Transient pressure analysis and interpretation for analytical composite model of CO2 flooding

Journal of Petroleum Science and Engineering 125 (2015) 128–135

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Transient pressure analysis and interpretation for analytical composite model of CO2 flooding Kun Su a,n, Xinwei Liao b, Xiaoliang Zhao b a b

Petroleum Exploration and Production Research Institute, SINOPEC, Beijing, China Petroleum Engineering Department, China University of Petroleum, Beijing, China

art ic l e i nf o

a b s t r a c t

Article history: Received 30 November 2012 Accepted 12 November 2014 Available online 2 December 2014

This work deals with the inverse problems of parameters estimation for mass transfer during CO2 injection in oil reservoir. Such method is based on well testing theory which monitors reservoir dynamic performance through transient pressure. In this paper, considering the skin and storage effect of well, three-region composite transient pressure analysis model for CO2 flooding is established by introducing pseudo pressure for real gas, and then non-dimensional transport equations for each region are derived and solved analytically by the method of Laplace transformation. According to the type curves of the model pressure, fluid flowing in reservoir can be divided into seven stages, and the flowing factors' effect on curve shape is discussed too. Then relevant parameters during mass transferring can be achieved by type curve matching field testing data. We also provide a new method for parameter matching during testing data interpretation through feature points' analysis, which facilitates the matching process and reduces ambiguity of parameters estimation. & 2014 Elsevier B.V. All rights reserved.

Keywords: transient pressure analysis three-region composite model analytical method feature points' analysis CO2 flooding

1. Introduction Global warming has made people pay increasing attention to carbon reduction in the atmosphere. Among all the methods, the method for injecting carbon into reservoir to enhance oil recovery (CO2-EOR) is favorable. In this way, carbon storage would be achieved during such process. In North America, where CO2-EOR is applied for many years, annual CO2 utilization is up to 50 million tons. In America, more than 250,000 barrels crude oil can be incremented by CO2 flooding comparing with base scenario (Global CCS Institute and Parsons Brinckerhoff, 2011), and in the future about 4–47 billion barrels oil can be economically produced as well as 8 million tons' carbon storage at the same time (Sharman, 2004). In China, CO2-EOR and storage projects are recently launched in small scale areas. But, under the support of government and oil companies, such activities grow rapidly. Now, demonstration project has been built up covering fundamental research, indoor experiment and pilot test. Well testing models of CO2 flooding was from the ones previously applied in thermal recovery (Eggenschwiler et al., 1979). Usually, well tests from thermal recovery projects are

n Corresponding author at: Petroleum Exploration and Production Research Institute, SINOPEC, Beijing, China. Tel.: þ 86 18618127941; fax: þ 861082311360. E-mail addresses: [email protected] (K. Su), [email protected] (X. Liao), [email protected] (X. Zhao).

http://dx.doi.org/10.1016/j.petrol.2014.11.007 0920-4105/& 2014 Elsevier B.V. All rights reserved.

analyzed using a radial, two-region composite reservoir model (Ambastha, 1988; Ambastha and Ramey, 1989). Onyekonwu and Barua presented a three-region composite model for thermal recovery and gave an analytical solution (Onyekonwu et al., 1984; Barua and Home, 1987). And then, similar researches on transient pressure behavior for CO2 flooding projects were proceeded based on those researches (MacAllister, 1987). Tang and Ambastha (1988) proved that three-region model was more applicable than two-region one when the interaction of fluid flow and phase behavior during CO2 flooding was considered. Ambastha and Ramey studied the effects of an intermediate region on the deviation time method and the pseudosteady state method, but neglecting the effects of wellbore storage and skin. According to this research, two-region model can provide acceptable result only when mobility ratio between inner and outer region is greater than 10, and three-region model is recommended in else conditions with low mobility ratio and pseudo-steady region (Ambastha and Ramey, 1992). So, three-region model was believed as more accurate pattern for transient pressure analysis during CO2 flooding. In the previous three-region, CO2 injection testing model once was described as CO2 bank, methane bank and oil bank (Ambastha and Ramey, 1988). The conclusions that methane bank is too compressible to be neglected during pressure analysis were got. However, from field experience, which is also analyzed in latter section, the intermediate region is far from pure methane but mixture of CO2 and oil, and adjustment of this region is necessary.

K. Su et al. / Journal of Petroleum Science and Engineering 125 (2015) 128–135

In this paper, a similar three-region model with newly defined intermediate region for CO2 flooding is developed, which is believed more reasonable than before. Also, considering wellbore storage effect and skin, such model is solved analytically in details in Laplace space, and more specific stages of flowing and thorough analysis are provided according to pressure type curves, which is different from those models introduced before in details. During interpretation section, a new method is introduced to reduce ambiguity of parameters, which is also firstly introduced for CO2 flooding to the best of our knowledge.

2. Model establishment and solution 2.1. Model establishment The CO2 flooding testing model is established based on CO2 flooding mechanism. As is well known, CO2 flooding is characterized as miscible and immiscible that is judged by the comparisons of reservoir pressure and minimum miscible pressure (MMP). Under either condition, in the CO2 sweeping areas, the mixture properties of CO2 and oil are functions of component saturation, temperature and initial viscosity, etc. None of its properties is constant but time and space related. So well testing model for CO2 injection should base on multi-region assumption. Miscibility is achieved between CO2 and oil through a process known as multi-contact miscible (MCM) flooding, which is characterized as component transfer between CO2 and crude oil. By this, oil volume is swelling and viscosity is lowering as the dissolution of CO2 into oil (Stalkup, 1984). In miscible condition, CO2 phase will completely disappear and interfacial tension of CO2–oil and capillary pressure is zero. So, in the CO2 sweeping area, displacing efficiency is high enough. The borehole zones can be treated as CO2 bank. For areas beyond CO2's sweeping, fluid properties remain in the original state, which can be treated as oil bank. In the middle of the two regions, the intermediate region should be redefined rather than simply treating as methane bank. In this paper, the middle region is under such assumption: the fluid in middle region will flow as single phase, and the properties are equivalent with the mixture of CO2 and oil. Such treatment is flexible to both miscible and immiscible conditions because fluid parameters can be calculated respectively. In order to simplify the model for calculation, fluid properties are homogeneous in each region, but different among them. Moreover, the well testing model should not be reduced to two regions by neglecting the existing of intermediate region, especially when such region is large enough to affect the reservoir pressure. Because when mobility ratio for two-region model is

129

low, their interface concentration gradient is not likely steep but gradually changing. Above all, three regions models for CO2 injection reservoir are defined as following: (1) Near-well-bore region (region-1), which is saturated by injected CO2 in gas phase. (2) Intermediate region (region-2), which is saturated by the mixture of CO2 and oil under miscible/immiscible conditions (Fig. 1). (3) Outer region (region-3), which is in original state as oil phase without sweeping of CO2. Moreover, new model is under assumptions as below (1) Reservoir is homogeneous in properties and infinite in size. (2) Reservoir is coexisting by oil and gas, and flowing in reservoir is isothermal. (3) Hydrocarbon components are distributed in oil and gas phase. (4) Adsorption effect is ignored. (5) The flow of oil and gas follow Darcy Law and diffusion effect is neglected. (6) Phase equilibrium is achieved instantaneously for each component in pore space. Considering viscosity μg and deviation factor Z of CO2 are functions of pressure during gas flowing, the pseudo-pressure is introduced (Al-Hussainy and Ramey, 1966) in transporting equations. The relevant parameters for pseudo-pressure calculation are in Appendix A. Then, the three-region composite model (Fig. 2) can be described analytically as
 1 ∂ ∂mðp1 Þ 1 ∂mðp1 Þ region1 : r ¼ ; r w r r r R1 ð1Þ r ∂r ∂r η1 ∂t in which; η1 ¼

krg ; ϕμg C g


  m p1 ¼ 2

Z

p2 pwf

ρg krg dp: μg



1 ∂ ∂mðp2 Þ 1 ∂mðp2 Þ r ¼ ; r ∂r ∂r η2 ∂t

region2 :



R1 r r r R2

ð2Þ



ϕ C t þ ρo  ρg ð∂So =∂pÞ ¼ ; in which; η2 k ðkrg =μg Þρg þ ðkro =μo Þρo ! Z p   ρg krg ρo kro m p2 ¼ þ dp μg μo p2 1

For miscible condition, fluid in region 2 is treated as single phase, of which parameters is treated as mixture of oil and CO2. Then

Fig. 1. Three-region model for CO2 flooding (left: immiscible, right: miscible).

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K. Su et al. / Journal of Petroleum Science and Engineering 125 (2015) 128–135

Region 3 : ψ 3D ¼

re

Region 1

1:842  10

3

qD

k μB





p3  pi



o

In which, r R1D ; r R2D is the dimensionless radius of region-1 and region-2, respectively; t D is the dimensionless time; p is the average pressure of reservoir, MPa; T SC is the temperature under standard condition (T SC ¼ 293:15 K), K; P SC is the pressure under standard condition (P SC ¼ 0:10325 MPa), MPa; qSC is the well production rate under surface condition, m3 =d; h is the formation thickness, m; subscripts 1, 2, 3 indicate region-1, region-2 and region-3, respectively; subscript D demonstrates dimensionless parameter. Considering the well bore storage and skin effect, three-region composite model for CO2 injection is rewritten as below Region 1 (gas):

rw

R1




h

R2

Region 2 Region 3

∂2 ψ 1D 1 ∂ψ 1D 1 ∂ψ  1D ; þ ¼ r D ∂r D C D e2s ∂ t D =C D ∂r 2D

1 r r D rr R1D

ð4Þ

h

rw

Region 2 (gas, oil): ∂ ψ 2D 1 ∂ψ 2D 1 ∂ψ  1D ; þ ¼ r D ∂r D ω12 C D e2s ∂ t D =C D ∂r 2D 2

R1 R2

∂2 ψ 3D 1 ∂ψ 3D 1 ∂ψ  3D ; þ ¼ r D ∂r D ω13 C D e2s ∂ t D =C D ∂r 2D

Fig. 2. Three-region composite model for CO2 flooding.

transporting equation reduced to the form as region 1, and viscosity is calculated as in Appendix A.1(c).

in which;

∂2 p3 1 ∂p1 1 ∂p3 ¼ þ η3 ∂t ∂r 2 r ∂r 1

η3

¼

ϕC o μ o kro

R2 r r rr e

ð3Þ

r ; rw

And for dimensionless time; t D ¼

r R1D ¼

R1 ; rw

r R2D ¼

R2 rw

krg t t η ¼ r 2w 1 ϕμg C g r 2w

And for dimensionless production rate  qsc μ psc TZ pi 2π kh pT SC

For region-1 and region-2, dimensionless pseudo-pressures are represented by ψ 1D ; ψ 2D . For region-3, dimensionless pressure is ψ 3D . Then we have     m pi  m p1   Region 1 : ψ 1D ¼ qD m pi Region 2 : ψ 2D ¼

    m pi  m p2   qD m pi

ð6Þ

in which, C D is the storage factor, m3/MPa; s is the skin factor. Inner boundary condition is set as constant injection rate and demonstrated as
 ∂ψ ∂ψ  1D jrD ¼ 1  r D 1D jrD ¼ 1 ¼ 1 ð7Þ ∂r D ∂ t D =C D

lim ψ 3D ðr D ; t D Þ ¼ 0

In equations above, r w is the well bore radius, m; r e is the reservoir radius, m; R1 ; R2 is the radius of region-1 and region-2, respectively, m; krg ; kro is the relative permeability of gas and oil phase, respectively, dimensionless; k is the absolute permeability, mm2; ϕ is the reservoir porosity; C t is the comprehensive compressibility, and C o ; C g is the compressibility for oil and gas, MPa-1; t is the demonstrates time, h; So is the oil saturation; pwf is the flowing bottom hole pressure, MPa; p2 is the pressure of intermediate region. To establish the equations in dimensionless form, we introduce

qD ¼

r R2D rr D r1

Outer boundary condition is infinite, which can be demonstrated as

:

For dimensionless radius; r D ¼

ð5Þ

Region 3 (oil):

re

region3 :

r R1D rr D rr R2D

rD -1

Interface conditions between regions are pressure continuity and mass balance Region (1–2):

ψ 1D ðr R1D ; t D Þ ¼ ψ 2D ðr R1D ; t D Þ;

λ1

∂ψ 1D ∂ψ j ¼ λ2 2D jrD ¼ rR1D ∂r D rD ¼ rR1D ∂r D

λ2

∂ψ 2D ∂P 3D j ¼ λ3 j ∂r D rD ¼ rR2D ∂r D rD ¼ rR2D

Region (2–3):

ψ 2D ðr R2D ; t D Þ ¼ ψ 3D ðr R2D ; t D Þ; in which, λ ¼ 1=M. 2.2. Solution of the model

Since Van Everdingen and Hurst (1949) first applied Laplace transformation method to solve seepage questions, it can provide more accurate results than that by numerical methods. Here, R þ1 define ψ D as ψ D ðzÞ ¼ 0 e  zt D ψ D ðt D Þdt D , then Eqs. (4)–(6) can be adapted as Region-1: d ψ 1D 1 dψ 1D z þ ¼ ψ ; r D dr D C D e2s 1D dr 2D 2

1 r r D o r R1D

ð8Þ

Region-2: d ψ 2D 1 dψ 2D z þ ¼ ψ ; r D dr D ω12 C D e2s 2D dr 2D 2

r R1D r r D o r R2D

ð9Þ

K. Su et al. / Journal of Petroleum Science and Engineering 125 (2015) 128–135

Region-3:

100

d ψ 3D 1 dψ 3D z þ ¼ ψ ; r D dr D ω13 C D e2s 3D dr 2D

131

pressure

2

r R2D r r D o 1

Similarly, boundary condition is transformed as
 dψ dψ Inner boundary;  WD jrD ¼ 1  r D 1D jrD ¼ 1 ¼ 1 dr D d t D =C D Outer boundary; lim ψ 3D ðr D ; t D Þ ¼ 0 Interfacial conditions between regions are Region (1–2): dψ dψ λ1 1D jrD ¼ rR1D ¼ λ2 2D jrD ¼ rR1D dr D dr D

Region (2–3):

λ2

dψ 2D dψ j ¼ λ3 3D jrD ¼ rR2D dr D rD ¼ rR2D dr D

Then solutions of equations above in Laplace space can be obtained as 1 þ sψ 0D   u ψ 0D þ C D u 1 þ sψ 0D

ψ WD ¼  in which 

pffiffiffiKI ab KI 1 1



ψ 0D u; Rf D ; M12 ¼ I12 u and I 12 ¼

pffiffiffi I1 u pffiffiffi; I0 u

KI cd ¼

KI 0 ¼

KI 1 ¼

1

1 0.1

2

3

4

5

6

7

pffiffiffi K1 u pffiffiffi I0 u

pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi M 12 = M 13 ÞI 0 ðcÞK 1 ðdÞ pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi K 1 ðcÞK 0 ðdÞ  ð M 12 = M 13 ÞK 0 ðcÞK 1 ðdÞ

tD/CD Fig. 3. Type-curves of radial, three-region composite reservoir for CO2 flooding ðC D e2S ¼ 500; r R1D ¼ 300; r R2D ¼ 330; M 12 ¼ 2; M 13 ¼ 4; ω12 ¼ 10; ω13 ¼ 20Þ.

(1) The first one is the period dominated by well-bore storage effect, when two curves coincide with each other and the slope of straight-line part equals 1. (2) Pressure derivative curve sees descending after extreme value in the second period, which reflects the transition flow from well bore to region-1. (3) During the third period, radial flow in region-1 is illustrated by horizontal line in type-curve. (4) The fourth one is an ascending line which means transition flow from region-1 to region-2. (5) Similar to period 3rd, radial flow in region-2 is achieved in the fifth period and horizontal-line part is observed in curve. (6) Descending curve in sixth stage suggests transition flow from region-2 to region-3. (7) In the last stage, radial flow in outer region is reached, which is represented in pressure derivative curve by another horizontal-line with value equals 0:5M 13 .

I 1 ðcÞK 0 ðdÞ þ ð

pffiffiffiffiffiffiffiffiffiffiffiffi c ¼ r R2D M 12 u; KI ab ¼

KI ab KI 1 þ1

pffiffiffi K0 u pffiffiffi ; I0 u

0.5M13

0.01 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

r D -1

ψ 2D ðr R2D ; t D Þ ¼ ψ 3D ðrR2D ; t D Þ;

ψwD,dψwD/dtD

10

in which, ω12 ; ω13 are storage ratios between regions, respectively, which defined as         ω12 ¼ ω1 =ω2 ¼ ϕC t 1 = ϕC t 2 ; ω13 ¼ ω1 =ω3 ¼ ϕC t 1 = ϕC t 3 ;

ψ 1D ðr R1D ; t D Þ ¼ ψ 2D ðrR1D ; t D Þ;

pressure derivatives

ð10Þ

pffiffiffiffiffiffiffiffiffiffiffiffi d ¼ r R2D M 13 u

pffiffiffiffiffiffiffiffiffi   I 1 ðaÞ KI cd K 0 ðbÞ þ I 0 ðbÞ þð1= M 12 ÞI 0 ðaÞ KI cd K 1 ðbÞ þ I 1 ðbÞ pffiffiffiffiffiffiffiffiffi   K 1 ðaÞ KI cd K 0 ðbÞ þ I 0 ðbÞ ð1= M 12 ÞK 0 ðaÞ KI cd K 1 ðbÞ  I 1 ðbÞ

pffiffiffi a ¼ r R1D u;

b ¼ r R1D

pffiffiffiffiffiffiffiffiffiffiffiffi M 12 u

in which, I 0 ; I 1 are the zero order and one order Bessel function, respectively; K 0 ; K 1 are the zero-order and one-order imaginary variation Bessel function, respectively; u is the Laplace parameter; M 12 ; M 13 are the mobility ratio between regions.

3. Type-curve analysis and interpretation method 3.1. Analysis and discussion On the basis of above results, numerical solutions in real space can be obtained through the Stehfest reversion method (Stehfest, 1970). And then, two type-curves, which are called double logarithmic pressure curve and semi-log pressure derivative curve, can be drawn to illustrate the flow process of CO2 flooding in infinite reservoir. As shown by pressure derivative type-curve in Fig. 3, the flow process in reservoir for CO2 flooding includes seven stages.

Further, curves' shape is affected by parameters in certain laws. In the following part, key parameters in ranges from Table 1 are discussed. (1) Parameters combination C D e2s is firstly introduced and proved its obvious correlation with pressure derivative (Earlougher and Robert, 1974). Here in Fig. 4, type curves are drawn when such parameter's values are equal to 100, 500 and 1000. As shown in the first graph, the first extreme value is increasing as the parameter value. At the same time, the elapsed time for second extreme value is getting shorter, so is it for the minimum value. That phenomenon demonstrates that periods after storage effect period are advanced as the storage parameter increases. In the second graph, curves are of the similar shape and straight-line parts are paralleling with each other. For models considering influence of borehole storage and skin effect, both CD and S are among parameters need to be calculated. Definitely, curve character will provide information to solve them, which is introduced in latter section. (2) Effect of M 12 =M 13 on type-curves is shown in Fig. 5 when the ratio equals 2:4, 5:10 and 10:20, respectively. Conclusions can be easily made from the first graph that mobility ratio only affects second extreme value. As the mobility ratio increases, the value of M12 is increasing. Given that parameters of pure CO2 in region 1 can be treated as constants, conclusion can be made that viscosity of fluid in region 2 is increasing. And fluid property in intermediate region is getting worse and viscosity is getting higher. In such case, the

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K. Su et al. / Journal of Petroleum Science and Engineering 125 (2015) 128–135

Table 1 Properties of fluid in gas, supercritical and liquid state (Vesovic et al., 1990; Span and Wagner, 1996). Properties

Gaseous

Supercritical

Liquid

Density (  103 kg/m3) Diffusion coefficient (m2/s) Viscosity (Pa s)

0.062 (0.1–0.4)  10  4 (1–3)  10  5

0.2–0.9 (0.2–0.7)  10  7 (1–9)  10  5

0.6–1.6 (0.2–2.0)  10  9 (0.2–3)  10  3

100

100

Cd*exp(2S)=100

2:4

Cd*exp(2S)=500 2nd extreme value

Cd*exp(2S)=1000

5:10 10

10:20 1st extreme value

dψwD/dtD

dψwD/dtD

10

1

1

Minimum value 0.1

0.1

0.01 1.E-02

1.E+00

1.E+02

1.E+04

0.01 1.E-02

1.E+06

1.E+00

60

Cd*exp(2S)=100

1.E+06

2:4 120

Cd*exp(2S)=500 Cd*exp(2S)=1000

100

50

5:10 10:20

80

40

ψwD

ψwD

1.E+04

140

80 70

1.E+02

tD/CD

tD/CD

30

60

20

40 10

20

0 1.E-02

1.E+00

1.E+02

1.E+04

1.E+06

tD/CD 2S

Fig. 4. Effect of CDe on type-curves ðr R1D ¼ 300; r R2D ¼ 500; M 12 ¼ 2; M 13 ¼ 10; ω12 ¼ 10; ω13 ¼ 100Þ.

elapsed time for radial flow in region-2 will get longer and latter periods are all postponed. Also, the slope of the straight-line part differs from each other in the second graph, which indicates the effect of mobility on curve shapes of semi-log pressure. (3) It is well known that the area of intermediate region greatly affects type-curves in thermal three-region model. In this section, three different areas with constant radius ratio between region-1 and region-2 are simulated as shown in Fig. 6. As region area grows, periods after storage effect is advanced in time, but no variance is made on extreme values. For second graph, similar phenomenon is shown as in Section 1, which implies that the slope of straight-line is irrelevant to region area. (4) Fluid in each region is quite different in properties. On the aspect of compressibility, pure gas in region-1 is largest, followed by mixture of oil and gas in region-2 and oil bank in region-3. So, storage ratio of every region ω ¼ ϕC t should be discussed about its effects on type-curves. Here three groups of ω12 =ω13 values are discussed, and in such condition, ω12 is increased as ω12 =ω13 varies from 2:4, 5:10 to 10:20, which represents the conditions of increasing gas saturation. As is shown in Fig. 7, pressure derivative curve is slightly changed except that second extreme value is

0 1.E-02

1.E+00

1.E+02

1.E+04

1.E+06

tD/CD Fig. 5. Effect of M12/M13 on type curves ðC D e2S ¼ 500; r R1D ¼ 200; r R2D ¼ 220; ω12 ¼ 2; ω13 ¼ 10Þ.

increasing. For straight-lines in second graph, first part is getting steeper but second one keeps paralleling. (5) In Fig. 8, type curves under miscible and immiscible conditions are illustrated, respectively. During wellbore storage effect period, both curves coincide with each other till first extreme value. This character corresponds with the fact that at the beginning, storage effect dominates the curve shape and first extreme value. For the period of transition from well bore to region-1, curves develop in different ways which indicate various flooding mechanism under different conditions. As shown in pressure derivative curve, for miscible flooding, the elapsed time of radial flow in region-1 is quite longer than that under immiscible condition, which means larger area of region-1 than the latter. This also verifies a much more complex process of miscibility. CO2 will firstly extract the light components in the oil and then intermediate components after subsequent injected CO2 is reached. Such multi-contact process needs much larger distance than immiscible process to form a pure region near the well bore. What's more, in immiscible

K. Su et al. / Journal of Petroleum Science and Engineering 125 (2015) 128–135

100

133

10 100:110 200:220 500:550

1 dψD/dtD

dψwD/dtD

10

1

2:4 0.1

5:10

0.1

10:20 0.01

1.E-02

1.E+00

1.E+02

1.E+04

1.E+06

1.E+04

1.E+06

0.01 1.E-02

tD/CD

1.E+00

1.E+02 tD/CD

1.E+04

1.E+06

90 80 70

ψwD

60

100-110 200:220

500:550

50 40 30 20 10 0 1.E-02

1.E+00

1.E+02

tD/CD

condition, CO2 dissolves in oil and makes it higher compressible than miscible fluid. Such effect leads to high pressure conductivity in immiscible reservoir and radial flow periods of region-2 and region-3 is earlier than that in miscible condition. This mechanism can also explain that gas breakthrough in oil wells is earlier than that in miscible flooding as observed in field experience. Above all, we can conclude that the treatment of region 2 under miscible and immiscible conditions is practicable in this paper. 3.2. Interpretation method of type curve In former sections, type-curves of CO2 flooding can finally be drawn based on model establishment and analytical solutions. While in field experience, observed curves can also be obtained by changing well working system. Both of the curves can be matched by certain methods to determine reservoir parameters, and such process is commonly defined as inverse problem in well testing. Matching methods mainly consist of plots matching and automatic matching. The accuracy of plots matching is limited by plot number in graph, while automatic matching tends to be in poor convergence. Moreover, three-region composite model for CO2 flooding inherently brings severe interpretation ambiguity, which limit the application of conventional matching method. In fact, it has been proved theoretically (Ran and Liao, 2002) that for homogeneous reservoir, first and second derivatives of dimensionless pressure are of linear function with C D e2s , which has been also demonstrated previously for newly-established model. In this way, researches on feather points can facilitate the interpretation process. Pressure derivatives curves is depicted in Fig. 9 when C D e2s equals 100, 500, 1000, 2000 and 5000. As is shown, three feature points,

Fig. 7 . Effect of ω12/ω13 on type curves . ðC D e2S ¼ 500; r R1D ¼ 200; r R2D ¼ 220; M 12 ¼ 2; M 13 ¼ 10Þ. 100

10

ψwD ,dψ wD /dt D

Fig. 6. Effect of rR1D/rR2D on type-curves ðC D e2S ¼ 500; r R1D ¼ 200; r R2D ¼ 220; ω12 ¼ 2; ω13 ¼ 10Þ.

1

immiscible,pressure immiscible,pressure derivative miscible,pressure

0.1

miscible,pressure derivative 0.01 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

tD/CD Fig. 8. Type-curves comparisions under miscible and immsicible conditons ðC D e2S ¼ 500Þ.

including two maximum points and one minimum point are observed. Here their elapsed times and values on y-axis are marked as T 1max ; T 2max; T 1min and P 1max' ; P 2max' ; P 1min' , respectively. Further investigation from the figure can be made that T 2max and P 1max' suggest more obvious relations with C D e2s . Finally, significant linear correlation (R2 4 0:99) is demonstrated as Fig. 10 and expressions as below lgðT 2max Þ ¼ 6:248 0:962 lgðC D e2s Þ   lg p1max' ¼  0:141 þ 0:123 lgðC D e2s Þ

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K. Su et al. / Journal of Petroleum Science and Engineering 125 (2015) 128–135

  value of  feature point on both curves can be read as t D =C D ; pD and Δp; t , respectively. So, parameters can be calculated according to conventional expression as follows:
 qμB pD Reservoir permeability; K ¼ 1:842  10  3 h Δp

100 CD*exp(2s)=100

CD*exp(2s)=500

CD*exp(2s)=2000

CD*exp(2s)=5000

CD*exp(2s)=1000

dψwD/dtD

10

Wellbore storage factor; C ¼ 7:2π

1

Kh

1

μ ððt D =C D Þ=ΔtÞ

The value of C D e2s has been obtained as described above, skin factor s can be derived as   C D e2S 1 S ¼ ln 2 CD

0.1

0.01 1.E-02

1.E+02

1.E+00

1.E+04

4. Conclusions

1.E+06

tD/CD Fig. 9. Feature points under different storage factors C D e2S ðr R1D ¼ 300; r R2D ¼ 330; M 12 ¼ 2; M 13 ¼ 10; ω12 ¼ 10; ω13 ¼ 100Þ.

5 4.5

lg(T2max)

4 3.5

y = -0.9627x + 6.2481 R² = 0.9994

3 2.5 2 1

1.5

2

2.5

3

3.5

4

lg(CD*exp(2S)) 0.35 0.3

lg(p'1max)

0.25

0.2

y = 0.123x - 0.1416 R² = 0.9963

0.15 0.1

1. The CO2 dissolution in oil leads to reducing viscosity and swelling volume, and in certain conditions miscibility is achieved by multicontact. Such mechanism make fluid properties and flow pattern complex, which make composite model necessary for CO2 well testing. On the aspect of fluid composition, near-bore area is definitely saturated with pure gas and non-sweeping area is original state as oil. For intermediate region, fluid can be simplified as the mixture of CO2 and oil, of which properties depends on specific composition and miscible condition. This region obviously contributes to entire pressure behavior and greatly differs with other two regions in fluid properties, so that such region should be treated separately in model establishment. 2. During the establishment of theoretical model, well-bore storage and skin effect on pressure is considered, and the real gas is described by introducing pseudo-pressure. Then dimensionless seepage equations for each region are built analytically and solved according Laplace transformation method. BHP derivation for miscible and immiscible condition is identical via this model, but different in properties calculation of intermediate region. 3. Numerical solutions in real space can be obtained through Stehfest reversion method and then type curves are drawn in Cartesian coordinate system. From pressure derivative curve, flow process in reservoir can be divided into seven periods, corresponding seven different flow stages in three regions. Through analysis of parameters' influence on type curve shape, conclusions are made that: first extreme value increases as C D e2s grows, while M 12 =M 13 and ω12 =ω13 impact second extreme value. In addition, the elapsed times of radial flow in region-1 and region-2 are affected by both C D e2s and r R1D =r R2D . 4. Miscible flooding and immiscible flooding have different type curve patterns. When CO2 and oil achieve miscibility, pressure conduction is affected by viscosity and concentration gradient. While in immiscible flooding, fluid compressibility is higher 5

0 1.5

2

2.5

3

3.5

4

lg(CD*exp(2S)) Fig. 10. Correlation between feature points and CDe2S. (a) Correlation between T2max and C D e2S . (b) Correlation between p01max and C D e2S .

and then, the value of C D e2s can be calculated according either of them. The matching process is performed by compiling software on computer. Firstly, both of the observed and theoretical curves are depicted on one coordinate system. Secondly, move the real curve to keep horizontal part (radial flow of region-3) being of value 0:5M 13 and the slope of storage period equaling 1. Lastly, move the feature points of theoretical curve to coincide with the ones of observed curve. Then, coordinate

Viscosity of mixture, mPa.s

0.05

4 3 2 1 0

0

0.2

0.4

0.6

0.8

1

CO2 saturation Fig. 11. Mixture viscosity under different CO2 concentration (μg ¼ 0.0365 MPa s, μo ¼ 4.1 MPa s).

K. Su et al. / Journal of Petroleum Science and Engineering 125 (2015) 128–135

following:

and pressure spreads much more quickly, which lead to earlier radial flow periods in each region. 5. The analysis on feature points is helpful to well test interpretation in the three-region composite model. In pressure derivative curve, both first extreme value and elapsed time of second extreme value have obvious linear correlations with C D e2s in double logarithmic coordinate system. Combined with conventional interpretation method, well and reservoir characters are finally determined. Such new method can be generally applied for models considering well-bore storage and skin effect.

a ¼ aðT c ÞαðT r ; ωÞ;

Acknowledgment

αðT r ; ωÞ ¼ 1 þ k 1 T 0:5 r

This study was supported by National 973 Basic Research Program of China (Grant 2011CB707302) and Chinese National Major Science and Technology Projects (Grant 2011ZX05016-006 and 2011ZX05009004-001). The supports are gratefully acknowledged. Appendix A. Calculation of CO2 parameters

135

    Z 3  ð1 BÞZ 2 þ A 3B2  2B Z  AB  B2 B3 ¼ 0 and A¼

ap R2 ðT þ 273Þ2

h

b ¼ 0:0778

;

B¼

bp RðT þ 273Þ

aðT c Þ ¼ 0:45724



i2

;

R2 ðT c þ 273Þ2 pc

k ¼ 0:37464 þ 1:54226ω  0:26992ω2

RðT c þ 273Þ pc

in which, A, B are constants; a is the attraction parameter; b is the Van der Waals volume; α is the scaling factor; k is the characteristic parameter; ω is the acentric factor; T c is the critical temperature, 1C; pc is the critical pressure, MPa; T r is the pseudo-critical temperature, 1C.

A.1. Viscosity of CO2 CO2 viscosity under different pressures and temperatures is usually obtained by tables or graphs. However, such method is not convenient when applied in analytical process. So, approximate analytical expression of viscosity is more favorable. a. Under standard condition, viscosity of CO2 can be calculated as sffiffiffiffiffiffiffiffiffiffiffiffi MT μ ¼ 2:6693  10  3 2 n d Ω in which, μ is the viscosity, MPa s; M is the molecular weight, g/mol; T is the temperature, K; d is the molecular effective diameter, Å, d¼3.897 while T¼300–1000 K, and d¼3.996 while n T¼ 80–300 K; Ω is the function of characteristic temperature defined as K n ¼ KT=ε, and ε=K ¼ 2=3 when T¼300–1000 K, and ε=K ¼ 190 when T¼80–300 K. b. In supercritical condition, viscosity of CO2 is calculated as

η ¼ 0:399

e3:83ðT b =TÞ V

in which, η is the viscosity, MPa s; T b is the boiling point when p ¼0.1 MPa, for CO2, T b ¼ 194:75 K; T is the temperature, K; V is the volume of 1 g liquid, cm3. c. For mixture of CO2 and oil, viscosity can be calculated in concentration-weighted method (Fig. 11)       ln μm ¼ sCO2 ln μg þ ð1  sCO2 Þln μo in which, sCO2 is the concentration of CO2 in mixture, μg is the viscosity of gaseous CO2, and μo is the initial viscosity of crude oil. Assume that CO2 flooding is isothermal process, so temperature's impact on viscosity is neglected. A.2. Deviation factor Z of CO2 Gas deviation factor is always acquired from tables, figures and empirical formula method and equation of state (EOS) method. Similarly with viscosity, look-up in tables and figures is difficult to apply in programming. Here PR EOS method is chosen as

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