+{\alpha }_{3}RPM+{\alpha }_{4}TRQ+{\alpha }_{5}Bd+{\alpha }_{6}SPP+{\alpha }_{7}MW$ (14)

In the above equation another drilling parameter i.e. mud weight was introduced. The analysis was carried out using the same procedure as it was done initially in the B&Y linear regression analysis and a very desirable result was obtained.

The table above shows the generated coefficient from Statistical Package for Social Sciences Software for both initial B&Y and new B&Y regression analysis. Comparison was made between the actual ROP and predicted ROP. The actual ROP was the ROP contained in the original well Data (Well A Data) while the predicted ROP was the ROP calculated using B&Y models (Table 1). The comparison was done between the well depths of 1000ft to 9000ft at 200ft depth interval. Percentage error at each depth interval was calculated and finally average percentage error was also calculated. A graph of ROP was plotted against well depth, depth on the horizontal axis in feet (ft.) and ROP on the vertical axis in feet per hour (ft./hr.) on the vertical axis. From the graph, the actual ROP was considered as the reference plot represented with blue dotted point (Table 2). The predicted ROP are represented with lines in difference colors (Table 3). B&Y line graph seem to correlate very well in many section of the well, this shows that this model performed very well (Figure 1).

Figure 1 Actual & predicted ROP vs. well depth.

Predicted B&Y model was further modeled by including additional drilling parameter (i.e. mud weight) into the initial regression equation and analyzed. The result shows that the error was minimized from 0.03% to 0.0003%. Actual ROP, Predicted ROP of initial B&Y and modified B&Y in feet per hour was plotted against well depth measured in feet. From the fig, the actual ROP was considered as the reference plot represented with dotted point. The initial and modified B&Y predicted ROP are represented with lines (Figure 2). Initial and modified B&Y line graph seem to correlate very well in many section of the well, this shows that the modified B&Y model performed very well even more than the initial B&Y Model. The error difference between initial & and new predicted ROP using B&Y model can be can be clearly illustrated from the chart in Figure 3.

Figure 2 Actual, initial B&Y and new B&Y ROP vs. well depth.

Figure 3 Average percentage error vs. Initial B&Y and new B&Y.

B&Y model has been tested with Niger Delta well data for ROP prediction, the result shows that the model performed very well by producing a little amount of error of about 0.03% and the error was further minimized to 0.0003% after inclusion of additional drilling parameter. The model can estimate ROP as function of several drilling parameters such as WOB, RPM, Mud weight, Standpipe Pressure, Torque, flow rate, mud weight etc. with a reasonable accuracy.

The result can also provide a guide for next drilling operation near the drilled well within the Niger Delta basin and the predicted values can be used as a reference to obtain optimum drilling performance and therefore reduce cost and time of drilling operation.

We want to thank the Nigerian Petroleum Development Company (NPDC) for providing us with the data we used in the work.

There are no conflicts of interest.

None.

1. Kate Van Dyk. Fundamentals of Petroleum. 4th ed. 1997.
2. Adam T Bourgoyne Jr. Applied Drilling Engineering. 1991.
3. Morten Adamsen Husvaeg. ROP Modeling And Analysis. Masters Thesis; 2015.
4. Teale R. The Concept of Specific Energy in Rock Drilling. Int J Rock Mechanics Mining Science. 1965;2(1):57–73.
5. Bahari MH, Moharrami B. Burgoyne And Young Model Co. efficient Using Generic Algorithm to Predict Drilling Rate. Journal of Applied Science. 2008;8(17).
6. Bourgoyne,A.T And Young. A Multiple Regression Approach To Optimal Drilling And Abnormal Pressure Detection. Society of Petroleum Engineers SPE Reprint series. 1999;49:27–40.
7. Ramadan A. Mathematical Modeling of Drilling Foam Flows”, IMV ProJect, Ergun Kuru, University of Alberta season Arild, Statoil; 2000.
8. Maurer WC. The perfect Cleaning Theory of Rotary Drilling. Journal of Petroleum Technology. 1962.
9. Michele LW Tuttle, Ronald R Charpentier, Michael E Brownfield. The Niger Delta Petroleum System. Niger Delta province Nigeria, Cameroon and Equatorial Guniea, Africa; 2015.
10. Fatoke. Sequence Stratigraphy of the Pliocene -Pleistocene Strata and Shelf margin delta of the eastern Niger Delta. Nigeria (Ph.D.) University of Houston; 2010.
11. Kurfe Udis, MfonAkpan, OkechukwuAgbosi. Estimation of over pressure in onshore Niger Delta using Wireline data. IJSR. 2013;4:438.
12. Adam T Burgoyne, Keith K, Milkam Martin E, et al. Applied Drilling Engineering SPE text series. 1986;2.
13. Samera M Hamad-Allahh, Ali A Ismael. Application of Mathematical Drilling model on southern Iraq oil fields. University of Baghdad/College of Engineering, Petroleum Engineering Department. I Journal of Engineering. 2018;3(14).
14. Cesar Maltos de Salles Soares. Development and Application of a new system to Analyze Field Data and Compare Rate of Penetration (ROP) model. Master’s Thesis, University of Texas; 2015.
15. Andreas Nascimento, David Tamas Kutas, Asad Elmgerbi. Mathematical Modeling Applied to Drilling Engineering. An Application of Burgoyne and Yong ROP model to a presalt case study. Hindawi publishing corporation mathematical problems in Engineering. 2015; ID631290.
16. Eten T. Real time-optimization of drilling parameter during drilling operation [P.hd thesis], middle East Technical University; 2015.
17. Malik Alsenwar. NCS Drilling Data Based ROP Modeling and its Application. Master Thesis Faculty of Science & Technology, University of Stavenger; 2017.
18. Fuad Mammadov. Developing Drilling Optimization Program for Galle and Method. M.Sc. Thesis; Istanbul Technical University; 2010.