2. Mechanical specific energy model development

#### 2.1. Key models of mechanical specific energy

Mechanical specific energy (MSE) has been defined as the mechanical work done to excavate a unit volume of rock. Teale in 1965 initially proposed the MSE model for rotating drilling system [6].

$$MSE = \frac{WOB}{A\_b} + \frac{120\pi \cdot RPM \cdot T}{A\_b \cdotROP} \tag{1}$$

In the above model, torque at the bit is a main variable. Although torque at the bit can be easily measured in the laboratory and with Measurement While Drilling (MWD) systems in the field, the majority of field data is in the form of surface measurement. While in the absence of reliable torque at the bit measurements, the calculation of MSE based on this model contains even large sources of error. Therefore, it is only used qualitatively as a trending tool.

which the ROP is maximized [5]. Although these methods have enhanced drilling performance, they do not provide an objective assessment of the true potential ROP, only the founder point of the current system. Actually the process of optimizing drilling parameters should be not only drilling system specific but also formation specific. MSE is defined as the mechanical work done to excavate a unit volume of rock, it could provide an objective assessment of the drilling efficiency and an objective tool to identify the bit founder. The initial MSE model for rotating drilling system was proposed by Teale in 1965 [6]. In this model, as the majority of field data is in the form of surface measurements, which results in MSE's calculation containing even large sources of error. Then numerous investigators were motivated to develop more accurate models. These models include those presented by Pessier and Fear [7], Dupriest and Koeteritz [5], Armenta [8], Mohan et al. [9], Cherif [10], Mohan et al. [11] and they have been widely used in bit selection, drilling efficiency quantification, drilling performance monitoring, drilling performance optimization, ROP improvement and so on. Although the MSE obtained from these models are more and more precisely model the actual downhole drilling in vertical wells, currently there are few effective MSE models to precisely model the actual downhole drilling in directional or horizontal wells due to the majority of field data is in

Moreover, in recent years, PDM has gained widespread use in the hard formation drilling to improve ROP. In rotating drilling with PDM, the power section of PDM converts hydraulic energy of mud flow into mechanical rotary power, the surface rotation is superimposed on downhole motor rotation. During slide drilling, bit rotation is generated only from the PDM as drilling fluid is pumped through the drill string. However, the PDM's performance is controlled by the combination of the rotor/stator lobe configuration, and the direct measurement of PDM rotary speed and torque in down hole has proven difficult. Therefore, currently there are also few effective MSE models to precisely model the actual downhole drilling for rotating

In this chapter, MSE models respectively for directional or horizontal drilling and rotating drilling with PDM are established based on the evaluation of key MSE models and the analysis on PDM performance, meanwhile methods for drilling performance prediction and optimiza-

Mechanical specific energy (MSE) has been defined as the mechanical work done to excavate a unit volume of rock. Teale in 1965 initially proposed the MSE model for rotating drilling

120π � RPM � T

Ab � ROP (1)

MSE <sup>¼</sup> WOB Ab þ

the form of surface measurements.

tion based on MSE technologies are presented.

2.1. Key models of mechanical specific energy

2. Mechanical specific energy model development

drilling with PDM.

134 Drilling

system [6].

In 1992, Pessier and Fear provided a simple method of the calculation of torque at bit while in the absence of reliable torque measurements and optimized Teale's model [7].

$$\begin{aligned} MSE &= \text{WOB} \cdot \left( \frac{1}{A\_b} + \frac{13.33 \cdot \mu\_b \cdot RPM}{D\_b \cdotROP} \right) \\ \mu\_b &= 36 \frac{T}{D\_b \cdot WOB} \end{aligned} \tag{2}$$

The above model's parameters are easy to be obtained on the ground, and its calculation precision has been improved, as a result, it has a common usage in the drilling industry. In this model, the torque of bit is calculated through WOB. However, WOB is always read based on the surface measurement, which is not the bottom hole real WOB. As for directional and horizontal drilling, there is a great difference between the bottom hole real WOB and the WOB of surface measurement [12]. And every bit has a certain mechanical efficiency in drilling even for the new bits, thus Pisser's model has a limited application and also exists a certain error in MSE calculation.

Given the bit had a certain mechanical efficiency in the actual drilling process, Dupriest, Cherif and Amadi defined a mechanical efficiency on the base of Teale model [5, 10, 13].

$$MSE = E\_m \cdot \left(\frac{\text{WOB}}{A\_b} + \frac{120 \cdot \pi \cdot RPM \cdot T}{A\_b \cdot ROP} \right) \tag{3}$$

Dupriest and Koederitz thought peak bit efficiencies are always in the 30–40% range, therefore thought the mechanical efficiency were 35% [5]. However, this is a controversial issue due to the bits' mechanical efficiency depending on a variety of factors, and it may vary greatly from the assumed 35%. Cherif argued that the mechanical efficiency were 26–64% instead of 35% [10]. In directional and horizontal drilling, the MSE values may eventually become several times the formation CCS due to torsional friction. So Amadi and Iyalla thought the mechanical efficiency were 12.5% in directional and horizontal drilling [13]. Actually the mechanical efficiency is not only bit specific but also formation specific, and it may vary greatly from bit to bit and formation to formation, so it must be determined according to the real drilling conditions. Therefore, the model also has certain limitations.

Recently some researchers think that hydraulic energy also aids in actual drilling for certain formations, then they add the hydraulic term to the MSE function as [9, 11].

$$MSE = \frac{WOB}{A\_b} + \frac{120\pi \cdot RPM \cdot T}{A\_b \cdotROP} + \frac{\beta \cdot AP\_b \cdot Q}{A\_b \cdotROP} \tag{4}$$

Hydraulic energy has a great influence on drilling efficiency, but its role is complex. In conventional rotating drilling, bit hydraulics mainly accounts for the removal of cuttings from the bottom hole by jet-erosion, and the jet from bit nozzles could hardly aid in rock-broken especially in the deep and hard formations. Therefore, the MSE model is suitable for high pressure jet drilling and soft formation drilling.

F2

F2 00

0 -F1 F2 <sup>0</sup> � F<sup>2</sup> 00 ¼ F<sup>1</sup>

WOBb > 0:

Obviously, "F2

where:

F2 0

F2

Eq. (7) minus Eq. (8), we get

0 -F2

lower end of the bend, "F1

(2) In straight sections.

<sup>00</sup> , then gives.

WOBb ¼ 0:

WOBb > 0:

Apparently, "F2

the lower end, "F1

Eq. (13) minus Eq. (14), we get

<sup>0</sup> � F2

<sup>0</sup> � F1

the upper end. So we may express Eq. (15) as follows

<sup>0</sup> ¼ fð Þþ α<sup>2</sup> F<sup>1</sup>

¼ fð Þþ α<sup>2</sup> F<sup>1</sup>

WOBb at the upper end of the bend. So we may express Eq. (9) as follows

In straight sections, the drag model is as follows in the process of drilling

If WOBb ¼ 0, assuming that the force at the upper end is F1

. If WOBb > 0, using that the force at the upper end is F1

F2 <sup>0</sup> ¼ F<sup>1</sup>

F2 00 ¼ F<sup>1</sup> 00

> F2 <sup>0</sup> � F<sup>2</sup> 00 ¼ F<sup>1</sup>

Fi<sup>2</sup> ¼ Fi<sup>1</sup> � e

00

<sup>0</sup> ð Þ� � fð Þ α<sup>1</sup> e

� fð Þ α<sup>1</sup> � <sup>e</sup>

<sup>0</sup> � F<sup>1</sup> <sup>00</sup> <sup>e</sup>

<sup>00</sup>" is the internal force of drill string produced by bottom hole WOBb at the

<sup>00</sup>" is the internal force of drill string produced by bottom hole

α<sup>2</sup> � α<sup>1</sup> ¼ Δα ¼ Δγ (11)

<sup>0</sup> <sup>þ</sup> <sup>w</sup> � <sup>Δ</sup><sup>s</sup> � <sup>μ</sup> sin <sup>α</sup> � cos <sup>α</sup> (13)

<sup>þ</sup> <sup>w</sup> � <sup>Δ</sup><sup>s</sup> � <sup>μ</sup> sin <sup>α</sup> � cos <sup>α</sup> (14)

<sup>F</sup><sup>2</sup> <sup>¼</sup> <sup>F</sup><sup>1</sup> <sup>þ</sup> <sup>w</sup> � <sup>Δ</sup><sup>s</sup> � <sup>μ</sup> sin <sup>α</sup> � cos <sup>α</sup> (12)

0

<sup>0</sup> � F<sup>1</sup>

<sup>00</sup>" is the internal force of drill string produced by bottom hole WOBb at

<sup>00</sup>" is the internal force of drill string produced by bottom hole WOBb at

�μ αð Þ <sup>2</sup>�α<sup>1</sup>

Drilling Performance Optimization Based on Mechanical Specific Energy Technologies

�μ αð Þ <sup>2</sup>�α<sup>1</sup> (8)

http://dx.doi.org/10.5772/intechopen.75827

�μ αð Þ <sup>2</sup>�α<sup>1</sup> (9)

�μ αð Þ <sup>2</sup>�α<sup>1</sup> (10)

, and the force at the lower end is

<sup>00</sup> , and the force at the lower end is

<sup>00</sup> (15)

(7)

137

In the above MSE models, MSE's calculation containing even large sources of error due to the majority of field data is in the form of surface measurements. Especially in directional and horizontal drilling, WOB and torque of surface measurement differs greatly from bottom hole actual WOBb and torque [12]. Therefore, few of the above MSE models can precisely model the actual downhole drilling in directional or horizontal wells. Moreover, in rotating drilling with PDM, the surface rotation is superimposed on downhole motor rotation [14]. During slide drilling, bit rotation is generated only from the PDM as drilling fluid is pumped through the drill string. However, the direct measurement of PDM rotary speed and torque in down hole has proven difficult, so few of the above MSE models can also precisely model the actual downhole drilling for rotating drilling with PDM.

#### 2.2. Mechanical specific energy model of directional or horizontal drilling

#### 2.2.1. Model of bottom hole WOBb

Undersection trajectory of directional well or horizontal well can reduce drag greatly compared to a conventional tangent section due to well friction. Therefore, there is a great difference between surface measured WOB and bottom hole WOBb at the bit. The surface measured WOB is actually the bottom hole WOBb acting on the ground. Therefore, by analyzing the internal force of drill string produced by bottom hole WOBb in each well section, we can get the formula between the surface measured WOB and bottom hole WOBb.

#### (1) In bends section.

In 2008, Aadnoy formulated the drag model in bends and straight sections [15]. In the process of drilling, assuming the string contacts lower side, so the drag model in bends section is as follows

$$F\_2 = f(a\_2) + (F\_1 - f(a\_1)) \cdot e^{-\mu(a\_2 - a\_1)}\tag{5}$$

where:

$$f(a) = \frac{w \cdot R}{1 + \mu^2} \left\{ \left( 1 - \mu^2 \right) \sin a + 2\mu \cos a \right\} \tag{6}$$

If WOBb ¼ 0, assuming that the force at the upper end of the bend is F1 0 , and the force at the lower end of the bend is F2 0 . If WOBb > 0, using that the force at the upper end of the bend is F1 <sup>00</sup> , and the force at the lower end of the bend is F2 <sup>00</sup> , then gives

WOBb ¼ 0:

Drilling Performance Optimization Based on Mechanical Specific Energy Technologies http://dx.doi.org/10.5772/intechopen.75827 137

$$F\_2 ' = f(a\_2) + \left( (F\_1 ' - f(a\_1)) \cdot \varepsilon^{-\mu(a\_2 - a\_1)} \right) \tag{7}$$

WOBb > 0:

Hydraulic energy has a great influence on drilling efficiency, but its role is complex. In conventional rotating drilling, bit hydraulics mainly accounts for the removal of cuttings from the bottom hole by jet-erosion, and the jet from bit nozzles could hardly aid in rock-broken especially in the deep and hard formations. Therefore, the MSE model is suitable for high

In the above MSE models, MSE's calculation containing even large sources of error due to the majority of field data is in the form of surface measurements. Especially in directional and horizontal drilling, WOB and torque of surface measurement differs greatly from bottom hole actual WOBb and torque [12]. Therefore, few of the above MSE models can precisely model the actual downhole drilling in directional or horizontal wells. Moreover, in rotating drilling with PDM, the surface rotation is superimposed on downhole motor rotation [14]. During slide drilling, bit rotation is generated only from the PDM as drilling fluid is pumped through the drill string. However, the direct measurement of PDM rotary speed and torque in down hole has proven difficult, so few of the above MSE models can also precisely model the actual

Undersection trajectory of directional well or horizontal well can reduce drag greatly compared to a conventional tangent section due to well friction. Therefore, there is a great difference between surface measured WOB and bottom hole WOBb at the bit. The surface measured WOB is actually the bottom hole WOBb acting on the ground. Therefore, by analyzing the internal force of drill string produced by bottom hole WOBb in each well section, we can get

In 2008, Aadnoy formulated the drag model in bends and straight sections [15]. In the process of drilling, assuming the string contacts lower side, so the drag model in bends section is as

�μ αð Þ <sup>2</sup>�α<sup>1</sup> (5)

0

, and the force at the

<sup>1</sup> <sup>þ</sup> <sup>μ</sup><sup>2</sup> <sup>1</sup> � <sup>μ</sup><sup>2</sup> sin <sup>α</sup> <sup>þ</sup> <sup>2</sup><sup>μ</sup> cos <sup>α</sup> (6)

. If WOBb > 0, using that the force at the upper end of the bend is

, then gives

00

F<sup>2</sup> ¼ fð Þþ α<sup>2</sup> ð Þ� F<sup>1</sup> � fð Þ α<sup>1</sup> e

pressure jet drilling and soft formation drilling.

downhole drilling for rotating drilling with PDM.

2.2.1. Model of bottom hole WOBb

(1) In bends section.

lower end of the bend is F2

follows

136 Drilling

where:

F1 00

WOBb ¼ 0:

2.2. Mechanical specific energy model of directional or horizontal drilling

the formula between the surface measured WOB and bottom hole WOBb.

w � R

If WOBb ¼ 0, assuming that the force at the upper end of the bend is F1

fð Þ¼ α

0

, and the force at the lower end of the bend is F2

$$F\_2^{\prime\prime} = f(a\_2) + \left(F\_1^{\prime\prime} - f(a\_1)\right) \cdot e^{-\mu(a\_2 - a\_1)}\tag{8}$$

Eq. (7) minus Eq. (8), we get

$$F\_2{"-} - F\_2{"-} ^\prime = \left(F\_1{"-} - F\_1{"-} ^\prime\right) e^{-\mu(a\_2 - a\_1)} \tag{9}$$

Obviously, "F2 0 -F2 <sup>00</sup>" is the internal force of drill string produced by bottom hole WOBb at the lower end of the bend, "F1 0 -F1 <sup>00</sup>" is the internal force of drill string produced by bottom hole WOBb at the upper end of the bend. So we may express Eq. (9) as follows

$$F\_{i2} = F\_{i1} \cdot e^{-\mu(\alpha\_2 - \alpha\_1)} \tag{10}$$

where:

$$
\alpha\_2 - \alpha\_1 = \Delta \alpha = \Delta \gamma \tag{11}
$$

(2) In straight sections.

In straight sections, the drag model is as follows in the process of drilling

$$F\_2 = F\_1 + w \cdot \Delta s \cdot \left(\mu \sin a - \cos a\right) \tag{12}$$

If WOBb ¼ 0, assuming that the force at the upper end is F1 0 , and the force at the lower end is F2 0 . If WOBb > 0, using that the force at the upper end is F1 <sup>00</sup> , and the force at the lower end is F2 <sup>00</sup> , then gives.

WOBb ¼ 0:

$$F\_2' = F\_1' + \overline{w} \cdot \Delta \mathbf{s} \cdot \left(\mu \sin a - \cos a\right) \tag{13}$$

WOBb > 0:

$$F\_2^{''} = F\_1^{''} + w \cdot \Delta s \cdot \left(\mu \sin \alpha - \cos \alpha\right) \tag{14}$$

Eq. (13) minus Eq. (14), we get

$$\left|F\_{2}{}^{\prime} - F\_{2}{}^{\prime}\right. = F\_{1}{}^{\prime} - F\_{1}{}^{\prime} \tag{15}$$

Apparently, "F2 <sup>0</sup> � F2 <sup>00</sup>" is the internal force of drill string produced by bottom hole WOBb at the lower end, "F1 <sup>0</sup> � F1 <sup>00</sup>" is the internal force of drill string produced by bottom hole WOBb at the upper end. So we may express Eq. (15) as follows

$$F\_{i2} = F\_{i1} \tag{16}$$

Therefore, in the straight sections, internal force produced by bottom hole WOBb in each crosssection of the drill string is the same. As for straight sections, α<sup>2</sup> � α<sup>1</sup> ¼ Δα ¼ 0, so Eq. (16) is the same as Eq. (10). Therefore, Eq. (10) is also suitable for straight section.

(3) Formula between WOB and WOBb.

In the horizontal well, on the surface

$$F\_i = F\_{i1} = \text{WOB}, \ a\_1 = 180^\circ, \ \gamma = 0^\circ \tag{17}$$

At the bit

$$F\_i = F\_{i2} = \text{WOB}\_b \quad a\_2 = 180^\circ + \gamma\_b \tag{18}$$

and

$$
\Delta \mathfrak{a} = \Delta \mathfrak{v} = \mathfrak{v}\_b \tag{19}
$$

Insert Eqs. (17), (18) and (19) into Eq. (10), then we get the formula between WOB and WOBb in horizontal well [12].

$$\text{WOB}\_b = \text{WOB} \cdot e^{-\mu \gamma\_b} \tag{20}$$

Usually the bit sliding coefficient of friction is assumed to be of an average value of 0.3 and 0.85

Figure 1. Relationship between weight on the bit ratio and bottom hole deviation angle (μ<sup>b</sup> is set to 0.35) [12].

WOB and torque are key variables in MSE calculation. In directional or horizontal drilling, they are greatly inflated for well friction. Eqs. (20) and (22) are the model of bottom hole WOBb and model of bottom hole torque at the bit, which are modified by wellbore wall friction coefficient and bottom hole inclination. They can fit the bottom hole's actual working conditions. However, it has also been observed, from lab data under confined bottom hole pressure, that MSE is often substantially higher than the rock CCS, even when the bit is apparently drilling efficiently, for bit has a certain mechanical efficiency in the actual drilling process even for a new bit [5]. Finally, substitute Eqs. (20) and (22) in Teale model (Eq. (1)) and consider the mechanical efficiency (Em) of the new bit, we can get a new model of MSE which can be shown as [12].

> Ab þ

The bit sliding coefficient (μb) of friction is assumed to be of an average value of 0.3 and 0.85 for rollercone and PDC bits respectively [16]. The drill string sliding coefficient (μ) of friction is

WOBb ¼ WOB � e

13:33 � μ<sup>b</sup> � RPM

Drilling Performance Optimization Based on Mechanical Specific Energy Technologies

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139

Db � ROP (23)

�μγ<sup>b</sup> (24)

[16] for rollercone and PDC bits respectively.

2.2.3. Mechanical specific energy model of directional or horizontal well

MSE <sup>¼</sup> Em � WOBb � <sup>1</sup>

Figure 1 shows the relationship between weight on the bit ratio and bottom hole inclination, it indicates that there is a big difference between the surface measured WOB and bottom hole WOBb for horizontal well drilling.

#### 2.2.2. Model of bottom hole torque at the bit

Torque at the bit can be measured with MWD systems in the field. However, the majority of field data is in the form of surface measurements, it usually uses of surface torque to calculate MSE, which results in the value of MSE eventually is inflated by torsional friction. In horizontal drilling, the baseline trend of MSE may become several times the rock confined compressive strength (CCS). For this reason, Pessier and Fear introduced a bit-specific coefficient of sliding friction to express torque as a function of WOB, which has been widely used to compute MSE values in the absence of reliable torque measurements [7].

$$T = \int\_0^{D\_b/2} \int\_0^{2\pi} \rho^2 \frac{4\mu\_b \text{WOB}}{\pi D\_b^2} d\rho d\theta = \int\_0^{D\_b/2} \frac{8\mu\_b \text{WOB}}{D\_b^2} \rho^2 d\rho = \frac{\mu\_b \cdot \text{WOB} \cdot D\_b}{3} \tag{21}$$

In Eq. (21), WOB is changed with WOBb. Then we get the model of bottom hole torque at the bit [12].

$$T\_b = \frac{\mu\_b \cdot \text{WOB}\_b \cdot D\_b}{3} = \frac{\mu\_b \cdot \text{WOB} \cdot e^{-\mu \gamma\_b} \cdot D\_b}{3} \tag{22}$$

Figure 1. Relationship between weight on the bit ratio and bottom hole deviation angle (μ<sup>b</sup> is set to 0.35) [12].

Usually the bit sliding coefficient of friction is assumed to be of an average value of 0.3 and 0.85 [16] for rollercone and PDC bits respectively.

#### 2.2.3. Mechanical specific energy model of directional or horizontal well

Fi<sup>2</sup> ¼ Fi<sup>1</sup> (16)

Fi <sup>¼</sup> Fi<sup>1</sup> <sup>¼</sup> WOB, <sup>α</sup><sup>1</sup> <sup>¼</sup> <sup>180</sup><sup>∘</sup> , <sup>γ</sup> <sup>¼</sup> <sup>0</sup><sup>∘</sup> (17)

Fi <sup>¼</sup> Fi<sup>2</sup> <sup>¼</sup> WOBb, <sup>α</sup><sup>2</sup> <sup>¼</sup> <sup>180</sup><sup>∘</sup> <sup>þ</sup> <sup>γ</sup><sup>b</sup> (18)

Δα ¼ Δγ ¼ γ<sup>b</sup> (19)

�μγ<sup>b</sup> (20)

<sup>r</sup><sup>2</sup>d<sup>r</sup> <sup>¼</sup> <sup>μ</sup><sup>b</sup> � WOB � Db

<sup>3</sup> (22)

<sup>3</sup> (21)

Therefore, in the straight sections, internal force produced by bottom hole WOBb in each crosssection of the drill string is the same. As for straight sections, α<sup>2</sup> � α<sup>1</sup> ¼ Δα ¼ 0, so Eq. (16) is

Insert Eqs. (17), (18) and (19) into Eq. (10), then we get the formula between WOB and WOBb in

Figure 1 shows the relationship between weight on the bit ratio and bottom hole inclination, it indicates that there is a big difference between the surface measured WOB and bottom hole

Torque at the bit can be measured with MWD systems in the field. However, the majority of field data is in the form of surface measurements, it usually uses of surface torque to calculate MSE, which results in the value of MSE eventually is inflated by torsional friction. In horizontal drilling, the baseline trend of MSE may become several times the rock confined compressive strength (CCS). For this reason, Pessier and Fear introduced a bit-specific coefficient of sliding friction to express torque as a function of WOB, which has been widely used to compute MSE

> ðDb=<sup>2</sup> 0

In Eq. (21), WOB is changed with WOBb. Then we get the model of bottom hole torque at the

8μbWOB D2 b

<sup>3</sup> <sup>¼</sup> <sup>μ</sup><sup>b</sup> � WOB � <sup>e</sup>�μγ<sup>b</sup> � Db

WOBb ¼ WOB � e

the same as Eq. (10). Therefore, Eq. (10) is also suitable for straight section.

(3) Formula between WOB and WOBb. In the horizontal well, on the surface

At the bit

horizontal well [12].

WOBb for horizontal well drilling.

T ¼

bit [12].

ðDb=<sup>2</sup> 0

ð<sup>2</sup><sup>π</sup> 0

2.2.2. Model of bottom hole torque at the bit

values in the absence of reliable torque measurements [7].

<sup>r</sup><sup>2</sup> <sup>4</sup>μbWOB πD<sup>2</sup> b

Tb <sup>¼</sup> <sup>μ</sup><sup>b</sup> � WOBb � Db

drdθ ¼

and

138 Drilling

WOB and torque are key variables in MSE calculation. In directional or horizontal drilling, they are greatly inflated for well friction. Eqs. (20) and (22) are the model of bottom hole WOBb and model of bottom hole torque at the bit, which are modified by wellbore wall friction coefficient and bottom hole inclination. They can fit the bottom hole's actual working conditions. However, it has also been observed, from lab data under confined bottom hole pressure, that MSE is often substantially higher than the rock CCS, even when the bit is apparently drilling efficiently, for bit has a certain mechanical efficiency in the actual drilling process even for a new bit [5]. Finally, substitute Eqs. (20) and (22) in Teale model (Eq. (1)) and consider the mechanical efficiency (Em) of the new bit, we can get a new model of MSE which can be shown as [12].

$$MSE = E\_m \cdot WOB\_b \cdot \left(\frac{1}{A\_b} + \frac{13.33 \cdot \mu\_b \cdot RPM}{D\_b \cdotROP}\right) \tag{23}$$

$$\text{WOB}\_b = \text{WOB} \cdot e^{-\mu \gamma\_b} \tag{24}$$

The bit sliding coefficient (μb) of friction is assumed to be of an average value of 0.3 and 0.85 for rollercone and PDC bits respectively [16]. The drill string sliding coefficient (μ) of friction is assumed 0.25 to 0.4, usually use the value of 0.35 [17, 18]. The mechanical efficiency (Em) of a new bit can be got by core samples' laboratory studies, or inversed by adjacent wells logging data.

Tideal ¼ 3:066 � ΔPm � q (25)

Drilling Performance Optimization Based on Mechanical Specific Energy Technologies

Tm ¼ Tideal � ΔT (27)

<sup>q</sup> (26)

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141

<sup>q</sup> (28)

2Fny

<sup>3</sup><sup>π</sup> (29)

<sup>h</sup>phΔPm þ

<sup>60</sup> � RPMm (31)

<sup>1714</sup> (32)

MHP ¼ η � HHP (33)

<sup>Δ</sup>Pm (30)

RPMideal <sup>¼</sup> <sup>Q</sup>

However, in actual drilling process, leakage and torque losses play important roles in the performance of a PDM. The actual rotary speed of the PDM is decreased by the slip flow through the seal line, and the actual torque is also decreased by the resisting torque due to mechanical friction, elastomeric friction and viscous shearing of drilling fluid. The actual PDM

RPMm <sup>¼</sup> <sup>Q</sup> � Qslip

<sup>h</sup>Lsμ þ Cf

DhnsΓ<sup>i</sup> tan α 12μLsLm

In Eqs. (29) and (30), many parameters are functions of motor geometry, property and even drilling conditions, some of them are difficult to be determined. Therefore, the prediction of Tm and RPMm has proven difficult. However, in PDM the mechanical power is converted by hydraulic horsepower, and it depends on the converting efficiency of the PDM. Then the mechanical power can be predicted based on its input hydraulic power. The mechanical

π 1 � i <sup>2</sup> 4 2ð Þ � <sup>i</sup> <sup>2</sup> <sup>D</sup><sup>2</sup>

i 1 � i

output torque and rotary speed can be estimated by

Torque losses is given by [20].

Slip flow is estimate as

<sup>Δ</sup><sup>T</sup> <sup>¼</sup> <sup>π</sup><sup>2</sup><sup>i</sup>

4 2 1ð Þ � <sup>i</sup> ð Þ <sup>2</sup> � <sup>i</sup> <sup>3</sup>

horsepower provided by PDM can be estimated by [21].

The hydraulic horsepower can be given as

Their relationship can be written as

RPMm <sup>δ</sup> <sup>D</sup><sup>3</sup>

MHP <sup>¼</sup> Tm

550

2π

HHP <sup>¼</sup> <sup>Q</sup> � <sup>Δ</sup>Pm

In Eqs. (32) and (33), the operating differential pressure drop across the motor, at a constant flow rate, can be measured by comparing off-bottom (zero torque) and on-bottom surface

Qslip <sup>¼</sup> πδ<sup>3</sup>

#### 2.3. Mechanical specific energy model for rotating drilling with PDM

According to the field experience, the bit's mechanical rotary energy has a much higher efficiency on rock breaking than the hydraulic energy. If the hydraulic energy of mud flow is converted into mechanical rotary power, it could improve ROP greatly. In the field, PDM has gained widespread use in the hard formation drilling to improve ROP. In rotating drilling with PDM, the power section of PDM converts hydraulic energy of mud flow into mechanical rotary power, the surface rotation is superimposed on downhole motor rotation (see Figure 2) [14]. Moreover, during slide drilling, bit rotation is generated only from the PDM as drilling fluid is pumped through the drill string. Due to the direct measurement of PDM rotary speed and torque in down hole has proven difficult, so currently there are few effective MSE models to precisely model the actual downhole drilling for rotating drilling with PDM.

#### 2.3.1. PDM performance

In PDM, the power section converts hydraulic energy of mud flow into mechanical rotary power. The output parameters of its mechanical horsepower are rotor torque and rotary speed, whereas differential pressure and mud flow rate are its operational parameters. However, the direct measurement of PDM rotary speed and torque in down hole has proven difficult. The key design parameter that relates PDM output parameters to its operational parameters is PDM unit displacement. It is defined as the mud volume required to revolve a PDM rotor shaft one revolution and can be found on PDM performance data sheets. Then the ideal PDM output torque and rotary speed can be defined by [19].

Figure 2. PDM converts hydraulic energy of mud flow into mechanical rotary power [14].

Drilling Performance Optimization Based on Mechanical Specific Energy Technologies http://dx.doi.org/10.5772/intechopen.75827 141

$$T\_{ideal} = \text{3.066} \cdot \Delta P\_m \cdot q \tag{25}$$

$$\text{RPM}\_{\text{ideal}} = \frac{\mathcal{Q}}{q} \tag{26}$$

However, in actual drilling process, leakage and torque losses play important roles in the performance of a PDM. The actual rotary speed of the PDM is decreased by the slip flow through the seal line, and the actual torque is also decreased by the resisting torque due to mechanical friction, elastomeric friction and viscous shearing of drilling fluid. The actual PDM output torque and rotary speed can be estimated by

$$T\_m = T\_{ideal} - \Delta T \tag{27}$$

$$RPM\_m = \frac{Q - Q\_{slip}}{q} \tag{28}$$

Torque losses is given by [20].

$$
\Delta T = \frac{\pi^2 \text{i}^4}{2(1-\text{i})(2-\text{i})^3} \frac{\text{RPM}\_m}{\delta} D\_h^3 L\_s \mu + C\_f \frac{\pi \left(1-\text{i}^2\right)}{4(2-\text{i})^2} D\_h^2 p\_h \Delta P\_m + \frac{2F\_n y}{3\pi} \tag{29}
$$

Slip flow is estimate as

assumed 0.25 to 0.4, usually use the value of 0.35 [17, 18]. The mechanical efficiency (Em) of a new bit can be got by core samples' laboratory studies, or inversed by adjacent wells logging

According to the field experience, the bit's mechanical rotary energy has a much higher efficiency on rock breaking than the hydraulic energy. If the hydraulic energy of mud flow is converted into mechanical rotary power, it could improve ROP greatly. In the field, PDM has gained widespread use in the hard formation drilling to improve ROP. In rotating drilling with PDM, the power section of PDM converts hydraulic energy of mud flow into mechanical rotary power, the surface rotation is superimposed on downhole motor rotation (see Figure 2) [14]. Moreover, during slide drilling, bit rotation is generated only from the PDM as drilling fluid is pumped through the drill string. Due to the direct measurement of PDM rotary speed and torque in down hole has proven difficult, so currently there are few effective MSE models

In PDM, the power section converts hydraulic energy of mud flow into mechanical rotary power. The output parameters of its mechanical horsepower are rotor torque and rotary speed, whereas differential pressure and mud flow rate are its operational parameters. However, the direct measurement of PDM rotary speed and torque in down hole has proven difficult. The key design parameter that relates PDM output parameters to its operational parameters is PDM unit displacement. It is defined as the mud volume required to revolve a PDM rotor shaft one revolution and can be found on PDM performance data sheets. Then the ideal PDM output

2.3. Mechanical specific energy model for rotating drilling with PDM

to precisely model the actual downhole drilling for rotating drilling with PDM.

Figure 2. PDM converts hydraulic energy of mud flow into mechanical rotary power [14].

data.

140 Drilling

2.3.1. PDM performance

torque and rotary speed can be defined by [19].

$$Q\_{slip} = \frac{\pi \delta^3 D\_h n\_s \Gamma\_i \tan \alpha}{12 \mu L\_s L\_m} \left(\frac{i}{1-i}\right) \Delta P\_m \tag{30}$$

In Eqs. (29) and (30), many parameters are functions of motor geometry, property and even drilling conditions, some of them are difficult to be determined. Therefore, the prediction of Tm and RPMm has proven difficult. However, in PDM the mechanical power is converted by hydraulic horsepower, and it depends on the converting efficiency of the PDM. Then the mechanical power can be predicted based on its input hydraulic power. The mechanical horsepower provided by PDM can be estimated by [21].

$$\text{MHP} = \frac{T\_m}{550} \left(\frac{2\pi}{60}\right) \cdot \text{RPM}\_m \tag{31}$$

The hydraulic horsepower can be given as

$$HHP = \frac{Q \cdot \Delta P\_m}{1714} \tag{32}$$

Their relationship can be written as

$$\text{MHP} = \eta \cdot \text{HHP} \tag{33}$$

In Eqs. (32) and (33), the operating differential pressure drop across the motor, at a constant flow rate, can be measured by comparing off-bottom (zero torque) and on-bottom surface standpipe pressures. Flow rate can also be easily obtained on the surface. The efficiency of a particular type of motor can be estimated based on data measured on test stands [22].

#### 2.3.2. A MSE model for rotating drilling with PDM

In rotary-drilling with PDM (see Figure 3), the mechanical work required to remove a unit volume of rock comes from the WOB, torque at bit provided by surface rotation and torque at bit provided by PDM rotation. The total mechanical work done by the bit in 1 h can be estimated as

$$\mathcal{W}\_t = \text{WOB}\_b \cdot \text{ROP} + \text{60} \cdot 2\pi \cdot \text{RPM}\_s \cdot T\_s + \text{60} \cdot 2\pi \cdot \text{RPM}\_m \cdot T\_m \tag{34}$$

In the above model, RPMs is bit rotary speed provided by surface rotation; Ts is torque at bit provided by surface rotation; RPMm is PDM output rotary speed; Tm is PDM output torque. As PDM is near above bit, bit rotary speed and torque provided by PDM can be nearly considered as PDM's output rotary speed and torque.

Please note that every bit has a mechanical efficiency for drilling when it is produced. The mechanical efficiency is mainly related to the bit's cutting structure and exists all along the drilling process [10, 11]. Given the mechanical efficiency of the new bit, the mechanical work required to break the rock drilled in 1 h can be nearly expressed as

$$\mathcal{W}\_V = \mathcal{W}\_t \cdot E\_m \tag{35}$$

The volume of rock drilled in 1 h is

MSE <sup>¼</sup> WV

directional and horizontal drilling [12].

hole motor can also be estimated as

bit. RPMs is drill pipe rotary speed.

drilling with PDM [14].

<sup>V</sup> <sup>¼</sup> Em �

Em �

Em �

MSE ¼ Em � WOB � e

through the drill string. The MSE can be estimated by [14].

MSE ¼ Em � WOB � e

expressed by

V ¼ Ab � ROP (36)

Drilling Performance Optimization Based on Mechanical Specific Energy Technologies

Ab � ROP (37)

http://dx.doi.org/10.5772/intechopen.75827

(38)

143

MSE has been defined as the mechanical work done to excavate a unit volume of rock. By combining Eqs. (34), (35) and (36), then the MSE for rotating drilling with PDM can be

However, the mechanical energy provided by the surface has a great transmission loss in horizontal and directional drilling. Chen et al. formulated a relationship between bottom hole WOB and the surface measured WOB and presented a method to calculate torque of bit in

> WOBb <sup>¼</sup> WOB � <sup>e</sup>�μsγ<sup>b</sup> <sup>μ</sup><sup>b</sup> <sup>¼</sup> <sup>36</sup> Ts

WOBb � ROP þ 60 � 2π � RPMs � Ts Ab � ROP

> 1 Ab þ

According to Eqs. (31), (32) and (33), the mechanical specific energy provided by the down

Finally, substitute Eqs. (39) and (40) into Eq.(37), we can get a new MSE model for rotating

For slide drilling, bit rotation is generated only from the PDM as drilling fluid is pumped

�μsγ<sup>b</sup> � <sup>1</sup> Ab þ

Note that ΔP<sup>m</sup> is the pressure drop across the PDM, and η is the efficiency of PDM but not the

13:33μ<sup>b</sup> � RPMs Db � ROP <sup>þ</sup>

Ab � ROP (41)

1155:2 � ηΔPmQ Ab � ROP (42)

Ab � ROP <sup>¼</sup> Em �

Then the mechanical specific energy provided by the surface can be estimated as

<sup>¼</sup> Em � WOB � <sup>e</sup>�μsγ<sup>b</sup>

60 � 2π � RPMm � Tm

�μsγ<sup>b</sup> <sup>1</sup> Ab þ

Db � WOB � e�μsγ<sup>b</sup>

13:33 � μ<sup>b</sup> � RPMs

1155:2 � ηΔPmQ

Db � ROP (39)

Ab � ROP (40)

1155:2 � ηΔPmQ

WOBb � ROP þ 60 � 2π � RPMs � Ts þ 60 � 2π � RPMm � Tm

Figure 3. Rotating drilling system with PDM [14].

Drilling Performance Optimization Based on Mechanical Specific Energy Technologies http://dx.doi.org/10.5772/intechopen.75827 143

The volume of rock drilled in 1 h is

standpipe pressures. Flow rate can also be easily obtained on the surface. The efficiency of a

In rotary-drilling with PDM (see Figure 3), the mechanical work required to remove a unit volume of rock comes from the WOB, torque at bit provided by surface rotation and torque at bit provided by PDM rotation. The total mechanical work done by the bit in 1 h can be

In the above model, RPMs is bit rotary speed provided by surface rotation; Ts is torque at bit provided by surface rotation; RPMm is PDM output rotary speed; Tm is PDM output torque. As PDM is near above bit, bit rotary speed and torque provided by PDM can be nearly

Please note that every bit has a mechanical efficiency for drilling when it is produced. The mechanical efficiency is mainly related to the bit's cutting structure and exists all along the drilling process [10, 11]. Given the mechanical efficiency of the new bit, the mechanical work

W<sup>t</sup> ¼ WOBb � ROP þ 60 � 2π � RPMs � Ts þ 60 � 2π � RPMm � Tm (34)

WV ¼ Wt � Em (35)

particular type of motor can be estimated based on data measured on test stands [22].

2.3.2. A MSE model for rotating drilling with PDM

considered as PDM's output rotary speed and torque.

Figure 3. Rotating drilling system with PDM [14].

required to break the rock drilled in 1 h can be nearly expressed as

estimated as

142 Drilling

$$V = A\_b \cdot ROP\tag{36}$$

MSE has been defined as the mechanical work done to excavate a unit volume of rock. By combining Eqs. (34), (35) and (36), then the MSE for rotating drilling with PDM can be expressed by

$$MSE = \frac{W\_V}{V} = E\_m \cdot \frac{WOB\_b \cdot ROP + 60 \cdot 2\pi \cdot RPM\_s \cdot T\_s + 60 \cdot 2\pi \cdot RPM\_m \cdot T\_m}{A\_b \cdot ROP} \tag{37}$$

However, the mechanical energy provided by the surface has a great transmission loss in horizontal and directional drilling. Chen et al. formulated a relationship between bottom hole WOB and the surface measured WOB and presented a method to calculate torque of bit in directional and horizontal drilling [12].

$$\begin{aligned} \, \text{WOB}\_b &= \text{WOB} \cdot e^{-\mu\_s \gamma\_b} \\ \, \mu\_b &= 36 \frac{T\_s}{D\_b \cdot \text{WOB} \cdot e^{-\mu\_s \gamma\_b}} \end{aligned} \tag{38}$$

Then the mechanical specific energy provided by the surface can be estimated as

$$\begin{aligned} E\_m \cdot \frac{\text{WOB}\_b \cdot \text{ROP} + 60 \cdot 2\pi \cdot \text{RPM}\_s \cdot T\_s}{A\_b \cdot \text{ROP}}\\ = E\_m \cdot \text{WOB} \cdot e^{-\mu\_s \gamma\_b} \left(\frac{1}{A\_b} + \frac{13.33 \cdot \mu\_b \cdot \text{RPM}\_s}{D\_b \cdot \text{ROP}}\right) \end{aligned} \tag{39}$$

According to Eqs. (31), (32) and (33), the mechanical specific energy provided by the down hole motor can also be estimated as

$$E\_m \cdot \frac{60 \cdot 2\pi \cdot RPM\_m \cdot T\_m}{A\_b \cdotROP} = E\_m \cdot \frac{1155.2 \cdot \eta \Delta P\_m Q}{A\_b \cdotROP} \tag{40}$$

Finally, substitute Eqs. (39) and (40) into Eq.(37), we can get a new MSE model for rotating drilling with PDM [14].

$$\text{MSE} = E\_m \cdot \left( \text{WOB} \cdot e^{-\mu\_s \gamma\_b} \left( \frac{1}{A\_b} + \frac{13.33 \mu\_b \cdot \text{RPM}\_s}{D\_b \cdot \text{ROP}} \right) + \frac{1155.2 \cdot \eta \Delta P\_m Q}{A\_b \cdot \text{ROP}} \right) \tag{41}$$

For slide drilling, bit rotation is generated only from the PDM as drilling fluid is pumped through the drill string. The MSE can be estimated by [14].

$$MSE = E\_m \cdot \left( \text{WOB} \cdot e^{-\mu\_s \gamma\_b} \cdot \frac{1}{A\_b} + \frac{1155.2 \cdot \eta \Delta P\_m Q}{A\_b \cdot ROP} \right) \tag{42}$$

Note that ΔP<sup>m</sup> is the pressure drop across the PDM, and η is the efficiency of PDM but not the bit. RPMs is drill pipe rotary speed.
