2. Mathematical model for gas injection rate

#### 2.1. The minimum required gas injection rate

It is shown in [18] that only the result given by Angel's minimum kinetic energy criterion has a trend that is consistent with field experience, although the minimum volumetric gas requirements are underestimated. We believe that this underestimation is partially because Angel's model does not consider the gas energy consumed on grinding cuttings from large size to small size in the borehole annular space. We propose the following equation to modify Angel's model (derivation is given in Appendix):

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

where

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

subsonic flow is identified by the critical downstream to upstream pressure ratio pdn

gives an expression of the maximum gas flow rate without causing sonic flow as:

Application procedure of FDJ is reported in the literature, for example, see [33].

in Well no. 2, Angel's model gave the minimum required gas injection rate of 66 Nm3

Site elevation (above mean sea level) 200 m Ambient pressure 0.1 MPa Ambient temperature 20�C Relative humidity 10% Geothermal gradient 3C/100 m Specific gravity of rock 2.7 water = 1 Hole section in Well no. 1 2840–3650 m Hole section in Well no. 2 2550–3305 m Bit diameter 215.9 mm Drill pipe outer diameter 127 mm Bit orifices 14.29 mm�3 Rate of penetration 18 m/h Rotary speed 50 rpm

Table 1. Basic data for the nitrogen drilling cases in the Daqing Field, China.

required gas injection rate of 69 standard cubic meter per minute (Nm3

2.3. Application examples

¼ 0:53 when k is 1.4 for air [32]. The choke flow coefficient C takes the maximum value of 1.2, according to [32]. Substituting C = 1.2, k = 1.4, and the critical pressure ratio of 0.53 into Eq. (4)

> Qgmax <sup>¼</sup> <sup>3</sup>:25Anpup ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Sg 9 <sup>5</sup> tup <sup>þ</sup> <sup>492</sup> <sup>q</sup> � �

If the operating gas injection rate is higher than this value, larger orifice area An should be utilized to expand the maximum permissible flow rate. If changing the orifice area An is not an option, a flow diverging joint (FDJ) should be employed at the shoulder of the drill collar.

Shale sections of two wells in the Daqing Field, China, were drilled with nitrogen. Basic data are shown in Table 1. For the hole section in Well no. 1, Angel's model predicted the minimum

The initial cuttings size was estimated on the basis of rate of penetration and rotary speed to be about 6 mm. The average cuttings size received at surface was observed to be about 1 mm. Assuming the major content of the shale is clay, its fragmentation energy is 6. 3 kWh/t. The nvalue in Eq. (1) reflects the amount of fragmentation energy from the lowing gas. An empirical n = 1/ 3 is assumed based on the observations that the average size of returned drill cuttings is significantly larger when the drilling string is not rotating while the gas booster is turned on.

pup <sup>¼</sup> <sup>2</sup> kþ1 � � <sup>k</sup> k�1

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

/min). For the hole section

/min.

(5)

167

$$\mathcal{W}\_{\mathcal{S}} = \frac{100f\_{\mathcal{S}}D\_{h}^{2}W\_{i}\rho\_{s}h\_{\text{ROP}}}{Q\_{\mathcal{S}^{0}}} \left(\frac{1}{\sqrt{d}} - \frac{1}{\sqrt{D}}\right) \tag{3}$$

r<sup>g</sup> and vg in Eq. (1) are dependent on gas-flow-rate through bottom hole pressure; therefore, this equation has to be solved for Qg<sup>0</sup> numerically.

#### 2.2. The maximum permissible gas injection rate

The excessive gas flow rate through bit can cause several problems including borehole erosion, hole deviation, and ice-balling of drill bit [2]. These problems are usually associated with the sonic flow condition at bit. The temperature of gas at bit can be much lower than expected under sonic flow conditions. This low temperature is due to the Joule-Thomson cooling effect, i.e., a sudden gas expansion below the bit orifice causes a significant temperature drop. The temperature can easily drop to below ice point, resulting in ice-balling of the bit if water exists. Even though the temperature can still be above the ice point, it could be below the dew-point of water vapor, resulting in the formation of liquid water which promotes mud ring problems in the annulus. If natural gas is used as the drilling fluid, it can form gas hydrates with water around the bit, i.e., hydrate balling. The temperature at the bit orifice downstream may be predicted by assuming an isentropic process for an ideal gas flowing through bit orifices [2]. The bit upstream temperature may be lower than the geothermal temperature at the bit depth because the downstream gas cools the bit body, and the bit body, in turn, cools the upstream gas. The process can continue until a dynamic equilibrium with geothermal and gas temperatures is reached at the bottom of the hole. Ref. [31] presented an analytical method for predicting borehole enlargement due to low-pressure and low-temperature effects. In additional to the borehole erosion, hole deviation and ice-balling, the sonic flow condition can also cause pipe sticking problem [2].

The flow equation for subsonic flow is given by [2]:

$$Q\_{\mathcal{g}} = 5.6CA\_{n}p\_{up}\sqrt{\frac{k}{S\_{\mathcal{g}}(k-1)\left(\frac{9}{5}t\_{up} + 492\right)} \left[ \left(\frac{p\_{du}}{p\_{up}}\right)^{\frac{2}{5}} - \left(\frac{p\_{du}}{p\_{up}}\right)^{\frac{k+1}{k}} \right]}\tag{4}$$

Eq. (4) relates the upstream pressure to the down-stream pressure only in subsonic flow conditions. This relation was first presented in [32]. The boundary between the sonic flow and subsonic flow is identified by the critical downstream to upstream pressure ratio pdn pup <sup>¼</sup> <sup>2</sup> kþ1 � � <sup>k</sup> k�1 ¼ 0:53 when k is 1.4 for air [32]. The choke flow coefficient C takes the maximum value of 1.2, according to [32]. Substituting C = 1.2, k = 1.4, and the critical pressure ratio of 0.53 into Eq. (4) gives an expression of the maximum gas flow rate without causing sonic flow as:

$$Q\_{\text{gmax}} = \frac{3.25 A\_n p\_{up}}{\sqrt{S\_{\text{g}} \left(\frac{9}{5} t\_{up} + 492\right)}}\tag{5}$$

If the operating gas injection rate is higher than this value, larger orifice area An should be utilized to expand the maximum permissible flow rate. If changing the orifice area An is not an option, a flow diverging joint (FDJ) should be employed at the shoulder of the drill collar. Application procedure of FDJ is reported in the literature, for example, see [33].

#### 2.3. Application examples

model does not consider the gas energy consumed on grinding cuttings from large size to small size in the borehole annular space. We propose the following equation to modify Angel's

> <sup>n</sup> <sup>¼</sup> Wg 1 <sup>2</sup> rg0v<sup>2</sup> g0

<sup>h</sup>WirshROP

r<sup>g</sup> and vg in Eq. (1) are dependent on gas-flow-rate through bottom hole pressure; therefore,

The excessive gas flow rate through bit can cause several problems including borehole erosion, hole deviation, and ice-balling of drill bit [2]. These problems are usually associated with the sonic flow condition at bit. The temperature of gas at bit can be much lower than expected under sonic flow conditions. This low temperature is due to the Joule-Thomson cooling effect, i.e., a sudden gas expansion below the bit orifice causes a significant temperature drop. The temperature can easily drop to below ice point, resulting in ice-balling of the bit if water exists. Even though the temperature can still be above the ice point, it could be below the dew-point of water vapor, resulting in the formation of liquid water which promotes mud ring problems in the annulus. If natural gas is used as the drilling fluid, it can form gas hydrates with water around the bit, i.e., hydrate balling. The temperature at the bit orifice downstream may be predicted by assuming an isentropic process for an ideal gas flowing through bit orifices [2]. The bit upstream temperature may be lower than the geothermal temperature at the bit depth because the downstream gas cools the bit body, and the bit body, in turn, cools the upstream gas. The process can continue until a dynamic equilibrium with geothermal and gas temperatures is reached at the bottom of the hole. Ref. [31] presented an analytical method for predicting borehole enlargement due to low-pressure and low-temperature effects. In additional to the borehole erosion, hole deviation and ice-balling, the sonic flow condition can also cause pipe sticking problem [2].

Qg<sup>0</sup>

ffiffiffiffiffiffiffiffiffiffiffi 1 þ n � � p <sup>2</sup>

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

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 4

pdn pup

 !<sup>2</sup> k

vuuut (4)

� pdn pup

 !<sup>k</sup>þ<sup>1</sup> k

3 5

k

<sup>5</sup> tup <sup>þ</sup> <sup>492</sup> � �

Eq. (4) relates the upstream pressure to the down-stream pressure only in subsonic flow conditions. This relation was first presented in [32]. The boundary between the sonic flow and

Sgð Þ <sup>k</sup> � <sup>1</sup> <sup>9</sup>

(1)

(2)

(3)

model (derivation is given in Appendix):

this equation has to be solved for Qg<sup>0</sup> numerically.

2.2. The maximum permissible gas injection rate

The flow equation for subsonic flow is given by [2]:

Qg ¼ 5:6CAnpup

where

166 Drilling

1 2 rgv<sup>2</sup> <sup>g</sup> <sup>¼</sup> <sup>1</sup> 2 rg<sup>0</sup> vg<sup>0</sup>

Wg <sup>¼</sup> <sup>100</sup><sup>f</sup> <sup>g</sup>D<sup>2</sup>

Shale sections of two wells in the Daqing Field, China, were drilled with nitrogen. Basic data are shown in Table 1. For the hole section in Well no. 1, Angel's model predicted the minimum required gas injection rate of 69 standard cubic meter per minute (Nm3 /min). For the hole section in Well no. 2, Angel's model gave the minimum required gas injection rate of 66 Nm3 /min.

The initial cuttings size was estimated on the basis of rate of penetration and rotary speed to be about 6 mm. The average cuttings size received at surface was observed to be about 1 mm. Assuming the major content of the shale is clay, its fragmentation energy is 6. 3 kWh/t. The nvalue in Eq. (1) reflects the amount of fragmentation energy from the lowing gas. An empirical n = 1/ 3 is assumed based on the observations that the average size of returned drill cuttings is significantly larger when the drilling string is not rotating while the gas booster is turned on.


Table 1. Basic data for the nitrogen drilling cases in the Daqing Field, China.


carried out a comprehensive study including an integration design, technological process investigation, cuttings transport analysis, separation and filter equipment selection, and con-

New Development of Air and Gas Drilling Technology http://dx.doi.org/10.5772/intechopen.75785 169

The general idea of the GRS is to separate gas effectively from the gas-liquid-solid mixtures returned from the well and re-inject the gas back into the well. At the same time, cuttings and fluids are discharged after the separation. During the natural gas drilling process, the separated gas can be released to the gas gathering system in the field. The integrated design of the

When nitrogen is used as the drilling fluid, a low-capacity nitrogen generator is employed to supply nitrogen gas and inject it into the well. If the well is deep, compressors or boosters may be used to provide the required injection pressure. When the gas pressure and volume reach the required value for recycling, the gas drilling process can be initiated. Because the nitrogen gas is recycled, only a low-capacity nitrogen generator is required for supplying a small amount of nitrogen gas to make up the losses due to leakage and to meet the requirement of

Compressors and boosters: The compressors and boosters used in the gas recycling system are the standard equipment used in conventional gas drilling operations. After filtration, the

clean nitrogen gas is introduced into the system through parallel connections.

trol system design.

3.1. System description

process is illustrated in Figure 1.

Figure 1. A sketch of the gas recycling system.

additional gas volume in the wellbore as depth increases. The major equipments in the GRS are described as follows:

Table 2. Comparison of model-calculated data and field observations.

On the one hand, the latter phenomenon is due to the weak ability of cuttings transportation of gas, i.e., the cuttings have to be fine enough so as to be returned from the bottom. On the other hand, the excessive gas injection rate is non-commercial and may lead to the ice-balling of drill bit induced by Joule Thompson effect as we have mentioned in the previous text. Moreover, field practice and theoretical analysis have shown that the returned debris in gas drilling is extremely fine, regardless of strata types [34]. Utilizing this value in the new model expressed by Eq. (1) gives the minimum required gas injection rate of 85 and 82 Nm<sup>3</sup> /min for the two hole sections in Well no. 1 and Well no. 2, respectively. The maximum permissible gas injection rates were calculated by Eq. (5) to be 144 and 134 Nm<sup>3</sup> /min for the two hole sections in Well no. 1 and Well no. 2, respectively.

Table 2 shows a comparison of model-calculated data and field-applied gas injection rates. Using the new model and a design factor of 1.15, The designed gas injection rate was 1.1585, or 97.8 Nm3 /min. For the hole section in Well no. 1. The section was drilled with a fixed compressor capacity of 120 Nm3 /min, which is between the minimum required gas rate of 85 Nm3 /min and the maximum permissible gas rate of 144 Nm3 /min, with no problem of hole cleaning and hole enlargement. Using the new model and the same design factor of 1.15, the designed gas injection rate was 1.1582, or 94.3 Nm3 /min for the hole section in Well no. 2. The section was drilled smoothly with a fixed compressor capacity of 95 Nm3 /min, which is between the minimum required gas rate of 82 Nm3 /min and the maximum permissible gas rate of 134 Nm3 /min. This comparison indicates a good consistency between the model-predicted optimum range of gas injection rates and the field-observed problem-free gas injection rates.
