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Carbonation of wellbore cement by CO2 diffusion from caprock
International Journal of Greenhouse Gas Control 3 (2009) 731–735
Contents lists available at ScienceDirect
International Journal of Greenhouse Gas Control journal homepage: www.elsevier.com/locate/ijggc
Carbonation of wellbore cement by CO2 diffusion from caprock George W. Scherer a,b,*, Bruno Huet b a b
Princeton University, Civil & Env. Eng./PRISM, Eng. Quad. E-319, Princeton, NJ 08544, USA Schlumberger Riboud Product Center, 1 rue Becquerel, BP 202, 92142 Clamart Cedex, France
A R T I C L E I N F O
A B S T R A C T
Article history: Received 8 March 2009 Received in revised form 8 August 2009 Accepted 11 August 2009 Available online 24 September 2009
To evaluate the risk of corrosion of cement by geosequestered CO2, samples are being retrieved from wells placed in natural CO2 deposits [e.g., Crow et al., 2009]. If the cement passing through the cap rock is carbonated, it may indicate that annular gaps or cracks have allowed carbonic acid to come into contact with the cement. However, it must be recognized that the pore water in the cap rock has become saturated with CO2 over geological time. After the well is placed, the CO2 will diffuse toward the cement and react with it. A simple analysis of the diffusion kinetics demonstrates that carbonation depths of millimeters to centimeters can be expected from this reaction within the lifetime of a well, in the absence of any cracks or gaps. Therefore, the occurrence of carbonation in cement sealing natural CO2 deposits must be interpreted with caution. ß 2009 Elsevier Ltd. All rights reserved.
Keywords: Shale Diffusion Carbonation
1. Introduction Samples of cement were recovered by Crow et al. by taking sidewall cores from a 30-year-old well that had been placed into a natural CO2 deposit in Colorado (Crow et al., 2008; Crow et al., 2009). The cement cores showed a degree of carbonation that decreased with increasing distance from the boundary with the CO2 deposit. The extent of the reaction is indicated in the diagram in Fig. 1. This was interpreted to mean that a leak must have been present at the cement–formation or cement–casing interface to bring the CO2 into contact with the cement. The purpose of the present note is to determine whether the carbonation might have been caused by reaction with CO2 that was dissolved in the brine in the pores of the shale caprock. If so, then the presence of carbonation does not necessarily mean that any leak (such as an annulus or crack) was present. Given the uncertainties about the properties of the formation and the details of the geochemistry, we aim only to provide an approximate analysis that indicates the potential importance of the CO2 content of the pore fluid in the caprock. 2. Saturation of the shale All of the following data regarding the structure of the formation and the chemistry of the brine and cement come from
* Corresponding author at: Princeton University, Civil & Env. Eng./PRISM, Eng. Quad. E-319, Princeton, NJ 08544, USA. E-mail address: [email protected] (G.W. Scherer). 1750-5836/$ – see front matter ß 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijggc.2009.08.002
Crow et al. (2008, 2009), which provides a very detailed description of the procedures employed. The stratigraphy of the site is shown in Fig. 2 and the brine composition is shown in Table 1. Semiquantitative results for the distribution of crystalline phases in the cement were obtained by X-ray diffraction. The data for carbonate phases (calcite, aragonite, and vaterite) are summarized in Table 2 and Fig. 3. The carbonate content of the cement averages about 70% near the cap rock/reservoir boundary and about 30% near the Carneros/Greenhorn boundary. At the time the well was completed (1976), the reservoir contained 96% CO2 at a pressure of 10 MPa (1480 psi) and a temperature of 58 8C (1368F); water saturation was 26.7% and porosity 14.4%. From the time that the Graneros shale came into contact with the gas, the CO2 was diffusing into the pore solution. At the bottom surface (z = 0 in Fig. 4, the solution would be in equilibrium with CO2 at the formation pressure, so the mole fraction of CO2 in the water is estimated to be 0.0185 using Spycher’s EOS for the CO2/H2O system (Spycher et al., 2003); this corresponds to a CO2 concentration of about 1 mol/l. Table 1 indicates that the salinity of the pore water is low, so its salinity is neglected in evaluating the CO2 solubility in the pore water. Assuming that the porosity of the shale is 10% and that the pores are fully saturated with solution, the concentration of dissolved CO2 would be C0 0.1 mol/l of formation. At the boundary with the Greenhorn formation (z = L 40 m), we can assume that the amount of CO2 in the pore water was near zero, because that layer contains calcite that would dissolve and consume most of the carbonic acid. Indeed, Table 1 shows that the bicarbonate concentration in the Greenhorn formation is 14.07 mg/ kg 0.23 mmol/l, which is three orders of magnitude lower than
732
G.W. Scherer, B. Huet / International Journal of Greenhouse Gas Control 3 (2009) 731–735 Table 1 Brine composition of sample taken at 45450 mean depth (in the Greenhorn shale/ limestone formation). The pH was measured at depth and found to be 6.1 (data taken from Crow et al., 2008, 2009). Anion
Concentration (mg/kg)
Cation
Concentration (mg/kg)
Cl Br SO4 HCO3
403.4 4.34 118.47 14.07
Li Na NH4 K Ca Mg Fe
0.370 276.33 2.79 126.23 357.24 18.49 80.67
Specific gravity Resistivity at 778F Salinity TDS + TSS TDS
Fig. 1. ‘‘Pagoda’’ diagram showing abundances of minerals in cement at the cement– formation interface as a function of depth. The width of the column reflects mineral abundance. Actual sample locations are indicated at right-hand side. Calcite, aragonite, and vaterite are various forms of CaCO3 that result from reaction of cement with CO2 (From Crow et al. (2008, 2009)).
0.999 362 V cm 728.8 3294 mg/kg 2714 mg/kg
Table 2 Carbonate content (%) of cement cores (data taken from Crow et al., 2008, 2009). Sample
Depth
Place
Core Core Core Core Core Core Core Core Core Core Core Core Core Core Core 1.1 2.2 2.2 2.2 2.2 3.1 4.1 4.1 5.1 5.1
4458.0 4458.0 4458.0 4528.0 4528.0 4528.0 4528.0 4547.0 4547.0 4547.0 4547.0 4560.0 4560.0 4560.0 4560.0 4650.0 4682.0 4682.0 4682.0 4682.0 4701.0 4713.0 4713.0 4722.0 4722.0
fm. fm. cas. fm. cas. fm. core cas. core fm. fm. core cas. cas. core fm. fm. core fm. core cas. core fm. fm. fm. core cas. cas. core fm. fm. cas. fm. cas.
7 7 7 3 3 3 3 4 4 4 4 13 13 13 13
Calcite 8.1 7.4 7.9 10.8 18.9 43.8 35.4 7.2 17.4 10.3 46.6 9.7 21.3 27.1 48.4 58.8 60.0 52.3 35.7 62.7 19.2 13.9 15.6 58.6 38.7
Aragonite
15.5 1.0 9.9
6.9 7.4 7.0 20.3 18.4 67.4 68.6 67.9 21.9 39.8
Vaterite
Total carbonate
12.0 4.1
20.1 11.5 7.9 64.2 26.7 62.5 67.7 7.2 17.4 10.3 72.2 16.2 38.3 71.5 55.7 65.7 67.4 59.3 56.0 81.1 86.6 82.5 83.5 80.5 78.5
37.9 6.8 8.8 32.3 0.0 0.0 0.0 25.6 6.5 17.0 44.4 7.3
Semi-quantitative X-ray diffraction results for cement samples arranged by depth. Abbreviations: fm., formation side of cement core; cas., casing side of cement core; fm. core, whole core surface.
Fig. 2. Stratigraphy of the site where the samples were collected (From Crow et al. (2008, 2009)).
the concentration at the bottom of the caprock. Therefore, we can estimate the concentration, C(z, t), of dissolved CO2 in the Granerous shale by solving the diffusion equation for flow through a finite plate with concentration C(0, t) = C0 at the bottom (reservoir interface) and C(L, t) = 0 at the upper (Greenhorn) surface. A low concentration at the upper boundary would also be expected if the diffusivity of the overlying layer were much greater than that of the caprock. (The greater the CO2 concentration chosen for the upper boundary, the greater the degree of carbonation of the cement that is predicted. Therefore, the assumption that C(L, t) = 0 is the most conservative for evaluating the effect of dissolved CO2 in the pores of the caprock. For example, if the overlying layer were less permeable than the shale, then a zero-flux boundary condition would be appropriate at the upper boundary. That would result in a uniform concentration of dissolved CO2 in the shale over a period of time somewhat shorter than that found using the zeroconcentration condition. The main difference would be that the subsequent carbonation of the cement would be more uniform with depth.)
G.W. Scherer, B. Huet / International Journal of Greenhouse Gas Control 3 (2009) 731–735
Fig. 3. Total carbonate phases (calcite + aragonite + vaterite) detected in cement cores by XRD. Data from Table 1 (originally from Crow et al., 2008,2009).
733
Fig. 5. Distribution of dissolved CO2 in Graneros shale versus time. A linear concentration profile develops by the time Dt/L2 0.3; here D = diffusivity in caprock, L = thickness of caprock.
5) 1011 m2/s on a Muderong shale (permeability 1 nanodarcy, porosity 20%). Schloemer and Krooss (1997) found methane diffusion coefficients in shale to fall in the range from 1.4 1011 to 4.5 1010 m2/s. Taking a conservative estimate of D 1011 m2/s, and given L 40 m, the time to achieve a linear concentration drop through the Graneros shale would be about (0.3)(402)/(1011) 4.8 1013 s 1.6 million years. Since this natural deposit certainly existed for a longer time than this, we can assume that the CO2 concentration varied linearly through the Graneros shale at the time when the well was completed. The time to saturate the shale will be longer, if the stone contains components that react with the CO2. In that case, we would replace Eq. (1) with
@C @2 C @S ¼D 2 @t @t @x
(3)
where S is the concentration of immobilized solute. The latter is controlled by a reversible reaction described by Fig. 4. (a) The cap rock over the CO2 deposit is a layer Graneros shale about 40 m thick that is overlain by the Greenhorn limestone/shale layer. Over geological time, the CO2 diffuses into the pore solution of the shale; at the Greenhorn boundary, the CO2 is expected to react with the limestone, so the amount of dissolved CO2 will be low at the upper boundary of the Graneros shale. (b) After the well is installed, dissolved CO2 diffuses from the shale toward the cement, where it reacts.
In this case, the diffusion equation is
@C @2 C ¼D 2 @t @x
(1)
where D is the diffusion coefficient for CO2 in the shale. Given the initial condition C(z, 0) = 0, the solution is 1 h Cðz; tÞ z 2X ð1Þn z i Dt exp n2 p2 2 sin np 1 (2) ¼1 þ C0 L p n¼1 n L L As shown in Fig. 5, the concentration evolves into a linear distribution by the time that Dt/L2 0.3. To evaluate this result, we need a value for the rate of diffusion in the Granerous shale, which has not been measured, so we must take estimates from the literature. An abstract by Busch et al. (2007) indicates a range from 109 to 1011 m2/s for shales, and a subsequent paper by the same group (Busch et al., 2008) measured a value of D = (3–
@S ¼ lC mS @t
(4)
where l and m are parameters controlling the rates of the forward and backward reactions. The solution to this equation is given in Appendix 1, where it is shown that the final concentration distribution is still linear, and the time to reach equilibrium is increased by a factor of 1 + l/m. The equilibrium concentration of immobilized carbonate is S(u, 1)/C(u, 1) = l/m, so even if that ratio is unity the time to equilibrate the shale is only doubled. Therefore, we can assume that the shale layer is saturated with carbonate and has a linear distribution of dissolved CO2 in the pore liquid. Of course, the linearity of the final distribution depends on homogeneity of the formation, which allows the diffusivity and the concentration of immobilization sites to be constant in space. The irregular distribution of carbonation that is evident in Fig. 3 may reflect the presence of heterogeneities in the caprock that are not taken into account in our treatment. 3. Diffusion from the shale to the cement After the well was completed, the CO2 would begin to diffuse from the surrounding shale, leading to carbonation of the cement. Our goal is to estimate the amount of carbonation that could have
G.W. Scherer, B. Huet / International Journal of Greenhouse Gas Control 3 (2009) 731–735
734
occurred during the 30 years of contact between the shale and the cement, assuming that no annulus was present. We will see that the CO2 would be depleted from a zone of tens of cm around the well. Since the vertical gradient in concentration ranges from C0 to 0 over 40 m, the radial gradient is orders of magnitude steeper. Therefore, we are justified in ignoring vertical diffusion, and approximating the diffusion from the shale to the cement as being purely radial. The diffusion equation in this case is ! @C @2 C 1 @C ¼D þ (5) @t @r 2 r @r
Table 3 Constants in Eq. (12). n
an
1 2 3 4 5 6 7
7.0898 3.1416 0.5908 0.3927 0.4471 0.1992 0.02819
where r is the radial coordinate. The initial condition is C(r, z, 0) = C0(1 z/L) and the boundary conditions are C(r0, z, t) = CI and C(1, z, t) = C0(1 z/L), where r = r0 is the surface of the cement and CI is the concentration of CO2 in the brine in contact with the cement. The solution to this problem is given by Crank (1975): Cðr; z; tÞ C 0 ð1 z=LÞ C I C 0 ð1 z=LÞ " ! # Z 2 1 Dt J ðyr=r 0 ÞY 0 ðyÞ J 0 ðyÞY 0 ðyr=r 0 Þ dy (6) 1þ exp 2 y2 0 p 0 r0 y J02 ðyÞ Y02 ðyÞ where J0 and Y0 are the Bessel functions of order zero of the first and second kinds, respectively. The form appropriate for early times (which is the present case) is rffiffiffiffiffi Cðr; z; tÞ C 0 ð1 z=LÞ r0 r r0 erfc pffiffiffiffiffiffi þ (7) C I C 0 ð1 z=LÞ r 2 Dt where the ellipsis indicates higher order terms. The flux, J, of CO2 to the cement is given by @C Jðz; tÞ ¼ 2pr 0 D (8) @r r¼r0 and the amount of CO2 that has been transported to the surface of the cement by time t is Z t Mðz; tÞ ¼ Jðz; tÞ dt (9) 0
Fig. 6. M is the amount of CO2 crossing the surface of the cement at r = r0 where the surface concentration is Ci; f(z) = 1 z/L is the concentration distribution of CO2 in the shale at the time when the well is installed. Here, r0 = outer radius of cement; D = diffusivity in caprock; L = thickness of caprock.
is found from a material balance:
pðr02 rC2 ÞC Ca ¼ Mðz; tÞ or (assuming that Ci 0) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C0 z 1 gðtÞ rC r0 1 L pC Ca
(14)
(15)
Using Eqs. (6) and (8) in Eq. (9), the mass of CO2 entering the cement is found to be z C I gðtÞ (10) Mðz; tÞ r02 C 0 1 L
The depth of carbonation is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! C0 z r0 rC r0 1 1 1 gðtÞ L pC Ca
where
For times greater than Dt=r02 1, it is necessary to use the exact solution given by Crank, which involves an integral that must be evaluated numerically. That solution was used to calculate the depth of the carbonated layer, which is shown in Fig. 7. Over a period of 30 years, if the diffusivity is D = 1011 m2/s, the depth of carbonation will be about 0.6 cm at the bottom of the shale, and decrease linearly toward the top of the shale layer; if D = 1010 m2/s, the depth of carbonation will be 5 times larger (3.2 cm at the bottom). Thus, it is quite feasible to find centimeters of carbonated cement as a result of diffusion from the shale, even if there are no cracks or annuli. However, one should keep in mind the assumptions of the proposed scenario for carbonation: (1) carbonate phases form first at the cement/formation interface; (2) the carbonate layer grows towards the casing/cement interface; (3) calcium silicate hydrate is completely carbonated in the carbonate layer (as required by the assumption of local equilibrium). A more detailed reactive transport model would be required to predict the spatial distribution of carbonation in the cement, but the field data do not presently include that level of detail for comparison. In particular, Crow et al. do not report the radial distribution of carbonate phases or radial degree of carbonation of calcium silicate hydrates.
gðtÞ ¼
8
p
Z
1
(
"
1 exp
0
!
Dt 2 y r02
#)
dy y3 J02 ðyÞ Y02 ðyÞ
(11)
An approximation that is valid up to Dt=r02 10 is 7 X Dt an 2 gðtÞ r0 n¼1
!n=2 (12)
where the constants are given in Table 3. This result is shown in Fig. 6. The diameter of the casing was 7 in., so the outer radius of the cement sheath would be around r0 = 10 cm. Given D 1011 m2/s and t = 30 years 109 s, Dt=r02 1, so z CI (13) Mðz; 30 yearsÞ 9:65r02 C 0 1 L To estimate the depth of carbonation, we suppose that the amount of CO2 entering the cement reacts with the calcium-bearing phases of the cement matrix. For a 50/50 blend of fly ash and Portland cement, the concentration of calcium is about CCa = 2.58 mol/l. The carbonation penetrates to a depth where the radius is rC(t), which
(16)
G.W. Scherer, B. Huet / International Journal of Greenhouse Gas Control 3 (2009) 731–735
735
Using Eq. (18) to eliminate S from Eq. (17), we obtain a differential equation whose solution is ˜ sÞ ¼ Cðu;
C 0 sinh½kð1 uÞ sinh½k s
(20)
where L = ltd, M = mtd, and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sðs þ L þ MÞ k¼ sþM
(21)
The final concentration distribution can be found directly from the transformed solution: Cðu; 1Þ ¼ lim Cðu; u Þ ¼ lim sC˜ ¼ C 0 ð1 uÞ t!1
Fig. 7. Thickness of carbonated layer, r0 rC (in cm), as a function of normalized height above the reservoir, z, and dimensionless time; in this simulation, r0 = 10 cm is the outer radius of the cement; rC = inner radius of carbonated zone; L = thickness of cap rock; D = diffusivity of cap rock.
s!0
(22)
Thus, the distribution of dissolved CO2 is linear, even when the reaction is occurring. Similarly,
LC 0 lC 0 ð1 uÞ ¼ ð1 uÞ Sðu; 1Þ ¼ lim sS˜ ¼ s!0 M m
(23)
4. Conclusions
so the immobilized solute is also linearly distributed through the shale layer.
To evaluate the risk of corrosion of abandoned wells by carbonic acid, samples are being retrieved from wells placed in natural CO2 deposits. If the cement is found to be carbonated, it could be interpreted to mean that annular gaps or cracks are present. The purpose of the present analysis is to evaluate the amount of carbonation that might be expected, simply owing to the CO2 dissolved in the pore water of the surrounding formation. Over geological time, the caprock may become saturated with carbonic acid, which then diffuses toward the well and reacts with the cement. A simple analysis is presented which shows that carbonation depths of centimeters can be expected, even in the absence of cracks or annuli. Therefore, carbonation of cement retrieved from wells in natural CO2 deposits must be interpreted with caution.
Eq. (20) can only be inverted numerically, but a very useful approximation is obtained for long times, by considering the case where s is small so that
This work was supported by the Carbon Mitigation Initiative (CMI) project at Princeton University, sponsored by BP and Ford Motor Company.
The solution in the case where the CO2 reacts with the shale is found using the Laplace transform. Introducing the reduced variables u = z/L, td = L2/Dc, u = Dct/L2 and taking the transform with respect to u, Eqs. (3) and (4) become (17)
and sS˜ ¼ LC˜ M S˜
(18)
where s is the transform parameter (Hildebrand, 1962). The transformed boundary conditions are ˜ sÞ ¼ C 0 =s Cð0; ˜ sÞ ¼ 0 Cð1;
(24)
In this case, the transform can be inverted using the method discussed in Crank (1975), with the result: 2 2 1 X Cðu; tÞ 2ð1Þn sin½npð1 uÞ n p t t 1uþ exp > 0:3 ; C0 n p t t n¼1 (25)
t¼
L2 l 1þ D m
(26)
Thus, the chemical reaction delays equilibration by the factor S(u, 1)/C(u, 1) = l/m. References
Appendix A. Reversible carbonation of shale
@2 C˜ sS˜ @u2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sð1 þ L=MÞ
where the relaxation time is
Acknowledgments
sC˜ ¼
k
(19)
Busch, A., Alles, S., Krooss, B.M., Dewhurst, D., 2007. Potential of caprocks as CO2 storage reservoirs. Geophysical Research Abstracts 9, 06734 (SRef-ID: 16077962/gra/EGU2007-A-06734 (European Geosciences Union 2007)). Busch, A., Alles, S., Gensterblum, Y., Prinz, D., Dewhurst, D.N., Raven, M.D., Stanjek, H., Krooss, B.M., 2008. Carbon dioxide storage potential of shales. International Journal of Greenhouse Gas Control 2, 297–308. Crank, J., 1975. The Mathematics of Diffusion, 2nd ed. Clarendon, Oxford, p. 87. Crow, W., Williams, D.W., Carey, J.W., Celia, M., Gasda, S., 2008. CO2 capture project field study of a wellbore from a natural CO2 reservoir. In: Carbon Capture and Sequestration Conference, May 5–8. Crow, W., Williams, D.W., Carey, J.W., Celia, M., Gasda, S., 2009. Wellbore integrity analysis of a natural CO2 producer. International Journal of Greenhouse Gas Control, under review. Hildebrand, F.B., 1962. Advanced Calculus for Applications. Prentice-Hall, Englewood Cliffs, NJ. Schloemer, S., Krooss, B.M., 1997. Experimental characterisation of the hydrocarbon sealing efficiency of cap rocks. Marine and Petroleum Geology 14 (5), 565–580. Spycher, N., Pruess, K., Ennis-King, J., 2003. CO2–H2O mixtures in the geological sequestration of CO2. I. Assessment and calculation of mutual solubilities from 12 to 100 8C and up to 600 bar. Geochimica et Cosmochimica Acta 67, 3015– 3031.
Contents lists available at ScienceDirect
International Journal of Greenhouse Gas Control journal homepage: www.elsevier.com/locate/ijggc
Carbonation of wellbore cement by CO2 diffusion from caprock George W. Scherer a,b,*, Bruno Huet b a b
Princeton University, Civil & Env. Eng./PRISM, Eng. Quad. E-319, Princeton, NJ 08544, USA Schlumberger Riboud Product Center, 1 rue Becquerel, BP 202, 92142 Clamart Cedex, France
A R T I C L E I N F O
A B S T R A C T
Article history: Received 8 March 2009 Received in revised form 8 August 2009 Accepted 11 August 2009 Available online 24 September 2009
To evaluate the risk of corrosion of cement by geosequestered CO2, samples are being retrieved from wells placed in natural CO2 deposits [e.g., Crow et al., 2009]. If the cement passing through the cap rock is carbonated, it may indicate that annular gaps or cracks have allowed carbonic acid to come into contact with the cement. However, it must be recognized that the pore water in the cap rock has become saturated with CO2 over geological time. After the well is placed, the CO2 will diffuse toward the cement and react with it. A simple analysis of the diffusion kinetics demonstrates that carbonation depths of millimeters to centimeters can be expected from this reaction within the lifetime of a well, in the absence of any cracks or gaps. Therefore, the occurrence of carbonation in cement sealing natural CO2 deposits must be interpreted with caution. ß 2009 Elsevier Ltd. All rights reserved.
Keywords: Shale Diffusion Carbonation
1. Introduction Samples of cement were recovered by Crow et al. by taking sidewall cores from a 30-year-old well that had been placed into a natural CO2 deposit in Colorado (Crow et al., 2008; Crow et al., 2009). The cement cores showed a degree of carbonation that decreased with increasing distance from the boundary with the CO2 deposit. The extent of the reaction is indicated in the diagram in Fig. 1. This was interpreted to mean that a leak must have been present at the cement–formation or cement–casing interface to bring the CO2 into contact with the cement. The purpose of the present note is to determine whether the carbonation might have been caused by reaction with CO2 that was dissolved in the brine in the pores of the shale caprock. If so, then the presence of carbonation does not necessarily mean that any leak (such as an annulus or crack) was present. Given the uncertainties about the properties of the formation and the details of the geochemistry, we aim only to provide an approximate analysis that indicates the potential importance of the CO2 content of the pore fluid in the caprock. 2. Saturation of the shale All of the following data regarding the structure of the formation and the chemistry of the brine and cement come from
* Corresponding author at: Princeton University, Civil & Env. Eng./PRISM, Eng. Quad. E-319, Princeton, NJ 08544, USA. E-mail address: [email protected] (G.W. Scherer). 1750-5836/$ – see front matter ß 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijggc.2009.08.002
Crow et al. (2008, 2009), which provides a very detailed description of the procedures employed. The stratigraphy of the site is shown in Fig. 2 and the brine composition is shown in Table 1. Semiquantitative results for the distribution of crystalline phases in the cement were obtained by X-ray diffraction. The data for carbonate phases (calcite, aragonite, and vaterite) are summarized in Table 2 and Fig. 3. The carbonate content of the cement averages about 70% near the cap rock/reservoir boundary and about 30% near the Carneros/Greenhorn boundary. At the time the well was completed (1976), the reservoir contained 96% CO2 at a pressure of 10 MPa (1480 psi) and a temperature of 58 8C (1368F); water saturation was 26.7% and porosity 14.4%. From the time that the Graneros shale came into contact with the gas, the CO2 was diffusing into the pore solution. At the bottom surface (z = 0 in Fig. 4, the solution would be in equilibrium with CO2 at the formation pressure, so the mole fraction of CO2 in the water is estimated to be 0.0185 using Spycher’s EOS for the CO2/H2O system (Spycher et al., 2003); this corresponds to a CO2 concentration of about 1 mol/l. Table 1 indicates that the salinity of the pore water is low, so its salinity is neglected in evaluating the CO2 solubility in the pore water. Assuming that the porosity of the shale is 10% and that the pores are fully saturated with solution, the concentration of dissolved CO2 would be C0 0.1 mol/l of formation. At the boundary with the Greenhorn formation (z = L 40 m), we can assume that the amount of CO2 in the pore water was near zero, because that layer contains calcite that would dissolve and consume most of the carbonic acid. Indeed, Table 1 shows that the bicarbonate concentration in the Greenhorn formation is 14.07 mg/ kg 0.23 mmol/l, which is three orders of magnitude lower than
732
G.W. Scherer, B. Huet / International Journal of Greenhouse Gas Control 3 (2009) 731–735 Table 1 Brine composition of sample taken at 45450 mean depth (in the Greenhorn shale/ limestone formation). The pH was measured at depth and found to be 6.1 (data taken from Crow et al., 2008, 2009). Anion
Concentration (mg/kg)
Cation
Concentration (mg/kg)
Cl Br SO4 HCO3
403.4 4.34 118.47 14.07
Li Na NH4 K Ca Mg Fe
0.370 276.33 2.79 126.23 357.24 18.49 80.67
Specific gravity Resistivity at 778F Salinity TDS + TSS TDS
Fig. 1. ‘‘Pagoda’’ diagram showing abundances of minerals in cement at the cement– formation interface as a function of depth. The width of the column reflects mineral abundance. Actual sample locations are indicated at right-hand side. Calcite, aragonite, and vaterite are various forms of CaCO3 that result from reaction of cement with CO2 (From Crow et al. (2008, 2009)).
0.999 362 V cm 728.8 3294 mg/kg 2714 mg/kg
Table 2 Carbonate content (%) of cement cores (data taken from Crow et al., 2008, 2009). Sample
Depth
Place
Core Core Core Core Core Core Core Core Core Core Core Core Core Core Core 1.1 2.2 2.2 2.2 2.2 3.1 4.1 4.1 5.1 5.1
4458.0 4458.0 4458.0 4528.0 4528.0 4528.0 4528.0 4547.0 4547.0 4547.0 4547.0 4560.0 4560.0 4560.0 4560.0 4650.0 4682.0 4682.0 4682.0 4682.0 4701.0 4713.0 4713.0 4722.0 4722.0
fm. fm. cas. fm. cas. fm. core cas. core fm. fm. core cas. cas. core fm. fm. core fm. core cas. core fm. fm. fm. core cas. cas. core fm. fm. cas. fm. cas.
7 7 7 3 3 3 3 4 4 4 4 13 13 13 13
Calcite 8.1 7.4 7.9 10.8 18.9 43.8 35.4 7.2 17.4 10.3 46.6 9.7 21.3 27.1 48.4 58.8 60.0 52.3 35.7 62.7 19.2 13.9 15.6 58.6 38.7
Aragonite
15.5 1.0 9.9
6.9 7.4 7.0 20.3 18.4 67.4 68.6 67.9 21.9 39.8
Vaterite
Total carbonate
12.0 4.1
20.1 11.5 7.9 64.2 26.7 62.5 67.7 7.2 17.4 10.3 72.2 16.2 38.3 71.5 55.7 65.7 67.4 59.3 56.0 81.1 86.6 82.5 83.5 80.5 78.5
37.9 6.8 8.8 32.3 0.0 0.0 0.0 25.6 6.5 17.0 44.4 7.3
Semi-quantitative X-ray diffraction results for cement samples arranged by depth. Abbreviations: fm., formation side of cement core; cas., casing side of cement core; fm. core, whole core surface.
Fig. 2. Stratigraphy of the site where the samples were collected (From Crow et al. (2008, 2009)).
the concentration at the bottom of the caprock. Therefore, we can estimate the concentration, C(z, t), of dissolved CO2 in the Granerous shale by solving the diffusion equation for flow through a finite plate with concentration C(0, t) = C0 at the bottom (reservoir interface) and C(L, t) = 0 at the upper (Greenhorn) surface. A low concentration at the upper boundary would also be expected if the diffusivity of the overlying layer were much greater than that of the caprock. (The greater the CO2 concentration chosen for the upper boundary, the greater the degree of carbonation of the cement that is predicted. Therefore, the assumption that C(L, t) = 0 is the most conservative for evaluating the effect of dissolved CO2 in the pores of the caprock. For example, if the overlying layer were less permeable than the shale, then a zero-flux boundary condition would be appropriate at the upper boundary. That would result in a uniform concentration of dissolved CO2 in the shale over a period of time somewhat shorter than that found using the zeroconcentration condition. The main difference would be that the subsequent carbonation of the cement would be more uniform with depth.)
G.W. Scherer, B. Huet / International Journal of Greenhouse Gas Control 3 (2009) 731–735
Fig. 3. Total carbonate phases (calcite + aragonite + vaterite) detected in cement cores by XRD. Data from Table 1 (originally from Crow et al., 2008,2009).
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Fig. 5. Distribution of dissolved CO2 in Graneros shale versus time. A linear concentration profile develops by the time Dt/L2 0.3; here D = diffusivity in caprock, L = thickness of caprock.
5) 1011 m2/s on a Muderong shale (permeability 1 nanodarcy, porosity 20%). Schloemer and Krooss (1997) found methane diffusion coefficients in shale to fall in the range from 1.4 1011 to 4.5 1010 m2/s. Taking a conservative estimate of D 1011 m2/s, and given L 40 m, the time to achieve a linear concentration drop through the Graneros shale would be about (0.3)(402)/(1011) 4.8 1013 s 1.6 million years. Since this natural deposit certainly existed for a longer time than this, we can assume that the CO2 concentration varied linearly through the Graneros shale at the time when the well was completed. The time to saturate the shale will be longer, if the stone contains components that react with the CO2. In that case, we would replace Eq. (1) with
@C @2 C @S ¼D 2 @t @t @x
(3)
where S is the concentration of immobilized solute. The latter is controlled by a reversible reaction described by Fig. 4. (a) The cap rock over the CO2 deposit is a layer Graneros shale about 40 m thick that is overlain by the Greenhorn limestone/shale layer. Over geological time, the CO2 diffuses into the pore solution of the shale; at the Greenhorn boundary, the CO2 is expected to react with the limestone, so the amount of dissolved CO2 will be low at the upper boundary of the Graneros shale. (b) After the well is installed, dissolved CO2 diffuses from the shale toward the cement, where it reacts.
In this case, the diffusion equation is
@C @2 C ¼D 2 @t @x
(1)
where D is the diffusion coefficient for CO2 in the shale. Given the initial condition C(z, 0) = 0, the solution is 1 h Cðz; tÞ z 2X ð1Þn z i Dt exp n2 p2 2 sin np 1 (2) ¼1 þ C0 L p n¼1 n L L As shown in Fig. 5, the concentration evolves into a linear distribution by the time that Dt/L2 0.3. To evaluate this result, we need a value for the rate of diffusion in the Granerous shale, which has not been measured, so we must take estimates from the literature. An abstract by Busch et al. (2007) indicates a range from 109 to 1011 m2/s for shales, and a subsequent paper by the same group (Busch et al., 2008) measured a value of D = (3–
@S ¼ lC mS @t
(4)
where l and m are parameters controlling the rates of the forward and backward reactions. The solution to this equation is given in Appendix 1, where it is shown that the final concentration distribution is still linear, and the time to reach equilibrium is increased by a factor of 1 + l/m. The equilibrium concentration of immobilized carbonate is S(u, 1)/C(u, 1) = l/m, so even if that ratio is unity the time to equilibrate the shale is only doubled. Therefore, we can assume that the shale layer is saturated with carbonate and has a linear distribution of dissolved CO2 in the pore liquid. Of course, the linearity of the final distribution depends on homogeneity of the formation, which allows the diffusivity and the concentration of immobilization sites to be constant in space. The irregular distribution of carbonation that is evident in Fig. 3 may reflect the presence of heterogeneities in the caprock that are not taken into account in our treatment. 3. Diffusion from the shale to the cement After the well was completed, the CO2 would begin to diffuse from the surrounding shale, leading to carbonation of the cement. Our goal is to estimate the amount of carbonation that could have
G.W. Scherer, B. Huet / International Journal of Greenhouse Gas Control 3 (2009) 731–735
734
occurred during the 30 years of contact between the shale and the cement, assuming that no annulus was present. We will see that the CO2 would be depleted from a zone of tens of cm around the well. Since the vertical gradient in concentration ranges from C0 to 0 over 40 m, the radial gradient is orders of magnitude steeper. Therefore, we are justified in ignoring vertical diffusion, and approximating the diffusion from the shale to the cement as being purely radial. The diffusion equation in this case is ! @C @2 C 1 @C ¼D þ (5) @t @r 2 r @r
Table 3 Constants in Eq. (12). n
an
1 2 3 4 5 6 7
7.0898 3.1416 0.5908 0.3927 0.4471 0.1992 0.02819
where r is the radial coordinate. The initial condition is C(r, z, 0) = C0(1 z/L) and the boundary conditions are C(r0, z, t) = CI and C(1, z, t) = C0(1 z/L), where r = r0 is the surface of the cement and CI is the concentration of CO2 in the brine in contact with the cement. The solution to this problem is given by Crank (1975): Cðr; z; tÞ C 0 ð1 z=LÞ C I C 0 ð1 z=LÞ " ! # Z 2 1 Dt J ðyr=r 0 ÞY 0 ðyÞ J 0 ðyÞY 0 ðyr=r 0 Þ dy (6) 1þ exp 2 y2 0 p 0 r0 y J02 ðyÞ Y02 ðyÞ where J0 and Y0 are the Bessel functions of order zero of the first and second kinds, respectively. The form appropriate for early times (which is the present case) is rffiffiffiffiffi Cðr; z; tÞ C 0 ð1 z=LÞ r0 r r0 erfc pffiffiffiffiffiffi þ (7) C I C 0 ð1 z=LÞ r 2 Dt where the ellipsis indicates higher order terms. The flux, J, of CO2 to the cement is given by @C Jðz; tÞ ¼ 2pr 0 D (8) @r r¼r0 and the amount of CO2 that has been transported to the surface of the cement by time t is Z t Mðz; tÞ ¼ Jðz; tÞ dt (9) 0
Fig. 6. M is the amount of CO2 crossing the surface of the cement at r = r0 where the surface concentration is Ci; f(z) = 1 z/L is the concentration distribution of CO2 in the shale at the time when the well is installed. Here, r0 = outer radius of cement; D = diffusivity in caprock; L = thickness of caprock.
is found from a material balance:
pðr02 rC2 ÞC Ca ¼ Mðz; tÞ or (assuming that Ci 0) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C0 z 1 gðtÞ rC r0 1 L pC Ca
(14)
(15)
Using Eqs. (6) and (8) in Eq. (9), the mass of CO2 entering the cement is found to be z C I gðtÞ (10) Mðz; tÞ r02 C 0 1 L
The depth of carbonation is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! C0 z r0 rC r0 1 1 1 gðtÞ L pC Ca
where
For times greater than Dt=r02 1, it is necessary to use the exact solution given by Crank, which involves an integral that must be evaluated numerically. That solution was used to calculate the depth of the carbonated layer, which is shown in Fig. 7. Over a period of 30 years, if the diffusivity is D = 1011 m2/s, the depth of carbonation will be about 0.6 cm at the bottom of the shale, and decrease linearly toward the top of the shale layer; if D = 1010 m2/s, the depth of carbonation will be 5 times larger (3.2 cm at the bottom). Thus, it is quite feasible to find centimeters of carbonated cement as a result of diffusion from the shale, even if there are no cracks or annuli. However, one should keep in mind the assumptions of the proposed scenario for carbonation: (1) carbonate phases form first at the cement/formation interface; (2) the carbonate layer grows towards the casing/cement interface; (3) calcium silicate hydrate is completely carbonated in the carbonate layer (as required by the assumption of local equilibrium). A more detailed reactive transport model would be required to predict the spatial distribution of carbonation in the cement, but the field data do not presently include that level of detail for comparison. In particular, Crow et al. do not report the radial distribution of carbonate phases or radial degree of carbonation of calcium silicate hydrates.
gðtÞ ¼
8
p
Z
1
(
"
1 exp
0
!
Dt 2 y r02
#)
dy y3 J02 ðyÞ Y02 ðyÞ
(11)
An approximation that is valid up to Dt=r02 10 is 7 X Dt an 2 gðtÞ r0 n¼1
!n=2 (12)
where the constants are given in Table 3. This result is shown in Fig. 6. The diameter of the casing was 7 in., so the outer radius of the cement sheath would be around r0 = 10 cm. Given D 1011 m2/s and t = 30 years 109 s, Dt=r02 1, so z CI (13) Mðz; 30 yearsÞ 9:65r02 C 0 1 L To estimate the depth of carbonation, we suppose that the amount of CO2 entering the cement reacts with the calcium-bearing phases of the cement matrix. For a 50/50 blend of fly ash and Portland cement, the concentration of calcium is about CCa = 2.58 mol/l. The carbonation penetrates to a depth where the radius is rC(t), which
(16)
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735
Using Eq. (18) to eliminate S from Eq. (17), we obtain a differential equation whose solution is ˜ sÞ ¼ Cðu;
C 0 sinh½kð1 uÞ sinh½k s
(20)
where L = ltd, M = mtd, and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sðs þ L þ MÞ k¼ sþM
(21)
The final concentration distribution can be found directly from the transformed solution: Cðu; 1Þ ¼ lim Cðu; u Þ ¼ lim sC˜ ¼ C 0 ð1 uÞ t!1
Fig. 7. Thickness of carbonated layer, r0 rC (in cm), as a function of normalized height above the reservoir, z, and dimensionless time; in this simulation, r0 = 10 cm is the outer radius of the cement; rC = inner radius of carbonated zone; L = thickness of cap rock; D = diffusivity of cap rock.
s!0
(22)
Thus, the distribution of dissolved CO2 is linear, even when the reaction is occurring. Similarly,
LC 0 lC 0 ð1 uÞ ¼ ð1 uÞ Sðu; 1Þ ¼ lim sS˜ ¼ s!0 M m
(23)
4. Conclusions
so the immobilized solute is also linearly distributed through the shale layer.
To evaluate the risk of corrosion of abandoned wells by carbonic acid, samples are being retrieved from wells placed in natural CO2 deposits. If the cement is found to be carbonated, it could be interpreted to mean that annular gaps or cracks are present. The purpose of the present analysis is to evaluate the amount of carbonation that might be expected, simply owing to the CO2 dissolved in the pore water of the surrounding formation. Over geological time, the caprock may become saturated with carbonic acid, which then diffuses toward the well and reacts with the cement. A simple analysis is presented which shows that carbonation depths of centimeters can be expected, even in the absence of cracks or annuli. Therefore, carbonation of cement retrieved from wells in natural CO2 deposits must be interpreted with caution.
Eq. (20) can only be inverted numerically, but a very useful approximation is obtained for long times, by considering the case where s is small so that
This work was supported by the Carbon Mitigation Initiative (CMI) project at Princeton University, sponsored by BP and Ford Motor Company.
The solution in the case where the CO2 reacts with the shale is found using the Laplace transform. Introducing the reduced variables u = z/L, td = L2/Dc, u = Dct/L2 and taking the transform with respect to u, Eqs. (3) and (4) become (17)
and sS˜ ¼ LC˜ M S˜
(18)
where s is the transform parameter (Hildebrand, 1962). The transformed boundary conditions are ˜ sÞ ¼ C 0 =s Cð0; ˜ sÞ ¼ 0 Cð1;
(24)
In this case, the transform can be inverted using the method discussed in Crank (1975), with the result: 2 2 1 X Cðu; tÞ 2ð1Þn sin½npð1 uÞ n p t t 1uþ exp > 0:3 ; C0 n p t t n¼1 (25)
t¼
L2 l 1þ D m
(26)
Thus, the chemical reaction delays equilibration by the factor S(u, 1)/C(u, 1) = l/m. References
Appendix A. Reversible carbonation of shale
@2 C˜ sS˜ @u2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sð1 þ L=MÞ
where the relaxation time is
Acknowledgments
sC˜ ¼
k
(19)
Busch, A., Alles, S., Krooss, B.M., Dewhurst, D., 2007. Potential of caprocks as CO2 storage reservoirs. Geophysical Research Abstracts 9, 06734 (SRef-ID: 16077962/gra/EGU2007-A-06734 (European Geosciences Union 2007)). Busch, A., Alles, S., Gensterblum, Y., Prinz, D., Dewhurst, D.N., Raven, M.D., Stanjek, H., Krooss, B.M., 2008. Carbon dioxide storage potential of shales. International Journal of Greenhouse Gas Control 2, 297–308. Crank, J., 1975. The Mathematics of Diffusion, 2nd ed. Clarendon, Oxford, p. 87. Crow, W., Williams, D.W., Carey, J.W., Celia, M., Gasda, S., 2008. CO2 capture project field study of a wellbore from a natural CO2 reservoir. In: Carbon Capture and Sequestration Conference, May 5–8. Crow, W., Williams, D.W., Carey, J.W., Celia, M., Gasda, S., 2009. Wellbore integrity analysis of a natural CO2 producer. International Journal of Greenhouse Gas Control, under review. Hildebrand, F.B., 1962. Advanced Calculus for Applications. Prentice-Hall, Englewood Cliffs, NJ. Schloemer, S., Krooss, B.M., 1997. Experimental characterisation of the hydrocarbon sealing efficiency of cap rocks. Marine and Petroleum Geology 14 (5), 565–580. Spycher, N., Pruess, K., Ennis-King, J., 2003. CO2–H2O mixtures in the geological sequestration of CO2. I. Assessment and calculation of mutual solubilities from 12 to 100 8C and up to 600 bar. Geochimica et Cosmochimica Acta 67, 3015– 3031.