Accepted Manuscript Integrated study of gas condensate reservoir characterization through pressure transient analysis Jiawei Li, Gang Zhao, Xinfeng Jia, Wanju Yuan PII: S1875-5100(17)30308-6 DOI: 10.1016/j.jngse.2017.07.017 Reference: JNGSE 2251 To appear in: Journal of Natural Gas Science and Engineering Received Date: 7 January 2017 Revised Date: 20 July 2017 Accepted Date: 24 July 2017 Please cite this article as: Li, J., Zhao, G., Jia, X., Yuan, W., Integrated study of gas condensate reservoir characterization through pressure transient analysis, Journal of Natural Gas Science & Engineering (2017), doi: 10.1016/j.jngse.2017.07.017. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Transcript
Accepted Manuscript
Integrated study of gas condensate reservoir characterization through pressuretransient analysis
Jiawei Li, Gang Zhao, Xinfeng Jia, Wanju Yuan
PII: S1875-5100(17)30308-6
DOI: 10.1016/j.jngse.2017.07.017
Reference: JNGSE 2251
To appear in: Journal of Natural Gas Science and Engineering
Received Date: 7 January 2017
Revised Date: 20 July 2017
Accepted Date: 24 July 2017
Please cite this article as: Li, J., Zhao, G., Jia, X., Yuan, W., Integrated study of gas condensatereservoir characterization through pressure transient analysis, Journal of Natural Gas Science &Engineering (2017), doi: 10.1016/j.jngse.2017.07.017.
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service toour customers we are providing this early version of the manuscript. The manuscript will undergocopyediting, typesetting, and review of the resulting proof before it is published in its final form. Pleasenote that during the production process errors may be discovered which could affect the content, and alllegal disclaimers that apply to the journal pertain.
Characterization through Pressure Transient Analysis
Jiawei Lia,b*, Gang Zhaoa , Xinfeng Jiac, Wanju Yuana
a Department of Petroleum Systems Engineering, University of Regina, Regina, Canada, S4S 0A2 b* School of Civil Engineering, University of Queensland, Brisbane, Australia, 4072 c College of Petroleum Engineering, China University of Petroleum (Beijing), Beijing, China, 100086 * Corresponding author: Jiawei Li
Address: School of Civil Engineering, University of Queensland, St Lucia, Brisbane, Queensland, Australia, 4072.
Abstract
This paper presents an alternative semi-analytical model which is able to integrate PVT properties of gas with dynamic pressure domain during production process. A modified three-region radial composite model is developed to evaluate the gas condensate reservoir, taking account of different gas flow behaviors and pressure dependent properties, such as the compressibility factor. The governing equation of the pressure diffusion process is highly nonlinear due to the complex dependence of coefficients on pressure. The linearization of the non-linear partial differential equation describing the complicated gas flowing in a reservoir is handled by application of reasonable definitions of pseudo-pressure and pseudo-time for each region, which is integrated into the reservoir system through physical continuity of changing phases with PVT properties. Modified forms of total compressibility factor are proposed by valid theoretical developments.
Results show that gas compositions of a gas condensate reservoir have significant
effects on the fluid flow behaviors. Different proportions of 5C , 6C and 7+C are
simulated with constant makeup of 2CO , 2N and 1 4C�
, showing that small changes in
composition of heavier components make distinct differences in the flow behaviors, as reflected on the liquids dropout curves. In addition, total compressibility factor varies with fluid compositions instead of remaining constant in a gas condensate reservoir.
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Through integrating PVT behaviors, the three-region model is able to accurately describe transient pressure behaviors of a production well in a gas condensate reservoir, which clearly characterizes three regions on the pressure derivative curves. A series of sensitivity studies are conducted to analyze the differences among the three regions.
Keywords: Gas condensate reservoir; PVT properties; Flow behaviors; Semi-analytical method
1. Introduction
Liquids rich reservoirs, including gas condensate, shale gas, tight gas and coal seam gas, play an increasing important role in energy supply in the world. Most known gas-condensate reservoirs are discovered in the ranges of 5000 feet to 10000 feet deep, 3000 psi to 8000 psi and 200 ℉ to 400 ℉ (Roussennac, 2001). At a temperature between critical temperature and cricondentherm temperature, condensation drops out from gas when the pressure falls below the dew point pressure. The reservoir fluids will separate into: a gas phase and an oil phase (excluding water phase). With pressure decreasing, more condensate oil will drop out and reach a maximum volume. Gas condensate fluids can be divided into lean, medium-rich, rich, depending on the range of their condensate to gas ratio (Kgogo et al., 2010). A lean system may yield approximately 10 STB/MMscf (2 % maximum condensate), and a rich system could yield as much as 20 % condensate, 300 STB/MMscf (Kamath, 2007).
Many field investigations have been conducted over last decades to understand the flow behaviors in gas condensate reservoirs. Behrenbruch et al. (1984) interpreted results from well testing gas condensate reservoirs by comparing theory and field cases. In a very high temperature KAL-5 gas condensate well in the Moslavacka Gora formation in Yugoslavia, a successful hydraulic simulation was performed (Economides et al., 1989). In 1991, S∅gnesand discussed the effect of a retrograde condensate blockage on long-term well performance of vertically fractured gas condensate wells and presented a method to correct the effect of condensate blockage by using the concept of time-dependent skin factor. Raghavan et al. (1995) considered practical factors in analysis of gas condensate wells and made two conclusions: it is possible to relate relative permeability values to pressure and use the resulting analogue to evaluate pressure-buildup tests in a quantitative manner; the saturation profile at shut in governs the shape of the pressure buildup trace and the success of the two-phase analogue is dependent on the ability to estimate this profile. Diamond et al. (1996) developed a method to estimate probabilistic well deliverability in the Britannia gas condensate field based on log and core data. Marhaendrajana et al. (1999) proposed a rigorous and coherent approach for the analysis of well test data from a multi-well reservoir system: all of the available well test data from the giant Anrun gas field (Sumatra, Indonesia). Kool et al. (2001) outlined the metrology and procedure to obtain a representative formation fluid sample that may be used for compositional and pressure-volume-temperature (PVT) analysis. A modified
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black-oil model was tested against a fully compositional model, and the performances of both models were compared by using various production and injection scenarios for a rich gas condensate reservoir (Izgec et al., 2005). Goktas and Thrasher (2011) introduced a methodology for determining gas-oil relative permeability curves using well performance data from retrograde condensate wells.
Due to the complex behaviors and multi-phase flow in gas condensate reservoirs, it is difficult to interpret production well performance in gas condensate reservoirs. Many researches have been done in order to solve the problem. In 1949, Muskat found that a condensate bank builds up around the producing well once the bottomhole pressure falls below the dew point pressure. Kniazeff et al. (1965) identified that two more regions other than the condensate bank exist in the reservoir from the numerical simulations. The radial model that considers the flow of individual components and account for component mass transfer between phases was used to predict the performance of a producing well in a reservoir containing a rich gas condensate reservoir (Roebuck et al., 1969). Fussell (1973) modified the radial model developed by Roebuck et al. (1969) to study long-term single well performance in three condensate reservoirs. O’Dell et al. (1965) presented a simple method based on steady state flow concepts that can be used to quickly estimate the deliverability from the well. A unique relationship between pressure and saturation was developed by Boe et al. (1989). Jones and Raghavan (1988) used a fully implicit model to simulate the well responses in a gas condensate system by modifying the steady-state theory. Thompson et al. (1992) presented an analytical solution for well testing in gas condensate reservoirs. Fevang et al. (1995) proposed the three-region model to model the well deliverability in a gas condensate reservoir. Whitson et al. (1999) showed that the relative permeability in gas condensate systems should include three parts by considering capillary number effect and non-Darcy flow effect. Gringarten et al. (2000) showed that three regions exist with different liquid saturations when pressure falls below the dew point pressure. A novel approach was introduced in the use of two-phase pseudo-pressure for the interpretation of gas condensate well test data in naturally fractured reservoirs (Mazloom et al., 2005). In addition, a method to characterize condensate bank was proposed by Bozorgzadeh et al. (2006). A Fetkovich method was chosen to evaluate the reservoir productivity and the well future production performance in conjunction with well test analysis based on real drawdown test data (Zheng et al., 2006). Clarkson et al. (2015) summarized analytical, semi-analytical and empirical methods for gas condensate well forecasting. Zeng and Zhao (2008) presented a semi-analytical method for studying non-Darcy flow on transient pressure behaviors. Zhu et al. (2012) applied this method for evaluating an early-period SAGD process. This method has been extended to simulate fracture conductivities and discrete fracture system (Zeng and Zhao, 2012; Luo and Wang, 2014; Zhang and Yang, 2014).
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In this paper, we discuss effects of different compositions on a gas condensate system and show how to integrate the PVT properties into a modified semi-analytical model for transient pressure analysis in gas condensate reservoirs.
2．．．．Mathematical Model
2.1 Fluid Properties
Fluid behaviors of a gas condensate reservoir depend on not only pressure but also compositions. In Table 1, Composition #1 represents a typical fluid composition from Senoro field (Suwono et al., 2012) and is used as a base case, from which three simplified compositions are derived. The differences among four compositions are the mole
fractions of 5C , 6C and 7+C (The properties of 7+C is given in Table 2). Because 5C , 6C
and 7+C are the main source of heavy ends. The total mole fractions of 5C , 6C and 7C +
are assumed to be constant in Composition #2. In Composition #2, 5C is the most of the
total mole fractions, whereas in Composition #3 and #4, 6C and 7+C are the most parts
respectively. Equation of State (EOS) is the core of PVT simulation. With known compositions,
volumes, and temperatures, the PVT properties of a single-phase liquid/gas is expressed by using a Peng–Robinson Equation of State (PR-EOS) in this study. For those in the two-phase region, a flash calculation is required. In a typical EOS flash calculation, pressure is known and EOS calculates phase volumes. A commercial simulator (WinProp, CMG) is used to perform two-phase calculation on the basis of PR-EOS. The phase diagrams of four different compositions are displayed in Figure 1.
2.2 Model assumption
Based on Darcy’s Law and a mass conservation equation, the governing equation for the flow of gas component in a gas condensate reservoir is yielded, which is derived on the basis of following assumptions:
1. Thickness of the reservoir is constant. 2. Gravity is ignored. 3. Temperature is constant. 4. Darcy Law is applicable. 5. Only gas phase is considered. 6. Capillary pressure is ignored. 7. No water exists.
2.3 Model Demonstration
A radial single-porosity reservoir with an infinite outer boundary is considered and a production well is located at the center of the reservoir. Figure 2 shows a schematic of a production well in a radial composite reservoir. The bottomhole pressure is under the dew point pressure. Three regions are developed in the reservoir. The region boundaries
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remain constant during a short well test period. The pressure response caused by gas flow is evaluated because main compositions comprising a gas condensate reservoir are gas phase. The movement of gas consists of two parts: the flow of gas and the flow of gas components in a liquid phase.
Real gas properties are computed as functions of pressure under isothermal conditions through PR-EOS. In a gas condensate reservoir that is composed of three regions (Fevang and Whitson, 1996), although gas and oil coexist, the flow of oil phase is ignored due to its small amount and poor mobility, and only the flow of gas phase is considered. The diffusivity equations of gas are written as:
( )gg rg go ro 1gg g go o
g o
k k p1k + r = S + S
r r r t
ρρρ
µ µφ
ρ
∂∂ ∂∂ ∂∂
(rw < r < r1) (1)
( )gg rg 2gg g go o
g
k p1k r = S + S
r r r t
ρφ
µρρ∂∂ ∂
∂ ∂
∂
(r1 < r < r2) (2)
( )gg rg 3gg g
g
k p1k r = S
r r r t
ρρφ
µ ∂
∂ ∂∂ ∂ ∂
(r2 < r < ∞) (3)
where
ggg
g
=B
ρρ
(4)
g
ogo s= R
Bρ
ρ (5)
Submitting Equations (4) and (5) into Equations (1), (2) and (3) to eliminate gρ
from both sides:
rg gro o1
s sg g o o g o
k Sk Sp1+ R r = + R
r r B B r t k B B
φµ µ
∂∂ ∂ ∂ ∂ ∂
(6)
rg g o1
sg g g o
k S Sp1r = + R
r r B r t k B B
φµ
∂∂ ∂ ∂ ∂ ∂
(7)
rg g3
g g g
k Sp1r =
r r B r t k B
φµ
∂∂ ∂ ∂ ∂ ∂
(8)
The initial and boundary conditions are:
rg ros gsc
g g o o
k k p2pkh + R r = q
B B rµ µ ∂ ∂
(r = rw, t ≥ 0) (9)
ip P= (r → ∞, t ≥ 0) (10)
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ip P= (rw < r < ∞, t = 0) (11)
where oS and gS are the oil and gas saturation, respectively, dimensionless; oµ and gµ
are oil and gas viscosities, respectively, mPa.s; rok and rgk are the oil-phase and gas-
phase relative permeabilities, respectively, dimensionless; oB and gB are the oil and gas
formation volume factors, respectively, dimensionless; sR is the solution gas–oil ratio,
gρ is the molar density of gas at standard conditions, g/mol; ggρ is the molar density of
gas in gas phase, g/mol; goρ is the molar density of solution gas in oil phase, g/mol; wr ,
1r and 2r are the radii of wellbore, inner region, and condensate bank, respectively, m; t
is the time variable, s; iP is the initial reservoir pressure, Pa; φ is the porosity,
dimensionless; k is the absolute permeability, m2; gscq is the gas flow rate at standard
conditions, m3/s; and h is pay-zone thickness, m.
The oil/gas relative permeability orok and o
rgk are functions of pressure. The gas/oil
relative permeability data are calculated by the Corey power-law relationship (Corey, 1954; Brooks and Corey, 1966; Ali et al., 1997):
1
on
o o orro ro
or wc gc
S Sk k
S S S
−= − − − (12)
1
gn
g gcorg rg
or wc gc
S Sk k
S S S
−= − − −
(13)
where orok and o
rgk are maximum relative permeability for oil and gas, respectively, orS is
oil residual saturation, gcS is gas critical saturation, wcS is water critical saturation, gn
and on are exponents ranging from 1 to 6.
In the near-wellbore region (Region 1), there is a phenomenon that gas relative
permeability increases due to low interfacial tensions at high gas flow rates (Gondouin et
al., 1967). This is caused by the high capillary number, and is also called ‘positive
coupling’ (Boom et al., 1995; Henderson et al., 2000). The definition of capillary number
that is the ratio of viscous to capillary force is given by Moore and Slobod (1955):
c
vN
µσ
= (14)
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where cN is capillary number, ν is the velocity, µ is the viscosity and σ is the capillary
force.
The dependence of gas relative permeability on capillary number is proposed as (Whitson et al., 1999):
( )1rg I rgI I rgMk f k f k= + − (15)
1
1
1
maxrgM rg
rg
ro
k kk
k
−=
+
(16)
( )0.65
1
1I
c
fNα
=+
(17)
The inertial high velocity gas flow in gas condensate reservoirs is one source of additional pressure drop. On the basis of the Forchheimer equation, the non-Darcy factor is shown, which is combined with Equation (13) for gas relative permeability calculation (Forchheimer, 1901; Whitson et al., 1999):
1
1 rgND eff gm g
g
kkF vβ ρ
µ
−
= + (18)
1eff rgkβ β −= ⋅
(19)
where If is the immiscibility factor, rgIk is immiscible gas relative permeability, rgMk is
miscible gas relative permeability, NDF is the non-Darcy factor, β is the Forchheimer
constant, effβ is the effective Forchheimer constant, gmρ is gas mass density, and gv is
the velocity of the gas phase. As shown in Equations (6)–(8), the governing equations are non-linear due to the
complex dependence of gas on pressure. In order to linearize these equations, pseudo-pressure and pseudo-time are defined in this study based on the work of Raghavan et al. (1972):
1
w
rg ro1 sP
g g o g
P k km = + R dp
B Bµ µ
∫ (20)
1
2 dew rg
g g
P
P
km = dp
Bµ∫ (21)
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( ) 1
e
dew
P
3 rg wi Pg g
m = k S dpBµ∫ (22)
0
rg ros
g g o ga1
1
t
i t
k k+ R
B Bt = dt
c
k
µ µφ∫ (23)
0
rg
t g ga2
i t2
k
Bt dt
c
k
φµ
= ∫ (24)
0
rg
t g ga3
i t3
k
Bt = dt
c
k
µφ∫ (25)
where m and at are the pseudo-pressure and the pseudo-time, respectively; tc is the
total compressibility, 1/psi; and number 1, 2, 3 in the subscripts of the above variables correspond to the three regions, respectively.
The derivation of the total compressibility1tc is appended. Apparently, the total
compressibility factors are implicit functions of pressure. Applying the pseudo-pressure and pseudo-time to Equations (6)-(8), the governing
equations can be reformed as:
1 1 1
a1
m mr =
r r r t
∂
∂∂∂ ∂ ∂
(26)
2
2
21
a
m mr =
r r r t
∂
∂∂∂ ∂ ∂
(27)
3
3
31
a
m mr =
r r r t
∂
∂∂∂ ∂ ∂
(28)
After transformed, the partial differential equations for three flow regions should be considered as a whole in order to pursue the solutions to the mathematical model. Since one uniform expression of pseudo-time instead of three different ones can be made (Xiao et al., 2013; Acosta et al., 1994), this study obtains the following forms under the conditions in terms of two-phase pseudo-variables:
gsc1
w
qm=
r 2pkhr
∂∂
(r = rw, t ≥ 0) (29)
im m= (r → ∞, ta3 ≥ 0) (30)
im m= (rw < r < ∞, ta3 = 0) (31)
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2.3.1 Semi-analytical Solution
The non-linear governing equations are linearized through the application of pseudo-pressure and pseudo-time. Set Region 1 as the reference region, three dimensionless equations can be written as:
1 D 1 D 1
DD D D aD
m mr =
r r r t
∂ ∂∂∂ ∂ ∂
(32)
2
21
21 1 D D
DD D D R aD
m mr =
r r r C t
∂
∂∂∂ ∂ ∂
(33)
3
31
31 1 D D
DD D D R aD
m mr =
r r r C t
∂
∂∂∂ ∂ ∂
(34)
where Dm is the dimensionless pseudo-pressure, aDt is the dimensionless pseudo-time,
Dr is dimensionless radius, RC is the diffusivity ratio (Zhao et al., 2002).
1
1
i2 t 2R21
i t
ku c
C =k
u c
φ
φ
(35)
3
1
1
i3 t 3R 1
i t
k
u cC =
k
u c
φ
φ
(36)
The diffusivity ratio can be derived from transmissibility ratio TC and storability
ratio SC :
T21R21
S21
CC =
C (37)
T31R31
S31
CC =
C (38)
The expression of the transmissibility ratio TC and the storability ratio SC are
shown below:
2T21
1
kh
C =kh
µ
µ
(39)
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3T31
1
kh
C =kh
µ
µ
(40)
( )( )
i2 t 2S21
i 11 t
c hC =
c h
φφ
(41)
( )( )3 i3 t 3
S 1i 11 t
c hC =
c h
φφ
(42)
After Laplace transformation, the general solutions to Equations (32)-(34) are respectively given as:
( ) ( )0010011
srKBsrIAmDDD
+= (43)
+
=
21
0
02
21
0
022
R
D
R
DD
C
s
rKB
C
s
rIAm (44)
=
31
0
033
R
DD
C
s
rKBm (45)
where 0
S is the Laplace variable.
2.4 Multi-region Model
Zeng et al. (2008) have developed a semi-analytical model to examine the transient pressure behavior of vertical wells with non-Darcy flow in the reservoir. This semi-analytical model was also applied to evaluate the early-period SAGD by interpreting the temperature falloff data (Zhu et al. 2012). A semi-analytical model is built on the basis of three-region model to investigate the performance of gas condensate reservoirs.
The whole reservoir is divided into many sub-segments (Figure 3). The
mathematical model for sub-segment i, Ni ≤≤1 , can be written as (Zeng et al., 2008):
==
∂∂
−
==
∂∂
−
=
∂∂
∂∂
+++
11
1
,
,
1
DiDDi
D
Di
D
DiDDi
D
Di
D
Dii
D
Di
D
DD
rrq
r
m
r
rrq
r
m
r
ms
r
m
r
rr
(46)
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where i
s is the Laplace variable in sub-segment i , Di
r and 1+Di
r are inner and outer
boundary radius for sub-segment i, Di
m is the dimensionless pseudo-pressure for sub-
segment i, Di
q is dimensionless flow rate in region i.
The analytical solution in the Laplace domain for segment i is:
( ) ( )0000
srKBsrIAmDiDiDi
+= (47)
Applying the boundary conditions helps generate the coefficients, i
A andi
B . Finally,
the dimensionless pseudo-pressure in sub-segment i can be written as a linear equation in terms of flow rate:
1++=DiiDiiDi
qFqEm (48)
where i
E and i
F are combinations of Bessel functions of local Laplace variable.
Combining all sub-segments generates a linear tri-diagonal system.
CBA
rrr=⋅ (49)
with
=A
r
⋅⋅⋅⋅⋅⋅
⋅⋅⋅
−
−−−−−
nnnn
nnnnnn
AA
AAA
AAA
AA
,1,
,11,12,1
3,22,21,2
2,11,1
, =B
r
⋅⋅⋅
−
n
n
B
B
B
B
,1
1,1
2,1
1,1
, =C
r
⋅⋅⋅
−
n
n
C
C
C
C
1
2
1
(50)
ji
A,
is a function of flow rates, ji
B,
represents flow rates, i
C is the residual.
2.4.1 Semi-analytical Solution
Combining the three-region model with the multi-region model, the flow rates of all segments can be calculated by applying Stehfest inversion method (Stehfest, 1970). Then the bottomhole pressure can be generated by applying the known flow rates of the first sub-segment, as is shown in the following equation:
2111 DDwD
qFqEm += (51)
wD
m is the bottomhole pressure, 1D
q and 2D
q are flow rates on the boundary of the Sub-
segment 1, respectively.
3. Results Analysis
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3.1 Liquid Volume
Figure 4 illustrates that the liquid (condensates) volume changes as a function of pressure in a gas condensate system for Composition #1. When pressure falls below the dew point pressure, the condensate drops out and its quantity cumulates within a certain pressure range. After the dropout reaches a maximum value, its volume decreases because further pressure reduction permits the heavy molecules to vaporize.
In Figure 5, the cumulative liquid volumes show obvious differences due to the change of compositions. The liquid volume of Composition #4 is much higher than those of the other three compositions because it has a larger proportion in heavy ends. It is worthwhile to notice that a dew point pressure changes significantly with compositions. It inclines upward with the increase of heavier hydrocarbons.
3.2 Compressibility Factor Z
Gas compressibility factor (Z-factor) can be calculated directly through PR-EOS given gas compositions. It is found that Z-factor varies strongly with pressure (Figure 6). In addition, it also depends on the intermolecular forces of gases. At a low pressure, attraction is the dominant force among gas molecules, which leads to a smaller gas compressibility factor. With the increase of pressure, repulsion will become the dominant force that will result in an increasing Z-factor after the space between gas molecules reduces to a critical value.
3.3 Viscosity
Figure 7 shows the viscosity versus pressure profiles, which have small differences because the main compositions for Composition #1, #2, #3 and #4 are similar. The Pederson correlation is expected to give better liquid viscosity prediction for light and mediums oils than the JST model (Suwono et al., 2012). Therefore, the modified Pederson (Pederson et al., 1987) is applied in the following calculations.
3.4 Total Compressibility
Applying the compositions of a gas condensate reservoir mentioned in Table 1, Figure 8 shows the total compressibility for Composition #1. It increases slightly with pressure in a linear trend. Since the pressure range for condensate dropout varies with compositions, a relative pressure range is set here in order to compare the trends of total compressibilities. In Figure 9, the total compressibilities for Composition #1 and #2 almost overlap each other. It increases slightly for Composition #3 and drastically for Composition #4.
3.5 Mathematical Model Validation
Few direct validation of semi-analytical model is readily available due to the lack of real data. Here the commercial software Kappa is used for model validation because it can provide accurate analytical solutions for a homogeneous model and a two-region
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radial composite model. The number of the semi-analytical sub-segments (N ) is 80 and
the dimensionless radius (D
r ) of every sub-segment is 10 (This can guarantee the
appearance of radial flow in the known dimensionless time). When ordering the
transmissibility ratios 13121
==TT
CC , the diffusivity ratios (21R
C and31R
C ) are assumed
to be equivalent to the mobility ratios while the stability ratios 13121
==SS
CC . Then
the three-region model will become a homogeneous model. The dimensionless pseudo-pressure and dimensionless pseudo-pressure derivative generated from the semi-analytical model are compared with those generated from Kappa. Figure 10 shows the simulation results of the semi-analytical model match those of Kappa in terms of the dimensionless pseudo-pressure and dimensionless pseudo-pressure derivative
respectively. When 5.03121
==TT
CC , 13121
==SS
CC , and the diffusivity ratios 21R
C
and 31R
C are both equal to 0.5 , the dimensionless pseudo-pressure and dimensionless
pseudo-pressure derivative curves are also identical (Figure 11). Figure 10 and Figure 11 illustrate that the semi-analytical model has accurate well performance compared with Kappa. For Figure 12, the dimensionless pseudo-pressure and dimensionless pseudo-time are transformed into the values of pseudo-pressure and pseudo-time. Then, the real pressure and real time are calculated from pseudo-pressure and pseudo-time directly. Figure 12 shows that comparison of real pressure is almost same, proving that the pseudo-pressure and pseudo-time in semi-analytical model are accurate and reasonable. The identical results have validated this semi-analytical model built on the basis of the three-region model.
3.6 Effect of Transmissibility Ratio
The differences between each region in a gas condensate reservoir are expressed in terms of transmissibility and storability ratios. The effects of transmissibility are discussed here. The transmissibility ratio includes three parameters: permeability ratio, viscosity ratio and reservoir thickness. Generally, the thickness of a reservoir is considered to be constant. The permeability ratio is positively correlated to the transmissibility ratio and the viscosity is negatively correlated to the transmissibility ratio on the basis of the definitions.
The number of the semi-analytical sub-segments (N ) is 80 and the dimensionless
radius (D
r ) of every sub-segment is 10. The storability ratios equal to 1
( 13121
==SS
CC ). The values of the boundary between Region 1 and Region 2 (1D
r )
and the boundary between Region 2 and Region 3 (2D
r ) are constant.
In Figure 13, the dimensionless pseudo-pressure curves vary from each other due to
different values of transmissibility between Region 1 and Region 2 (21T
C ). Larger value
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of 21T
C leads to larger pressure drop. This is because the transmissibility in Region 2 is
better than that of Region 1, which means fluids flow more easily in Region 2. Finally, the slopes of the dimensionless pseudo-pressure curves are the same due to the fact that the fluids flow ability in Region 3 are the same as others. Figure 14 shows totally differences on dimensionless pseudo-pressure derivative curves in the area of Region 2, which are caused by the transmissibility ratios. A larger distinction between Region 1 and Region 2 results in a larger variance. When pressure disturbance reaches the boundary
between Region 2 and Region 3 (2D
r ), the dimensionless pseudo-pressure derivative
curves gradually become the same line with a value of 0.5, which means that they reach radial flow. This is because the transmissibility ratios between Region 1 and Region 3 are same in these cases.
For Figure 15 and Figure 16, the transmissibility between Region 1 and Region 2 is
kept constant ( 5.021
=T
C ) for three different values of the transmissibility between
Region 1 and Region 3. Another case is the homogenous reservoir ( 13121
==TT
CC ),
which is used as the standard. In Figure 15, three different regions can be identified clearly from the dimensionless pseudo-pressure curves. The first distinction on the dimensionless pseudo-pressure curves can be used to identify the existence of Region 2.
The second distinction among three dimensionless pseudo-pressure curves ( 5.021
=T
C )
is caused by the different transmissibility ratios between Region 1 and Region 3. Same as
Figure 10, a larger value of transmissibility between Region 1 and Region 3 (31T
C ) leads
to a larger pressure drop. Figure 16 also reflects the differences of each region. When
pressure disturbance reaches the boundary between Region 1 and Region 2 (1D
r ), the
standard case ( 13121
==TT
CC ) can be classified into three other cases due to different
transmissibility ratios. Then, the distinction appears when the pressure disturbance
reaches the boundary between Region 2 and Region 3 (2D
r ) due to different
transmissibility ratios (31T
C ). And finally, radial flow appears for every pseudo-pressure
derivative curve.
3.7 Effect of Storability Ratio
The storability ratio is the other part that forms the diffusivity ratio. Based on the definition, the storability ratio includes the initial porosity ratio, the total compressibility ratio and the reservoir thickness ratio. The reservoir thickness ratios are generally considered to be constant, which equal to 1. The initial porosities in Region 1, 2 and 3 are different due to the condensate dropping out from the gas. And the total compressibility of every region is different from the others. As a result, the storability ratio is positively correlated to the total compressibility ratio and the initial porosity ratio. Due to the fact
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that the storability ratio is negatively correlated to the diffusivity ratio, the total compressibility has direct effects on the diffusivity ratio.
For Figure 17 and Figure 18, the number of the semi-analytical sub-segments (N )
is 80 and the dimensionless radius (D
r ) of every sub-segment is 25. The transmissibility
ratios are defined to be constant ( 13121
==TT
CC ). In addition, the value of storability
between Region 1 and Region 3 (21S
C ) also equals to 1.
In Figure 17, the dimensionless pseudo-pressure curves show small differences in the area of Region 2, which is caused by the different values of storability ratios for Region 2. Differences on the dimensionless pseudo-pressure curves exist only when
pressure disturbance reaches the boundary between Region 1 and Region 2 (1D
r ) and the
boundary between Region 2 and Region 3 (2D
r ). Figure 18 shows larger values of the
storability ratios between Region 1 and Region 2 lead to higher humps when the pressure turbulence reaches Region 2 and Region 3. Three dimensionless pseudo-pressure derivatives will finally reach the radial flow after certain humps. This is because the transmissibility ratios between each region equal to 1. For Region 2, the radial flow does not appear because the length of Region 2 is short. The following Figure 19 will show the existence of radial flow in Region 2.
For Figure 19, the number of the semi-analytical sub-segments is 200 and the
dimensionless radius (D
r ) of every sub-segment is 25. The transmissibility ratios are
defined to be constant ( 13121
==TT
CC ). Figure 19 is used to prove that storability
ratios have direct effects on the humps on the dimensionless pseudo-pressure curves. The area of Region 2 should be long enough to guarantee the appearance of radial flow. For
the case ( 1,8.03121
==SS
CC ), when pressure disturbance reaches the boundary
between Region 1 and Region 2 (1D
r ), a hump appears due to difference from the
storability ratio (21S
C ) and then the dimensionless pseudo-pressure derivative curve
reaches to a value of 0.5 during the period of radial flow area due to the same
transmissibility ( 121
=T
C ). When the pressure disturbance reaches the boundary between
Region 2 and Region 3 (2D
r ), another hump appears which is opposite to the first hump.
This is also caused by the difference from the storability ratio ( 1,8.03121
==SS
CC ).
Finally, the radial flow period is reached. The other two cases ( 25.1,13121
==SS
CC
and 8.03121
==SS
CC ) can be taken as the application of two region composite model
and be used to validate the results from the case ( 1,8.03121
==SS
CC ).
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For Figure 20, the number of the semi-analytical sub-segments (N ) is 80 and the
dimensionless radius (D
r ) of every sub-segment is 10. The transmissibility ratios are
defined to be constant ( 13121
==TT
CC ). The storability ratios are also given the same
value ( 1.03121
==SS
CC ). Figure 17 shows that the dimensionless pseudo-pressure
derivative curves are the same for different values of the boundary between Region 2 and
Region 3 (2D
r ). There is a hump existing when the pressure disturbance reaches the
boundary between Region 1 and Region 2 (1D
r ). The increase of the boundary between
Region 2 and Region 3 has no effect on the appearance of humps at late period. This is because the storability ratio for Region 2 and Region 3 are same
( 1.03121
==SS
CC ).This can be used to validate the semi-analytical model because
transmissibility ratio and storability ratio between Region 2 and Region 3 are totally the same, not affected by the boundary.
4. Conclusion
A semi-analytical model for predictions of flow behaviors in gas condensate reservoirs is developed in this paper. The model accounts for the PVT properties of gas condensate systems and properties of reservoir rocks, which are expressed as functions of pressure. A modified three-region model incorporating a multi-region model is built to describe the gradual changes of permeability and saturation induced by the pressure drop in gas condensate reservoirs. Applications of proper definitions of pseudo-pressure and pseudo-time have successfully linearized the diffusivity equations for three-region model. The model is then validated against simulation results from Kappa, showing excellent agreement in these cases. In sensitivity analysis, compositions of a gas condensate reservoir have direct effects on fluids properties and heavier compositions have more significant effect compared with lighter compositions. The total compressibility depending on pressure is an important factor associated with the model. The effects of transmissibility ratios are assessed and reflected on the type curves. In addition, storability ratios cause humps on the type curves, having a tight relationship with pressure responses. The properties of fluids flow and reservoir rock in gas condensate reservoirs, including relative permeability, saturations, viscosities, reservoir thickness and total compressibility, are expressed in transmissibility and storability ratios. The model provides a new method for pressure transient analysis in gas condensate reservoirs and a solid foundation for the further research of liquids rich shale gas reservoirs.
Nomenclature
gB gas formation volume factor, RB/scf
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oB oil formation volume factor, RB/STB
b van der Waals covolume
fc formation compressibility, 1/psi
tc total system compressibility, 1/psi
RC diffusivity ratio
TC transmissibility ratio
SC storability ratio
h thickness of reservoir, ft
0I modified Bessel function of zero order
0K modified Bessel function of zero order
k permeability, md
rgk gas relative permeability
rok oil relative permeability
L molar fraction of liquid
m∆ the value of pseudo-pressure
im reference pseudo-pressure
m pseudo-pressure
Dm dimensionless pseudo-pressure
N number of sub-segments
dewP dew point pressure, psi
eP reservoir external boundary pressure, psi
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iP initial reservoir pressure, psi
1P boundary pressure between Region 1 and Region 2, psi
wfP wellbore flowing pressure, psi
Di
q dimensionless flow rate in region i
gsc
q standard gas flow rate
pR producing gas/oil ratio, scf/STB
sR solution gas/oil ratio, scf/STB
r radius, ft
Dr dimensionless radius
gS gas saturation
oS oil saturation
wiS initial water saturation
wcS water critical saturation
t time, h
at pseudo-time
aDt dimensionless pseudo-time
gv velocity of gas phase, ft/s
V molar fraction of vapor
0V molar volume
Z compressibility factor
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Greek Letters
gρ density of gas at standard conditions, �� ⁄
ggρ density of gas in gas phase, �� ⁄
gmρ gas density, �� ⁄
goρ density of solution gas in oil phase, �� ⁄
oµ oil viscosity, cp
gµ gas viscosity, cp
φ porosity, fraction
Subscript
D dimensionless variable
g gas phase
i initial condition
o oil phase
1, 2, 3 Region 1,2,3
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Table 1 Four Different Compositions for Gas Condensate Reservoir (Suwono et al., 2012)
Component Composition #1 (mol. %)
Composition #2 (mol. %)
Composition #3 (mol. %)
Composition #4 (mol. %)
CO2 1.0808 1.5000 1.5000 1.5000 N2 0.9093 1.5000 1.5000 1.5000
The variation of porosity with pressure can be expressed using the formation compressibility as:
( )1i f ic p pφ φ = + − (A-3)
Submitting Equation (A-3) into Equation (A-2) yields:
( )11g g g go o i o o
s s f s f i sg o g o g o g o
S S S SS S S SR R c R c p p R
k p B B p B B k B B p B B
φφ φ ∂ ∂ ∂
⋅ + + ⋅ + = ⋅ + + + − ⋅ + ∂ ∂ ∂ (A-4)
As is shown in Equation (A-4), the definition of 1tc is expressed as follows:
( )1 1g go ot f s f i s
g o g o
S SS Sc c R c p p R
B B p B B
∂ = ⋅ + + + − ⋅ + ∂
(A-5)
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Highlights
� A semi-analytical model for predictions of flow behaviors in gas condensate
reservoirs is developed � The model takes into account the reservoir system through physical continuity of
changing phases with PVT properties � Applications of proper definitions of pseudo-pressure and pseudo-time have
successfully linearized the diffusivity equations � Compositions of a gas condensate reservoir have direct effects on PVT properties
and transmissibility and storability ratios derive from PVT properties � The total compressibility depending on pressure is an important factor associated
