Hydraulics analysis of the U-tubing effect in a riserless drilling system

As a newly developed marine drilling technology, riserless drilling (RD) shows obvious advantages for use in deep-water and ultra-deepwater petroleum resources exploitation. However, the available published hydraulic analysis models for RD are rare or imperfect, especially regarding the lack of accounting for the out-of-hole time. To analyse the actual flow characteristics of drilling mud in an RD system after the surface mud pumping is complete, this paper presents a new hydraulic simulation model of the U-tubing effect. The specific calculation methods of pressure loss in the drilling string and the wellbore annulus at different flow regimes (turbulent flow, transition flow, and laminar flow) are provided based on the Herschel-Bulkley (HB) rheological model. A case study shows that a critical volumetric flow rate (CVFR) exists for a specific RD system to just cause the standpipe pressure to become zero. Only when the initial pump circulation rate is greater than the CVFR can it ensure the normal operation of the RD. Otherwise, the fluid mud level inside the drilling string will drop, resulting in the u-tubing effect. In the first few seconds after the surface pump is shut down, the wellbore transient flow rate is influenced strongly by the initial circulation rate. When the initial flow rate is less than the CVFR, the transient flow rate is accelerated during the first few seconds, which is not the expected behaviour. The influence of the drilling string size, fluid density and rheological parameters of the drilling fluid on the transient flow rate and the Bottom Hole Pressure (BHP) are also considered. The results proved that the smaller values of the rheological parameters are beneficial for reducing the fluctuation of the BHP during the U-tubing process.


Introduction
Petroleum reserves on land will decline because of the continuous exploration and production of natural gas and oil resources. With the new theories, new technologies and new operations of drilling developed in recent years [1], the huge marine petroleum resources have gradually become the new sources of growth of energy development. Traditional offshore drilling is conducted using a larger diameter riser, which limits offshore drilling from extending to deep-water and ultra-deepwater areas. The reason for this limitation is described by Choe [2][3] (1998,1999), Stair (2002), Schubert (2003), Carter (2005) and Mirrajabi (2010).
As an attractive dual-gradient offshore drilling technology, RD is beneficial for reducing the drilling cost, shortening the well construction period and lowering the selection standards of the drilling rig. The RD concept was first developed by Watkins (1969) to reduce rotating blowout preventer (BOP) wear and balance the subsea internal and external well pressures. In an RD system, the return mud is forced by the subsea pump to the surface through a smaller diameter pipe instead of  (Lima, 1998;Choe, 1998;Schubert, 2003) [4][5], the main function of the subsea pump is to maintain the wellhead pressure in the annulus to be equal to the hydrostatic pressure of seawater at the seafloor.
When making connections, taking tripping operations, or performing other no-drilling operations, the surface pump will stop working in an RD system. Because of the pressure imbalance inside the drilling string and the well annulus, the U-tubing effect occurs. A dynamic equilibrium governing equation is used by Choe (1998Choe ( , 1999 and Jonggeun (2004) to analyse the transient flow rate and the corresponding mud level inside the drilling string during this process. However, the Power-Law model (without yield stress) is used in the hydraulic calculation. In addition, the specific calculation methods for the pressure loss in the drilling string and the wellbore annulus as well as for the bit pressure loss are not provided.
Based on the analysis, this paper presents a new hydraulic model for analysis of the U-tubing effect in an RD system based on the Herschel-Bulkley (HB) model (Platzer, 1996). The specific calculation methods of pressure loss are also provided; moreover, the parameters affecting the U-tubing process are also investigated. This research provides a guide for engineers to control the circulation rate to within a reasonable range of values before shutting down the surface pump. This study also provides some advice on how to weaken the fluctuation of BHP in an RD system when the U-tubing effect occurs.

Model
As mentioned above, the hydraulics of an RD is quite different from that of a conventional riser drilling system. To balance the internal and external well pressures, the subsea pump must continue to operate during the whole well construction process, regardless of whether drilling or no drilling operations are underway.  The drilling fluid rheological property is closely related to the calculation of the hydraulic parameters. According to earlier scholars [6][7][8][9] (Horton, 2005;Erge, 2015), the Bingham plastic model (Hanks, 1967), Power-law model (Wallick, 1969), and HB model are the most frequently used models to describe drilling fluid rheological properties. This paper selects a synthetic-based drilling fluid from a Gulf of Mexico well (White, 1997). The properties of the drilling fluid and the measured data are listed in Table 1 and Table 2 Table 3 and plotted in Figure 1 As shown in Figure 1, the predictions of the HB model have the best agreement with measured values among the three models. Figure 2 shows a Hydraulic illustration of a riserless drilling system. The average error in Table 3 also indicates that the HB model can precisely describe the thixotropy and shear thinning behaviour for this synthetic-based drilling fluid.

Normal circulating state
The flow route of the drilling mud in an RD is shown in Figure 2. During normal well drilling, the pressure equilibrium relationship can be established as follows:  where m,w m w -3 w Δp = (ρ -ρ h *10 )g . Flow pressure losses Δp f,p and Δp f,a are two important hydraulic parameters for an RD; these losses change with drilling string size, circulation rate, and mud rheology. These two parameters in Eq. (4) are calculated by the methods presented as follow [10], respectively, in which three flow regimes (laminar flow, transition flow and turbulent flow) are considered.
Flow pressure loss in the drilling string. Shear stress at pipe wall  wp and friction coefficient f are the two key parameters in calculating the flow pressure loss in the drilling string. The specific calculation equations are listed as follows: This is the general formula of the circulation rate and the shear stress at the pipe wall based on the HB model, which is suitable for different flow regimes. Eq. (5) comes from Fan (2014a) [11]. We apply the generalised Reynolds number from Fan (2014b) [12] in which an equivalent pipe diameter is used for the pipe flow: To distinguish the different flow regimes, the criteria of the flow regimes from Khataniar [13] (1994) and Schuh [14] According to the criteria, the friction coefficient f at the different flow regimes is provided by Reed [15] (1993) and Dodge [16] (1959).
The final flow pressure loss in the drilling string is: Flow pressure loss in the annulus.The process of calculating the pressure loss in the annulus is similar with that in the drilling string. Based on the HB model, the general relation of the circulation rate Q and the shear stress in the annular wall  wp is provided by Peng [17] (2013) as follows: The criteria of flow regimes in the annulus and the friction coefficient f a are almost the same as that in the drilling string. The only differences are the computing formulas of Re gp vs. Re ga and n' vs. n' a . ρ v D (19) As mentioned above, hydrostatic pressure inside the drilling string is higher than that of the annular wellhead at the sea floor. According to Eq. (4), when the circulation rate is very small, p spp has negative values. Meanwhile, Δp f,p , Δp b , and Δp f,a are functions of the circulation rate; these three parameters increase with increasing volumetric flow rate. Thus, there must be a volumetric flow rate that just causes the value of p spp to be equal to zero; this volumetric flow rate is defined as the critical volumetric flow rate (CVFR). Only when the circulation flow rate exceeds the CVFR can the normal operation of the RD be ensured. Table 4 lists the basic parameters of a deep-water well. The water depth is 3000 m, and the well depth is 9000 m. The CVFR is 30.25 L/s, based on the default data from Table 4. According to Eq. (4), the CVFR is influenced by the drilling string size, water depth and well depth, and mud rheology, among other factors. Figure 3 displays the CVFR vs. well depth for different water depths.   Table 4. As expected, when the water depth remains unchanged, the deeper well depth requires the lower CVFR to maintain ordinary operation. When the well depth below the mud line is fixed, the deeper water depth requires the higher CVFR to maintain normal drilling operation. Table 4. Basic parameters of a deep-water well.

U-tubing effect
As mentioned above, when the surface mud pump is shut down or the circulation flow is less than the CVFR, the mud level inside drilling string will drop because of gravity until a new equilibrium is reached; this phenomenon is the so-called U-tubing effect. During this process, the dynamic pressure equilibrium can be expressed as follows: where h x is the current fluid mud level above the mud line and Δp * f,p , Δp * f,a , and Δp * b are the current pressure loss in the drilling string, the pressure loss in the wellbore annular, and the bit pressure loss, respectively.
Because of the acceleration pressure loss Δp* acc in the wellbore, the true circulation rate is no longer equal to the pump displacement. Undoubtedly, these dynamic parameters vary with time. Note that the pressure losses *

Case discussion
Define the time interval Δt = 1 s and the final calculation error of the circulation rate ΔQ = 10 -5 m 3 /s. Based on the equations and calculation procedures mentioned above, the initial parameters shown in Table 4 are used to analyse the whole U-tubing process.     Figure 6 describes the transient pressure loss versus time after the mud pump is shut down. The pressure loss in the drilling string, wellbore annulus and bit pressure loss are all increased within the initial 13 seconds due to the increase in the circulation rate. Drilling mud in both the drilling string and the wellbore annulus is in turbulent flow within the first 273 seconds. Fluid in the annulus turns into transition flow during the period from 273 to 382 seconds and then goes into laminar flow. Note that the flow regime change occurs earlier in the annulus than that in the drilling string. Within 1500 seconds, approximately 64.79% of the system pressure loss occurs in the drilling string, and the amounts of pressure losses in the annulus and the bit are 32.58% and 2.63%, respectively. After the first 16 seconds, the volumetric flow rate and the mud level in the drilling string of the three cases become nearly the same: 29.67 L/s and 2953 m. Afterwards, drilling mud undergoes a free fall inside the drilling string and the variation of the flow rate of the three cases is always almost the same. During this period, the flow rate decreases sharply until the flow regime changes from turbulent to transition flow, and it decreases exponentially to zero when the flow regime transforms to laminar flow. The transient flow rate curves in Figure 7 indicate that different initial circulation rates have little influence on the total time for achieving a new equilibrium after the surface pump is shut down.  Similarly, the corresponding BHP curves during the U-tubing process are shown in Figure 8. The curves of the transient BHP are similar to the curves of the transient flow rate in Figure 7. The difference is that the smaller initial circulation rate corresponds to the higher BHP within the first 16 seconds. This difference occurs because when initial flow rate (25.2 L/s) is less than the CVFR (30.25 L/s), the flow rate in the wellbore is accelerated and Δp acc has positive values. However, when the initial flow rate (35.3 L/s) is greater than the CVFR, the flow rate in the wellbore decreases and Δp acc has negative values. In addition, the difference of bottom hole pressure of the three cases also decreases over time.

Parametric analysis of the U-tubing process
According to the analysis above, the main influence factors of the transient volumetric flow rate and the BHP during the U-tubing process are the drilling string size, the drilling fluid density, and the rheological parameters of drilling fluid [18][19][20][21][22].

Influence of the drilling string size on the U-tubing effect
We study on the transient volumetric flow rate vs. time for different drilling string sizes. For the same initial circulation rate and mud rheological property during the U-tubing process, the larger the drilling string size, the faster is the transient volumetric flow rate. The time points of the flow regime changes and the total time for a new equilibrium are almost the same for the three cases. In addition, the difference in transient flow rate during the first few seconds in Figure 9 is caused by the difference in the results compared to the initial circulation rate (31.5 L/s) with the CVFR of the three cases. The CVFR of 114.3 mm, 127 mm and 139 mm OD drilling strings are 23.5 L/s, 30.2 L/s and 35.4 L/s, respectively. Figure 10 illustrates the BHP vs. time when the U-tubing effect occurs for different drilling string sizes.The larger drilling string size corresponds to the higher BHP, with the other parameters fixed. This phenomenon occurs because the transient volumetric flow rate in a larger size drilling string is higher than that in a smaller size drilling string, as shown in Figure 9. The results in Figure 10 indicate that the smaller drilling string size is beneficial for reduction of the fluctuation of the bottom hole pressure during the U-tubing effect, with the other conditions remaining unchanged.     Figure 11 shows the change in the transient flow rate with time for different drilling fluid densities. When the drilling string size and the initial circulation rate are fixed, the transient volumetric flow rate of the heavier drilling fluid is higher than that of the lighter drilling fluid. In addition, the transformation points of the flow regimes (both turbulent to transition flow and transition to laminar flow) occur earlier in the lighter drilling fluid situation. However, the total time required to achieve a new equilibrium of the three cases differ little. The differences in the BHP of the three cases are obvious, as shown in Figure 12. When the variation amplitude of the drilling fluid density is 6.45%, the change rate of the BHP is from 4.95 % to 5.26 %. The results in Figure 12 indicate that the BHP is sensitive to the change in drilling fluid density when the U-tubing effect occurs.

Influence of the drilling fluid rheological parameters on the U-tubing effect
We study on the variation of transient volumetric flow rate vs. time for different rheological parameters (YS, K and n, respectively) after the surface pump is shut down. Although the transient volumetric flow rate decreases with time, the variation trends are not the same for different conditions. In addition, the transformation points of the flow regime changes occur earlier with the increase in the rheological parameters YS, K and n.    The corresponding BHP curves of Figures 13, 14, 15, 16, 17 and 18, respectively. In general, the greater the values of the rheological parameters YS, K and n, the greater is the BHP during the Utubing process. When the increments of YS, K and n are 61.07%, 22.38% and 4.48%, respectively, the variations of BHP are from 0.491% to 1.07%, 0.065% to 0.568% and 0.029% to 0.325%, respectively. For an RD system, the fluctuation of the bottom hole pressure during the U-tubing process can be reduced by using drilling fluids with smaller rheological parameters (YS, K and n).

Conclusions
This paper established a new hydraulics model of the U-tubing process in an RD system. The pressure loss in the drilling string and the wellbore annulus are calculated under three flow regimes (turbulent flow, transient flow and laminar flow) based on the HB model. The critical volumetric flow rate is introduced as the key parameter to analyse the U-tubing effect. The influence of the initial circulation flow rate, drilling string size, fluid density and rheological parameters of the drilling fluid on the transient volumetric flow rate and the BHP are fully studied. The main conclusions are as follows: (1) Only when the initial flow rate is greater than the critical volumetric flow rate can the normal operation of the riserless drilling system be ensured. Otherwise, the U-tubing effect will appear.
(2) Different initial flow rates have little influence on the total time required to achieve a new equilibrium, when the other parameters are fixed during the U-tubing process. However, the differences in the transient volumetric flow rate are very obvious in the first few seconds for different situations.
(3) After the surface pump is shut down, when the initial circulation rate is less than the CVFR, the transient flow rate will increase in the first few seconds. While the initial circulation rate is greater than the CVFR, the transient flow rate will decrease in the first few seconds.