It is regarded the isotherm movement of gases through a porous and isotropic medium towards a central well, taking into account the deviation from the perfect gases law and the viscosity pressure variation. The resulting model, completed with the specific limit conditions, is approached through a numerical method of solving and is applied to the wells through which the gas fields are exploited at constant flow. For the current exploitation rates of flow the variation curves of gas pressure at top of the well have been determined, thus resulting the flow maximum value after which the extraction process becomes unstable.
When you are citing the document, use the following link http://essuir.sumdu.edu.ua/handle/123456789/1154

Albulescu, Mihai

Ionescu, Eugen Mihai

Trifan, Cornel

Trifan, C.

Albulescu, M.

Ionescu, E.M.

Electronic Sumy State University Institutional Repository

Economics related to exploitation of gas wells at constant  pressure

It is regarded the isotherm movement of gases through a porous and isotropic medium towards a central well, taking into account the deviation from the perfect gases law and the viscosity pressure variation. The resulting model, completed with the specific limit conditions, is approached through a numerical method of solving and is applied to the wells through which the gas fields are exploited at constant flow. For the current exploitation rates of flow the variation curves of gas pressure at top of the well have been determined, thus resulting the flow maximum value after which the extraction process becomes unstable.
When you are citing the document, use the following link http://essuir.sumdu.edu.ua/handle/123456789/1154

 «Вісник СумДУ», №1(73), 2005 25 ECONOMICS RELATED TO EXPLOITATION OF GAS WELLS AT CONSTANT FLOW   Cornel Trifan;  Mihai Albulescu;  Eugen Mihai Ionescu Dr Ing. Conferenţiar, Universitatea “Petrol-Gaze” Ploieşti   1 INTRODUCTION The problem of gas flow through porous mediums with application to exploitation of fields shows serious difficulties even in the simple case of plane-quaquaversal flow through a homogenous and isotropic porous medium. Indeed, even if it accepted that the flowing phenomenon is usually isothermal, the pressure differential equation is non-linear and as for the gas behaviour both the deviation from the perfect gases low and the viscosity pressure variation should be taken into account. The mathematical model described in this study takes into account all the above aspects, therefore the use of a numerical method of solving is required. Such method is applied in the case of a permeable field exploited at various rates of flow. For each of these the variation of gas pressure at top of the well is to be determined.   2 MATHEMATICAL MODEL The process of gas plane-quaquaversal isothermal flow through a homogenous and isotropic porous medium is simulated by equation [1]  p m ppZ k t Z             (1) where -Hamiltonian, m and k – porosity, i.e. the permeability of porous medium(constants), p – field gases pressure, and  and Z – dynamic viscosity, i.e. the factor of gas deviation from the perfect gas model, both depending on pressure. Equation (1) may also be written as    2221 1p p p d pr r t dp rp                 (2) where the function  and   result from the relations    pZ ; 21m dZZ pk dpz      (3) and depend on gas pressure only. Equation (2) represents the mathematical model of the process of gas flow through a circular field towards a central well, its solution providing the gas pressure distribution in time according to radius. Introducing a new dimensionless variable by    lnln crR  ; srrR ; ccsRRR   (4) sR  being the well radius and cR  the field outline, as well as the low pressure P(,t) by    ( , ),cp tP tp    (5) cp  being the critical pressure, equation (2) becomes [2]    22 22 cP P Pa R bt              (6) where the functions  and  result from the expressions  «Вісник СумДУ», №1(73), 2005 26    1 1 dZdPP Z P     ;  (7)    1 1 dZ 1 dd dPP Z P P     ,  (8) and the coefficients a and b are calculated with    2 20 lns ccma R Rkp ; 18 cbp .  (9)  Equation (6) simulating the process of gas flow towards a central well into a homogenous and isotropic circular field is a differential equation with partial derivative of 2nd  degree of parabolic type and non-linear. It admits no analytic solution, but a computer assisted numerical approach.   3 INITIAL AND LIMIT CONDITIONS The initial condition transposes mathematically the fact that the gas pressure in entire field has the value pz at the initial moment, i.e.     ,0 zzcPP PP    (10)  The limit conditions are necessary in terms of field exploitation method and also of its characteristic. Hence, in case the field is exploited at constant rate of flow, the condition of maintaining constant the gas velocity at well outlet is to be set according to the well exploitation flow:   0sQPA , (11) sQ  being the gas rate of flow, and A a constant resulted from     2 s s c Ns s N s cHR kp p TAZ p T R  (12)  where H – thickness of carrier bed, sp  – gas pressure at top of the well,  sT  – gas temperature at top of the well, s  – dynamic viscosity corresponding to this pressure, sZ  – compresibility factor corresponding to this pressure, Np  – normal pressure, and NT  – normal temperature. The second limit condition refers to the field outline. Thus, in the case of closed field with permeable outline(with water pushing), the condition of maintaining constant the pressure zp  is necessary on such outline     1, zzcpP t Pp    (13)  and in the case of closed field with impermeable outline, the condition of maintaining zero flow on such outline is necessary, i.e.    10P  (14)     «Вісник СумДУ», №1(73), 2005 27 4 NUMERICAL METHOD We will transform the continuous spectrum C: [0 ≤  ≤ 1, 0 ≤  t ≤ T ] into the point discrete lattice Rij :[ =(i-1)h, t=jini ], where  i=1n,  j=0m are the spatial index and the temporal index, respectively, and h and  are the spatial pitch and temporal pitch, respectively, and n and m are their numbers. Thus, instead of exact values of pressure P(, t)  we will consider the discrete approximate values  ,ji i jP P t .  Making use of calculus scheme with finite quotients of Hyman Kaplan implicit type [3], known as stable unconditionally and absolutely convergent, as well as end approximations proposed by West, Garvin and Sheldon [2], equation (8) becomes a system of equations generated by the scheme    1 11 1 1 11 12j jj j j j ji i i i i i iP C P P C P D            (15) for i = 2n, where      2 2 11 1h ij ji c iahC R P   (16)     21 11 11 14j jj ji i i ibD P P P        (17)    11 11 11 jj j ii i j ji iZpP VP Z       (18)    1 111 1 11 j jj i ii j j ji i iZp VpPP Z V         (19)    1 1j ji iV P  ;  1 1ddj ji iVp PP  , (20)    1 1j ji iZ Z P  ;  1 1ddj ji iZZp PP  . (21) completed with   jiP Pz ; 11jnP Pz   (22)   1 1 11 2 323 4j j jhQP P PA       (23)   1 111 13 4 0j jjn n nP P P      (24)  Here the values of low pressure at temporal level j are considered as known, and unknown at temporal level j+1, respectively.   5 NUMERICAL SIMULATION OF FIELD EXPLOITATION In order to solve the system of linear algebraic equations generated by the scheme with finite quotients (15) we will mark with P0(i) the distribution of low pressure corresponding to known moment, j, and with P(i) the distribution of low pressure corresponding to unknown moment, j+1. The two approximate solutions successively obtained by solving the algebraic system generated by the calculus scheme adopted will be marked with P1(i) and P2(i), respectively; obviously i will take values from 1 to n+1. To facilitate the compiling of the calculus program we will define the supporting functions Z(X), Zp(X), V(X), Vp(X) that will help us calculate the value sets of deviation factor Z(i), Zp(i) and dynamic viscosity V(i), Vp(i), respectively, corresponding to the distribution of low pressure at moments required by running the calculus program, i.e. for P0(i), P1(i)or P2(i). Now the values corresponding to expressions C(i) and D(i) can be calculated based on calculus procedures. The conditions (22), (23) and (24), respectively are obviously written as  «Вісник СумДУ», №1(73), 2005 28    0P i Pz ;  1P Ps ;  1P n Pz   (25)        23 1 4 2 3hQP P PA     (26)        3 1 4 1 0P n P n P n      (27)  In the case of permeable closed field exploited at constant pressure, the calculus scheme (15) With  the conditions (25) generate the following system of equations     1P Ps  (28)            1 2 1P i C i P i P i B i        , 2i n   (29)    1 zP n P   (30) where          0B i C i P i D i   , 2i n   (31)  As the resulting algebraic system has a coefficient matrix of Jacobi type, i.e. tridiagonally, its solving becomes easy following the use of Thomas process [2]. Thus, we will firstly determine    ' 1 0a  ;  ' 1 sb P  (32) after which the following can be calculated for 2i n         2a i C i     ;        0b i C i P i D i    (33)       1'' 1a ia i a i ;        ' 1'' 1b i b ib ia i a i   (34) and finally, because  1 zb n P  , the solution of the system of equation is     1 zP n P  ;        1 ' 1 ' 1 2P n i b n i a n i P n i           (35)   5 CALCULUS EXAMPLE We will consider a closed impermeable circular gas field having the outline radius of 200 m, 40 m thickness, 20% porosity and 10 mD permeability. The pressure of field gases is of 140 bar, and their temperature of 27 C. The field is exploited by a well of 0,1 m radius. For pressure depending on gas viscosity and deviation factor, respectively, we accept the relations:      21 0,02 0,004P P P     (36)     21 0,062 0,004Z P P P    (37)  experimentally determined from field curves. We will consider a spatial pitch of digitization lattice of 0,01 a temporal pitch of 60 s, and 10-5 acceptable error of iterative calculus. In diagram 1 the pressure curves at top of the well are showed for various values of flow extracted. It can be observed the extraction process becoming unstable for a certain flow value.   «Вісник СумДУ», №1(73), 2005 29  Diagram 1   ABSTRACT  It is regarded the isotherm movement of gases through a porous and isotropic medium towards a central well, taking into account the deviation from the perfect gases law and the viscosity pressure variation. The resulting model, completed with the specific limit conditions, is approached through a numerical method of solving and is applied to the wells through which the gas fields are exploited at constant flow. For the current exploitation rates of flow the variation curves of gas pressure at top of the well have been determined, thus resulting the flow maximum value after which the extraction process becomes unstable.   BIBLIOGRAPHY  1. T.Oroveanu, Scurgerea fluidelor multifazice prin medii poroase.Editura Academiei, Bucureşti 1966. - Р.74-78. 2. C-Tin Рopa. Modelarea numerică a exploatării zăcămintelor de hidrocarburi, în Ingineria zăcămintelor de hidrocarburi, (coordonator Al.Soare). - Vol. 2. - Editura tehnică, Bucureşti, 1981. - P.412-418. 3. C. Trifan. Contribuţii la studiul curgerii gazelor prin conducte lungi. Teză de doctorat, Ploieşti, 1983.   Надійшла до редакції 10 березня 2004р.  4050607080901001101200 4 8 12 16 20 23 27 31 35 39 43 47 51 55 59 62 66 70Presiunea, bar Timpul, ore q=16 q=15  q=14 q=13 

Economics related to exploitation of gas wells at constant flow

SocioEconomic Challenges Journal (Sumy State University)

 «Вісник СумДУ», №1(73), 2005 25 
ECONOMICS RELATED TO EXPLOITATION 
OF GAS WELLS AT CONSTANT FLOW 
 
 
Cornel Trifan;  Mihai Albulescu;  Eugen Mihai Ionescu
 
Dr Ing. Conferenţiar, Universitatea “Petrol-Gaze” Ploieşti 
 
 
1 INTRODUCTION 
The problem of gas flow through porous mediums with application to exploitation of 
fields shows serious difficulties even in the simple case of plane-quaquaversal flow through 
a homogenous and isotropic porous medium. Indeed, even if it accepted that the flowing 
phenomenon is usually isothermal, the pressure differential equation is non-linear and as for 
the gas behaviour both the deviation from the perfect gases low and the viscosity pressure 
variation should be taken into account. 
The mathematical model described in this study takes into account all the above aspects, 
therefore the use of a numerical method of solving is required. Such method is applied in 
the case of a permeable field exploited at various rates of flow. For each of these the 
variation of gas pressure at top of the well is to be determined. 
 
 
2 MATHEMATICAL MODEL 
The process of gas plane-quaquaversal isothermal flow through a homogenous and 
isotropic porous medium is simulated by equation [1] 
 
p m p
p
Z k t Z
    
     
   
 (1) 
where -Hamiltonian, m and k – porosity, i.e. the permeability of porous 
medium(constants), p – field gases pressure, and  and Z – dynamic viscosity, i.e. the factor 
of gas deviation from the perfect gas model, both depending on pressure. Equation (1) may 
also be written as  
  
22
2
1 1p p p d p
r r t dp rp
 
 
    
     
    
  (2) 
where the function  and   result from the relations  
  
p
Z


 ; 
2
1m dZ
Z p
k dpz

 
  
 
  (3) 
and depend on gas pressure only. Equation (2) represents the mathematical model of the 
process of gas flow through a circular field towards a central well, its solution providing the 
gas pressure distribution in time according to radius. Introducing a new dimensionless 
variable by  
  
ln
ln c
r
R
  ; 
s
r
r
R
 ; cc
s
R
R
R
   (4) 
sR  being the well radius and cR  the field outline, as well as the low pressure P(,t) by 
   
( , )
,
c
p t
P t
p

    (5) 
cp  being the critical pressure, equation (2) becomes [2] 
   
22 2
2 c
P P P
a R b
t

 

   
    
   
   (6) 
where the functions  and  result from the expressions 
 «Вісник СумДУ», №1(73), 2005 26 
   
1 1 dZ
d
P
P Z P
 
 
  
 
;  (7) 
   
1 1 dZ 1 d
d d
P
P Z P P



 
   
 
,  (8) 
and the coefficients a and b are calculated with 
   2 20 lns c
c
m
a R R
kp

 ; 
1
8 c
b
p
 .  (9) 
 
Equation (6) simulating the process of gas flow towards a central well into a 
homogenous and isotropic circular field is a differential equation with partial derivative of 
2nd  degree of parabolic type and non-linear. It admits no analytic solution, but a computer 
assisted numerical approach. 
 
 
3 INITIAL AND LIMIT CONDITIONS 
The initial condition transposes mathematically the fact that the gas pressure in entire 
field has the value pz at the initial moment, i.e. 
 
   ,0 zz
c
P
P P
P
    (10) 
 
The limit conditions are necessary in terms of field exploitation method and also of its 
characteristic. Hence, in case the field is exploited at constant rate of flow, the condition of 
maintaining constant the gas velocity at well outlet is to be set according to the well 
exploitation flow: 
  
0
sQP
A 



, (11) 
sQ  being the gas rate of flow, and A a constant resulted from 
 
 
  
2 s s c N
s s N s c
HR kp p T
A
Z p T R


  (12) 
 
where H – thickness of carrier bed, sp  – gas pressure at top of the well,  
sT  – gas temperature at top of the well, s  – dynamic viscosity corresponding to this 
pressure, sZ  – compresibility factor corresponding to this pressure, Np  – normal 
pressure, and NT  – normal temperature. 
The second limit condition refers to the field outline. Thus, in the case of closed field with 
permeable outline(with water pushing), the condition of maintaining constant the pressure 
zp  is necessary on such outline 
 
   1, zz
c
p
P t P
p
    (13) 
 
and in the case of closed field with impermeable outline, the condition of maintaining zero 
flow on such outline is necessary, i.e. 
 
  
1
0
P
 



 (14) 
 
 
 
 «Вісник СумДУ», №1(73), 2005 27 
4 NUMERICAL METHOD 
We will transform the continuous spectrum C: [0 ≤  ≤ 1, 0 ≤  t ≤ T ] into the point discrete 
lattice Rij :[ =(i-1)h, t=jini ], where  i=1n,  j=0m are the spatial index and the 
temporal index, respectively, and h and  are the spatial pitch and temporal pitch, 
respectively, and n and m are their numbers. Thus, instead of exact values of pressure P(, 
t)  we will consider the discrete approximate values  ,ji i jP P t .  
Making use of calculus scheme with finite quotients of Hyman Kaplan implicit type [3], 
known as stable unconditionally and absolutely convergent, as well as end approximations 
proposed by West, Garvin and Sheldon [2], equation (8) becomes a system of equations 
generated by the scheme 
   1 11 1 1 11 12j jj j j j ji i i i i i iP C P P C P D            (15) 
for i = 2n, where 
   
 
 
2 2 1
1 1
h i
j j
i c i
ah
C R P


   (16) 
    
2
1 11 1
1 1
4
j jj j
i i i i
b
D P P P        (17) 
   
1
1 1
1 1
1 jj j i
i i j j
i i
Zp
P V
P Z


 
 
 
  
 
 (18) 
   
1 1
1
1 1 1
1 j jj i i
i j j j
i i i
Zp Vp
P
P Z V

 

  
 
   
 
 (19) 
   1 1j ji iV P  ;  1 1
d
d
j j
i iVp P
P
  , (20) 
   1 1j ji iZ Z P  ;  1 1
d
d
j j
i i
Z
Zp P
P
  . (21) 
completed with 
  jiP Pz ; 
1
1
j
nP Pz

   (22) 
  1 1 11 2 3
2
3 4j j j
hQ
P P P
A
       (23) 
  1 111 13 4 0
j jj
n n nP P P
 
     (24) 
 
Here the values of low pressure at temporal level j are considered as known, and 
unknown at temporal level j+1, respectively. 
 
 
5 NUMERICAL SIMULATION OF FIELD EXPLOITATION 
In order to solve the system of linear algebraic equations generated by the scheme with 
finite quotients (15) we will mark with P0(i) the distribution of low pressure corresponding 
to known moment, j, and with P(i) the distribution of low pressure corresponding to 
unknown moment, j+1. The two approximate solutions successively obtained by solving 
the algebraic system generated by the calculus scheme adopted will be marked with P1(i) 
and P2(i), respectively; obviously i will take values from 1 to n+1. 
To facilitate the compiling of the calculus program we will define the supporting 
functions Z(X), Zp(X), V(X), Vp(X) that will help us calculate the value sets of deviation 
factor Z(i), Zp(i) and dynamic viscosity V(i), Vp(i), respectively, corresponding to the 
distribution of low pressure at moments required by running the calculus program, i.e. for 
P0(i), P1(i)or P2(i). Now the values corresponding to expressions C(i) and D(i) can be 
calculated based on calculus procedures. 
The conditions (22), (23) and (24), respectively are obviously written as 
 «Вісник СумДУ», №1(73), 2005 28 
   0P i Pz ;  1P Ps ;  1P n Pz   (25) 
       
2
3 1 4 2 3
hQ
P P P
A
     (26) 
       3 1 4 1 0P n P n P n      (27) 
 
In the case of permeable closed field exploited at constant pressure, the calculus scheme 
(15) 
With  the conditions (25) generate the following system of equations 
 
   1P Ps  (28) 
           1 2 1P i C i P i P i B i        , 2i n   (29) 
   1 zP n P   (30) 
where 
         0B i C i P i D i   , 2i n   (31) 
 
As the resulting algebraic system has a coefficient matrix of Jacobi type, i.e. 
tridiagonally, its solving becomes easy following the use of Thomas process [2]. Thus, we 
will firstly determine 
   ' 1 0a  ;  ' 1 sb P  (32) 
after which the following can be calculated for 2i n   
 
     2a i C i     ;        0b i C i P i D i    (33) 
   
   
1
'
' 1
a i
a i a i

 
;  
   
   
' 1
'
' 1
b i b i
b i
a i a i
 

 
 (34) 
and finally, because  1 zb n P  , the solution of the system of equation is 
 
   1 zP n P  ;        1 ' 1 ' 1 2P n i b n i a n i P n i           (35) 
 
 
5 CALCULUS EXAMPLE 
We will consider a closed impermeable circular gas field having the outline radius of 
200 m, 40 m thickness, 20% porosity and 10 mD permeability. The pressure of field gases 
is of 140 bar, and their temperature of 27 C. The field is exploited by a well of 0,1 m 
radius. For pressure depending on gas viscosity and deviation factor, respectively, we 
accept the relations: 
 
    21 0,02 0,004P P P     (36) 
    21 0,062 0,004Z P P P    (37) 
 
experimentally determined from field curves. 
We will consider a spatial pitch of digitization lattice of 0,01 a temporal pitch of 60 s, 
and 10
-5
 acceptable error of iterative calculus. 
In diagram 1 the pressure curves at top of the well are showed for various values of flow 
extracted. It can be observed the extraction process becoming unstable for a certain flow 
value. 
 
 «Вісник СумДУ», №1(73), 2005 29 
 
Diagram 1 
 
 
ABSTRACT 
 
It is regarded the isotherm movement of gases through a porous and isotropic medium towards a central well, 
taking into account the deviation from the perfect gases law and the viscosity pressure variation. The resulting 
model, completed with the specific limit conditions, is approached through a numerical method of solving and is 
applied to the wells through which the gas fields are exploited at constant flow. For the current exploitation rates 
of flow the variation curves of gas pressure at top of the well have been determined, thus resulting the flow 
maximum value after which the extraction process becomes unstable. 
 
 
BIBLIOGRAPHY 
 
1. T.Oroveanu, Scurgerea fluidelor multifazice prin medii poroase.Editura Academiei, Bucureşti 1966. - Р.74-
78. 
2. C-Tin Рopa. Modelarea numerică a exploatării zăcămintelor de hidrocarburi, în Ingineria zăcămintelor de 
hidrocarburi, (coordonator Al.Soare). - Vol. 2. - Editura tehnică, Bucureşti, 1981. - P.412-418. 
3. C. Trifan. Contribuţii la studiul curgerii gazelor prin conducte lungi. Teză de doctorat, Ploieşti, 1983. 
 
 
Надійшла до редакції 10 березня 2004р. 
 
40
50
60
70
80
90
100
110
120
0 4 8 12 16 20 23 27 31 35 39 43 47 51 55 59 62 66 70
P
re
s
iu
n
e
a
, 
b
a
r 
Timpul, ore 
q=16 
q=15  
q=14 
q=13 


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Economics related to exploitation of gas wells at constant flow

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