Transient Analysis of Coalbed Methane Flow in a Coupled Reservoir-Wellbore System

Physical and Numerical Simulation of Geotechnical Engineering
10th Issue, Mar. 2013
Transient Analysis of Coalbed Methane Flow in a Coupled
Reservoir-Wellbore System
ZHANG Xianmin, TONG Dengke
School of Petroleum Engineering, China University of Petroleum, Dongying 257061, P.R.China
[email protected]
ABSTRACT: Considering the effects of friction and kinetic energy change, a wellbore model was
developed for the coalbed methane flow based on conservation of mass and momentum, and then a
coupled reservoir-wellbore model for the coalbed methane flow was presented by the comprehensive
research on the wellbore flow, reservoir flow and the deformability of coal seams. The coupled
mathematical model was solved by the finite-difference method, and in a coal seam with constant
pressure outer boundary, the curves of the wellhead pressure and density for the constant-rate
production well were plotted along with the time.
KEYWORDS: Coalbed Methane, Wellbore, Coupled flow, Finite Difference
1 INTRODUCTION
In the process of gas reservoir development, gas first
flows into the wellbore from the underground reservoir,
and then is produced from the wellbore. Thus in fact, the
gas flow is composed of the reservoir flow and the wellbore
flow. Dias [1] investigated the influence of thermal effects
and the presence of the wellbore volume on the
bottom-hole pressure response of a drawdown test by a
coupled wellbore-reservoir simulator; while in view of
high-temperature and high-pressure gas reservoirs, Hasan [2]
and Kabir [3] established a coupled wellbore-reservoir
transient model for estimating the bottom-hole pressure and
temperature efficiently from measured wellhead pressure
and temperature; Subsequently, Fan etc.[4] presented a
semi-analytic coupled wellbore/reservoir model, and the
model can describe the general wellbore effects, especially
the temperature effects on the high-temperature gas well
pressure buildup tests. But these models not suitable for the
horizontal wells, so considering the coupled effects of the
reservoir and the horizontal wellbores, Vicente etc.[5]
developed a new three-dimensional and fully implicit
numerical simulator. In China, there were scholars[6]-[8]
established the coupled mathematical models to predict the
distributions of the wellbore pressure and temperature in
the wellbore by conserving mass, momentum and energy of
single-phase gas.
Unlike the conventional natural gas, coalbed methane is
mainly stored as adsorbed gas on the internal surface of
coal matrix, the amount stored depending on a number of
coal properties like rank, moisture, ash content, temperature
and pressure of reservoirs. Although many scholars [9]-[13]
presented a variety of the coalbed methane reservoir
numerical simulation models, all of these models didn’t
consider the gas flow effects in the wellbore. This paper
established a coupled wellbore-reservoir model and finite
difference numerical model of the single-phase coalbed
methane, and gave the variation curves of gas pressure,
density and flow velocity of well head with time when the
coalbed methane wells have constant terminal rate
© ST. PLUM-BLOSSOM PRESS PTY LTD
production under the condition of constant pressure outer
boundary.
2 MATHEMATICAL MODEL
The following assumptions have been made: (1) the well
has been producing at a constant rate for a long time such
that steady-state flow has been achieved inside the wellbore.
(2) The gas flow in the reservoir obeys the Darcy’s law. (3)
The gas flow in the wellbore and reservoir are both
isothermal process. (4) Free gas is the real gas.
2.1 Wellbore flow model
Gas flow in the wellbore is quite different from gas flow
in the reservoir. The wellbore-pressure transient
propagation is dominated by inertial effects, as well as
gravity and friction effects. According to the mass and
momentum conservation law in the gas flowing process,
the general mass balance equation and momentum balance
equation of the gas steady flow in the wellbore [6] can be
expressed as:
 g
t

   g vg 
   g vg 
t
z

0
   g vg 2 
z

(1)
f g  g vg 2
pw
 g g 
0
z
2D
(2)
Where vg is gas velocity, m/s; g is gas density, kg/m3; g
is the acceleration of gravity, m/s2; pw is the wellbore
pressure, MPa; D is tubing diameter, m; the friction factor
fg is a function of roughness of the tubing wall e, inner
diameter of tubing D and Reynolds number NRe, and
calculated by the following equation:

 e
21.25  
f g  1.14  2 lg 

0.9  
 D N Re  

2
(3)
2.2 Reservoir flow model
The porosity  and permeability k along with the
variation in pressure is expressed as follows [9]:
Transient Analysis of Coalbed Methane Flow in a Coupled Reservoir-Wellbore System
DOI: 10.5503/J.PNSGE.2013.10.009
  0 e
 c f  p0  pr 
 
k  k0 

 0 
or Qwb  A g vg
(4)
(5)
or
Vm
1
  Vm  VE ( pr ) 
t

(8)
pr (r , t ) r r  p0 or p(r , t )
(16)
The equations are strongly nonlinear, thus it is difficult to
solve the system simultaneously. The finite difference
method is used to solve the equation system sequentially
and iteratively. The reservoir model is solved by the
Newton-Raphson method to provide the bottomhole
pressure for the wellbore model. According to the condition
of constant production rate at the well head, the wellbore
flow discretized equations are shown as follows.
 v   v 
n 1
g g j
n
g g j
t
2
2
2
pwnj 11  pwnj 1  g vg  j 1   g vg  j n1 f g  g vg  j 1/2


 g j1/2 g 
0
z
z
2D
n 1
n 1
n 1
Using the obtained bottomhole pressure as the boundary
condition of the wellbore model, the distributions of
pressure, gas density and velocity in the wellbore are
calculated.
(10)
The equations (1)—(10) together with the following
initial-boundary conditions constitute the complete
mathematical model of coalbed methane flowing in a
coupled reservoir-wellbore system.
Initial conditions:
3 RESULTS AND DISCUSSION
Considering a well in the center of a homogeneous,
isotropic circular reservoir with a constant-pressure outer
boundary, sensitivity studies have been performed to
indicate the effects of gas production rate on the gas
pressure, density and velocity at the wellhead. The coal
seam properties selected for case studies are shown in Table
1:
(11)
Boundary conditions at the well head:
pw ( z, t ) z 0  const
0
r  re
2.3 Numerical solution
Where VL is Langmuir volume, m3/m3; pL is Langmuir
pressure, MPa.
The gas density g may be determined by the real gas
state equation:
pr (r, t ) t 0  p0 , vg ( z , t ) t 0  0
r
Where A is the flow cross-sectional area, m2; H is the
wellbore depth, m; re is the reservoir’s outer boundary
position, m; Qwb is the wellbore mass velocity, kg/s; Qres is
the mass velocity of coalbed methane flowing from coal
bed to wellbore, kg/s.
(9)
pl M
,(l  w, r )
ZRT
(15)
zH
Boundary conditions at the outer limits of the reservoir:
Where g is the gas viscosity, mPas; qvm is the desorbed
gas flow rate through coal matrix, kg/ (m3d); Vm is the
average matrix gas concentration, m3/m3;  is the
adsorption time, d; and the equilibrium matrix gas
concentration VE (pr) is described by a Langmuir isotherm:
VL pr
pr  pL
Qres  Qwb
(6)
(7)
(14)
w
e
Vm
t
g 
(13)
pr (r , t ) r r  pw ( z, t ) z  H
1  
k pr 

 g 
 r g
  qvm 
r r 
 r 
t
VE ( pr ) 
 const
Conditions at the well-reservoir junction:
3
Where 0 is the initial fracture porosity, f; cf is the pore
compressibility, 1/MPa; p0 is the initial reservoir pressure,
MPa; k0 is the initial fracture permeability, 10-3m2.
The basic flowing equations in the coalbed methane
reservoir are:
qvm    g
z 0
(12)
Table 1 Parameters for base case
Parameters
Buried depth, m
Net pay thickness, m
Reservoir pressure, MPa
Formation porosity, %
Formation permeability, 10-3 m2
Pore compressibility, 1/MPa
Langmuir pressure, MPa
Langmuir volume, m3/ m3
Adsorption time, d
Gas relative density
Wellbore radius, m
The changes of wellhead gas pressure, density and
Value
540
2.83
3.445
4.0
1.80
0.034
1.054
18.60
10.5
0.67
0.08
velocity along with gas production rate are shown from
40
Physical and Numerical Simulation of Geotechnical Engineering
10th Issue, Mar. 2013
Figs. 1~3. We can see that the wellhead pressure and gas
density are continuously decreased with the production
time prolongation. At the same time, the wellhead gas
pressure decreases with the increasing gas production, due
to the increased friction pressure. Correspondingly, the gas
density changes are also the same with the gas pressure
changes. In addition, when considering the wellbore
coupled flow effects, pressure and gas density at the
wellhead are both lower than when neglecting the coupled
effects. However, the gas velocity at the wellhead is quite
the contrary. The steady-state flow can finally be achieved
inside the wellbore, and the wellhead pressure essentially
maintains the same after the flow achieves stability.
Fig.1 Variation of wellhead pressure with time
Fig.2 Variation of wellhead gas density with time
Fig.3 Variation of wellhead velocity with time
41
Transient Analysis of Coalbed Methane Flow in a Coupled Reservoir-Wellbore System
DOI: 10.5503/J.PNSGE.2013.10.009
[4].
Fan L., Lee W.J., Spivey J.P. Semi-Analytical Model for
Thermal Effect on Gas Well Pressure Buildup Tests. paper
SPE56612 presented at the SPE Annual Technical
Conference and Exhibition, 3–6 October, 1999:1-16.
[5]. Vicente R., Sarica C. A Numerical Model Coupling
Reservoir and Horizontal Well Flow Dynamics: Transient
Behavior of Single-Phase Liquid and Gas Flow. SPE 65508,
presented at the 2000 SPE/Petroleum Society of CIM
International Conference on Horizontal Well Technology,
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4 CONCLUSIONS
The following conclusions can be drawn from the result
analysis:
(1) The wellbore flow, reservoir flow and the coal
deformation are considered as a coupled system during the
coalbed methane production process, and then a coupled
wellbore-reservoir model was established by conserving
mass and momentum of single-phase coalbed methane in
the wellbore.
(2) In a coal seam with constant-pressure outer boundary
conditions, the changing curves of the wellhead pressure,
gas density and velocity along with the time were given for
a constant-rate production well, and the reservoir-wellbore
coupling effect on the above parameters was obvious.
REFERENCES
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[2].
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42