A. The minimum energy criterion for hole cleaning considering cuttings grinding

According to Angel [3], the gas stream at bottom hole should be powerful enough to have at least a kinetic energy given by the following expression:

$$\frac{1}{2}\rho\_{\mathcal{S}}v\_{\mathcal{S}}^2 = \frac{1}{2}\rho\_{\mathcal{S}^0}v\_{\mathcal{S}^0}^2\tag{A.1}$$

where W is the energy requirement for crushing an individual particle, x is particle diameter, c is a proportionality coefficient, and a is a diameter index. When the particle is ground from its initial diameter D to its final diameter d, Eq. (A.3) can be integrated to obtain a relation:

In gas drilling, the initial cuttings equivalent diameter is usually in the order of 10 mm. The final cuttings equivalent diameter is normally greater than 0.5 mm. This particle size range falls into the category of Bond Crack Propagation where b ≈ 0:5. Eq. (A.4) then degenerates to.

> 10ffiffiffi d <sup>p</sup> � <sup>10</sup> ffiffiffiffi D p

where W is the energy requirement for crushing an individual particle, kWh; Wi = k/10

π <sup>6</sup> <sup>D</sup><sup>3</sup> h i � � �

WirsD<sup>3</sup> <sup>1</sup>

where w is the energy requirement for crushing an individual particle, J; r<sup>s</sup> is the density of

The number of drill cuttings (m) created by the drill bit in a unit volume of gas at standard condition can be estimated on the basis of rate of cuttings volume generation (Qc), volume of

> Qc Vc Qg<sup>0</sup>

> > <sup>h</sup>hROP

m ¼

Qc <sup>¼</sup> <sup>c</sup>πD<sup>2</sup>

ffiffiffi d <sup>p</sup> � <sup>1</sup> ffiffiffiffi D p

1 ffiffiffi d <sup>p</sup> � <sup>1</sup> ffiffiffiffi D p

db � <sup>1</sup> Db

� � (A.4)

New Development of Air and Gas Drilling Technology http://dx.doi.org/10.5772/intechopen.75785 177

� � (A.5)

� � (A.6)

� � (A.7)

4 60 ð Þ (A.9)

(A.8)

<sup>W</sup> <sup>¼</sup> <sup>k</sup> <sup>1</sup>

W ¼ Wi

represents the fragmentation energy determined in standard test [35], kWh/t. If the energy requirement is expressed in Joule per particle, Eq. (A.5) becomes:

<sup>907</sup> Wi <sup>r</sup><sup>s</sup>

<sup>w</sup> <sup>¼</sup> <sup>3</sup>:<sup>97</sup> � 103

<sup>w</sup> <sup>¼</sup> <sup>3</sup>:<sup>6</sup> � <sup>106</sup>

where <sup>b</sup> <sup>¼</sup> <sup>a</sup> � 1 and <sup>k</sup> <sup>¼</sup> <sup>c</sup>

or

solid particle, kg/m<sup>3</sup>

.

individual cuttings (Vc), and gas injection rate (Qg0):

The rate of cuttings volume generation is expressed as:

where Dh is the hole diameter, m; hROP is the rate of penetration, m/hr.

a�1.

The right-hand-side of Eq. (A.1) is equal to 142 J/m<sup>3</sup> .

Angel's energy criterion underestimates the gas flow rate requirement for hole cleaning possibly because it does not consider the gas energy consumed on grinding cuttings from large size to small size in the borehole annular space. We propose the hypothesis that gas stream should have at least the kinetic energy of.

$$\frac{1}{2}\rho\_{\mathcal{S}}v\_{\mathcal{S}}^2 = \frac{1}{2}\rho\_{\mathcal{S}^0}v\_{\mathcal{S}^0}^2 + W\_{\mathcal{S}}\tag{A.2}$$

where Wg is the gas energy spent on grinding cuttings, J/m<sup>3</sup> .

Gas drilling produces drill cuttings of dust-like. The fine sizes of the solid particles are believed to be resulted from many times of collisions of drill cuttings to the borehole wall and drill string. If this is true, the energy spent on the collision must be from the flowing gas and the rotating drill bit and drill string. Consider the work done during the collision. Charles' equation for grinding energy has been widely used in the powder grinding industry [35]:

$$dW = -c\mathbf{x}^{-a}d\mathbf{x} \tag{A.3}$$

where W is the energy requirement for crushing an individual particle, x is particle diameter, c is a proportionality coefficient, and a is a diameter index. When the particle is ground from its initial diameter D to its final diameter d, Eq. (A.3) can be integrated to obtain a relation:

$$\mathcal{W} = k \left( \frac{1}{d^{\overline{b}}} - \frac{1}{D^{\overline{b}}} \right) \tag{A.4}$$

where <sup>b</sup> <sup>¼</sup> <sup>a</sup> � 1 and <sup>k</sup> <sup>¼</sup> <sup>c</sup> a�1.

In gas drilling, the initial cuttings equivalent diameter is usually in the order of 10 mm. The final cuttings equivalent diameter is normally greater than 0.5 mm. This particle size range falls into the category of Bond Crack Propagation where b ≈ 0:5. Eq. (A.4) then degenerates to.

$$W = W\_i \left(\frac{10}{\sqrt{d}} - \frac{10}{\sqrt{D}}\right) \tag{A.5}$$

where W is the energy requirement for crushing an individual particle, kWh; Wi = k/10 represents the fragmentation energy determined in standard test [35], kWh/t.

If the energy requirement is expressed in Joule per particle, Eq. (A.5) becomes:

$$w = \frac{3.6 \times 10^6}{907} W\_i \left[ \rho\_s \left( \frac{\pi}{6} D^3 \right) \right] \times \left( \frac{1}{\sqrt{d}} - \frac{1}{\sqrt{D}} \right) \tag{A.6}$$

or

1. Based on the modified energy criterion, the minimum required gas injection rate for hole cleaning is nearly proportional to the grinding energy contributed by the flowing gas. 2. The range of the optimum required nitrogen gas injection rate given by the newly devel-

3. The first open-loop field test on the GRS was successful. The purity of the post-separation gas is superior to the atmospheric air in terms of particle concentration. The filtered gas met the requirement of gas compressor and circulation in the well. The success of the test has laid a good foundation for future development of the system. The GRS has been proven to be a viable and feasible innovation for reducing the cost of gas drilling. It has a huge potential to be applied to the gas drilling operations including nitrogen drilling and natural gas drilling. This technology is predicted to have a huge impact on reducing the

A. The minimum energy criterion for hole cleaning considering cuttings

According to Angel [3], the gas stream at bottom hole should be powerful enough to have at

Angel's energy criterion underestimates the gas flow rate requirement for hole cleaning possibly because it does not consider the gas energy consumed on grinding cuttings from large size to small size in the borehole annular space. We propose the hypothesis that gas stream should

Gas drilling produces drill cuttings of dust-like. The fine sizes of the solid particles are believed to be resulted from many times of collisions of drill cuttings to the borehole wall and drill string. If this is true, the energy spent on the collision must be from the flowing gas and the rotating drill bit and drill string. Consider the work done during the collision. Charles' equa-

tion for grinding energy has been widely used in the powder grinding industry [35]:

dW ¼ �cx�<sup>a</sup>

.

<sup>g</sup><sup>0</sup> (A.1)

<sup>g</sup><sup>0</sup> þ Wg (A.2)

dx (A.3)

.

1 2 rgv<sup>2</sup> <sup>g</sup> <sup>¼</sup> <sup>1</sup> 2 rg0v<sup>2</sup>

1 2 rgv<sup>2</sup> <sup>g</sup> <sup>¼</sup> <sup>1</sup> 2 rg0v<sup>2</sup>

where Wg is the gas energy spent on grinding cuttings, J/m<sup>3</sup>

oped mathematical model is consistent with field experience.

cost of gas drilling and improving drilling performance.

least a kinetic energy given by the following expression:

The right-hand-side of Eq. (A.1) is equal to 142 J/m<sup>3</sup>

have at least the kinetic energy of.

grinding

176 Drilling

$$w = 3.97 \times 10^3 W\_i \rho\_s D^3 \left(\frac{1}{\sqrt{d}} - \frac{1}{\sqrt{D}}\right) \tag{A.7}$$

where w is the energy requirement for crushing an individual particle, J; r<sup>s</sup> is the density of solid particle, kg/m<sup>3</sup> .

The number of drill cuttings (m) created by the drill bit in a unit volume of gas at standard condition can be estimated on the basis of rate of cuttings volume generation (Qc), volume of individual cuttings (Vc), and gas injection rate (Qg0):

$$m = \frac{\frac{Q\_c}{V\_c}}{Q\_{\mathcal{S}^0}}\tag{A.8}$$

The rate of cuttings volume generation is expressed as:

$$Q\_c = \frac{c\pi D\_h^2 h\_{ROP}}{4(60)}\tag{A.9}$$

where Dh is the hole diameter, m; hROP is the rate of penetration, m/hr.

Assuming cuttings sphericity 1.0, the volume of individual cuttings is.

$$V\_c = \frac{4\pi}{3} \left(\frac{D}{2}\right)^3 \tag{A.10}$$

d final diameter, m D initial diameter, m Dh hole diameter, m

hROP rate of penetration, m/h

Sg gas specific gravity, air = 1 tup upstream temperature, C

15 m/s

\*, Yulong Yang<sup>1</sup>

r<sup>s</sup> density of solid particle, kg/m<sup>3</sup>

\*Address all correspondence to: lijun446@vip.163.com

1 China University of Petroleum, Beijing, China

2 University of Louisiana at Lafayette, USA

tions of AIME. 1958;213:180-185

Greek symbols

Author details

Jun Li<sup>1</sup>

References

pdn downstream pressure, MPa absolute pup upstream pressure, MPa absolute

<sup>r</sup><sup>g</sup> gas density at bottom hole condition, kg/m3

vg gas velocity at bottom hole condition, m/s

Fg fraction of grinding energy contributed by the flowing gas, dimensionless

Qg0 the minimum required gas volumetric flow rate at standard condition, Nm<sup>3</sup>

Wi fragmentation energy, 6.30 kWh/t for clay and 12.74 kWh/t for limestone

vg0 Angel's gas velocity at standard condition for hole cleaning (0. 1 MPa, 15C),

[1] Gray KE. The cutting carrying capacity of air at pressures above atmospheric. Transac-

/min

179

New Development of Air and Gas Drilling Technology http://dx.doi.org/10.5772/intechopen.75785

k heat capacity ratio of gas (≈ 1.4 according to Guo and Liu [2])

Wg the energy spent on grinding cuttings by the gas stream, J/m<sup>3</sup>

<sup>r</sup>g<sup>0</sup> gas density at standard condition (0. 1 MPa, 15<sup>C</sup>), 1.22 kg/m3

, Boyun Guo<sup>2</sup> and Gonghui Liu<sup>1</sup>

Substituting Eqs. (A.9) and (A.10) into Eq. (A.8) results in

$$m = \frac{D\_h^2 h\_{ROP}}{40 Q\_{\\$^3} D^3} \tag{A.11}$$

The energy requirement for grinding all particles in a unit volume of gas is then expressed as.

$$\mathcal{W} = mw \tag{A.12}$$

It is understood that crushing energy should be from rotating drill bit, drill string, and the flowing gas. Assuming the fraction of the crushing energy from the flowing gas is fg, we have

$$\mathcal{W}\_{\mathcal{S}} = f\_{\mathcal{g}} m w \tag{A.13}$$

Substitution of Eqs. (A.7) and (A.11) into Eq. (A.13) yield:

$$\mathcal{W}\_{\mathcal{S}} = \frac{100 f\_{\mathcal{g}} D\_h^2 \mathcal{W}\_i \rho\_s h\_{ROP}}{Q\_{\mathcal{s}0}} \left( \frac{1}{\sqrt{d}} - \frac{1}{\sqrt{D}} \right) \tag{A.14}$$

Substituting Eq. (A.14) into Eq. (A.2) results in:

$$\frac{1}{2}\rho\_{\mathcal{S}}v\_{\mathcal{S}}^2 = \frac{1}{2}\rho\_{\mathcal{S}^0}v\_{\mathcal{S}^0}^2 + \frac{100f\_gD\_h^2W\_i\rho\_{\mathcal{S}}h\_{\text{ROP}}}{Q\_{\mathcal{S}^0}}\left(\frac{1}{\sqrt{d}} - \frac{1}{\sqrt{D}}\right) \tag{A.15}$$

To make the model easy to be adopted in existing computer models, this equation can be rewritten in the same form of Angel's equation as:

$$\frac{1}{2}\rho\_{\mathcal{S}}v\_{\mathcal{S}}^2 = \frac{1}{2}\rho\_{\mathcal{S}^0} \left(v\_{\mathcal{S}^0}\sqrt{1+n}\right)^2\tag{A.16}$$

where

$$m = \frac{\mathcal{W}\_{\mathcal{S}}}{\frac{1}{2}\rho\_{\mathcal{S}^0}v\_{\mathcal{S}^0}^2} \tag{A.17}$$

#### Nomenclature

An total nozzle area, mm<sup>2</sup>

C choke flow coefficient (≈1.2 according to Guo and Liu [2])

