Journal of Petroleum Science and Engineering 205 (2021) 108739
Contents lists available at ScienceDirect
Journal of Petroleum Science and Engineering
journal homepage: http://www.elsevier.com/locate/petrol
The coupled model of wellbore temperature and pressure for variable
gradient drilling in deep water based on dynamic mass flow
Ruiyao Zhang a, Jun Li a, b, *, Gonghui Liu a, b, Hongwei Yang a, Ting Yue a, Jiangshuai Wang a
a
b
China University of Petroleum-Beijing, Beijing, 102249, China
Beijing University of Technology, Beijing, 100192, China
A R T I C L E I N F O
A B S T R A C T
Keywords:
Dynamic mass flow
Variable gradient drilling
Separation efficiency
Indoor experiment
Heat and mass transfer
Wellbore temperature and pressure
Variable gradient drilling in deep water is a new drilling method of using separators to separate hollow glass
spheres (Referred to as HGS, it is the glass microbeads with low density less than 1.0 kg/m3, and the worse
thermal conductivity) from the drill string directly into the annulus, which can form multiple gradients of drilling
fluid density in annulus. This method can better deal with the problem of narrow pressure window in deep water
drilling. At present, there are few studies on the wellbore temperature and pressure distribution of this tech­
nology, and the separation efficiency of the existing separator is too low to achieve variable gradient. The
purpose of this paper is to develop a new separator to improve the separation efficiency, and to study the coupled
wellbore temperature and pressure field of variable gradient drilling based on dynamic mass flow. Firstly, indoor
simulated experiments were used to study the separation efficiency of the newly developed filter separator, and
based on the separation efficiency, wellbore flow theory and the first law of thermodynamics, the dynamic mass
flow at the location of the separator were fully considered. Considering the influence of wellbore temperature
and pressure on thermophysical parameters, a coupled model of wellbore temperature and pressure was
established based on dynamic mass flow. Then the model was discretized and solved by finite difference method
and cyclic iterative algorithm. Further, this model was verified based on the existing models and drilling site
data. Finally, the effects of key parameters in variable gradient drilling on the annulus temperature, pressure and
fluid density were studied. The results show that the filter separator has high feasibility, and the separation
efficiency can reach 95%–98.5%. Compared with conventional drilling, distribution curve of the temperature,
pressure and the mixture density in annulus of the filter separator section has obvious inflection points in var­
iable gradient drilling, and the number of inflection points is equal to the number of separators. What’s more, the
number and location of separators, the HGS volume fraction, pump flow rate, and the HGS density have sig­
nificant effects on the annulus temperature and pressure. This research can significantly improve the feasibility
of variable gradient drilling in deep water and provide a theoretical reference for further research in this
direction.
1. Introduction
The rapid development of the world economy has gradually
increased energy consumption. In recent years, the global energy has
been in short supply. At the same time, the environment is gradually
deteriorating due to the use of petroleum resources. The above reasons
have prompted scientists, industrialists and research communities to
study new technologies to increase oil production or develop new
alternative energy sources. As there are abundant fuel resources such as
oil and combustible ice stored in the oceans around the world, in recent
year, the main battlefield of exploration and development has gradually
been aimed at deep water areas.(L. R. Raymond, 1969; D. W. Marshall
et al., 1982; R. Vigneswaran et al., 2021). During deep water drilling, the
complex temperature field environment formed by low temperature of
seawater and high temperature of formation has a significant impact on
fluid characteristics, wellbore pressure and wellbore stability (M. M.
Abdelhafiz et al., 2020; H. Yang et al., 2019; Lin R et al., 2013; Khan NU
et al., 2016; D. Sui et al., 2018). At the same time, the existence of
narrow safe density window has also brought great risks and challenges
to pressure control in deep water drilling (Bhandari J et al., 2015; Sun X
* Corresponding author.
E-mail addresses: 1432679705@qq.com (R. Zhang), lijun446@vip.163.com (J. Li), lgh1029@163.com (G. Liu), 1539073021@qq.com (H. Yang), 1242316193@
qq.com (T. Yue), 1183472804@qq.com (J. Wang).
https://doi.org/10.1016/j.petrol.2021.108739
Received 24 November 2020; Received in revised form 25 March 2021; Accepted 26 March 2021
Available online 20 April 2021
0920-4105/© 2021 Elsevier B.V. All rights reserved.
R. Zhang et al.
Journal of Petroleum Science and Engineering 205 (2021) 108739
Fig. 1. Experimental system and filter separator.
et al., 2018; Dokhani V et al., 2016; C. S. Kabir, 1996; M. Yang et al.,
2016). Therefore, the application of conventional drilling technology in
deep water has encountered a bottleneck, which often causes well kick
and lost circulation, which significantly reduces drilling efficiency (Song
D et al., 2016; Li M et al., 2015).
In order to solve this technical problem, many scholars have pro­
posed new drilling methods for wellbore pressure control. By summa­
rizing the existing technology at present, the relatively promising
method is variable gradient drilling method by injecting the HGS (H.
Yang et al., 2019; LI J et al., 2016; WANG Jiangshuai et al., 2020). The
key factor of variable gradient drilling is to connect the separator on the
drill string based on the traditional drilling technology. Then the mixed
fluid of drilling fluid and low density HGS is injected from the drill
string. When the mixture enters the separator, the HGS will be separated
and directly enter the upper annulus from the drill string. The drilling
fluid enters the lower drill string and finally returns to the lower annulus
via the drill bit. Because the drilling fluid in the upper annulus is diluted
by the low density HGS, different density gradients are formed in the
upper and lower annulus. If multiple separators are installed on drill
string, then multiple gradients of mixture density in annulus are
realized. Consequently, the different gradient of the mixture density in
annulus can be achieved by adjusting the injected HGS volume fraction
and the position and number of the separator, which is the so-called
variable gradient drilling. Compared with traditional drilling, the dis­
tribution curve of wellbore pressure in variable gradient drilling is no
longer a single linear, but a broken line or curve-like distribution.
In 2003, the company Maurer (MAURER W C et al., 2003) first
proposed the idea of multi gradient drilling, and pointed out that the key
point of this technology was to install the HGS separation device. Yin
et al. (Yin, 2007; Yin et al., 2012) used the centrifugal force of the
cyclone separator to separate the HGS and established a mathematical
model of wellbore pressure based on separation efficiency. The results
illustrated that cyclone separator can optimize the distribution of well­
bore pressure, so that the profile line of annulus pressure was within the
density window. In 2016, Liao Chao (LIAO,2016) added a draft tube at
the axis of the cyclone separator to further optimize the structure. Nu­
merical simulation results show that the separation efficiency is less than
30%. In 2019, Wang et al. (WANG et al., 2019) established a mathe­
matical model of differential pressure at bottom-hole during dual
gradient drilling. However, the separation efficiency of the cyclone
2
R. Zhang et al.
Journal of Petroleum Science and Engineering 205 (2021) 108739
Fig. 2. The principle of indoor simulated experiment of variable gradient drilling.
separator was only 40%. It was difficult to achieve dual gradient of
drilling fluid density, so the conclusions were not accurate enough. In
2019, Yang (YANG et al., 2019) established a mathematical model of
wellbore temperature and pressure under the assumption that the HGS
can fully enter the annulus. The results show that the position of the
separator and the HGS volume fraction will have a significant effect on
the annulus temperature and pressure.
By summarizing the research status in this direction, there are still
some deficiencies in variable gradient drilling technology. On the one
hand, the separation efficiency of the separator is low, which makes it
impossible to achieve the purpose of variable gradient drilling. On the
other hand, few scholars have conducted comprehensive studies on the
variation law of wellbore temperature and pressure in variable gradient
drilling.
Therefore, a new type of downhole separator named filter separator
was developed in this paper to improve separation efficiency. And its
effectiveness was verified through indoor simulated experiments. Then,
considering the dynamic mass flow process when the filter separator
separates the HGS into the annulus, and a mathematical model of the
coupled wellbore temperature and pressure for variable gradient drilling
was established based on dynamic mass flow. Further, drilling site data
and existing models was utilized to verify this model. Finally, the
influence of key parameters in variable gradient drilling, such as posi­
tion and number of filter separator, HGS volume fraction, HGS density,
pump flow rate on the temperature, pressure, the mixture density in
annulus was studied. The filter separator can significantly improve the
feasibility of the application of variable gradient drilling technology.
Simultaneously, the study of wellbore temperature and pressure for
variable gradient drilling can provide an important reference for the
theoretical research of this technology.
2. Physical model
2.1. Experimental equipment
As shown in Fig. 1(a), it is an indoor simulated experiment system of
variable gradient drilling, which mainly includes a control cabinet
(including software interface and controller), simulated drill string and
annulus, filter separator, hydraulic pump, injection and discharge
pipelines, air valves, mixing tank and reservoir. The control cabinet is
mainly used to adjust the flow rate of the hydraulic pump and the switch
of the air valve. As shown in Fig. 1(b), it is the two dimensional structure
of the filter separator including upper joint, lower joint, filter structure
(including spherical filter plug and metal filter screen) as well as the
first, second, and third stage outer cylinder. Among them, the three stage
outer cylinder is assembled through two combinations of threads and
bolts, which can ensure the stability of the tool. The metal filter screen is
covered on the spherical filter plug and fixed by bolts. Three small holes
with the same inner diameter are evenly distributed on the spherical
filter plug to ensure the smooth passage of drilling fluid. Three separa­
tion openings distributed equidistantly on the sleeve can ensure that the
separated HGS enter the annulus from the inner cavity smoothly without
causing blockage of the flow channel.
The significant advantage of the filter separator is the use of the
principle of porous media. As long as the pore size of the filter screen is
smaller than the diameter of the HGS, filtration and separation can be
basically achieved. This breaks through the technical bottleneck of the
existing cyclone separator that the HGS cannot be separated due to the
small inner diameter of the cyclone cavity and insufficient centrifugal
force. During the experiment, the filter separator was connected to the
Table 1
Experimental conditions and related parameters.
Experimental conditions
Parameters
The height of experiment
system
Mud circulation system
5m
Filter separator
The composition of waterbased drilling fluid
HGS volume fraction
Diameter of HGS
Density of HGS
temperature and pressure
Rated displacement 100 m3/h, rated pressure 0.2
MPa
Length is 1082 mm, outer diameter is 125 mm
Seawater+0.2% Na2CO3+2% starch+0.5% coating
agent+0.3% XC+1%～2% sulfonated asphalt+5%
KCl+9%NaCl+3% polyamine+3% anti-sludge
package lubrication Agent
From 5% to 30%
From 0.2 mm to 0.8 mm
From 150 kg/m3 to 850 kg/m3
Room temperature and atmospheric pressure
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R. Zhang et al.
Journal of Petroleum Science and Engineering 205 (2021) 108739
Fig. 3. Separation efficiency under different influence conditions.
simulated drill string. The upper inlet was connected to the injection
pipeline, separation port of the filter separator was connected to the
outlet pipeline, the fluid returned from the simulated annulus directly
enters the underflow port.
fluid. The rest of the drilling fluid can normally pass through the filter
structure and then return to the lower simulated annulus, thus realizing
the separation between the HGS and drilling fluid. The separated HGS in
the reservoir 1 were filtered, recovered, and dried, and the separation
efficiency of the filter separator can be obtained by comparing the HGS
content of the injected and the collected. Finally, flow rate of the high
pressure pump, the HGS volume fraction, the density and diameter of
HGS was changed to test the corresponding separation efficiency.
Further, repeat the above process, the variation law of separation effi­
ciency under different conditions can be obtained.
2.2. Experimental principle
Based on the above experimental equipment, the principle of the
indoor simulated experiment for variable gradient is shown in Fig. 2.
The drilling fluid was a pure water-based drilling fluid with high tem­
perature resistance, which can still maintain good rheological properties
from a low-temperature to high-temperature environment. The experi­
mental conditions and related parameters are shown in Table 1.
The experiment process was as follows: firstly, mixed the HGS and
the drilling fluid evenly in the mixing tank. Secondly, turn on the high
pressure pump and valves 1, 2, 3, and 4. Thirdly, the mixture flow from
the injection line through the upper inlet, and then into the filter sepa­
rator. Due to the filter structure (including metal filter screen and
spherical filter plug) inside the filter separator, according to the filtering
principle of porous media (Yoon W J et al., 2011; Mahesh Lavanis et al.,
1998), it can be known that the HGS can be separated if the diameter of
HGS is larger than the pore size of the metal filter screen. Fourthly, the
HGS will temporarily stay on the spherical outer surface of the filter
structure, and then enter the simulated annulus from the separation port
of the filter separator under the impact of the small part of the drilling
2.3. Experimental results
Through the above experiments, the experimental results were
shown in Fig. 3 (a) and (b). It can be seen from Fig. 3 (a) that with the
increase of the pump flow rate and the HGS volume fraction, firstly the
separation efficiency of the filter separator increased rapidly, then the
increase rate decreased and finally tends to remain basically unchanged.
Because the principle of the filter separator is the same to that of the
porous media, the separated HGS will temporarily stay on the spherical
outer surface of the filter structure. The larger the pump flow rate was,
the stronger the scouring effect on the separated HGS, which was helpful
for HGS to enter the annulus. As the HGS volume fraction increased, the
contact area between the drilling fluid and the HGS reduced, then the
drag effect of the drilling fluid on the HGS decreased, which was more
Fig. 4. Physical model of wellbore for variable gradient drilling in deep-water.
4
R. Zhang et al.
Journal of Petroleum Science and Engineering 205 (2021) 108739
Fig. 5. Physical model of heat and mass transfer under dual gradient drilling conditions.
(1) Ignored the impact of cuttings on the wellbore temperature and
pressure as well as the thermophysical parameters of the drilling
fluid. What’s more, the influence of temperature and pressure on
the specific heat of drilling fluid and HGS as well as the viscosity
of drilling fluid were not considered.
(2) Ignored the fluctuating pressure when the HGS enters the annulus
from the separation port of the filter separator.
(3) The temperature of the drilling fluid at the same section of
wellbore was the same.
(4) Ignore the slip between the HGS and the drilling fluid.
beneficial for the separation.
As shown in Fig. 3(b), it is the variation of separation efficiency with
different diameter and density of HGS. As the diameter of HGS
increased, the separation efficiency increased slowly. When the diam­
eter of HGS reached a certain value, the separation efficiency no longer
increased. As the HGS density increased, the separation efficiency pre­
sented a decreasing trend. Because the diameter of HGS increased, the
larger the contact area between the drilling fluid and the single HGS, the
stronger the "scouring" effect of the HGS, which was beneficial for the
HGS to be separated. As the HGS density increased, the gravity of HGS
increased, and the sedimentation effect became more obvious, so it was
not conducive for the separated HGS to enter the annulus smoothly.
However, it can be seen from Fig. 3 (a) and (b) that as the pump flow
rate, the volume fraction and diameter of HGS increased, the separation
efficiency didn’t not increase after reaching a certain value. Owing to
the adhesion and aggregation occurring in the entire system, some HGS
cannot be separated into the annulus smoothly, therefore, the experi­
mental error caused by the part of the HGS staying in the circulation
system cannot be eliminated.
3.1. Mathematical model of wellbore heat transfer during dual gradient
drilling
Taking the filter separator as a reference, the drill string and the
annulus were divided into three parts. The drill string was divided into
an upper part, the middle part (the filter separator section) and a lower
part. Similarly, the annulus was divided into an upper part, the middle
part (a separator section) and a lower part. The physical model of heat
and mass transfer in drill string and annulus are shown in Fig. 5 (a) and
(b) respectively.
3. Mathematical model
After the HGS was separated, it will directly enter the annulus from
the separation port of the filter separator, so the "dynamic mass flow"
formed from drill string to the annulus at the location of the filter
separator, which made an effect on the heat and mass transfer in drill
string and the annulus. Therefore, according to the different number of
filter separators, variable gradient drilling was divided into dual
gradient drilling (include one filter separator) and multi gradient dril­
ling (include two or more filter separators). The larger the number of
filter separators, the more key parameters we need to control. On the
contrary, it increases the difficulty of wellbore pressure control owning
to the improper control and complex downhole conditions. So only the
wellbore temperature and pressure for variable gradient drilling under
the condition of two separators and below was studied in this paper. The
wellbore temperature and pressure of the other conditions can be ob­
tained by the same method, so it will not be repeated here. The physical
model of wellbore for variable gradient drilling was established, as
shown in Fig. 4. Then mathematical model of the coupled wellbore
temperature and pressure was established based on the physical model.
In order to develop the mathematical model of the coupled wellbore
temperature and pressure for variable gradient drilling, the following
assumptions were made:
3.1.1. Inside the drill string
The main heat transfer processes that occurred inside the drill string
include the convective heat transfer between the drilling fluid and the
inner wall of the drill string, and the heat conduction between the inner
and outer wall of the drill string. So, the heat transfer processes of the
upper and lower drill string are basically the same. However, at the
position of filter separator, owing to the separated HGS entering the
annulus directly, there was variable heat and mass transfer between the
drill string and the annulus of the separator section. Therefore, the upper
and lower drill string as well as the drill string of filter separator section
were divided into two parts for research, respectively.
3.1.1.1. Heat transfer equation in the upper and lower drill string. If the
micro-element body of a certain drilling fluid in the upper and lower
drill string was the research object, the heat variation includes the
following parts. InΔttime, the heat flowing into and out of the microelement body, the heat generated by the convective heat exchange be­
tween the mixture inside drill string and in the annulus, and the heat
generated by the flow friction of the mixture, as shown in equation (1).
Each part in equation (1) successively represents the change of heat
5
R. Zhang et al.
Journal of Petroleum Science and Engineering 205 (2021) 108739
in the micro-element body, the heat flowing into, the heat flowing out,
the convective heat exchange with the mixture in annulus, and the heat
generated by flow friction duringΔttime. According to the first law of
thermodynamics, each part of equation (1) satisfies the relationship
shown in equation (2), and the heat transfer equation of the upper and
lower drill string can be obtained after sorting, as shown in equation (3).
(
)
⎧
∂ ρm cm Tp
2
⎪
Q
=
π
d
ΔyΔt
⎪ dp,v
pi
⎪
∂t
⎪
⎪
⎪
⎪
⎪
⎪
Q
=
c
ρ
A
ΔyΔtT
⎪
dp,i
m
1
p,y
m
⎨
(1)
Q
=
c
ρ
⎪
dp,o
m m A1 ΔyΔtTp,y+Δy
⎪
⎪
⎪
⎪
[
]
⎪
⎪
⎪ Qpa = − π dp Upa,m Ta − Tp ΔyΔt
⎪
⎪
⎩
Qf = Qca,m ΔyΔt
A1
)
(
3.1.2.1. Heat transfer equation in the upper or lower annulus. The heat
transfer process in the upper and the lower annulus can be expressed by
equation (7). Each part in equation (7) successively represents the
change of heat in the micro-element body, the heat flowing into, the heat
flowing out, the convective heat exchange with wellbore or inner wall of
casing tube, the convective heat exchange with outer wall of drilling
string and the heat generated by flow friction duringΔttime. Similarly,
according to the first law of thermodynamics, each part of equation (7)
should satisfy equation (8), and the heat transfer equations of upper and
lower annulus can be obtained as shown in equation (9) by further
sorting.
(
)
)
⎧
∂ ρm cm Ta,y
π( 2
2
ΔyΔt
=
−
d
d
Q
⎪
a,v
po
⎪
⎪
4 w
∂t
⎪
⎪
⎪
⎪
⎪
Q
=
c
ρ
A
ΔyΔtT
⎪
a,i
m
2
a,y+Δy
m
⎪
⎪
⎪
⎪
⎨
Qa,o = cm ρm A2 ΔyΔtTa,y
(7)
⎪
⎪
⎪
⎪
Q
=
−
π
d
U
(T
−
T
)ΔyΔt
a,w
w aw,m
w
a
⎪
⎪
⎪
⎪
(
)
⎪
⎪
⎪
T
Q
=
−
π
d
U
−
T
ap
po ap,m
a
p ΔyΔt
⎪
⎪
⎩
(2)
Qdp,v = Qdp,i − Qdp,o − Qpa + Qf
(
the annulus from the separation port of the filter separator directly.
Therefore, it was necessary to divide the upper and lower annulus as
well as the annulus of filter separator section into two parts for research.
)
(
)
∂ ρm cm Tp
∂ ρ cm Tp
= Qm m
+ πdpi Upa,m Ta − Tp + Qca,m (i = 1, 3)
∂t
∂y
(3)
where m = 1, 3stand for the upper and lower drill strings of the filter
separator respectively.
3.1.1.2. Heat transfer equation in the drill string of the filter separator
section. In the drill string of filter separator section, when the separated
HGS entered the annulus from the separation port of the separator, there
was heat and mass transfer during this process. The rest of the heat
transfer process was basically the same as that in the upper and lower
drill strings, and the heat variation of each part was shown in equation
(4). Each part in equation (4) successively represents the change of heat
in the micro-element body, the heat flowing into, the heat flowing out,
the heat carried by the HGS entering the annulus, the convective heat
exchange with the mixture in annulus, and the heat generated by flow
friction duringΔttime.
According to the first law of thermodynamics, each part of equation
(4) satisfies equation (5), and then the heat and mass transfer equations
in the drill string of the filter separator section can be obtained, as shown
in equation (6).
(
)
(
)
⎧
∂ ρm cm Tp
∂ ρs1 cs1 Tpa
π
Qdp,v = dpi2 ΔyΔt
− dpi2 ΔyΔt
⎪
⎪
⎪
4
∂t
∂t
⎪
⎪
⎪
⎪
⎪
Q
=
ρ
c
A
ΔyΔtT
⎪ dp,i
p,y
m m 1
⎪
⎪
⎪
⎪
⎨
Qdp.o = ρm cm A1 ΔyΔtTp,y+Δy
(4)
⎪
⎪
⎪
⎪ Qs,out = cs1 ρs1 A1 ΔyΔtTa,y
⎪
⎪
⎪
⎪
(
)
⎪
⎪
⎪
⎪
⎪ Qpa = − π dpo Upa,m Ta,y − Tp,y ΔyΔt
⎩
Qf = Qca,m ΔyΔt
Qa,v = Qa,i − Qa,o − Qa,w − Qpa + Qf
where m = 4, 6 stands for the upper and lower annulus respectively.
A2
(
A1
)
[
(
∂ ρm cm Tp
∂ ρ cs1 Tp − Ta
− A1 s1
∂t
∂t
(
)
+ π dpi Upa,m Ta − Tp + Qca,m
)]
(9)
3.1.2.2. Heat transfer equation in the annulus of the filter separator
section. Because the separated HGS will enter the annulus directly from
the separation port, heat and mass transfer occurred during this process,
and the rest of the heat transfer process was basically the same as that of
the upper and lower annulus. Therefore, the heat change of each part
was shown in equation (10). Each part in equation (10) successively
represents the change of heat in the micro-element body, the heat
flowing into, the heat carried by the separated HGS, the heat flowing
out, the convective heat exchange with wellbore or inner wall of casing
tube, the convective heat exchange with outer wall of drilling string and
the heat generated by flow friction duringΔttime.
In the same way, according to the first law of thermodynamics, it
should satisfy equation (11), and the heat and mass transfer equations in
the annulus of the filter separator section can be obtained as shown in
equation (12).
(
)
(
)
)
)
⎧
∂ ρm cm Ta,y
∂ ρs cs Tpa
π( 2
π( 2
2
2
⎪
Q
ΔyΔt
ΔyΔt
=
−
d
−
d
d
+
d
a,v
⎪
po
po
⎪
4 w
∂t
4 w
∂t
⎪
⎪
⎪
⎪
⎪
⎪
Q
a,i1 = cm ρm A2 ΔyΔtTa,y+Δy
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨ Qa,i2 = cs1 ρs1 A2 ΔyΔtTa,y+Δy
(5)
(
= Qm
(
)
∂(ρm cm Ta )
∂(ρ cm Ta )
= Qm m
+ πdpo Uap,m Ta − Tp + πdw Uaw,m (Tw − Ta )
∂t
∂y
+ Qca,m
Qf = Qca,m ΔyΔt
Qdp,v = Qdp,i − Qdp,o − Qs,out − Qpa + Qf
(8)
)
∂ ρm cm Tp
− A1 cs1 ρs1 Ta
∂y
(6)
wherem = 2, stand for drill string of the filter separator section.
Qa,o = cm ρm A2 ΔyΔtTa,y
⎪
⎪
⎪
⎪
⎪
⎪
⎪
Qa,w = − πdw Uaw,m (Tw − Ta )ΔyΔt
⎪
⎪
⎪
⎪
⎪
(
)
⎪
⎪
⎪
Qap = − πdpo Uap,m Ta − Tp ΔyΔt
⎪
⎪
⎩
Qf = Qca,m ΔyΔt
3.1.2. In annulus
During the drilling circulation, the mixture in annulus produced
convective heat exchange with the outer wall of the drill string as well as
the inner wall of the casing tube or wellbore, and the flow friction of the
mixture generated a certain amount of heat. In addition to the heat
transfer process described above, the annulus of the filter separator
section also included heat and mass transfer caused by the HGS entering
Qa,v = Qa,i1 + Qi2 − Qa,o + Qpa − Qaw + Qf
6
(10)
(11)
R. Zhang et al.
Journal of Petroleum Science and Engineering 205 (2021) 108739
Fig. 6. Physical model of heat and mass transfer under multi gradient drilling conditions.
[
(
transfer equation of drill string of the filter separator 1 can be obtained
for multi gradient drilling.
(
)
[
(
)]
(
)
∂ ρ cM Tp
∂ ρ cs1 Ta − Tp
∂ ρ cM Tp
− A1 s1
= Qm M
− A1 cs1 ρs1 Ta
A1 M
∂t
∂t
∂y
(14)
(
)
+πdpi Upa,M Ta − Tp + Qca,M
)]
∂ ρ cs1 Tp − Ta
∂(ρm cm Ta )
∂(ρ cm Ta )
+ A2 s1
= Qm m
+ A2 cs1 ρs1 Ta
∂t
∂t
∂y
(
)
− πdpo Upa Ta − Tp + πdw Uaw,m (Tw − Ta )
A2
+ Qca,m
(12)
where m = 5 stands for the annulus of the filter separator section.
whereM = 3represents drill string of filter separator 1 section during
multi gradient drilling.
3.2. Mathematical model of wellbore heat and mass transfer during multi
gradient drilling
3.2.1.3. Heat transfer equation in the drill string of filter separator 2
section. The heat transfer process of drill string of the filter separator 2
section was the same as that of the drill string of the filter separator 1,
except that the physical parameters of the HGS were different. There­
fore, referring to equation (14), the heat and mass transfer equation can
be obtained as (15).
(
)
[
(
)]
(
)
∂ ρ cM Tp
∂ ρ cs2 Ta − Tp
∂ ρ cM Tp
A1 M
+ A1 s2
= Qm M
− A1 cs2 ρs2 Ta
∂t
∂t
∂y
(15)
(
)
+ π dpi Upa,M Ta − Tp + Qca,M
Compared with dual gradient drilling, because two separators were
installed on the drill string, the drill string was divided into three parts
according to the heat transfer process of the multi gradient drilling. The
first part was the upper and lower drill string as well as the drill string
between the two separators. The second part was the drill string of the
separator 1 section, and the third part was the drill string of the sepa­
rator 2; In the same way, the annulus was also divided into three parts.
Therefore, a physical model of wellbore heat and mass transfer for multi
gradient drilling was established as shown in Fig. 6 (a) and (b).
whereM = 4represents the drill string of filter separator 2 section during
multi gradient drilling.
3.2.1. Inside the drill string
3.2.1.1. The heat transfer equation of the upper and lower drill strings as
well as the drill string between the two separators. During the multi
gradient drilling, except for the increase in heat and mass transfer
caused by the increase of the number of separators, the rest of the heat
transfer process was basically the same as the type of heat transfer under
dual gradient drilling conditions.
Therefore, referring to equation (3), the heat transfer equation of the
upper and lower drill strings as well as the drill string between the two
separators can be obtained under multi gradient drilling conditions as
shown in equation (13).
(
)
(
)
(
)
∂ ρ cM Tp
∂ ρ cM Tp
A1 M
(13)
= Qm M
+ π dpi Upa,M Ta − Tp + Qca,M
∂t
∂y
3.2.2. In annulus
3.2.2.1. The heat transfer equation of the upper and lower annulus as well
as the annulus between the two separators. According to the heat transfer
equation (9) in the upper and lower annulus during dual gradient dril­
ling, the heat transfer equation (16) in the upper and lower annulus as
well as the annulus between the two separators for multi gradient dril­
ling can be obtained.
A2
(
)
∂(ρM cM Ta )
∂(ρ cM Ta )
= Qm M
+ πdpo Uap,M Ta − Tp + πdw Uaw,M (Tw − Ta )
∂t
∂y
+ Qca,M
(16)
where M = 1, 2represent the upper and lower drill string during multi
gradient drilling.
where M = 5, 6represents the upper annulus and the lower annulus
during multi gradient drilling.
3.2.1.2. Heat transfer equation in the drill string of filter separator 1
section. In the same way, referring to equation (6), the heat and mass
3.2.2.2. The heat transfer equation in the annulus of the filter separator 1.
7
R. Zhang et al.
Journal of Petroleum Science and Engineering 205 (2021) 108739
Fig. 7. Physical model of wellbore pressure calculation of variable gradient drilling.
For the same reason, equation (12) can be referred to obtain equation
(17) of heat and mass transfer in the annulus of the filter separator 1
section for multi gradient drilling.
[
(
)]
∂ ρ cs1 Tp − Ta
∂(ρ cM Ta )
∂(ρ cM Ta )
A2 M
+ A2 s1
= Qm M
+ A2 cs1 ρs1 Ta
∂t
∂t
∂y
(17)
(
)
− πdpo Uap Ta − Tp + π dw Uaw,M (Tw − Ta ) + Qca,M
ρ4 = εψρs1 + (1 − εψ )ρf , ρ6 = ρ3
where m = 1, 3, 4, 6 represents the upper drill string, lower drill string,
upper annulus, and lower annulus respectively.
3.3.1.2. Initial densityρM of the mixture in annulusduring multi gradient
drilling
⎧
ρ = ρ ε1 + ρ ε2 + (1 − ε)ρf
⎪
⎪
⎪ ρ1 = (1s1− ψ εs2 ρ + ε ρ
⎪
)( 1 s1
⎪
2 s2 ) + (1 − (1 − ψ )ε)ρf
⎪
⎨ 2
ρ4 = ρs2 ε2 + (1 − ε)ρf
(21)
⎪ ρ5 = ρs1 ε1 ψ + (1 − εψ )ρf
⎪
⎪
⎪
ρ = ρ2 ; ρ3 = ρ1 ; ρ7 = ρ5
⎪
⎪
⎩ 6
ρ8 = ρs2 ε2 ψ + (1 − εψ )ρf , (ε1 + ε2 = ε)
where M = 7represents the annulus of separator 1 section for multi
gradient drilling.
3.2.2.3. The heat transfer equation in the annulus of the filter separator 2.
For multi gradient drilling, heat transfer equation of the annulus of the
filter separator 1 and 2 section was basically the same as equation (17),
except that different thermophysical parameters of HGS separated by
separator 1 and 2. So, the heat and mass transfer equation was shown in
equation (18).
[
A2
(
∂ ρ cs2 Tp − Ta
∂(ρM cM Ta )
+ A2 s2
∂t
∂t
)]
= Qm
(20)
whereM = 1, 2, 3, ...8represents the upper drill string, the lower drill
string, drill string of the separator 1 section, drill string of the separator 2
section, the upper annulus, the lower annulus, the annulus of the
(
)
∂(ρM cM Ta )
+ A2 cs2 ρs2 Ta − πdpo Uap,M Ta − Tp
∂y
+ π dw Uaw,M (Tw − Ta )
(18)
+ Qca,M ΔyΔt
separator 1 section, the annulus between the separator 2 and the sepa­
rator 1.
where M = 8represents the annulus of separator 2 section during multi
gradient drilling.
3.3.1.3. Mathematical model of density of the mixture coupled with tem­
perature and pressure. Because the thermophysical parameters of the
mixture, temperature and pressure affected each other during the dril­
ling circulation (Arnold, 1990; Espinosa-Paredes and Garcia-Gutierrez,
2004; Yang et al., 2013), the multivariate nonlinear regression anal­
ysis method was used to process the experimental data of water-based
drilling fluid of McMordie et al. (McMordie W C et al., 1982) and to
obtain the model of density of the mixture coupled temperature and
pressure, as shown in equation (22).
3.3. Auxiliary equation
3.3.1. Density equation
For variable gradient drilling, the density of the mixture in annulus
will vary with the number of separators, the HGS volume fraction, the
separation efficiency of filter separator, and the density of HGS.
Therefore, according to the above parameters, the initial density of the
mixture in drill string and annulus were obtained for variable gradient
drilling.
3.3.1.1. Initial densityρm of the mixture in annulus during dual gradient
drilling
ρ1 = ρs1 ε + ρf (1 − ε), ρ3 = ρs1 [(1 − ψ )ε] + ρf [1 − (1 − ψ )ε]
3.3.2. Mathematical model of annulus pressure
The physical model of wellbore pressure for variable gradient
(19)
8
R. Zhang et al.
Journal of Petroleum Science and Engineering 205 (2021) 108739
Fig. 8. Model verification.
[
ρ(P, T) = ρ0 exp 4.9224 × 10− 10 (P − P0 ) − 9.6877 × 10− 19 (P − P0 )2 −
− 6
2
− 1.7432 × 10 (T − T0 ) + 4.9168 × 10
− 13
]
(T − T0 )(P − P0 ) ,
3.2196 × 10− 4 (T − T0 )
(ρ0 = ρm or ρM )
drilling in deep water was shown in Fig. 7 shows. Because the upper part
of the separator was light fluid, and the lower part was heavy fluid
(Relative to the fluid density in the upper annulus). Taking the separator
as a reference, the annulus was divided into the light fluid section and
the heavy fluid section. For dual gradient drilling, according to the
different positions of arbitrary target points A and B, the calculation
formulas for annulus pressure were shown as (i) and (ii) in formula (23).
(22)
In the same way, for multi gradient drilling, according to the position of
any target point in the annulus, the calculation formulas for annulus
pressures at arbitrary target points C, D, and E were obtained by (i), (ii)
and (iii) in equation (24), respectively. The calculation model of pres­
sure drop used in the in this paper refers to the literature (WANG et al.,
2019).
⎧
*
*
⎪
⎪
⎨ (i) [ρ4 (T, P)gh sin α +](ΔPfL h + Pcp), h ≤ L[ − Hs− b
][
(
)]
*
*
*
Pdh,m = (ii) ρ4 (T, P)g + ΔPfL L − Hs− b sin α + ρ6 (T, P)g + ΔPfW h − L* − Hs−* b sin α + Pcp
⎪
⎪
⎩
, h > L* − Hs−* b
(23)
]
[
⎧
(i) ρ5 (T,P)g+ΔPfL1 (L* − Hs− b − hL2 )sin α, (h ≤ L* − Hs− b − hL2 )
⎪
[
⎪
⎪ (ii) ρ (T,P)g+ΔP ](L* − H − h )sin α + [ρ (T,P)g+ΔP ](L* − H − h )sin α
⎪
fL1
s− b
L2
fL1
s− b
L2
⎪
5
⎪
]
[5
⎨
+ ρ7 (T,P)g+ΔPfL2 hL2 sin α, (L* − Hs− b − hL2 < h ≤ L* − Hs− b )
[
] *
] *
[
P*dh,M =
⎪
fL1 (L − Hs− b − hL2 )sin α
⎪
⎪ (iii) [ρ5 (T,P)g+ΔPfL1 ](L − Hs− b (− hL2 )sin α +) ρ5 (T,P)g+ΔP
⎪
*
⎪
⎪
⎩ + ρ7 (T,P)g+ΔPfL2 hL2 sin α + ρ8 g+ΔPfW1 [h− (L − Hs− b − hL2 )− hL2 ]sin α +Pcp ,
(h > L* − Hs− b )
(24)
9
R. Zhang et al.
Journal of Petroleum Science and Engineering 205 (2021) 108739
3.3.3. Other auxiliary equations
3.4. Model verification
(1) The mass and momentum conservation equations
Based on the annulus pressure mathematical model of variable
gradient drilling and auxiliary equations obtained above, and using the
drilling site data in the literature (G.-O. Kaasa et al., 2012; Z. Xu et al.,
2018; A. Karimi Vajargah et al., 2015; J. Xie et al., 2014; E. Santoyo
et al., 2001) (as shown in Table 1), the wellhead temperature and
annulus pressure in this model were calculated. On the one hand, the
calculated wellhead temperature was compared with the measured
wellhead temperature. On the other hand, the annulus pressure calcu­
lated by this model was compared with the calculation results of several
typical models and test data of wellbore pressure in drilling site.
Fig. 8(a) shows the comparison between the calculated wellhead
temperature and measured data on drilling site. In the initial circulation
stage, because the wellhead temperature was affected by the heat
transfer process in the wellbore, and the temperature fluctuates to a
certain extent. So, the measurement data of wellhead temperature after
2 h’ circulation was selected to verify the calculation results of wellhead
temperature in this paper. It can be seen from Fig. 8(a) that there was a
certain fluctuation in the wellhead measured temperature during the
circulation, and the calculated wellhead temperature basically reached a
relatively constant valve during a period of time. Although there was a
certain error between the two, the relative error was within 3%, as
shown in Fig. 8(c), the accuracy requirements were satisfied.
As shown in Fig. 8(b), wellbore pressure of this model was in good
agreement with that of Wang’s model and measured data on drilling site,
but poor in agreement with that of Yin’s. This was because the Yin’s and
Liao’s models consider the influence of cuttings when calculating the
wellbore pressure, while the Wang’s model ignores that. In order to
better study the impact of HGS on wellbore temperature and pressure
based on variable mass flow, the impact of cuttings on wellbore tem­
perature and pressure was not considered in this model, as stated in
assumption (1). Therefore, the wellbore pressure in the Yin’s model and
the Liao’s model was greater than that in the Wang’s model and this
model. But the separation efficiency of the separator in the Liao’s model
was higher than that in the Yin’s model, so that the mixed fluid density
in the upper annulus (above separator) of the former was lower than that
of the latter. The fluid density in the lower annulus (below separator)
was the same, so the wellbore pressure of the Yin’s model at the same
well depth was higher than that of the Liao’s model. The Wang’s model
has been compared and verified with the existing classic models (Song
Xuncheng et al., 2011; He et al., 2015), and its model has good accuracy
and rationality. Compared with the Wang’s model, in this model, not
According to the flow process of the heavy drilling fluid in the lower
part of the filter separator, the mass conservation equation shown in
equation (25) was established (Li G et al., 2016; Arnold FC et al., 2004;
LAI et al., 2015). In the same way, the mass conservation equation of the
drilling fluid flow in the upper annulus of the filter separator can also be
obtained, and the form was consistent with equation (25).
∂(ρvAt)
∂(ρAdz)
dz = −
dt
∂z
∂t
(25)
The momentum conservation equation shown in equation (26) was
established for the heavy drilling fluid in the annulus under the filter
separator. Similarly, the momentum conservation equation of the light
drilling fluid flow process can also be obtained in the same form.
( )
[
]
∂ pf
∂(pA)
∂(ρAv2 dt)
∂(ρAvdz) /
−
− ρg sin θAdz − A
dz =
dz +
dt dt
∂z
∂z
∂t
dz
(26)
(2) The equation of radial heat conduction in formation (ZHANG
et al., 2019)
According to the equilibrium volume method and energy balance
equation, the thermophysical parameters of the fluid in the formation
can be calculated as the following equation (27). Thus, the radial heat
conduction equation in formation was further obtained, as shown in
equation (28).
′
λ = λφl + λ(1−
f
(ρc)eff
φ)
(27)
′
, (ρc) = φ(ρc)l + (1 − φ)(ρc)f
∂Ti (x, y, t)
∂2 Tf (x, y, t) λeff ∂Tf (x, y, t)
+
= λeff
2x
∂t
x
∂2 x
(28)
(3) Boundary conditions
Wellhead temperature can be regarded as the initial inlet tempera­
ture inside drill string, so its boundary condition can be expressed as
equation (29).
(29)
T0 (y = 0, t = 0) = Tin
According to the continuity of drilling fluid flow at bottom hole, the
bottom-hole temperature of the drilling fluid inside the drill string, the
wall of drill string and drilling fluid in the annulus was equation [29].
The boundary condition can be expressed as equation (30).
Table 2
Basic parameters for simulation.
(30)
T0 (y = H, t) = T1 (y = H, t) = T2 (y = H, t)
The temperature distribution in formation far from the wellbore was
shown in equation (31).
(31)
T(x → ∞, y, t) = Tsurf + Gh
(4) The axial temperature field of the seawater and the formation
(Locarnini RA et al., 2013)
]/
T3 = (y, t = 0) = Tsurf (200 − y) + 13.68y 200, y < 200
(32)
/(
T3 = (y, t = 0) = m2 + (m1 − m2 ) 1 + e(y−
m0 )/m3
)
, 200 < y < yml
(33)
The axial temperature in formation was mainly related to the tem­
perature gradient and depth, so the axial temperature distribution can
be expressed as equation (34).
Tf (y, t = 0) = T3 (y = yml , t = 0) + Gh, yml < y < h
(34)
10
Parameter
Value
Parameter
Value
Well depth, m
4000
1.02
Water depth, m
Mud density, kg/m3
1500
1.45
Mud viscosity, Pa•s
Surface temperature, ◦ C
0.03
20
Inlet temperature, ◦ C
Geothermal gradient,
◦
C/m
Reservoir permeability,
μm2
Sea density, kg/m3
15
0.035
Mud thermal conductivity, W/
(m•K)
Mud specific heat, J/(kg• ◦ C)
Sea thermal conductivity, W/
(m•K)
Sea specific heat, J/(kg• ◦ C)
Rock thermal conductivity, W/
(m•K)
Rock specific heat, J/(kg• ◦ C)
Steel thermal conductivity, W/
(m•K)
Steel specific heat, J/(kg• ◦ C)
Rock density
Drill string ID, mm
Riser ID, mm
2.64
108
508
Drill bit size, mm
Drill string OD, mm
215.9
127
0.03
1.02
Cement thermal conductivity, W/
(m•K)
Cement specific heat, J/(kg• ◦ C)
HGS density, kg/m3
HGS thermal conductivity, W/
(m•K)
HGS specific heat, J/(kg• ◦ C)
Separation efficiency,%
1600
0.6
4128
2.1
853
50
400
0.75
900
150–850
0.47
750
0.95
R. Zhang et al.
Journal of Petroleum Science and Engineering 205 (2021) 108739
Fig. 9. Comparison of annulus temperature and pressure under different conditions.
only was the dynamic mass flow process considered, but also the sepa­
ration efficiency of the filter separator in this model was much higher
than that of the cyclone separation in Wang’s model. Therefore, when
the separator was located at the same well depth, the HGS volume
fraction in the upper annulus in this model increased significantly,
which resulted in the decrease in fluid density in upper annulus as well
as an obvious decrease in wellbore pressure. Consequently, the heat in
the annulus near the bottom hole was transferred to the upper annulus,
resulting in lower temperature in the annulus near the bottomhole,
which made the drilling fluid density increase, and then the annulus
pressure became greater at the same well depth. However, the final
results of wellbore pressure for this model were closer to Wang’s model.
The calculation errors of temperature and pressure were respectively
analyzed below. As shown in Fig. 8(c), after the drilling fluid circulation
was stable, the error between the wellhead temperature calculated by
this model and that of the measured on the drilling site was the
maximum one of 2.25%, which met the accuracy requirements. As
shown in Fig. 8(d), It was the error analysis between the calculation
results of wellbore pressure of this model and that of Wang’s model. On
the one hand, the drilling fluid in the upper annulus absorbed more heat,
which caused the annulus increasing, then the drilling fluid density
reduced. On the other hand, the separation efficiency of filter separator
increased significantly, so the annulus pressure obviously reduced.
Finally, the error slightly increased, the maximum error was within 7%,
but the accuracy met the requirements by considering the above two
factors. And comparing this model with the wellbore pressure data on
the drilling site from 2000m to 4000 m, it can be seen that the maximum
error was below 5%.
In summary, the accuracy and rationality of this model were
generally verified through the comparison of measurement data on
drilling site and theoretical models. Because the basic principle of dual
gradient drilling was the same as that of multi gradient drilling, in the
same way, the correctness of mathematical model of annulus tempera­
ture and pressure for the multi gradient drilling can be verified, which
will not be repeated here.
The reliability and accuracy of this model were verified by the
theoretical model and actual test data on drilling site. Next, this model
was discretized by the finite difference method. The first-order space
uses the first-order upwind style, the first-order time uses the two-point
backward difference, and the second-order space uses the three-point
central difference. In order to ensure the accuracy and speed of calcu­
lation, a uniform grid was used for the wellbore and the formation near
the wellbore, and a non-uniform grid was used for the formation away
from the wellbore. Finally, the double loop iterative method was utilized
to solve this model, the specific discrete equation and solution process
are shown in Appendix A.
4. Results and discussions
Firstly, based on the drilling data in Table 2, wellbore temperature
and pressure of conventional drilling, dual gradient drilling without
considering dynamic mass flow, dual gradient drilling and multi
gradient drilling considering dynamic mass flow was calculated.
Further, according to mathematical model of coupled wellbore tem­
perature and pressure, the effect of different separator positions, volume
fraction and density of HGS, pump flow rate on wellbore temperature,
pressure and drilling fluid density in annulus were studied considering
dynamic mass flow for variable gradient drilling.
Fig. 9 (a) illustrated the annulus temperature distribution under
different drilling conditions. The distribution of annulus temperature
differed greatly below the mud line, while the difference above the mud
line was not obvious. The annulus temperature gradually decreased
from conventional drilling to dual gradient drilling and then to multi
gradient drilling at the same well depth. What’s more, there were
obvious inflection point in the annulus temperature at the position of
filter separator, and the inflection point was the same as the number of
separators. Compared with traditional drilling, because the low tem­
perature HGS directly entered the high temperature drilling fluid in the
annulus from the separator for variable gradient drilling, the annulus
temperature was lower regardless of whether the dynamic mass flow
was considered. For variable gradient drilling, when considering the
effect of dynamic mass flow, the annulus temperature at the same depth
was lower. Because when the influence of dynamic mass flow was not
considered, only the influence of the specific heat of the HGS on the heat
transfer process was taken into consideration. However, when consid­
ering the effect of dynamic mass flow, in addition to the effect of specific
heat, there was also the effect of the mass transfer process when HGS
entered form drill string in annulus on the annulus temperature. Because
the specific heat of the HGS was small, the heat absorption of the drilling
fluid from the formation is reduced, and the mass transfer process when
the low temperature HGS entered the annulus further reduced the
temperature of the drilling fluid. For the dual gradient drilling without
considering the dynamic mass flow, the abrupt range of the annulus
temperature at the separator was 25 ◦ C–49 ◦ C. When considering the
impact of dynamic mass flow, the abrupt range of the annulus temper­
ature at the separator was 15 ◦ C–34 ◦ C under dual gradient drilling
conditions, but the abrupt range of the annulus temperature at the
separator under multi gradient drilling conditions was 5 ◦ C–31 ◦ C. With
the successive variation of the above three conditions, the upper and
lower limits of the abrupt range of annulus temperature at the separator
were gradually decreasing.
As shown in Fig. 9(b), compared with traditional drilling, the dis­
tribution of wellbore pressure for variable gradient drilling presented
11
R. Zhang et al.
Journal of Petroleum Science and Engineering 205 (2021) 108739
Fig. 10. The effect of separator position on the annulus temperature and pressure of variable gradient drilling.
the distribution of a broken line rather than the distribution of a single
linear. For dual gradient drilling, there was an inflection point at 2000m
(the position of the separator) in the distribution curve of wellbore
pressure. For multi gradient drilling, there was an inflection point in the
wellbore pressure distribution curve at both 1000 m (the position of the
separator 1) and 2000m (the position of the separator 2). What’s more,
as the number of separators increased, the wellbore pressure gradually
decreased at the same well depth. In the case of traditional drilling, dual
gradient drilling and multi gradient drilling, the bottom hole pressure
was 58 MPa, 54 MPa and 52 MPa respectively. Because low density HGS
were injected into the annulus, the mix fluid density in upper annulus
significantly reduced, while the drilling fluid density in lower annulus
remained basically the same, which caused the wellbore pressure
decreasing. Meanwhile, there was an inflection point at the position of
the filter separator. When the number of separators increased, the
number of inflection points increased and the severer the distribution
curve bended. Compared with dual gradient drilling without dynamic
mass flow, the annulus temperature at the position of the separator for
dual gradient drilling with dynamic mass flow was significantly lower
(as shown in Fig. 8), hence the mixed fluid density will be greater.
Consequently, the annulus pressure at the position of filter separator
slightly increased from 28 MPa (no dynamic mass flow) to 30 MPa
12
R. Zhang et al.
Journal of Petroleum Science and Engineering 205 (2021) 108739
Fig. 11. The effect of HGS volume fraction on annulus temperature and pressure of variable gradient drilling.
(dynamic mass flow), but the annulus pressure of the other well depth
remained basically unchanged.
m, 3000 m and 3500 m). As the depth of the separator increased, the
bottom hole pressure gradually reduced from 54 MPa to 49 MPa.
Because the depth of the separator increased, the inflection point
gradually moved downward, so that the column length of light drilling
fluid in upper annulus gradually increased, and that of the heavy drilling
fluid in lower annulus gradually decreased, so that the annulus pressure
gradually reduced.
As shown in Fig. 10(c), as the number of separators increased, the
distribution curve of annulus pressure bended more significantly, and
the annulus pressure was lower for multi gradient drilling. Because the
number of separators increased, the column length of liquid column of
the light drilling fluid in upper annulus became longer, resulting in the
decrease in annulus pressure at the same well depth. Meanwhile, with
the increase of depth of the separator, the inflection points gradually
moved down, which also enhanced the length of the liquid column of the
low-density fluid in upper annulus, so the annulus pressure gradually
reduced from 37 MPa (separator 1 was at 2000m, separator 2 was at
2400 m) to 32.5 MPa (separator 1 was at 1800m, separator 2 was at
2000m).
As shown in Fig. 10 (d) and (e), the effect of the position of separator
on the drilling fluid density for dual gradient and multi gradient drilling,
respectively. It can be seen from Fig. 10 (d) that the upper part of dis­
tribution curve of the fluid density was light drilling fluid, and the lower
part of it was the heavy fluid. There was an obvious sudden change of the
drilling fluid density at the interface (at the position of separator). As
illustrated in Fig. 10 (e), the annulus was divided into the annulus above
separator 1, the annulus between separators 1 and 2, and the annulus
below separator 2 for multi gradient drilling. Because the HGS density
separated into the annulus was different, there were two inflection
points (as shown in reflection area) between the upper and lower
4.1. The effect of separator position
As presented in Fig. 10(a), it was the influence of different positions
of separators on the annulus temperature for multi gradient drilling. At
the positions of separators 1(1800m and 2000m) and 2 (2000m, 2200 m
and 2400 m), there was a sudden decrease in the annulus temperature.
Since the low-temperature HGS directly entered the annulus from the
separation port, the annulus temperature near the separation port
dropped significantly. The depth of separator 2 was greater than that of
separator 1 along the direction of borehole axis, so the temperature
difference between the annulus and the drill string was greater, resulting
in more significant mutations. Therefore, abrupt range of the annulus
temperature was from 25 ◦ C to 3 ◦ C at the position of separator 2 and
was from 7 ◦ C to 2 ◦ C at the position of separator 1 respectively. When
the depth of the separators increased continuously, the column length of
drilling fluid containing HGS was longer, making the heat transfer from
annulus to the surroundings slower. The column length of the original
drilling fluid in the lower annulus decreased, so that the heat absorbed
from the formation reduced. As a result, the lower annulus temperature
gradually decreased, while the upper annulus temperature gradually
increased. What’s more, there was an obvious temperature mutation at
the position of separator, the greater the well depth was, the more sig­
nificant the mutation will be.
As presented in Fig. 10(b), the effect of the position of separator on
the wellbore pressure for dual gradient drilling. The distribution curve
of annulus pressure presented a broken line distribution, and the in­
flection point was located at the position of the separator (2000m, 2500
13
R. Zhang et al.
Journal of Petroleum Science and Engineering 205 (2021) 108739
Fig. 12. The effect of pump rate on the annulus temperature and pressure during variable gradient drilling.
annulus. As the depth of the separator increased, the length of liquid
column of the light drilling fluid in upper annulus gradually increased,
while that of the heavy drilling fluid in lower annulus gradually
decreased. Consequently, the inflection points gradually moved down.
Moreover, when the initial drilling fluid density was the same, the
mutation range of the drilling fluid was from 1450 kg/m3 to 1200 kg/m3
for dual gradient drilling and was from 1450 kg/m3 to 1000 kg/m3 for
multi gradient drilling, respectively.
Fig. 11(c) presented the effect of HGS volume fraction on the annulus
pressure for multigradient drilling. The distribution curve of annulus
pressure also presented an inflection point at the position of separator
relative to reference line. As the volume fraction enhanced from 0% to
40%, the annulus pressure gradually decreased from 70 MPa to 50 MPa
at bottom hole. Because the HGS volume fraction increased, the mixed
fluid density in the upper annulus was lower, while the drilling fluid
density in lower annulus remained basically unchanged, so that the
annulus pressure gradually reduced.
As shown in Fig. 11(d) and (e), the influence of HGS volume fraction
on the drilling fluid density for dual gradient and multi gradient drilling,
respectively. As shown in Fig. 11 (d), when the position of the separator
was fixed at 2000m, the length of the liquid column in the upper and
lower annulus remained unchanged. There was an obvious sudden
change of the distribution curve of drilling fluid density at the interface
between the upper annulus and the lower one. As the HGS volume
fraction increased from 5% to 40%, the drilling fluid density in the upper
annulus decreased from 1400 kg/m3 to 1000 kg/m3, while the drilling
density in the lower annulus remained unchanged.
As illustrated in Fig. 11(e), there was also the inflection point of
distribution curve of drilling fluid density at the position of separator 1
and separator 2. When the HGS volume fraction increased from 0% to
40%, the drilling fluid density above separator 1 or between separators 1
and 2 constantly decreased from 1450 kg/m3 to 900 kg/m3. Hence, the
distribution curve of drilling fluid density above inflection point 1 and
inflection point 2 gradually shifted to the left.
4.2. The effect of HGS volume fraction
Fig. 11 (a) and (b) illustrated the effect of HGS volume fraction on the
annulus temperature for dual gradient and multi gradient drilling,
respectively. As the HGS volume fraction increased, the content of HGS
in the upper annulus increased, so that the specific heat of the mixed
fluid in the upper annulus reduced. Then the heat transferred from the
lower annulus to the upper annulus shrunk, so the annulus temperature
gradually decreased at the same well depth. However, as shown in
Fig. 11 (a), the HGS volume fraction mainly affected the small annulus
area above the separator from 1500 m to 2000m.
As shown in Fig. 11(b), compared with dual gradient drilling, the
number of separators increased for multi gradient drilling, so the in­
flection points on the distribution curve of annulus temperature
increased (from 1500 m to 2000m). With HGS volume fraction
increased, the annulus temperature in the upper and lower parts of the
separator did not change much, but the abrupt variation of annulus
temperature was from 24 ◦ C to 3.5 ◦ C at the position of separator 1 and
was from 2.5 ◦ C to 0.75 ◦ C at the position of separator 2.
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R. Zhang et al.
Journal of Petroleum Science and Engineering 205 (2021) 108739
Fig. 13. The effect of HGS density on the annulus temperature and pressure during variable gradient drilling.
4.3. The effect of pump flow rate
from 22.5 ◦ C to 2.5 ◦ C respectively. Therefore, the high temperature
fluid in the lower annulus returned to the upper annulus, the rise of
temperature in the upper annulus was not obvious.
Fig. 12(c) presented the effect of pump flow rate on the annulus
pressure for dual gradient drilling. There was an inflection point in the
annulus pressure at the position of separator. And as the pump flow rate
enhanced from 15L/s to 45L/s, the annulus pressure at bottom hole
gradually increased from 51 MPa to 60 MPa. And the larger the pump
flow rate, the more obvious the annulus pressure increased. Owing to
the increase of pump flow rate, the pressure loss during circulation in the
annulus gradually increased, so the annulus pressure increased at the
same well depth.
As shown in Fig. 12(d), it was the influence of pump flow rate on
annulus pressure for multi gradient drilling. In the same way, as the
pump flow rate enhanced from 15L/s to 45L/s, the annulus pressure at
bottom hole gradually increased from 30 MPa to 48 MPa. Compared
with dual gradient drilling, with the enhancement of pump flow rate, the
upper and lower limits of the variation range of bottom hole pressure
were significantly reduced. For multi gradient drilling, three gradients of
drilling fluid density were formed in the annulus. They were the first
light drilling fluid, the second light drilling fluid and the heavy drilling
fluid. Consequently, there were two inflection points on distribution
curve of the annulus pressure and the bending degree of the curve was
more obvious.
As shown in Fig. 12(a), as the pump flow rate increased from 15L/s to
45L/s, the annulus temperature near the bottom hole gradually
decreased. However, when the drilling fluid returned to 2800 m, the
annulus temperature began to gradually increase with the enhance in
pump flow rate. Because the pump flow rate enhanced, the drilling fluid
in the annulus near the bottom hole absorbed more heat from the for­
mation within the same time, which resulted in a drop in the formation
temperature and further reduced the annulus temperature. The heat
absorbed from the formation in the lower annulus was finally trans­
ferred to the upper annulus, so the temperature of the upper annulus
gradually increased at the same well depth. The annulus temperature at
the position of separator (2000m) suddenly decreased from 38 ◦ C to 5 ◦ C
due to the influence of the mass transfer process. In annulus above the
separator, the annulus temperature gradually increases as the high
temperature fluid in the lower annulus returns to the upper annulus.
When the drilling fluid reached the mud line at 1500 m, the heat was
gradually transferred to the seawater, so the temperature began to
decreased. When it approached the ground, the annulus temperature
gradually increases until close to the ground temperature.
As shown in Fig. 12(b), it was the influence of different pump flow
rate on the annulus temperature for multi gradient drilling. As the pump
flow rate increased from 15L/s to 45L/s, the lower annulus temperature
gradually decreased. When the pump flow rate was constant, there were
two heat and mass transfer processes between the drilling fluid in the
drill string and in annulus at the position of the separator 1 and sepa­
rator 2, so that the annulus temperature at the position of separator
occurred two cooling processes. The abrupt range of annulus tempera­
ture at separator 1 and separator 2 was from 5 ◦ C to 0.75 ◦ C and was
4.4. The influence of HGS density
As illustrated in Fig. 13, it was the influence of different HGS density
on the annulus pressure. As the HGS density increased from 150 kg/m3
to 850 kg/m3, the annulus pressure gradually increased at the same well
15
R. Zhang et al.
Journal of Petroleum Science and Engineering 205 (2021) 108739
Fig. 14. The research process of this paper.
depth. Besides, the bending degree of distribution curve of the annulus
pressure at the position of filter separator (2000m) was getting lower
and lower, but the bottom hole pressure gradually increased from 60
MPa to 65 MPa, as shown in Fig. 13(a). The mixture density in the upper
annulus gradually increased from 1100 kg/m3 to 1300 kg/m3 owing to
the increase in the HGS density, as shown in Fig. 13(c). However, the
mixture density in the lower annulus remained basically unchanged, so
the annulus pressure gradually increased at the same well depth.
Because the mixture density in upper annulus increased, the density
difference of the drilling fluid density between the upper annulus and
lower annulus reduced. What’s more, the slope difference of pressure
distribution curve between the upper and lower annulus decreased and
the degree of bending of the curve was smaller. As shown in Fig. 13(b),
the HGS with different density are separated into the annulus by the
separator 1 and separator 2, so there were three inflection points on the
pressure distribution curve. In the same way, there were three gradients
in the density of drilling fluid in annulus, two inflection points was
shown in the density distribution curve of the annulus fluid, as shown in
Fig. 13(d). As the density of HGS increased from 980 kg/m3 to 1350 kg/
m3 in the upper annulus, the annulus pressure gradually increased at the
same well depth. Compared with dual gradient drilling, the annulus
pressure in multi gradient drilling was lower when using a combination
of two HGS density and the bending degree of distribution curve of the
annulus pressure was greater. What’s more, the upper and lower limits
from 35 MPa to 37 MPa of the variation range of bottom hole pressure
were smaller.
numerically calculated based on drilling site data. Finally, the sensitivity
analysis of the calculation results of the model was carried out, and the
influence of the key parameters of variable gradient drilling, such as the
position of the separator, the HGS volume fraction, HGS density and the
pump flow rate on the wellbore temperature, pressure and the mixture
density was studied. The research process was shown in Fig. 14. The
following conclusions were obtained through the above research.
(1) The indoor simulation experiment illustrated that separation ef­
ficiency of filter separator increased with the enhancement of the
pump flow rate, volume fraction and diameter of HGS, and
decreased with increase of the HGS density. The separation effi­
ciency was from 95% to 98.5%. The filter separator significantly
improved the feasibility for variable gradient drilling.
(2) Since the influence of dynamic mass flow was taken into
consideration for variable gradient drilling, there were obvious
inflection points on the distribution curve of the annulus tem­
perature, pressure and the mixture density. Moreover, the
annulus temperature and pressure were lower than that of the
traditional drilling or non-dynamic mass flow at the same well
depth.
(3) For variable gradient drilling, the distribution curves of the
temperature, pressure and the drilling fluid density in annulus all
have inflection points at the separator. What’s more, the bending
degree of distribution curve for the multi gradient drilling was
more obvious than that for the dual gradient drilling. And
compared with dual gradient drilling, the annulus temperature,
pressure and the drilling fluid density for multi-gradient drilling
were smaller at same well depth.
(4) As the depth of the separator and the HGS volume fraction
increased, the temperature, pressure and the drilling fluid density
in annulus gradually decreased. And the change of the HGS vol­
ume fraction mainly affected the upper annulus above the sepa­
rator. With the enhancement of the pump flow rate, annulus
pressure gradually increased, while the annulus temperature near
the bottom hole gradually reduced, the annulus temperature near
the separator gradually increased. As the HGS density increased,
the annulus pressure and the drilling fluid density gradually
increased.
5. Conclusions and recommendations
In this paper, through the development of a new downhole filter
separator, the separation efficiency and the feasibility of variable
gradient drilling have been significantly improved, and its reliability has
been verified through indoor simulated experiment of variable gradient
drilling. Then based on the separation efficiency and considering the
dynamic mass flow of the filter separator, a mathematical model of the
coupled wellbore temperature and pressure for variable gradient drilling
in deep water was established. The test data of the drilling site and the
existing classic model was used to compare with this model, the accu­
racy and rationality of which were verified. Furthermore, this model was
16
R. Zhang et al.
Journal of Petroleum Science and Engineering 205 (2021) 108739
interests or personal relationships that could have appeared to influence
the work reported in this paper.
Credit author statement
Ruiyao Zhang: Conceptualization, Methodology, Software, Data
curation, Writing – original draft and Investigation. Jun Li: Data cura­
tion and Writing- Reviewing and Editing. Gonghui Liu: Supervision.
Hongwei Yang: Software and Validation.Ting Yue: Writing – original
draft. jiangshuai Wang: Supervision and Writing- Reviewing and
Editing.
Acknowledgments
Funding: This research was supported by the National natural sci­
ence foundation of China, “basic research on wellbore pressure control
in deep water oil and gas drilling and production” (51734010).
Declaration of competing interest
The authors declare that they have no known competing financial
Nomenclature
Qdp,v
Qdp,i
Qdp,o
Qpa
Qf
A1
cm
dpi
Tp
Ta
Δt
Uap
Δy
Qca
P0
T0
P*dh,m
the
the
the
the
the
the
the
the
the
the
the
the
the
the
the
the
the
the
the
the
the
the
the
the
the
the
the
the
the
the
the
the
the
the
the
the
the
the
change of heat in any control body unit in the drill string duringΔttime, J;
heat flowing into the control body unit in the drill string duringΔttime, J;
heat flowing out the control body unit in the drill string duringΔttime, J;
convection heat exchange of drilling fluid inside the drill string and in the annulus duringΔttime, J;
heat generated by the flow friction of the drilling fluid duringΔttime andΔylength, J;
effective cross-sectional area of drilling fluid flowing in the drill string, mm2
specific heat of the mixture in the wellbore during dual gradient drilling, J/(kg⋅◦C)
inner diameter of drill string, mm
temperature of the fluid in the drill string, ◦ C
temperature of the fluid in annulus, ◦ C
time interval of drilling fluid flow, s;
comprehensive convective heat transfer coefficient of drilling fluid between drill string and annulus, W/(m2⋅◦C)
length of the control unit in the direction of the borehole axis, m;
heat generated by the flow friction of the drilling fluid per unit time and unit length, J;
density of the mixture in the wellbore during dual gradient drilling, kg/m3
pump flow rate, m3/s
heat flowing out from the separation port of the separator, J;
specific heat of the hollow glass sphere separated in the first stage, J/(kg⋅◦C)
density of the hollow glass sphere separated in the first stage, kg/m3
change of heat in any control body unit in annulus duringΔttime, J;
heat flowing into the control body unit in annulus duringΔttime, J;
heat flowing out the control body unit in annulus duringΔttime, J;
convection heat exchange of drilling fluid in the annulus and formation duringΔttime,J;
effective cross-sectional area of drilling fluid flowing in annulus, mm2
comprehensive convective heat transfer coefficient of drilling fluid between the annulus and seawater or formation, W/(m2⋅◦C)
inner diameter of casing tube or wellbore, mm
specific heat of the mixture in the wellbore during multi gradient drilling, J/(kg⋅◦C)
density of the mixture in the wellbore during multi gradient drilling, kg/m3
specific heat of the hollow glass sphere separated in the second stage, J/(kg⋅◦C)
density of the hollow glass sphere separated in the second stage, kg/m3
total volume fraction of hollow glass sphere, %
volume fraction of the hollow glass sphere separated in the first stage, %
volume fraction of the hollow glass sphere separated in the second stage, %
separation efficiency of the filter separator, %
drilling fluid density,kg/m3
atmospheric pressure, MPa;ρ0 is the density of the mixed fluid, kg/m3
surface temperature, ◦ C
bottom-hole pressure while drilling during dual gradient drilling, MPa
ΔPfL
ΔPfL1
ΔPfL2
ΔPfW
g
hL2
h
Hs− b
L*
the
the
the
the
the
the
the
the
the
pressure drop of the upper annulus during dual gradient drilling, MPa
pressure drop in the upper annulus of the first stage separator during multi gradient drilling, MPa
pressure drop in the annulus between the first stage and the second stage separator during multi gradient drilling, MPa
pressure drop between the second stage separator and the bit during multi gradient drilling, MPa
acceleration of gravity, 9.8 m/s2
length of the liquid column between the two separators, m;
depth of any point in the annulus, m;
distance between the bottom of the separator and the drill bit, m;
real-time well depth, m;
ρm
Qm
Qs,out
cs1
ρs1
Qa,v
Qa,i
Qa,o
Qa,w
A2
Uaw
dw
cM
ρM
cs2
ρs2
ε
ε1
ε2
ψ
ρf
P*dh,M
the bottom-hole pressure while drilling during multi gradient drilling, MPa
17
R. Zhang et al.
Pcp
α
Journal of Petroleum Science and Engineering 205 (2021) 108739
the wellhead back pressure, MPa
the deflection angle, ◦
Appendix A. Model Solution
The model in this paper is solved by the finite difference method, in which the first-order and second-order spaces use the first-order upside-down
style and the three-point central difference respectively, and the time dispersion uses the two-point backward difference. The grid division of the
wellbore and the formation is shown in Fig.A1, whichxrepresents the radial direction and yrepresents the axial direction. jrepresents the radial grid
node andirepresents the axial grid node.
Fig. A1. Grid division of wellbore and formation
Appendix A.1 discretization of heat transfer model in the upper and lower drill string
(
)
(
)
∂ ρ ci Tp
∂(ρ ci Ti )
A1 i
= Qm i
+ πdpi Upa Ta − Tp + Qca,i (i = 1, 3)
∂t
∂y
(A-1)
makea = A1 ρi ci /Δt ,b = Qm ρi ci /Δy ,c = π dpi Uap,i
Then, the discretization equation is shown as(A-2):
(A-2)
n
(a + b + c)Ti,jn − bTi−n 1,j − cTi,j+1
= aTi,jn− 1 + (Qca )ni
Appendix A.2 discretization of the heat transfer equation in the drill string of the separator section
[
A1
(
∂ ρ cs Tp − Ta
∂(ρi ci Ti )
− A1 s
∂t
∂t
)]
= Qm
(
)
+πdpi Upa Ta − Tp + Qca,i (i = 2)
∂(ρi ci Ti )
− A1 cs ρs Ta
∂y
(A-3)
Makea = A1 ρi ci /Δt b = A1 ρs cs /Δt ,c = Qm ρi ci /Δy,d = A1 ρs cs ,e = πdpi Uap,i
Then, the discretization equation is shown as(A-4):
(A-4)
n
− cTi−n 1,j = aTi,jn− 1 + (Qca )ni
(a − b + c + e)Ti,jn + (b + d − e)Ti,j+1
Appendix A.3 discretization of heat transfer equation in upper or lower annulus
A2
(
)
∂(ρi ci Ta )
∂(ρ ci Ta )
= Qm i
+ πdpo Uap Ta − Tp + πdw Uaw,i (Tw − Ta ) + Qca,i (i = 4, 6)
∂t
∂y
(A-5)
Makea = A2 ρi ci /Δt ,b = Qm ρi ci /Δy ,c = π dpo Uap,i ,d = πdw Uaw,i
Then, the discretization equation is shown as(A-6):
(A-6)
n
n
n
n
n− 1
(a + b − c + d)Ti,jn − bTi+1,j
+ cTi,j−
1 − dTi,j+1 = aTi,j + (Qca )i
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R. Zhang et al.
Journal of Petroleum Science and Engineering 205 (2021) 108739
Appendix A.4 discretization of the heat transfer equation in the annulus of the separator section
[
A2
(
∂ ρ cs Tp − Ta
∂(ρi ci Ta )
+ A2 s
∂t
∂t
)]
= Qm
∂(ρi ci Ta )
+ A2 cs ρs Ta
∂y
(
)
− πdpo Upa Ta − Tp + πdw Uaw,i (Tw − Ta )
(A-7)
+ Qca,i (i = 5)
Makea = A2 ρi ci /Δt,b = A2 ρs cs /Δt,c = Qm ρi ci /Δy,d = A2 ρs cs ,e = πdpo Uap,i f = πdw Uaw,i
Then, the discretization equation is shown as(A-8):
(A-8)
n
n
n
n
n− 1
(a + b + c − d + e + f)Ti,jn − (b + e)Ti,j−
1 − cTi+1,j − fTi,j+1 = aTi,j + (Qca )i
Appendix A.5Schematic of solution procedure
Fig. A2. Schematic of solution procedure
19
R. Zhang et al.
Journal of Petroleum Science and Engineering 205 (2021) 108739
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