Pressure-transient analysis of CO2 flooding based on a compositional method

Journal of Natural Gas Science and Engineering 33 (2016) 30e36

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Pressure-transient analysis of CO2 flooding based on a compositional method Longlong Li a, Jun Yao a, *, Yang Li a, b, Minglu Wu a, Lei Zhang a a b

School of Petroleum Engineering, China University of Petroleum, Qingdao, 266580, China Department of Oilfield Exploration and Development, Sinopel, Beijing, 100029, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 23 December 2015 Received in revised form 27 April 2016 Accepted 29 April 2016 Available online 3 May 2016

To find a reliable and inexpensive method to estimate miscibility and other reservoir parameters, as well as to monitor the CO2 flood progress, this paper proposed a transient flow model based on a compositional model that considers the wellbore storage effect, skin factor and multiple-contact processes. The model solution is obtained with the finite volume method and by performing pressure transient analysis. The results demonstrate that the pressure derivative curve of the swept area radial flow rises first and later declines slightly; the pressure derivative curve of the unswept area radial flow rises first and later becomes flat. The CO2 flood front can be recognized when the derivative curve begins to rise after it declines slightly in the swept area radial flow regime. A numerical well test interpretation method can be established based on the model used in this paper to estimate the permeability, miscibility, and formation damage and to monitor the CO2 flood progress. © 2016 Elsevier B.V. All rights reserved.

Keywords: CO2 flooding Compositional model Monitor Transient flow

1. Introduction CO2 Enhanced Oil Recovery (EOR) is the green way to produce oil as it can recover the stranded or trapped oil by injecting the carbon emissions (CO2) from fossil-fueled power plants into the reservoir which can mitigate the greenhouse effect simultaneously (Gozalpour et al., 2005; NETL, 2010, 2006; Malik and Islam, 2000; Holtz et al., 2001; Jaramillo et al., 2009). This proven method has promising prospects as a result of technological advances, economic improvements, and environmental needs, and a study (NETL, 2011) conducted by DOE found that Next Generation CO2 EOR can provide 137 billion barrels of additional technically recoverable domestic oil. As the breakthrough of gas usually occurs, which leads to lower oil recovery and failure of the CO2 storage, useful methods should be found to monitor the CO2 flood progress and estimate the miscibility. The most common method is seismic monitoring (Araman et al., 2008; Kendall et al., 2003; Terrell et al., 2002), as Time-Lapse and Multicomponent seismic data analysis is an effective tool for monitoring CO2 injection through the detection of

* Corresponding author. E-mail addresses: [email protected] (L. Li), [email protected] (J. Yao), [email protected] (Y. Li), [email protected] (M. Wu), [email protected] (L. Zhang). http://dx.doi.org/10.1016/j.jngse.2016.04.062 1875-5100/© 2016 Elsevier B.V. All rights reserved.

changes in reservoir properties, such as porosity and fluid distribution. Although this method has been demonstrated to be accurate, it is expensive and not convenient enough to be used frequently. Another method is the Material Balance Equations (MBE) model (Tian and Zhao, 2008), which is convenient to use and inexpensive, but it is hard to obtain accurate results with this method, as it has many simplifications and assumptions. Well testing technology is commonly (Lee, 1982; Ozkan, 2001; Stratton, 2005, 2006; Zheng and Corbett, 2005; Yao and Wu, 2009; Fan et al., 2015) used to estimate the reservoir parameters and is a promising method to solve the above problem. MacAllister (1987) presented a procedure based on a three region composite model to analyze the CO2 and enriched-gas injection and production wells with emphasis on the real gas pseudo approach. Tang and Ambastha (1988) presented a three region analytical radial composite model to analyze the CO2 pressure transient; Su et al. (2015) established a three-region composite transient pressure analysis model for CO2 flooding, which considered the skin and storage effects of the well. Although the analytical composite model can describe the trend of the pressure change, it used too many simplifications and assumptions, which led to imprecise estimation of the parameters. In order to solve the above problem, an accurate model should be built and subsequently used to analyze the pressure transient of CO2 flooding. This paper describes a transient flow model that is

L. Li et al. / Journal of Natural Gas Science and Engineering 33 (2016) 30e36

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based on a compositional model which considers the effects of wellbore storage, skin effect and multiple-contact processes. The solution to the transient flow model is based on the finite volume method (FVM) followed by analysis of the transient pressure, which is used to divide the flow regime and find the CO2 flood front via the pressure derivative curve. In addition, the compositional transient flow model presented in this paper can provide effective technical support for the monitoring of CO2 flood progress and estimation for reservoir parameters. 2. Model description This section presents the mathematical model for the numerical simulation and the numerical well test, which can be used to describe the multiple-contact process involving interactions between the injected CO2 and the reservoir’s oil. 2.1. Compositional model The model assumes that there exists nc hydrocarbon components and a water component. The nc hydrocarbon component mass balances are

i h

v f ro So xi þ rg Sg yi
V$ ro xi ! v o þ rg yi ! v g þ qi ¼ vt

(1)

where subscript i is the index for mass components, i ¼ 1,…,nc,; subscripts o and g are phase indices for the oil and gas phases; r is phase molar density; S is phase saturation; x is molar fraction in oil phase; y is molar fraction in gas phase; f is the reservoir porosity; q is the sink/source per unit volume of reservoir. The water mass balance is


vðfrw Sw Þ
V$ rw ! v w þ qi ¼ vt

(2)

Fig. 1. Space discretization and geometry data in the finite volume method.

CVn and CVm, which are arbitrary, Anm is the area of the interface between two control volumes, lnm is the distance between the two control volume centroids, nnm is the unit normal to the interface inside CVn, fnm is the unit vector along the direction of the line joining the two control volume centroids. The discretized equations of hydrocarbon components and the water component can be obtained by volumetric integration and application of the divergence theorem for equations (1) and (2) over CVn. To assure numerical stability the time is discretized fully implicitly. For hydrocarbon component i,

X



tþ1
tþ1 tþ1 tþ1 tþ1 þ rg yi lg ðro xi lo Þtþ1 gtþ1 Pm nmþ1 gnm Pm
Pn 1 nm


Pntþ1 h

nmþ2

2

m2hn



þ ðVqi Þtþ1 n


itþ1 h
it Vf ro So xi þ rg Sg yi
Vf ro So xi þ rg Sg yi n

¼

! v b is the Darcy velocity of phase b:

kkrb VP ! vb¼


mb

xi ¼ 1

n

(7) For the water component,

(3)

where b is a phase index for the gas, oil or water phases; m is viscosity; k is the reservoir permeability; krb is the relative permeability of b phase; P is the reservoir pressure and the capillary pressure has been ignored in this model. The mole fraction constraint of the oil phase is nc X

Dt

(4)

i¼1

Xh m2hn

¼


i tþ1 tþ1 tþ1 ðrw lw Þnmþ1 gtþ1 þ ðVqw Þtþ1 nm Pm
Pn n 2

ðVfrw Sw Þtþ1
ðVfrw Sw Þtn n Dt

(8)

where subscript nm þ 1/2 denotes a proper averaging at the interface between CVn and CVm; hn donates all neighboring control volumes of n; l is the phase mobility defined as lb ¼ krb/mb for phase b; P is the pressure; t þ 1 is the current time step; and t is the previous time step; g is the transmissivity defined as

The mole fraction constraint of the gas phase is nc X

yi ¼ 1

(5)

gtþ1 nm ¼


A

nm nnm $f nm

lnm

pffiffiffiffiffiffiffiffiffiffiffi tþ1 kn km

(9)

i¼1

The saturation constraint is

So þ Sg þ Sw ¼ 1

(6) 2.3. Inner boundary model

2.2. Discretized governing equations The finite volume method is employed for space discretization. Fig. 1 shows the space discretization and geometry data. Fig. 1 shows the geometry of two neighboring control volumes,

During well testing, the sandface rate is unequal to the surface rate at the beginning as a result of wellbore storage. The inner boundary condition can be written as follows, considering the effect of wellbore storage:

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L. Li et al. / Journal of Natural Gas Science and Engineering 33 (2016) 30e36

Table 1 Composition and component properties of simulation. Component

Molar fraction

Pc (10
1MPa)

Tc (K)

vc (ft3/lbmole)

Acentric factor

Molar weight

CO2 C1 C2 C3 NC4 NC5 C6 C79 C1013 C1419 C2035 C36þ

0.000436 0.27215 0.004128 0.010484 0.02123 0.02002 0.022566 0.098746 0.10053 0.14514 0.16416 0.14041

73.88276 46.05379 48.85103 42.46621 37.97655 33.70966 30.11172 28.64828 17.6131 14.06276 10.89862 6.537931

304.7 190.6 305.43 369.8 425.2 469.6 507.5 577.9389 666.4667 748.0889 851.5389 1092.983

1.5057 1.5698 2.3707 3.2037 4.0847 4.9817 5.6225 8.3193 12.638 19.393 30.325 60.025

0.225 0.013 0.0986 0.1524 0.201 0.251 0.299 0.3165 0.4255 0.5768 0.7659 1.1313

44.01 16.043 30.07 44.097 58.124 72.151 84 145.16 223.26 353.51 554.55 1052

Table 2 Binary interaction parameters of simulation.

CO2 C1 C2 C3 NC4 NC5 C6 C79 C1013 C1419 C2035 C36þ

CO2

C1

C2

C3

NC4

NC5

C6

C79

C1013

C1419

C2035

C36þ

0.0000 0.0802 0.1144 0.1350 0.1350 0.1350 0.1350 0.1350 0.1350 0.1350 0.1350 0.1350

0.0802 0.0000 0.0000 0.0069 0.0248 0.0230 0.0212 0.0206 0.0167 0.0112 0.0044 0.0000

0.1144 0.0000 0.0000 0.0069 0.0248 0.0230 0.0212 0.0206 0.0167 0.0112 0.0044 0.0000

0.1350 0.0069 0.0069 0.0000 0.0248 0.0230 0.0212 0.0206 0.0167 0.0112 0.0044 0.0000

0.1350 0.0248 0.0248 0.0248 0.0000 0.0230 0.0212 0.0206 0.0167 0.0112 0.0044 0.0000

0.1350 0.0230 0.0230 0.0230 0.0230 0.0000 0.0212 0.0206 0.0167 0.0112 0.0044 0.0000

0.1350 0.0212 0.0212 0.0212 0.0212 0.0212 0.0000 0.0206 0.0167 0.0112 0.0044 0.0000

0.1350 0.0206 0.0206 0.0206 0.0206 0.0206 0.0206 0.0000 0.0167 0.0112 0.0044 0.0000

0.1350 0.0167 0.0167 0.0167 0.0167 0.0167 0.0167 0.0167 0.0000 0.0112 0.0044 0.0000

0.1350 0.0112 0.0112 0.0112 0.0112 0.0112 0.0112 0.0112 0.0112 0.0000 0.0044 0.0000

0.1350 0.0044 0.0044 0.0044 0.0044 0.0044 0.0044 0.0044 0.0044 0.0044 0.0000 0.0000

0.1350 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

PI ¼ ! 8  krg > kro krw
> > Pwf
Pi ¼ PI r þ r þ r q > b o g w < mo mg mw > > > > :

(10)

vPwf q
qb ¼ rCO2 C MCO2 vt

where q is the surface mass rate; MCO2 is the molar weight of CO2; qb is the sandface molar rate; rCO2 is the molar density of CO2 in bottom hole condition; C is the wellbore storage coefficient; Pwf is the flowing bottomhole pressure; Pi is the pressure of the control volume which contains the well; PI is the injectivity index which can be written as

Table 3 Input parameters of simulation. Parameter

Value

Unit

Reservoir radius Reservoir thickness Initial formation pressure Formation temperature Permeability Rock compressibility Porosity Well radius Water compressibility Initial water saturation Wellbore storage coefficient Skin factor CO2 injection rate CO2 injection time

300 10 30 120 5 2.3  10
4 0.2 10 4.2  10
4 0.3 0.01 1 20 60

m m MPa  C mD MPa
1 cm MPa
1 m3/MPa t/d d

2pkh lnðre =rw Þ þ S
1=2

(11)

where k is the permeability of the control volume which contains the well; h is the thickness of the reservoir; rw is the well bore radius; re is an equivalent radius of the control volume which contains the well; S is the skin factor. The inner boundary model is treated implicitly in this paper. 2.4. Vapor-liquid equilibrium calculation The vapor-liquid equilibrium calculation is of great importance in the compositional model, as it can be used to calculate the phase composition and thermodynamic properties and even the parameter which can be used to estimate the miscibility. Fugacities of a component in the two phases must be equal for a fluid system that is in thermodynamic equilibrium:

fiL ¼ fiV

i ¼ 1; …; nc

(12)

The Equation (12) can also be written as

4Li xi ¼ 4Vi yi

(13)

where superscript L and V are phase indices for the oil and gas phases; 4 is the fugacity coefficient defined as 4i ¼ fi =xi p. The fugacity coefficient can be computed from the equation of state, such as SRK and PR. An analysis of the stability of a single phase can determine whether the system will split into two-phases through the phase stability criteria introduced by Michelsen (Michelsen, 1982; , 2000). The vapor-liquid equilibrium is then Whitson and Brule

L. Li et al. / Journal of Natural Gas Science and Engineering 33 (2016) 30e36

33

Fig. 4. Pressure and pressure derivative curves.

where Ki is equilibrium ratio, defined as Ki ¼ yi/xi; FV is the mole ratio of feed and vapor. The commonly used methods to solve the R-R equation, the NewtoneRaphson method and the Successive Substitution Method are inefficient when the system is close to the critical point. To solve the R-R equation efficiently, this paper applies the accelerated and stabilized successive substitution Method (Risnes et al., 1981). The viscosity of the gas phase is calculated with the Lee, Gonzalez and Eakin correlation (Lee et al., 1966) and the viscosity of the oil phase is calculated with with the Lohrenz-Bray-Clark (LBC) correlation (Lohrenz et al., 1964). Next, solve Equations (1)e(6), (13) with Newton-Raphson method.

Fig. 2. Comparison of the model in this paper with CMG. (A) Bottom hole fluid rate versus time (B) Cumulative gas RC versus time.

2.5. Interfacial tension and relative permeability Apply the factor Fk introduced by Coats (Coats, 1980) to describe the miscibility:

calculated by solving the Rachford-Rice (R-R) equation (Rachford and Rice, 1952) below if the single phase is unstable. nc X

zi ðKi
1Þ ¼0 1 þ ðKi
1ÞFV i¼1

(14)

"



s Fk ¼ min 1; s0

0:25 # (15)

where s is the interfacial tension which could be calculated by the Macleod-Sugden correlation (Danesh, 1998; Weinaug and Katz, 1943) with composition data and parachor values:

s1=4 ¼

nc X


 Pchi xi ro
yi rg

(16)

i¼1

where Pchi is the parachor of hydrocarbon component i. The Fk is unity when there is only oil phase or gas phase for the hydrocarbon component. The relative permeability can be calculated by

Fig. 3. Pressure distribution after CO2 injection.

imm mis Kro ¼ Fk Kro þ ð1
Fk ÞKro

(17)

imm mis Krg ¼ Fk Krg þ ð1
Fk ÞKrg

(18)

where Kr is relative permeability; superscripts imm and mis represent the immiscible and miscible conditions, respectively; it is assumed that the relative permeability curves of the oil and gas phases under miscible conditions are straight lines; and the end points are modified by interfacial tension such that

34

L. Li et al. / Journal of Natural Gas Science and Engineering 33 (2016) 30e36

Fig. 5. Effect of CO2 injection time on the pressure and pressure derivative curves.

Fig. 8. Effect of the skin factor on the pressure and pressure derivative curves.

Fig. 6. Effect of injection rate on the pressure and pressure derivative curves.

Fig. 9. Effect of wellbore storage coefficient on the pressure and pressure derivative curves.

mis Socr ¼ Fk Simm ocr þ ð1
Fk ÞSocr

(19)

mis Sgcr ¼ Fk Simm gcr þ ð1
Fk ÞSgcr

(20)

where Socr is residual oil saturation; Sgcr is residual gas saturation.

3. Accurate validation and pressure-transient analysis In order to present an accurate validation and take a CO2 injection falloff test as an example on which to conduct pressuretransient analysis, Tables 1 and 2 show the composition data and thermodynamic properties of components of the sample oil which is used in the Eclipse PVTI Tutorials (Eclipse Technical Descrip, 2006). Table 3 shows the input parameters of the simulation. 3.1. Accurate validation Perform a numerical simulation that reservoir is 600 m$600 m, wellbore hole pressure is 42 MPa, of which the wellbore storage effect has been ignored. The comparison of the model in this paper with CMG (2012) is shown in Fig. 2, it demonstrates that the model used in this paper can match CMG and that the results are reliable. 3.2. Pressure-transient analysis

Fig. 7. Effect of permeability on the pressure and pressure derivative curves.

Fig. 3 shows the pressure distribution after CO2 injection, and Fig. 4 shows the pressure and pressure derivative curves of the falloff test, from which the radial flow can be divided into two flow regimes: swept area radial flow and unswept area radial flow. The pressure derivative curve, which could reflect the mean mobility of the pressure propagated area (caused by shut in), will move down

L. Li et al. / Journal of Natural Gas Science and Engineering 33 (2016) 30e36

with the increase of mobility. The pressure derivative curve of the swept area radial flow rises first and later declines a little. The pressure derivative curve is very low at the beginning because of the large mobility near the wellbore area, where hydrocarbons only exist in the gas phase and the CO2 content is very high; the reason why the derivative curve rises is that the mean mobility of the pressure propagated zone decreases when it is farther from the well bore as a result of the lower CO2 content and the existence of a hydrocarbon liquid phase; the slight decline in the derivative curve is mainly caused by the decreased mean pressure of the pressure propagated zone as time goes on, which reduces the mobility, and another reason is that it is a phenomenon caused by the decrease of storage capacity. In conclusion, the level of the pressure derivative curve when it is flat can be used to estimate the miscibility. The pressure derivative curve of the unswept area radial flow rises first and then becomes flat, for which the reason is that the mean mobility of the pressure propagated zone decreases as the mobility of the swept zone becomes larger than that of the unswept zone. The pressure derivative curve will decline when the pressure has propagated to the closed outer boundary. Fig. 5 shows the effect of CO2 injection time on the pressure and pressure derivative curves. The beginning time of the unswept area radial flow will be delayed with the increase of injection time. From that, the CO2 flood front can be recognized when the pressure derivative curve begins to rise after it has declined a little in the swept area radial flow regime. As a result, a numerical well test interpretation method can be established based on the model used in this paper to estimate the miscibility and the CO2 flood front. The peak of the pressure derivative curve in the unswept radial flow regime moves down with the increase of injection time, which means that the mean mobility of the reservoir has increased. Fig. 6 shows the effect of injection rate on the pressure and pressure derivative curves. The pressure and pressure derivative curves move up with the increase of injection rate, for which the reason is that the reservoir pressure increases with the increase of injection rate which makes the viscosities of the oil and gas phases higher. The beginning time of the unswept area radial flow will be delayed with the increase of injection rate which denotes a larger swept area. The pressure and pressure derivative curves are more sensitive to the injection rate for the swept area radial flow than for the unswept area radial flow as the viscosity of the gas phase is more sensitive to pressure than the oil phase. Fig. 7 shows the effect of permeability on the pressure and pressure derivative curves. The pressure and pressure derivative curves move down with the increase of permeability, and the beginning time of the swept area radial flow and the unswept area radial flow will be earlier. The pressure and pressure derivative curves are very sensitive to the permeability, which means that a numerical well test interpretation method can be established based on the model used in this paper to estimate the permeability. Fig. 8 shows the effect of the skin factor on the pressure and pressure derivative curves. The pressure and pressure derivative curves move up with the increase of skin factor and are very sensitive to the skin factor before the swept area radial flow regime. The beginning time of the swept area radial flow will be delayed with the increase of skin factor. Thus, a numerical well test interpretation method can be established based on the model used in this paper to estimate the formation damage. Fig. 9 shows the effect of wellbore storage on the pressure and pressure derivative curves. The pressure and pressure derivative curves move backward with the increase of wellbore storage coefficient, and they are very sensitive to the wellbore storage coefficient before the swept area radial flow regime, and slightly

35

sensitive during the swept area radial flow regime. The beginning time of the swept area radial flow will be delayed with the increase of wellbore storage coefficient. 4. Conclusions In this paper, a transient flow model is established for a CO2 injection well based on a compositional model considering the multiple-contact processes, wellbore storage effect and skin factor. There are two flow regimes during radial flow: swept area radial flow and unswept area radial flow. The pressure derivative curve of the swept area radial flow rises first and then declines a little; the pressure derivative curve of the unswept area radial flow rises first and then becomes flat. Therefore, the CO2 flood front can be recognized when the pressure derivative curve begins to rise after it declines a little in the swept area radial flow regime. The pressure and pressure derivative curves move up with the increase of injection rate and skin factor, move down with the increase of permeability, and move backward with the increase of wellbore storage coefficient; the beginning time of the swept area radial flow will be advanced with the increase of permeability, and delayed with the increase of skin factor and wellbore storage coefficient; the beginning time of the unswept area radial flow will be delayed with the increase of injection time and injection rate, and advanced with the increase of permeability. A numerical well test interpretation method can be established based on the model used in this paper to estimate the permeability, miscibility, and formation damage and to monitor the CO2 flood progress. Acknowledgments This work was financially supported by the Program for Changjiang Scholars and Innovative Research Team in University (IRT1294), the National Basic Research Program of China (“97300 Program) (GrantNo.2011CN201004), and China Postdoctoral Science Foundation (No. 2015M580621). Nomenclature A FV f K k kr nc P Ps q S t V ! v x y z 4 4

s r m l

interface area between two control volumes [cm2] mole ratio of feed and vapor [e] fugacity of component in oil or gas phase [10
1 MPa] equilibrium ratio of component [e] absolute permeability [mm2] relative permeability of phases [e] total number of hydrocarbon components [e] reservoir pressure [10
1 MPa] Parachor value [e] sink/source per unit volume of reservoir [mol/cm3/s] saturation of water, oil or gas phase [e] time [s] volume [cm3] Darcy velocity of water, oil or gas phase [cm/s] molar fraction in oil phase [e] molar fraction in gas phase [e] total molar fraction in hydrocarbon system of component [e] Fugacity coefficient [e] reservoir porosity [e] interfacial tension [dynes/cm] molar density of water, oil or gas phase [mol/cm3] viscosity [cP] phase mobility [mm2/cP]

36

h 0

L. Li et al. / Journal of Natural Gas Science and Engineering 33 (2016) 30e36

a set of neighboring grid blocks of a grid block [e] reference state

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