Vol.2, No.12, 1394-1399 (2010)
doi:10.4236/ns.2010.212170
Copyright © 2010 SciRes. Openly accessible at http:// www.scirp.org/journal/NS/
Natural Science
New method of predicting surge pressure apply to
horizontal well based on casson flow*
Yuxue Sun, Qiming Li#, Jingyuan Zhao
Petroleum Engineering Department; Northeast Petroleum University; Daqing, China; #Corresponding Author: userlqm@126.com
Received 14 September 2010; revised 20 October 2010; accepted 23 October 2010.
ABSTRACT
In order to predict the surge pressure caused in
the horizontal well drilling process, a new sim-
ple and applicable method has been established.
It is based on the general theory of hydrostatic
drilling fluid mechanics, and specifically de-
scribed the flowing physical model towards
surge pressure in horizontal well annulus, tak-
ing the effect of string eccentricity on the flow-
ing law of drilling fluid into consideration. Ac-
cording to the constitutive equation of cas-
son-mode under one-dimensional steady flow
and the equations of annular flow rate under
different drill string working conditions, this
paper introduced the flow rate computation
models of axial laminar flow in eccentric annu-
lus apply to horizontal well, of which the nu-
merical model was calculated by the program
called Mathematica, ultimately, a new model for
surge pressure prediction towards each interval
in horizontal well was put forward. Application
examples indicated that it can solve questions
easily and precisely, which presents important
meaning of guidance to the safety control while
horizont al well drilling.
Keywords: Surge Pressure; Hor izontal Well;
Eccentric Annulu s; Casson-Mode
1. INTRODUCTION
While pipe or casing string pulling and running in the
well, the drilling fluid adhesive to the string moves with
it and the motion of the string has to get over the viscous
force of fluid at the same time, which causes an addi-
tional pressure on the borehole face, named Surge Pres-
sure, an important effect factor related to the formation
stability [1-3]. Traditional computation model of surge
pressure is mostly applicable to vertical well [4,5]. The
model of horizontal well has also been studied recently,
however, it involves mathematic theory of comparative
complexity which leads to the inconvenient computation
and low efficiency of application, as a result, the pre-
dicting model apply to horizontal well still has more
space to develop [6-9].
This paper presents a method of predicting surge
pressure applicable to horizontal well based on casson
flow, because casson-mode is more precise to describe
the rheological behaviour of drilling fluid with high
shearing rate inside the pipe or around the bit nozzle
[10,11]. This new method conjugates the flow rate com-
putation models of axial laminar flow in eccentric annu-
lus and uses Mathematica software conducting the nu-
merical calculation to simplify the computational pro-
cedure, which can also help predict the surge pressure
caused in vertical well and directional well.
2. PHYSICAL FLOWING MODEL IN
HORIZONTAL WELL BORE
This model neglects the compressibility of the fluid
and well bore, it does consider the complexities of the
non-Newtonian flow of drilling fluid and choose cas-
son-mode as the rheological behavior of the fluid. It is
assumed that the drilling fluid is under isothermal lami-
nar flow in eccentric annulus with fixed-length in the
axis, and at the same time it considers the motion pa-
rameters of every spatial point in flow field to be
time-invariant, which can simplify the calculation with
negligible impact on the predicting result. This is be-
cause the previous predicting results under the hypothe-
sis of traditional models are a little more conservative,
that means the predicting magnitude is bigger than nor-
mal, but this prerequisite of steady flow can make an
effect of correction [12-15].
In the intervals of deviated hole and horizontal hole,
action of gravity makes the string diverge from the
borehole central axis, and the annulus between the string
and borehole tends to be eccentric, which will influence
*Project of Natural Science Foundation of Heilongjiang Province,
China: The Study on Membrane Drilling Fluid Technology
Y. X. Sun et al. / Natural Science 2 (2010) 1394-1399
Copyright © 2010 SciRes. Openly accessible at http://www.scirp.org/journal/NS/
1391395
the flowing pattern of the drilling fluid, that means the
assumption of concentric annulus generally accepted in
drilling annular hydraulics computation will not be ap-
plicable any more [16-19].
Figure 1 and Figure 2 present respectively the physi-
cal flowing model in eccentric annulus and the simpli-
fied model of flowing section in eccentric annulus. The
borehole radius is h, the outside radius of running
string is o, the inside radius of running string is i,
the hole deviation angle is α, the average annular flow-
ing velocity of drilling fluid is
D
d d
v, the velocity of run-
ning string is
p
v, the eccentric arm between the axis of
the string and that of the borehole is e, and the clearance,
at any circumferential angle θ, between the outside sur-
face of the string and the inside borehole wall is h,
222
hsin
ho
De decos

We choose a micro hexahedron unit at any circumfer-
ential angle θ in the annulus along the direction paral-
leled with the flowing. In Figure 3, τ is the shear stress
between fluid layers, L is the length of fluid along the
flowing direction, and Δp is the pressure drop within that
length L. In a steady-flow model, because the external
forces on the micro unit should balance we can get this
[20]:
p0
zy


 (1)
When the shape of flowing section along the flowing
direction is considered invariable and the fluid is incom-
pressible, we get this:
d
i
v
α
D
h
d
o
p
v
y
e
z
Figure 1. Physical flowing model in eccentric annulus.
do
y = 0
D
h
e
27
180°
90°
θ
h
Figure 2. Simplified model of flowing section in eccentric
annulus.
o
p
L
u
y
Outside wall of inner string
Micro-hexahedron unit
Inside wall of outer string
z
τ
h
2
Figure 3. Force analysis of annulus fluid.
pp
zL

(2)
So the surge pressure in the borehole annulus at the
hole deviation angle α is:
cos
sp
PL
(3)
3. COMPUTATIONAL MODEL
ESTABLISHING
3.1. Flow Rate Computation in Eccentric
Annulus
According to the coordinate system showed in Figure
3, casson flow’s rheological equation is:
2
1/2
1/2 1/2
c
du
dy
 

 


(4)
Y. X. Sun et al. / Natural Science 2 (2010) 1394-1399
Copyright © 2010 SciRes. Openly accessible at http:// www.scirp.org/journal/NS/
1396
Combine Eq.4 with Eq.1:
dp 2
c
p
uyy
dyLL c



 

 (5)
Integrate the above equation, then the point velocity
of fluid at any circumferential angle θ under different
values of y is:

2
2
1/2 3/2
1/2 3/2
u, 22 2
4
32
c
c
ph h
yy
L
ph
y
L



 


 
 



 

 
 


y
(6)
So the axial laminar flow rate model of casson flow in
eccentric annulus is:



22
00
2/2
00
QdQ
2,
h
o
vds
dhu ydyd




 (7)
3.2. Flow Rate Analysis under Practical
Working Conditions
Four types of working conditions are usually applied
on site and they are plugging string with pump opened,
plugging string with pump closed, opening string with
pump opened and opening string with pump closed.
There is much in common between the conditions of
plugging string with pump opened and opening string
with pump opened under which fluids are expelled from
mud pump inside the string and then influence the aver-
age flowing velocity in the well, therefore, these two
conditions can be merged into one [21-23].
We consider that workstring is rigid and drilling fluid
is steady-flow, and take into account the annulus flowing
velocity changes aroused by the adhesion effect of drill-
ing fluid. According to the relationship between the dis-
placement of drilling fluid expelled by the running string
in unit time and the flowrate in annulus, we can establish
the equilibrium equation to get the average flowing ve-
locity in annulus under three different working condi-
tions respectively:
1) Plugging string with pump closed:
2
22
1.5 ocp
ho
d
v
Dd



Kv
(8)
2) Opening string with pump closed:
22
222
1.5 oi cp
hoi
dd
vK
Ddd



v
(9)
3) Plugging or opening string with pump opened:

2
22 22
1.5 p
ocp
ho ho
Q
d
vKv
Dd Dd



 (10)
In the equations above:
p
Qis the output volume of
mud pump, c
K
is adhesion coefficient, dimensionless.
Refer to the experienced data and theoretical relationship
plate of adhesion coefficient and annular ratio(ratio of
string diameter) based on power law flow and casson
flow, c
K
is generally ranged from 0.4 to 0.5, and when
the annulus ratio 0h
dD is more deviated from 1, the
magnitude of is more close to 0.4.
c
According to the average annular velocity
Kv under
different working conditions and the annular section area
s,
22
sho
Dd

Qt
, we can work out the annular
flowrate , Qtvs
, corresponding to different
working conditions:

2
c
Q4.71 1
tch o
2
p
K
DKdv

(11)

22 22
22
222
Q4.71oi ho
tc
hoi
ddDd
hop
K
Ddv
Ddd



(12)
22
c
Q4.711]Q
tch op
KDK dv



p
(13)
3.3. Computation Model of Surge Pressure
After programming on Mathematica software to com-
pute Eq.7 and containing different value theory and el-
liptic integral, we finally got the mathematical model to
compute the flowrate of axial laminar flow of casson
flow in eccentric annulus (Eq.14), here we only give the
equation under the working condition of plugging string
with pump closed as an example, and the other two con-
ditions should be computed in the same way.
2
p28 2
Q33
288
B
53
oo
cc
e
ABed d
L
pBC C
L

3
3
 








 
 
(14)
There into:

4342 22.64 2.642.2
1
A2hooh ho
Dedd DeDed

3232.642.643.03
Bohoh o
dDdeDed
 
22
ho
CD de
2


22
2
22
2
2
h
h
h
hDe
e
EeD
D
e
ED
 
22
222 2
22
hh h
hh
ee
DDeK eDK
DD

 
 

 

 

Y. X. Sun et al. / Natural Scien c e 2 (2010) 1394-1399
Copyright © 2010 SciRes. Openly accessible at http://www.scirp.org/journal/NS/
1391397
K(m) is complete elliptic integral of the first kind and
E(m) is complete elliptic integral of the second kind, of
which values can be solved by programming on Mathe-
matica software.
Combine the worked out annulus flowrate corre-
sponding to specific working condition (Eqs.11-13) with
Eq.14, so the pressure gradient
p
L
can be cal-
culated, then put it into Eq.3, and the surge pressure
gradient at the hole deviation angle α in the well bore
annulus is got.
Until now, based on the above analysis and calcula-
tion, we can calculate the surge pressure gradient i of
each interval in horizontal well, even including the most
complicated well structure as showed in Figure 4.
P
The calculation equations of surge pressure for dif-
ferent well intervals and the total surge pressure expres-
sion at any well depth in horizontal well are presented as
follow.
Vertical interval, hole deviation angle α is 0˚,
0
is
Deviated interval (angle gaining interval and angle
dropping interval), hole deviation angle α changes
PP
from α1 to α2, 2
1
is
PPd
Hold angle interval, hole deviation angle α is in-
variant, PP
is
Horizontal interval, hole deviation angle α is 90˚,
90
is
Thereinto, “i” indicates the sequence number of the
interval location in horizontal well, i is the corre-
sponding annulus surge pressure when the sequence
number is i, i is the corresponding well depth of that
specific interval, so the magnitude of surge pressure at
PP
L
P
L
any well depth (i = n) is: .
1
n
ii
i
PP
4. SURGE PRESSURE PREDICTION
WITH ACTUAL EXAMPLE
There is a certain horizontal well filled with drilling
fluid which presents Casson-mode: τc = 1.51 (Pa), η =
15.5 (mPa.s). The Φ 244.48 (mm) casing pipe of which
inside diameter is 222.5 (mm) starts to build angle at a
vertical depth of 600 (m) and the initial hole deviation
angle α1 = 8˚. The Φ127 (mm) drilling pipe of which
inside diameter is 82 (mm) runs at the speed of 1 (m/s)
and starts to hold angle at a vertical depth of 2204.82
(m), where the hole deviation angle α2 = 84˚ and the well
depth is 2306.13 (m). Supposed the working condition is
plugging string with pump closed, try to calculate the
annular surge pressure at the pipe shoe (in the drilling
pipe annulus) and determine the corresponding addition
mud density.
Figure 4. Horizontal wellbore structure.
Some known calculation parameters are mentioned
above, they are: τc = 1.51 (Pa), η = 15.5 (mPa.s), 2h =
222.5 (mm), 2o
d = 127 (mm), 2i
d = 82 (mm),
D
p
vL
= 1
(m/s), α1 = 8˚, α2 = 84˚ and some length parameters: 1 =
600 (m), 2= 2204.82 (m) – 600 (m) = 1604.82 (m).
Then we can conduct the calculation as follow.
L
Firstly, the degree of eccentricity is generally taken as
0.5, so the eccentricity arm e = 0.5 * 0.5 * (Φ 244.48 – Φ
127) = 0.0239 (m). Then we substituted h{0.11125
(m)}, o{0.0635 (m)}, e{0.0239 (m)} into the corre-
sponding expression followed Eq.14 to get the value of A,
B and C in Eq.14: A = 0.000110297, B = –0.000937975,
C = 0.007979025.
D
d
Secondly, using Mathematica program to compute the
elliptic integral
2
2
h
e
ED



,
2
22
h
e
EeD



and
2
2
h
e
KD


.
After substituting and computing we got: E(0.0295) =
1.55914, E(–0.0304) = 1.58267, K(0.0295) = 1.58258,
K(–0.0304) = 1.55905, so
= 0.34691,
= 0.00019.
Thirdly, annular ratio {127 (mm)/222.5 (mm) = 0.57}
is comparatively small, so we considered the adhesion
coefficient Kc = 0.4 and we substituted the numerical
values of c
K
(0.4), h (0.11125), o (0.0635), D d
p
v(1)
in Eq.9, then we got the annular flow rate under the
condition of plugging string with pump closed: Qt=
4.71 * (0.4 * 0.111252 + 0.6 * 0.06352) * 1 = 0.034 (m3/s).
After calculating the annular flow rate and pa-
rameters such as A, B, C,
Qt
and
, we substituted them
into Eq.14, so we can got the pressure gradient
p
L
=
0.1917859 (kPa/m). According to what is presented in
Eq.3, the annular surge pressure Ps at hole deviation
angle α is: Ps = 0.1917859 * cosα.
At vertical interval within the depth of = 600 (m),
its surge pressure:
1
L
P1 = 110
*
s
PL PL
1
* = 0.1917859 (kPa/m) * 600
Y. X. Sun et al. / Natural Scien c e 2 (2010) 1394-1399
Copyright © 2010 SciRes. Openly accessible at http:// www.scirp.org/journal/NS/
1398
2
*
(m) = 115.072 (kPa).
At angle gaining interval within the depth of =
1604.82 (m), its surge pressure:
2
L
P2 = = 0.1917859 (kPa/m) *
2
1
22
*
s
PLPd L
(sin84° – sin8°)* 1604.82 (m) = 263.254 (kPa).
The total annular surge pressure value at the pipe shoe
is:
P = P1 + P2 = 115.072 (kPa) + 263.254 (kPa) =
378.326 (kPa).
The corresponding addition mud density is:
ρ = P/gH = 378.326 (kPa)/{9.8 (N/kg)*2204.82(m)} =
0.0175 (g/cm3).
5. CONCLUSIONS
Based on the rheological mode of Casson flow, the
flow rate computation models of axial laminar flow in
eccentric annulus apply to horizontal well were success-
fully established. Finally, we developed a new model of
predicting surge pressure imposed on different intervals
in horizontal well, of which the numerical model could
be calculated by the program called Mathematica con-
veniently. And the magnitude of the predicting surge
pressure provided a criterion in determining the addition
mud density.
After calculating the actual example using this new
model and comparing with traditional predicting method,
it is obvious that this new model can be computed easily
by the field engineers. Across the steps of derivation of
this new model, we concluded that it can calculate flexi-
bly, it provides a method of predicting surge pressures in
vertical well and directional well after being simplified.
And this new model can also direct the secure produc-
tion on location through predicting surge pressures under
different working conditions of drill string.
6. ACKNOWLEDGEMENTS
We would like to thank the drilling engineers Shurui Zhang and
Guobin Li from Daqing Oil Field for their supports to the field test and
operations
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