A Comprehensive Prediction Model of Hydraulic Extended-Reach Limit Considering the Allowable Range of Drilling Fluid Flow Rate in Horizontal Drilling

Hydraulic extended-reach limit (HERL) model of horizontal extended-reach well (ERW) can predict the maximum measured depth (MMD) of the horizontal ERW. The HERL refers to the well’s MMD when drilling fluid cannot be normally circulated by drilling pump. Previous model analyzed the following two constraint conditions, drilling pump rated pressure and rated power. However, effects of the allowable range of drilling fluid flow rate (Q min ≤ Q ≤ Q max) were not considered. In this study, three cases of HERL model are proposed according to the relationship between allowable range of drilling fluid flow rate and rated flow rate of drilling pump (Q r). A horizontal ERW is analyzed to predict its HERL, especially its horizontal-section limit (L h). Results show that when Q min ≤ Q r ≤ Q max (Case I), L h depends both on horizontal-section limit based on rated pump pressure (L h1) and horizontal-section limit based on rated pump power (L h2); when Q min < Q max < Q r (Case II), L h is exclusively controlled by L h1; while L h is only determined by L h2 when Q r < Q min < Q max (Case III). Furthermore, L h1 first increases and then decreases with the increase in drilling fluid flow rate, while L h2 keeps decreasing as the drilling fluid flow rate increases. The comprehensive model provides a more accurate prediction on HERL.

drilling pump and rated power of drilling pump on the HERL model; the allowable range of drilling fluid flow rate, an important hydraulic parameter range, was not taken into consideration.
Each drilling pump has a maximum output power, known as the rated power of drilling pump P r . Meanwhile, each drilling pump also possesses several cylinders with different diameters, and every cylinder has a certain allowable pressure, which is called the rated pressure of drilling pump p r . The drilling fluid flow rate Q under the conditions of P r and p r is called the rated flow rate of drilling pump Q r . In general, P r , p r and Q r have the following relationship.
When ≤ Q Q r , the pump pressure is restricted by the allowable pressure of cylinder, the maximum pump pressure can only reach the rated pump pressure of drilling pump p r . Then the pump power keeps increases with the increase in drilling fluid flow rate Q until = Q Q r , namely the rated power of drilling pump P r is reached, and the drilling fluid flow rate Q at this time is the rated flow rate of drilling pump Q r . In brief, p r is the major constraint condition when ≤ Q Q r . In contrast, the pump power is maintained at P r when > Q Q r , the pump pressure keeps decreasing as drilling fluid flow rate Q increase. In other words, P r becomes the main constraint condition when Q > Q r 6 .
As mentioned above, the rated pump pressure of drilling pump p r and the rated power of drilling pump P r as two constraint conditions of HERL model for horizontal ERW are provided under the conditions of Q ≤ Q r and Q > Q r respectively. However, the effects of allowable range of drilling fluid flow rate on the HERL model are not considered. During the drilling process, the drilling fluid flow rate Q has a theoretical range, namely the allowable range of drilling fluid flow rate. Specifically, too small Q cannot meet the needs of hole cleaning; however, if Q is too large, the bearing capacity of the drilled formation may be threatened. The allowable range of drilling fluid flow rate is expressed in Eq. (2).
where Q min is the lower limit of drilling fluid flow rate, L/s; Q max is the upper limit of drilling fluid flow rate, namely the upper limit considering the bearing capacity of drilled formation, L/s. The main purpose of this paper is to establish a more comprehensive and accurate model of HERL for horizontal ERW according to the relationship between the above allowable range of drilling fluid flow rate and the rated flow rate of drilling pump Q r . Moreover, the bearing capacity of existing hydraulic equipment can also be evaluated based on the established HERL model, avoiding the situation that the designed horizontal-section length exceeds the limit extension ability provided by the available drilling pump.

Results
HERL model. For a horizontal ERW, the lengths of vertical section and deviated sections can be obtained by an inclinometer before drilling into the horizontal section. Therefore, we mainly analyze the well's horizontal-section limit L h , which can be expressed in Eq. (3). where L h is the horizontal-section limit, m; L h1 is the horizontal-section limit based on rated pump pressure, m; L h2 is the horizontal-section limit based on rated pump power, m; p r is the rated pressure of drilling pump, MPa; P r is the rated power of drilling pump, kW; ∆p g is surface pipeline pressure drop, MPa; ∆p b is bit pressure drop, MPa; ∆p stv is the drill string pressure losses of vertical section, MPa; ∆p std is the drill string pressure losses of deviated sections, MPa; ∆p av is the annular pressure losses of vertical section, MPa; ∆p ads is annular pressure losses of small-inclination section, MPa; ∆p adl is annular pressure losses of large-inclination section, MPa; ∆ ∆ ( )  where Q r is rated flow rate of drilling pump, L/s. Case II:  HERL, especially the horizontal-section limit. The specific data of this well is listed in Tables 1 and 2 7 , and schematic overview of the horizontal ERW is illustrated in Fig. 1.
First of all, the authors assume that the fracture pressure in the horizontal section is identical, otherwise inconsistent comparison conditions will occur when the parameters sensitivity analysis is carried out. Meanwhile, the bearing capacity of drilled formation and the needs of hole cleaning should be considered to determine the allowable range of drilling fluid flow rate.
The specific calculation results show that the lower limit based on the needs of hole cleaning Q hc is 29.6 L/s, the lower limit considering the bearing capacity of drilled formation − Q df min is 27.1 L/s, and the upper limit of drilling fluid flow rate Q max is 39 L/s. Therefore, the allowable range of drilling fluid flow rate ranges from 29.6 L/s to 38.5 L/s. Moreover, according to the conditions given in Tables 1 and 2, the rated pressure of drilling pump p r is  Table 2. List of input data for modeling.
39 MPa, the rated power of drilling pump P r is 1323 kW, so the rated flow rate of drilling pump Q r is 34 L/s. Depending on the relationship between allowable range of drilling fluid flow rate and the rated flow rate of drilling pump Q r , the HERL model belongs to Case I, which can be determined by Eq. (4). Effects of drilling fluid flow rate on the horizontal-section limit based on rated pump pressure L h1 and the horizontal-section limit based on rated pump power L h2 are shown in Fig. 2a, which is also the schematic overview of the situation of ≤ ≤ Q Q Q r min m ax (Case I). As shown in the Fig. 2a, the horizontal-section limit based on rated pump pressure L h1 first increases and then decreases with increase in drilling fluid flow rate; meanwhile, the horizontal-section limit based on rated pump power L h2 keep decreasing as drilling fluid drilling fluid flow rate increases when ≤ ≤ Q Q Q r min m ax . The abscissa value of the intersection between these two curves is the rated flow rate of drilling pump Q r (34 L/s). The HERL, especially the horizontal-section limit is mainly dependent on L h1 when Q ranges from Q min to Q r ( ≤ ≤ Q Q Q r min ), which is indicated by the yellow dotted area in the Fig. 2a. However, the HERL especially the horizontal-section limit mainly depends on L h2 if Q ranges from Q r to Q max ( < ≤ Q Q Q r max ), which is indicated by the blue dotted area in the Fig. 2a. Furthermore, both L h1max (the maximum horizontal-section limit based on rated pump pressure) and L h2 max (the maximum horizontal-section limit based on rated pump power) are larger than L hmax (the maximum horizontal-section limit). L hmax can be obtained at Q r (34 L/s). Specifically, L h1max is 5270 m, L h2max is 5955 m, while L hmax is 5068 m. Considering the lengths of vertical section and deviated sections, each drilling fluid flow rate corresponds to a well's HERL, and the maximum HERL of the horizontal ERW is 7463 m, which can also be obtained at Q r (34 L/s).

Discussion
In order to analyze the effects of different parameters on the HERL especially the horizontal-section limit of horizontal ERW, parameters sensitivity analysis is discussed. Furthermore, results simulated by the established model are also compared with the results of the previous model that did not consider the allowable range of drilling fluid flow rate.  Table 3.
As shown in Table 3, the lower limit of drilling fluid flow rate Q min gradually increases and the upper limit of drilling fluid flow rate Q max keeps decreasing as ROP increases. In other words, the window of drilling fluid flow rate becomes narrower. The effects of different ROPs on L h1 and L h2 are shown in the Fig. 2b.
As shown in the Fig. 2b, L h1 first increases and subsequently decreases with the increase in drilling fluid flow rate, whereas L h2 keep decreases with the increase in drilling fluid flow rate. Moreover, both L h1 and L h2 have a negative correlation with ROP under the condition of identical drilling fluid flow rate, since the annular cuttings and the annular pressure losses increase with the increase in ROP. In addition, ROP has no effects on the rated flow rate of drilling pump Q r , and Q r = 34 L/s. The horizontal-section limit here can be determined by the Eq. (4) (Case I), and the maximum horizontal-section limit L hmax can be achieved at Q r with different ROPs.  Table 3. Table 3 shows that the lower limit of drilling fluid flow rate Q min decreases and the upper limit of drilling fluid flow rate Q max increases with the increase in drill pipe rotation speed N. In other words, the window of drilling fluid flow rate becomes wider. The effects of different drill pipe rotation speeds on L h1 and L h2 are shown in Fig. 2c.   Similarly, the Fig. 2c shows that L h1 begins to decrease as drilling fluid flow rate increases after L h1 reached its upper limit, whereas L h2 has a consistent negative correlation with drilling fluid flow rate. Moreover, the rotation of drill pipe is conductive to the efficiency of hole cleaning, as a result of which both L h1 and L h2 increase with the increase in drill pipe rotation speed N under the condition of identical drilling fluid flow rate. Furthermore, drill pipe rotation speed also has no effects on rated flow rate of drilling pump Q r . The horizontal-section limit here can also be determined by the Eq. (4) (Case I) and the maximum horizontal-section limit L hmax can be achieved at Q r = 34 L/s with different drill pipe rotation speeds.
Effect of rated pressure of drilling pump on HERL. The rated pressure of drilling pump p r , an important parameter of the HERL model, has great effects on the HERL of horizontal ERW especially the horizontal-section limit L h . First of all, allowable ranges of drilling fluid flow rate under different rated pump pressures are calculated and listed in Table 3. Table 3 shows that p r has no effects on hole cleaning and the bearing capacity of the drilled formation. Moreover, different p r correspond to different Q r . Specifically, Q r = 34 L/s when p r = 39 MPa, Q r = 39 L/s when p r = 34 MPa and Q r = 43 L/s when p r = 31 MPa. The situation of p r = 39 MPa is exactly the same as that in Fig. 2a. The situation of p r = 34 MPa is focused in this part, the horizontal-section limit L h can be calculated by Eq. (5) . The effects of different rated pump pressures on L h1 and L h2 are illustrated in Fig. 3, which is also the schematic overview of situation < < Q Q Q r min m ax (Case II). Figure 3 shows that L h1 first increases, but later decreases with the increase in drilling fluid flow rate, while L h2 keep decreasing as drilling fluid flow rate increase. Moreover, the higher p r corresponds to the greater L h1 when drilling fluid flow rates are the same. However, p r has no effects on L h2 .
According to the Eq. (5), the horizontal-section limit totally depends on L h1 when < < Q Q Q r min m ax (Case II), which is indicated by the yellow dotted area in the Fig. 3. Therefore, the maximum horizontal-section limit L hmax can be obtained at the drilling fluid flow rate Q when L h1max is achieved rather than at the rated flow rate of drilling pump Q r .

Effect of rated power of drilling pump on HERL.
Firstly, allowable ranges of drilling fluid flow rate under different rated pump powers P r are calculated. Which situation the HERL belongs to and which kind of HERL model needs to be adopted can be determined according to the relationships between these allowable ranges of drilling fluid flow rate and the rated flow rate of the drilling pump Q r . Table 3 shows that rated pressure of drilling pump P r has no effects on hole cleaning and the bearing capacity of drilled formation. The rated flow rate of drilling pump  Table 3. The situation of = P 1049kW r is focused in this part, which belongs to the situation of < < Q Q Q r min m ax (Case III), and the horizontal-section limit can be determined by Eq. (6). The effects of rated pressure of drilling pump P r on L h1 and L h2 are shown in Fig. 4a.
Similarly, the Fig. 4a shows that L h1 first increases and subsequently decreases with the increase in drilling fluid flow rate, whereas L h2 has a consistent negative correlation with drilling fluid flow rate. Moreover, the higher rated pressure of drilling pump P r means the greater L h2 with identical drilling fluid flow rate. However, P r has no effects on L h1 .
According to the Eq. (6), the horizontal-section limit L h depends entirely on L h2 when < < Q Q Q r min m ax , which is indicated by the yellow dotted area in the Fig. 4a. Therefore, the maximum horizontal-section limit L hmax can be obtained at the drilling fluid flow rate Q when L h2max is achieved rather than the rated flow rate of drilling pump Q r . The Fig. 4a shows that the horizontal-section limit L h at Q r is larger than L hmax when = P 1049kW r . Therefore, if the allowable range of drilling fluid flow rate is not considered, and taking the horizontal-section limit L h at Q r as the L hmax when = P 1049kW r , it will result in the designed horizontal-section length L h0 being larger than the horizontal-section limit L h that can be achieved, and causing safety hazards.

Effects of designed horizontal-section length on HERL.
In general, different designed horizontal-section lengths L h0 have great effects on drilling operations. In the part of application example, the designed horizontal-section length L h0 is 1500 m. Allowable ranges of drilling fluid flow rate under different designed horizontal-section lengths (1500 m, 3000 m, 6000 m) are listed in Table 3. Table 3 shows that the window of drilling fluid flow rate becomes narrower when the designed horizontal-section length L h0 increases. Effects of different designed horizontal-section lengths L h0 on the horizontal-section limit based on rated pump pressure L h1 and the horizontal-section limit based on rated pump power L h2 are illustrated in Fig. 4b.
As shown in the Fig. 4b, after the increase of L h1 in the first stage, it begins to decreases as drilling fluid flow rate increase, while L h2 keep decreasing as drilling fluid flow rate increases. When = L 3000 m h0 , the lower limit of drilling fluid flow rate Q min is 34.9 L/s, and the rated flow rate of drilling pump Q r is 34 L/s, which belongs to the situation of < < Q Q Q r min m ax (Case III). The horizontal-section limit L h here can be determined by Eq. (6), and it is mainly dependent on L h2 (seen from the Fig. 4b). The maximum horizontal-section limit L hmax can be achieved at the lower limit of drilling fluid flow rate Q min , rather than the rated flow rate of drilling pump Q r .
As shown in Table 3 Comparison of the established model and the previous model. In the previous model, since the allowable range of drilling fluid flow rate is not considered, the drilling fluid flow rate Q can be considered in the range of zero to infinity, including the rated flow rate of drilling pump Q r . The maximum horizontal-section limit L hmax is achieved at Q r when the allowable range of drilling fluid flow rate is not considered. In other words, the previous model can be taken as the Case I in the established model in this study. The previous model has been applied in the South China Sea. If the rated power of drilling pump P r is 1049 kW and other conditions remain the same as those in the application example, L hmax is 4438 m based on the previous model, which can be obtained at Q r of 26.9 L/s. But in fact, the lower limit of drilling fluid flow rate Q min is 29.6 L/s, namely < < Q Q Q r min m ax , which can also be considered as the Case III of the established model in this study. And L hmax is 3908 m based on the established model, which is achieved at Q min (Fig. 4a). Therefore, the designed measured depth cannot be drilled if the designed horizontal-section length L h0 is 4100 m, or even resulting in drilling hazards.
Therefore, the effects of allowable range of drilling fluid flow rate must be considered to establish the more comprehensive and accurate HERL model. Failure to consider the allowable range of drilling fluid flow rate may result in problems of wellbore cleaning or borehole instability, and it is also unclear that what is the main constraint condition for HERL model of horizontal ERW and which kind of HERL model should be adopted.
In this study, the allowable range of drilling fluid flow rate is taken into account to establish a more comprehensive and accurate HERL model of horizontal ERW. Depending on the relationship between the allowable range of drilling fluid flow rate and the rated flow rate of the drilling pump, three kinds of HERLs model are established. Specifically, both the horizontal-section limit based on rated pump pressure L h1 and the horizontal-section limit based on rated pump power L h2 should be considered when  The horizontal-section limit based on rated pump pressure L h1 first increases and subsequently decreases with the increase in drilling fluid flow rate, whereas the horizontal-section limit based on rated pump power L h2 keep decreases with the increase in drilling fluid flow rate. In addition, greater rated pump pressure drilling pump p r means greater L h1 . Similarly, the greater rated pump power P r corresponds to the greater L h2 . Moreover, both L h1 and L h2 show negative correlation with ROP but have positive correlation with drill pipe rotation speed N. However, both the ROP and the drill pipe rotation speed have no effects on the rated flow rate of drilling pump Q r . In order to achieve larger HERL, it is necessary to improve the rated pressure of drilling pump p r and rated power of drilling pump P r as much as possible; in addition, lower ROP and higher drill pipe rotation speed are also necessary.
For a horizontal ERW, the designed horizontal-section length L h0 should be less than the maximum horizontal-section limit L hmax It is prone to safety hazards if the designed horizontal-section length is longer than the maximum horizontal-section limit which can be achieved in actual drilling operation. Therefore, it is of great significance to accurately predict the HERL by selecting comprehensive and appropriate constraint conditions.

Method
Modified model of HERL. Constraint conditions. Three constraint conditions for HERL model are given by combining the previous studies and the allowable range of drilling fluid flow rate. They can be also expressed in Eq. (7).
(1) When several drilling pumps work together, the actual pump pressure cannot exceed anyone of these rated pump pressures; (2) When several drilling pumps work together, the actual pump power cannot exceed the sum of these rated pump power of all drilling pumps; For a horizontal ERW, we mainly analyze its horizontal-section limit L h . If merely one drilling pump is considered, its rated pressure is p r , the horizontal-section limit based on rated pump pressure L h1 under the first constraint condition can be expressed in Eq. (13). HERL based on rated pump power. According to the second constraint condition, the actual pump power cannot exceed the sum of these rated pump power of all drilling pumps when several drilling pumps work together. The second constraint condition can be expressed in Eqs (14) and (15). For a horizontal ERW, we mainly analyze its horizontal-section limit L h . Similarly, if merely one drilling pump is considered, its rated power is P r , the horizontal-section limit based on rated pump power L h2 under the second constraint condition can be expressed in Eq. (16). Upper limit. In general, greater drilling fluid flow rate often means better state of hole cleaning. However, there is an upper limit for drilling fluid flow rate since the too high drilling fluid flow rate poses a great threat to the bearing capacity of drilled formation due to the exceeded annular drilling fluid velocity. The upper limit of drilling fluid flow rate Q max , namely the upper limit considering the bearing capacity of drilled formation can be determined based on the open hole extended-reach limit (OHERL) theory. The OHERL theory can be summarized as follows. The horizontal ERW cannot extend without limitation, the drilled formation will be fractured if the bottom hole pressure exceeds the fracture pressure, which is a critical point, and it can be expressed in Eq. (17)  where ρ s is solids density, namely cuttings density, g/cm3; ρ m is drilling fluid density, g/cm3; C s is solid volumetric concentration, %; ∆p av is annular pressure losses of vertical section, MPa; ∆p adi are annular pressure losses of several deviated sections, MPa; ∆p ah annular pressure losses of horizontal section at the critical point, MPa.
In general, the horizontal-section limit based on OHERL theory should be larger than the designed horizontal-section length L h0 . The limit values of drilling fluid flow rate can be obtained when the horizontal-section limit based on OHERL theory equals L h0 , which is obtained from Eq. (18). The results show that there are two limit values of drilling fluid flow rate, the larger of which can be taken as the upper limit of drilling fluid flow rate Q max , namely the upper limit considering the bearing capacity of drilled formation.
Lower limit. If the drilling fluid flow rate is too small, the hole cleaning condition becomes worse. Moreover, both the annular pressure losses and the bottom hole pressure are increased, which will also pose a great threat to the drilled formation. Therefore, two factors should be considered to determine the lower limit of drilling fluid flow rate. On one hand, the lower limit considering the bearing capacity of drilled formation − Q df min can be obtained based on the above OHERL theory. On the other hand, considering the needs of hole cleaning, the lower limit based on the needs of hole cleaning Q hc can be obtained. The lower limit of drilling fluid flow rate Q min can be obtained by Eq. (19).