THERMAL AND HYDRODYNAMIC PHENOMENA ASSOCIATED WITH THE
GAS ASCENDING MOVEMENT THROUGH EXTRACTION WELL COLUMNS
Cornel Trifan, Mihai Albulescu, Eugen Mihai Ionescu
Dr Ing. Conferenţiar, Universitatea “Petrol-Gaze” Ploieşti
INTRODUCTION
In approaching the gas ascending movement through the extraction columns of wells,
we will start from the hypothesis of hydrodynamic parameters having the same values for
the entire flow section and depending exclusively on its elevation. Therefore we will
consider the following functions defined within the range 0H, where H is the well depth,
the origin of Oz axis being at bottom of the borehole: v(z) –velocity of gas particles, p(z) –
gas pressure along the pipe, T(z) –absolute temperature of gases along the pipe and (z) –
gas density along the pipe. As for the variation of adjacent formations temperature along
the extraction well, we will consider the geothermal gradient of 2 C, and the earth
temperature of 10 C. There is air in the annulus between the pipes and production string,
and the cement ring thickness depends on the size of bit.
MATHEMATICAL MODEL
The process of gas ascending flow through the well exploitation column is simulated by
the movement equation, continuity equation, next state equation, and thermal transport
equation[1]:
dv
1 dp f  v 2
v  g 

;
dz
 dz
2D
v  M
p
 ZRT

(1)
(2)
(3)
 1/    dp dT  K D
1  1

(4)
T


Tad  T 
cp  
T p  dz dz  cpQ


Here f is the coefficient of hydraulic resistance, D –column nominal bore, M – gas
specific weight rate in kg/(m2s), Z – factor of deviation from perfect gases law, cp – isobar
weight heat capacity of gases, K – thermal transmittance through the walls of conducting
tube towards adjacent formations, Q – gas cubical discharge through the column, and Tad is
the absolute temperature of adjacent formations resulted from the geothermal gradient and
considered at 1,5 m from the cement ring.
As the ascending drive of gas movement is turbulent – rough, the value of coefficient f
is to be calculated with Nikuradse’s formula
2

 D 
f  1,74  2l g   
(5)
 2k  

k being the absolute roughness of the inside wall of conducting tube. The value of deviation
factor Z is to be calculated using the Berthelot’s formula, which is considered as the most
accurate in the case of natural gases

9 p Tc  Tc2
(6)
Z 1
6 2  1 ,
128 pc T  T

where pc and Tc are the gas critical parameters. Considering a dry natural gas deposit, we
can accept for the critical parameters the methane values, i.e. pc = 46,5 bar and Tc = 190,5
K.
In order to calculate the coefficient K, it is taken into account the serially heat transfer
from gases to pipe walls by forced convection, through the tubing wall by conduction,
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«Вісник СумДУ», №13(72), 2004
through the air in the tubing annulus, through the wall of production string, through the
cementing column towards the adjacent formations and finally towards the environment.
2


 h 
D
D
D
1
1 t   c Di t
1
h



l n i c  i t l n c  l n 2
 4    1
(7)

K i
0
2a Det 2c Dec f  Dc
 Dc 


Here i and f are the coefficients of forced convection heat transfer from gases to the
tubing inside wall, and free convection heat transfer from the outside wall of cement
annulus to the well adjacent formations, and o, c, f and a are the coefficients of
conduction heat transfer through the steel of tubing and column walls, through cement,
through the air between column and tubing(convection here is not be considered), and
through adjacent formations, respectively(along the estimated distance of h=1,5 m).
Furthermore Dit, Det, t are the tubing inside diameter, outside diameter and thickness,
respectively, Dic, Dec,c –production string inside diameter, outside diameter and thickness,
respectively, and Dc– cement ring outside diameter.
The coefficient of forced convection heat transfer from gases to the tubing inside wall is
to be determined with the formula
N u
,
(8)

D
where Nu is Nusselt criterion, which is to be calculated with Ditts-Boelter formula specific
to the movement turbulent drive through pipes
(9)
Nu  0, 023Re0.8 Pr 0.4 ,
in which Re and Pr are Reynolds criterion and Prandtl criterion, respectively, resulted from
the criterion relations
 cpv
vD
; Pr 
,
(10)
Re 

v
where  is the conductivity of transported gases, v, cp and  –kinematic viscosity, isobar
heat capacity and density of gases, respectively. The equation of movement can be also
written as
 1 1 Z
1
p  dp  1 1 Z  dT
g
p2
f
 2
 
 2 2 2 2

 

p
Z

p
ZT
dz
T
Z

T
dz
2
D


M R
M R Z T


Introducing the dimensionless parameters
T
p
z
, T
,ζ 
,
P
Tc
pc
H
the movement equation is written as
 1 1 Z
p2 P  dP  1 1 Z  dT
pc2
P2
fH
 c
 
 gH

 


2
2
2
2
2
2

2D
M R Tc Z T
 P Z P M R Z T  dζ  T Z T  dζ
with the remark that formula (6) is now written as
Z 1

2
9 P 18-T
9 T2  6 Z
9 P 6
 Z

;
;


1


T 128
128 T  T2
T4
 P 128 T3
«Вісник СумДУ», №13(72), 2004

(11)
(12)
(13)
(14)
35
If we take into account the state equation (3) and calculate the isobar derivative of gas
specific volume, the energy equation becomes
1 
R
Z   dp dT  K D
1  T  Z  T


T  T  (15)

 cp 
p
T   dz dz  cpQ ad
which can be also written as
T Z dP cp P dT
4K
p


(16)
T  Tad 
Z T dζ
R ZT dz M RD ZT
Introducing the same dimensionless parameters (12) into this equation, the following
expression results
T Z dP cp P dT 4K H P


(17)
T  Tad 
Z T dζ R ZT dζ
M RD ZT
Tad being the dimensionless temperature of adjacent formations resulted from
Tad Ts  n

,
(18)
Tc
Tc
where =2 K is the geothermal gradient, n is the column length(hundreds of meters), and Ts
=293 K is the earth temperature. Thus, we have eliminated the functions v(z) and (z);in
order to determine the remainder functions p(z) and T(z), which has become P (), and
T(), we must solve the system of differential equations (13), (17). We will make use of the
quartic method Runge-Kutta for integrating the systems of ordinary differential
equations[3].
Tad 
NUMERICAL MODEL
In order to approach numerically the system of differential equations (13), (17), we will
write the movement equation (13) as
dP
dT
(19)
m1
 m2
 m0
d
d
where the functions m1, m2 and m0 are expressed as:
pc2
1 1 Z
P2
fH
1 1 Z
P
; m1  
, m2   


2
2
2
2
2
2
T
Z T
2
D
P
Z

P
Z
T
M R Tc Z T
M TcR
as well as the energy equation (17) as
dP
dT
e1
 e2
 e0 ,
dζ
dζ
where e0, e1 and e2 are expressed as:
c
4K H P
e0 
T - Tad  , e1  T Z , e2   p P
M RD ZT
Z T
R ZT
so that we can express the system solutions as
dP m0e2  m2e0
dT m1e0  m0e1

 Φ P, T ,

 Ψ P, T
dζ m1e2  m2e1
dζ m1e2  m2e1
m0  gH
pc2
(20)
(21)
(22)
(23)
For the numerical approach of the problem we will section the extracting column into n
sections of 100 m length(n = H/100). The lower elevation of section i is zi =(i-1)100 m, the
dimensionless elevation i =(i-1)/n, respectively. The values of the two dimensionless gas
dynamic parameters at
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«Вісник СумДУ», №13(72), 2004
this elevation will be P ζ i   Pi and T ζ i   Ti , respectively, and at the upper elevation
of the section P ζ i 1  Pi 1 and T ζ i 1  Ti 1 , respectively, calculated with the
quartic method Runge-Kutta .
CALCULUS EXAMPLE
Based on the above algorithm a Turbo/Pascal calculus program was issued, by means of
which pressure and temperature distribution have been determined along of 2000 m-length
well through wich 50.000 Nm3 of gas are extracted per day, with a 2 in tubing, 6 in
cementing column and 6 cm thickness of cement ring.
The variations of gas pressure and temperature, respectively, along the well tubing for
which the calculations have been carried out are showed hereinafter.
CONCLUSIONS
The mathematical model proposed in this study allows the calculation of gas pressure
and temperature in its ascending movement through the tubing, without making use of the
method proposed in [2].
150
60
145
50
140
Presiune
Presiunea, bar
40
130
125
30
120
20
115
110
Temperatura, gr. C
Temperatura
135
10
105
100
0
0
100
200
300
400
500
600
700
800
900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000
Adincim ea, m
ABSTRACT
It is regarded the ascending steady flow of natural gases through the extraction column, taking into account
the thermal transfer towards the adjacent formations, as well as the deviation from the perfect gases law. The
resulting differential equations being non-linear, a quartic integration numeric method Runge-Kutta is used.
BIBIOGRAPHY
1.
2.
3.
T. Oroveanu, V. David, Al. Stan, C. Trifan Colectarea, transportul, depozitarea şi distribuţia produselor
petroliere şi gazelor, Editura didactică şi pedagogică, Bucureşti, 1985, pg. 151-176.
N. Puşcoiu, Extracţia gazelor naturale, Editura tehnică, Bucureşti, 1970, p.257-284..
GH. Vraciu, A Popa, Metode numerice cu aplicaţii în tehnica de calcul, Editura Scrisul românesc, Craiova,
1982, pg.179-196.
Надійшла до редакції 10 березня 2004р.
«Вісник СумДУ», №13(72), 2004
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