ACCRETE - An explicit algorithm to solve kinetically constrained CO 2 -water-rock interactions

Transcription

1 Proceedngs of the 3rd WSEAS Internatonal Conference on Mathematcal Bology and Ecology, Gold Coast, Queensland, Australa, January 17-19, ACCRETE - An explct algorthm to solve knetcally constraned CO 2 -water-rock nteractons Abstract: HELGE HELLEVANG Department of Earth Scence Unversty of Bergen Allégaten 41, 5007 Bergen NORWAY BJØRN KVAMME Department of Physcs and Technology Unversty of Bergen Allégaten 55, 5007 Bergen NORWAY Increasng atmospherc concentratons of CO 2 are suspected to have caused a gradual warmng of the Earths surface. Underground storage of CO 2 n porous rocks or sedments s one feasble opton for reducng emssons. As CO 2 s njected nto the pore space, t dssolves n the reservor pore waters causng an acd corrosve envronment. Ths leads to dssoluton of the rock or sedment framework and precptaton of new mneral phases. The resultant carbonate mnerals are beleved to be a safe storage host over geologcal tmescales. To understand the magntude of these reactons at long tme scales we need to perform numercal smulatons. 3D reactve transport smulatons are most approprate gven the dynamcs and scales of njecton. Ths nvolves solvng a reacton term that commonly conssts of a stff system of ordnary dfferental equatons (ODE). Such systems can be solved wth mplct schemes lke a backward dfferental formulaton (BDF). Ths paper reports an alternatve explct algorthm to solve knetcally constraned mneral reactons n CO 2 -water-rock systems. The algorthm s used on a comprehensve mneral system that s representatve for a Utsra-lke reservor. Smulatons suggest that alumnum at low concentratons causes numercal nstabltes. By ntegratng the mneral reactons over short tme steps we are able to fnd an ndependent functon or constant for alumnum that facltate the calculatons and stll produces relable results. Usng ths method, a 1000 years sngle-grd smulaton can be completed n a few seconds or less. Ths demonstrates that t s possble to solve the stff ODE systems nvolved n the reactve transport equaton usng a smple explct algorthm. Key-words: CO2 storage, Utsra reservor, Slepner njecton ste, mneral knetcs, ACCRETE, stffness, ODE

2 Proceedngs of the 3rd WSEAS Internatonal Conference on Mathematcal Bology and Ecology, Gold Coast, Queensland, Australa, January 17-19, Introducton Measurements of atmospherc CO 2 concentraton durng the last decades have revealed a rapd ncrease, and dfferent strateges have been suggested of how to reduce such emssons. Examples are storage n salne aqufers, storage n abandoned ol- and gas felds, storage n deep unmnable coal seams, and storage combned wth enhanced recovery of hydrocarbon from actve ol and gas felds [1]. The probablty of leakage from a storage ste s dependent on several factors lke hydrodynamc condtons, numbers of wells penetratng the reservor, storage depth, and reservor ntegrty [2]. The probablty s also tme dependent n the sense that CO 2 s mmoblzed wth tme by dssoluton nto formaton waters and by reactons wth the sold rock or sedment framework. Informaton on the reactvty of CO 2 wth formaton water and mnerals, and the way ths affects moblty may be solved by reactve transport smulatons (e.g., [3]). A general reactve transport equaton for aqueous component u can be wrtten as: u t r +, (1) ( vu D u) = R( u) where v s a velocty feld, D s the dffuson tensor, and R s a reacton term dependent on the aqueous chemstry u r. The reacton term on the rght-hand sde of equaton (1) s ntmately connected to the transport operators on the left-hand sde. To ease the calculatons a tme-splt to separate the transport from the source term s usually performed. In ths way the transport operators are dscretszed n space and solved separately from the geochemcal reactons, and the source term becomes a set of ordnary dfferental equatons (ODEs) of the form: du dt r = R. (2) ( u) Reacton (2) covers both equlbrum reactons between aqueous components and between the CO 2 gas or lqud phase and the aqueous phase, as well as mneral reactons constraned by ther reacton rates. Equaton (2) must be solved for each grd pont n the spatal mesh. For large 3D reactve transport smulatons the number of grd ponts may exceed hundreds of thousands. Snce equaton (2) usually s attrbuted to some degree of stffness, the equaton represents a bottleneck n solvng the reactve transport equaton, and effectve algorthms, for solvng equaton (2) s strctly necessary. A common approach for such equatons s backward dfferental formulatons (BDF) whch have proven to be very robust. Ths paper presents the smple explct scheme used n the ACCRETE geochemstry code to solve CO 2 -waterrock nteractons. A 1000 years batch smulaton llustrates the use of the code on a typcal CO 2 storage settng. The ntal mneral composton, aqueous chemstry, and physcal condton resemble those found n the Utsra Reservor at the Slepner West njecton ste n the North Sea [4]. For ths knd of problems nvolvng alumnoslcate stablty coupled va aqueous alumnum and slca, and also carbonate stablty coupled va dssolved carbonate, only aqueous alumnum s shown to cause nstablty n the ODE solver for proper step szes. A soluton to ths specfc nstablty problem s here suggested.

3 Proceedngs of the 3rd WSEAS Internatonal Conference on Mathematcal Bology and Ecology, Gold Coast, Queensland, Australa, January 17-19, The ACCRETE algorthm ACCRETE s an acronym for Athena Carbon Capture and storage geochemstry module. The development of the code was ntated as the Athena multphase 3D flow research code, developed at the Unversty of Bergen, was expanded to cover reactve transport smulatons of geologcal storage of CO 2 [5, 6]. ACCRETE s executed as an external module for each Athena tme step and updates the geochemstry accordng to drectves gven by Athena. ACCRETE can also be run as a stand-alone batch geochemstry solver as n the case of ths report. The present ACCRETE verson smulates nteractons between 16 mneral phases, 14 aqueous solutes, H 2 O as a solvent, and CO 2 as a separate gaseous or supercrtcal phase. The CO 2 -water-rock solver s dvded nto two man solvers: (1) the aqueous phase specaton solver; and (2) the mneral solver. The two solvers are presented n the followng sectons. 2.1 Thermodynamcs The standard state adopted n ths study s that of unt actvty for pure mnerals and H 2 O at any temperature and pressure. For aqueous speces other than H 2 O, the standard state s unt actvty of speces n a hypothetcal 1 molal soluton referenced to nfnte dluton at any temperature and pressure. For gases, the standard state s for unt fugacty of a hypothetcal deal gas at 1 bar pressure. Thermodynamc data are extrapolated from data calculated wth the SUPCRT92 program [7] usng the dprons96.dat database. 2.2 Solvng the aqueous specaton Aqueous specaton s constraned by fve equlbrum reactons, and the demand for a mass- and charge balanced system. The frst equlbrum that needs to be satsfed s the dssoluton of gaseous or supercrtcal CO 2 nto the aqueous phase CO 2,g/l CO 2,aq. (3) Ths equlbrum s constraned by the bubblepont of CO 2 calculated from x Pyφ v = exp K H RT b CO P 2 ( 1), (4) where P s pressure, y s fracton CO 2 n the gas phase, φ s the fugacty coeffcent, K H s the Henrys law coeffcent for CO 2, v s the average partal molar volume of CO 2 n water at nfnte dluton, and T s temperature. Wth polynomals for both φ(t,p) and K H (T,salnty), expresson (4) provdes good estmates of CO 2 solublty usng a fast non-teratve procedure [8]. The next four equlbra provdes the specaton of aqueous carbon H 2 O + CO 2,aq HCO H +, (5) HCO 3 - CO H +, (6) and Na + + HCO 3 - NaHCO 3. (7) The two frst (5,6) represents the dssocaton of carbonc acd to bcarbonate and carbonate, whereas (7) s needed to reduce the actvty of bcarbonate and acheve a more accurate ph. The last equlbrum reacton (8) s dssocaton of water nto H + and OH -

4 Proceedngs of the 3rd WSEAS Internatonal Conference on Mathematcal Bology and Ecology, Gold Coast, Queensland, Australa, January 17-19, H + + OH - H 2 O. (8) In addton to the equlbrum reactons, the aqueous soluton must be charge balanced H n z n = 0, (9) H where n and z are moles and charge of aqueous speces. The fnal requrement s that the aqueous carbon mass balance s satsfed C ( n + n + n + n ) 0 CO = 2, aq HCO3 CO3 NaHCO3,(10) where C s total dssolved carbon. The aqueous specaton s solved teratvely by the requrement that the dfference between n H gven by expressons (9) and n H estmated from reacton (5) n H K1a a =, (11) a CO2, aq H2O γ HCO3 H s less than a predefned lmt ( 9) ( 11) 2 ( nh nh ) < αvaq, (12) where V aq s volume aqueous phase and α s Solvng mneral reactons The extent ξ of a mneral reacton over one tme step can be calculated from ( Ω ) t t ξ = k S 1, (13) where k and S are the knetc constant and reactve surface area of the reacton respectvely, t s the step sze, and the Ω- 1 term s the affnty of the reacton and provdes the reacton drecton. The affnty s gven by: Q Ω =, (14) K where K s the equlbrum constant and Q s the actvty product of the reacton gven by p Q =. (15) vr a r a v p r p The contrbuton of the mneral reactons to the aqueous components can be expressed as: t1 t 0 n k = n k + ξ ν, k, (16) where v s the stochometrc coeffcent for aqueous component k n mneral reacton. Superscrpts t0 and t1 ndcates the ntegraton tmestep t = t1 t0. The ntegraton s accepted f all k aqueous components at tme t1 have concentratons n t1 k 0. If ths s not the case, the ntegraton tme step s reduced and equaton (16) s solved agan. Because one or more of the equatons n the aqueous phase specaton s always affected by the mneral reactons, the specaton algorthm must be called for each ntegrated tme step n the mneral solver. The system s solved when the sum of all ntegraton tme steps equals the predefned smulaton tme. 3 Results and dscusson To show how the ACCRETE algorthms solve CO 2 -water-rock nteractons, the

5 Proceedngs of the 3rd WSEAS Internatonal Conference on Mathematcal Bology and Ecology, Gold Coast, Queensland, Australa, January 17-19, mneralogy, chemstry and physcal condtons from the Slepner West njecton ste were chosen (See Table 1 and 2). Three smulatons are presented here and nclude: (1) smulaton where the mneral reactons are ntegrated over short tme steps to avod nstabltes; (2) smulaton wth larger tme steps to llustrate where the nstablty occur and how t affect the smulated results; and (3) smulaton wth large tme steps wth a problem-spesfc soluton to the nstablty problem. The smulatons are all one-grd sngle-phase batch problems wth an aqueous phase n contact wth an nfnte CO 2 reservor. The smulatons cover 1000 years of CO 2 -water-rock nteractons. Table 1 Intal mneral volume fractons and knetc data for Utsra sand. Mneral nformaton s based on data reported by [4], whereas the knetc data (k and S) s from [5] and [6]. Note that knetc constant for K s lsted here. Ths value s recalculated to the ambent temperature usng an Arrhenus relatonshp. Mneral x Utsra k 298 S (mol/m 2 s) (cm 2 /cm 3 ) Calcte x x10 2 Magneste x x10 2 Sderte x x10 2 Dawsonte x x10 4 Albte x x10 2 Mcroclne x x10 3 Quartz x x10 2 Chalcedony x x10 4 Kaolnte x x10 4 Clnochlore- 14A x x10 4 Daphnte-14A x x104 Muscovte x x10 4 Phlogopte x x10 4 Annte x x10 4 Labradorte x x10 2 Gbbste x x10 3 Porosty Smulated CO 2 storage system The smulated system comprses chemstry, mneralogy and physcal propertes (T, P) that correspond to nformaton publshed from the Slepner CO 2 njecton ste, Norwegan North Sea (see [4]). The mneralogy s lsted n Table 1 together wth data used n the knetc expresson (13). Some mnerals are proxes for smlar mnerals observed n the sedments. The aqueous chemstry s based on data from the Slepner njecton ste presented n [3]. Orgnal data n [3] s slghtly modfed to provde a charge balanced soluton at the startng ph of 7.1 shown n Table 2. Table 2 Intal aqueous chemstry (molefractons). Component X (mol/mol) H + /ph 1.963x10-9 / 7.1 Ca CO 2,aq x HCO x10-5 Na Cl Al x CO x10-8 Mg SO 2,aq 3.514x10-9 K x10-6 Fe x10-5 NaHCO Temporal evoluton of mneral carbonates and aqueous components The frst smulaton ntegrates the mneral reactons over short tme steps to avod nstabltes n the stff mneral solver. Ths smulaton s slow, but provdes relable estmates of the reactvty of the system. Selected mneral data and aqueous chemstry of the smulaton are presented n Fg. 1.

6 Proceedngs of the 3rd WSEAS Internatonal Conference on Mathematcal Bology and Ecology, Gold Coast, Queensland, Australa, January 17-19, SO 2,aq Reacton (17) s one of several that leads to formaton of Fe-, Mg-, and NaAl(OH)- carbonates. The second part of Fg. 1 shows the temporal evoluton of aqueous components. As expected, ph falls to a steady value around 5.0, and the system reach a steady non-equlbrum state after a few tens of years. Fg. 1. Temporal evoluton of carbonate mnerals (upper) and dssolved aqueous speces (lower) for short tme steps n the ntegraton of mneral reactons. One of the man ssues concerned wth safe long-term storage of CO 2 s to quantfy the amount of stable sold carbonates that form. The mnerals presented n Fg. 1 are those carbonates ncluded n the present smulatons and the total amount of carbon formed. Carbonates form as a consequence of a lowerng of ph as CO 2 s njected, whch n turn trggers dssoluton of prmary mnerals present n the sedment. Ths s llustrated n equaton (17) for the dssoluton of phlogopte and formaton of magneste (MgCO 3 ). Fg. 2. Temporal evoluton of carbonate mnerals (upper) and dssolved aqueous speces (lower) for large tme steps n the ntegraton of mneral reactons. Phl.,s + 10H + + 6HCO 3 - MgCO 3,s + 3CO 2,aq + 9H 2 O + K + + Al 3+ (17)

7 Proceedngs of the 3rd WSEAS Internatonal Conference on Mathematcal Bology and Ecology, Gold Coast, Queensland, Australa, January 17-19, Stffness problems caused by Alfluctuatons The second smulaton llustrates stffness problems n the algorthm caused by ntegratng mneral reactons over large tme steps. Ths smulaton s fast but data produced are less relable. Results are presented n Fg. 2 that shows carbonate formaton and aqueous chemstry wth tme. The man dfference between the former smulaton and ths one s that sgnfcantly less sold carbonate forms. The explanaton s found n the aqueous chemstry. Whereas most aqueous components show a temporal evoluton smlar to the frst case, aqueous alumnum fluctuates. The reason for ths can be found by comparng the concentraton of alumnum wth the extent of alumnum-contanng reactons ξ ν, Al η = (18) t0 nal The concentraton of alumnum as a functon of η s presented n Fg. 3. Ths fgure shows that as η approach the lmt - 1 from rght, the alumnum at tme t1 may shft by orders of magntude. Snce 11 of the 16 ncluded reactons that represent a typcal CO 2 -water-rock smulaton, are dependent on the alumnum concentraton, both wth respect to reacton rate and drecton, a rapd change may nduce nstablty. In ths case, several fast reactng mnerals that are close to equlbrum, lke dawsonte, oscllate between saturaton states because of the rapdly changng aqueous alumnum. These rapd changes n saturaton states move the system n the rght drecton wth respect to the global thermodynamc drvng force, but at a slower rate than n a stable system. The reactvty of ndvdual mneral reactons, lke the formaton of dawsonte, are n addton wrongly estmated. Values of η for other aqueous speces are never close to -1, and alumnum s hence the only cause of nstablty n the system. Fg. 3. As η s approached from the rght hand sde the concentraton of alumnum at tme t1 drops rapdly. 3.4 A soluton to the present stffness problem For the present system, whch s representatve for a CO 2 -water-rock system wth a few proxes replacng the natural reservor mnerals, alumnum s dentfed as the cause of a major stffness problem. If we go back to the frst smulaton we notce that alumnum shows some varaton durng the frst 50 to 100 years, and gradually converge towards a steady value. Usng ths nformaton, we can attempt to remove the alumnum as a varable n the system and replace ts temporal evoluton wth an ndependent functon. Snce most of the 1000 years smulated shows constant alumnum concentraton, the frst attempt s to smulate the system usng a constant concentraton. From the frst smulaton we notce that alumnum reach a steady mole fracton of approxmately 2.0x An attempt usng ths value as a constant and ntegratng the mneral reactons over

8 Proceedngs of the 3rd WSEAS Internatonal Conference on Mathematcal Bology and Ecology, Gold Coast, Queensland, Australa, January 17-19, the same large tme steps as used n the second smulaton s presented n Fg. 4 and compared to the frst smulaton. The upper part of the fgure shows a drect comparson to the frst smulaton, whereas the lower part plots the dfference between smulated values. It s evdent that t s a very close agreement wth the two smulatons. The frst 100 years dawsonte formaton s overestmated, and also the total carbon formed. Ths corresponds to frst perod of the frst smulaton where alumnum shows most varaton. Durng the last 900 years the two smulatons converges to values that are ndstngushable for all mnerals. An even closer ft would be obtaned f a proper functon s constructed for the frst 100 years. Ths would be trval, but s not necessary f the focus s a long-term reactve system. tmes can probably be decreased further down to sub-seconds. The burden of solvng the reacton term n a large 3D system would then be feasble. For example, the penalty for ncludng the reacton term n the 1944 grds 3D system smulated n [5] and [6] could be less than 30 mnutes. We stll have to keep n mnd however that a mult-grd dynamc system s dfferent than the batch smulatons presented here. For example to fnd one functon approproate for all grds at all tmes may be mpossble. 3.5 Are large reactve transport smulatons possble wth the present algorthm? Wthout a soluton to the stffness caused by aqueous alumnum, a sngle grd smulatons that covers 1000 years of CO 2 - water-rock nteractons, lke the frst smulaton n the present report, requres two or three days of smulaton tme. A 1000 years large-scale reactve transport smulaton s therefore not achevable. On the other hand, by replacng the alumnum varable wth a proper ndependent functon, the runtme may decrease to a few seconds. For example, the executon tme for the thrd smulaton on a standard 32 Bt Lnux archtecture, ncludng extensve formatted reportng, s approxmately 50 seconds. By restrctng the amount of reportng, usng a nonformatted output, and ncreasng the ntegraton step sze further, executon Fg. 4. Comparson between the frst smulaton (R1) and the thrd smulaton (R2), the latter wth the varable alumnum replaced by a constant value.

9 Proceedngs of the 3rd WSEAS Internatonal Conference on Mathematcal Bology and Ecology, Gold Coast, Queensland, Australa, January 17-19, Concluson Solvng reactve transport equatons, lke equaton (1), commonly nvolve a reacton term that can cause some degree of stffness. Ths creates a bottleneck for solvng the reactve transport equatons. Ths work presents a typcal problem that nvolves solvng stff systems of ODE. Ths knd of problems s usually solved by some mplct routne lke a BDF solver. In ths report the ACCRETE geochemstry code attempts to solve the reacton term usng smple explct routnes. The report presents both panstakngly slow calculatons ntegratng the mneral reactons over short tme steps, and fast smulatons wth large tme step szes. By comparng the dfferent smulatons, aqueous alumnum s dentfed as the cause of nstablty for ths system. By replacng the varable alumnum wth an ndependent functon or constant, we can reproduce the long-term reactvty of CO 2 at a low CPU cost. A 1000 years snglegrd batch smulaton becomes possble n a few seconds or even less. Acknowledgements We are grateful to Ncola McLoughln for helpful comments on the language. Ths report was made possble by fnancal support from Norsk Hydro and the Research Councl of Norway through grant /210 and the BIODEEP project. References: [1] Gale, J., Geologcal storage of CO2: What do we know, where are the gaps and what more needs to be done? Energy, Vol. 29, 2004, [2] Bouchard, R., and Delaytermoz, A., Integrated path towards geologcal storage. Energy, Vol. 29, 2004, [3] Johnson, J.W., Ntao, J.J., and Knauss, K.G., Reactve transport modelng of CO 2 storage n salne aqufers to elucdate fundamental processes, trappng mechansms and sequestraton parttonng. In: Banes, S.J., and Worden, R.H. (eds), Geologcal Storage of Carbon Doxde. Geologcal Socety, London, Specal Publcatons, Vol. 233, 2004, [4] Chadwck, R.A., Zwegel, P., Gregersen, U., Krby, G.A., Holloway, S., and Johannessen, P.N., Geologcal reservor characterzaton of a CO 2 storage ste: The Utsra Sand, Slepner, northern North Sea. Energy, Vol. 29, 2004, [5] Hellevang, H., Khattr, S.K., Fladmark, G.E., and Kvamme, B., CO 2 storage n the Utsra Formaton ATHENA 3D reactve transport smulatons. Submtted to Basn Research. [6] Khattr, S.K., Hellevang, H., Fladmark, G.E., and Kvamme, B., Deposton of Green House Gases by Compostonal Smulator: Long Term Reactve Transport of CO 2 n the Sand of Utsra. Submtted to Journal of Transport n Porous Meda. [7] Johnson, J. W., Oelkers, E. H., Helgeson, H. C., SUPCRT92: a software package for calculatng the standard molal thermodynamc propertes of mnerals, gases, aqueous speces, and reactons from 1 to 5000 bar and 0 to 1000 C. Comput. Geosc. Vol. 18, No.7, 1992, [8] Hellevang, H., and Kvamme, B., ACCRETE Geochemstry solver for CO 2 -water-rock nteractons, Paper, GHGT8, Trondhem, Norway, June 19-22, 2006.