Prediction of Methane Solubility in Geopressured Brine Solutions by Application of Perturbation Theory.

Transcription

1 Louisiana State University LSU Digital Commons LSU Historical Dissertations and Theses Graduate School 1988 Prediction of Methane Solubility in Geopressured Brine Solutions by Application of Perturbation Theory. Yangtzu Chao Louisiana State University and Agricultural & Mechanical College Follow this and additional works at: Recommended Citation Chao, Yangtzu, "Prediction of Methane Solubility in Geopressured Brine Solutions by Application of Perturbation Theory." (1988). LSU Historical Dissertations and Theses This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Historical Dissertations and Theses by an authorized administrator of LSU Digital Commons. For more information, please contact

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4 Order N um ber P rediction o f m ethane solubility in geopressured brine solutions by application o f perturbation theory Chao, Yangtzu, Ph.D. The Louisiana State University and Agricultural and M echanical Col., 1988 UMI 300 N. Zeeb Rd. Ann Arbor, MI 48106

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6 P rediction of M ethane Solubility in G eopressured B rine Solutions by A pplication of P e rtu rb a tio n T heory A D issertation S u b m itted to the G rad u ate F acu lty o f the L ouisiana S tate U niversity and A gricultural and M echanical College in p a rtia l fulfillm ent of the requirem ents for the degree of D octor of Philosophy in T h e D ep artm en t of C hem ical Engineering by Y angtzu C hao B.S. T u n g h ai U niversity, T aiw an, 1976 M.S. Illinois In stitu te of T echnology, 1981 M ay 1988

7 A cknow ledgem ent I w ould like to express m y sincere g ratitu d e to m y m ajor professor, D r. A drain E. Johnson, Jr., who initiated m e in to th is reserach and guided m e w ith patience, encourgem ent, and u n d erstandin g th ro u g h o u t th is study. T h anks are due to Dr. A rm ando B. C orripio, Dr. A rth u r M. Sterling, Dr. F rank R. G roves, Dr. T ryfon C haralam popoulos, an d D r. Elvin C hoong for th eir tim e and cooperation in serving on m y g ra d u a te co m m ittee. T h e financial support from C hem ical Engineering D epartm ent a t LSU d uring m y g raduate stu d y here is deeply appreciated. I would also like to th ank D r. R alph W. Pike and the M ineral Research In stitu te for providing m e w ith fellow ship su p p o rt during several of the sum m er periods. Special th anks are due to M r. R ueyder Jen g for his assistance in word processing and figure plottin g in th is report. Also, th an k s to m y p aren ts and m y wife for the sacrifices they endured and the a b o u d an t love and confidence th ey shared w ith me. A bove all, th an k s and praises are due u n to the nam e of Jesus C h rist for His com fort, w isdom, love, and m any m any answ ering prayers th ro u g h o u t these years. i i

8 Table of Contents A cknow ledgem ent... ii Page T ab le of C onten ts... iii List of T ables... vii L ist o f Figures... ix A b stract... xii List of Sym bols... xiv C h ap ter 1. In tro d u ctio n T he G eopressured Energy Resource in T he U.S D O E-Sponsored Research on G eopressured Energy A nd M ethane Solubility G oal of T his Research A nd N ature of T he Problem C orrelation of H enry s Law C o n stan t Setchenow E quation E lectrostatic Theory Scaled P article T heory P ertu rb atio n T heory S um m ary C h ap ter 2. Previous W ork on C orrelation o f M ethane S olubility D a ta In tro d u ctio n E xperim ental W ork A nd E arly C orrelations C orrelation of H aas C orrelarion of B lount C orrelation of Coco A nd Johnson... 28

9 Page C h apter 3. D evelopm ent of G overing E q uations of P hase E q u alib ria for A C om plicated System of G ases, E lectrolytic Salts, A nd W ater System D escription Solubility of Gases in Liquids P ertu rb atio n A pproach In tro d u ctio n to S tatistical M echanics P ertu rb atio n T heory to O b tain C hem ical P o ten tial R elation betw een A p p aren t Ideal Solution F ugacity A nd C hem ical P o tential Expression of A pparent Ideal Solution F ugacity for G eopressured System... CO 3-4 D evelopm ent of T h e P artial M olar V olum e of A G as S o lu te D evelopm ent of T he Isotherm al C om pressibility o f T he P a rtia l M olar V olum e G as Solubility at High Pressure D issociation of W eak E lectrolyte C h apter 4. M ethod of Solution V apor Phase Liquid Phase Solution D ensity S atu ratio n V apor Pressure P o y n tin g F acto r Physical P aram eters C om puter A lgorithm iv

10 Page 4-5 A nalysis of Initial R esults A nd Decisions M ade to Im prove P redictions Possible Effect of W ater D ensity on M ethane Solubility Possible Effect of T em p eratu re on M ethane Solubility via th e L ennard-jones P aram eter a Possible Im provem ent of T h e M odel by A dding C harge-d ipole A nd D ipole-dipole In teractio n s P aram eters F ittin g C H 4 / H 20 System C II4 / H 20 /N a C l System C 2II6 / I I 20 System C 0 2 /H oo System S um m ary C h apter 5. R esults A nd C om parisons T h e Effect of T em p eratu re on M ethane Solubility for M ethane-w ater M ixtures T he Effect of Pressure on M ethane Solubility for M ethane-w ater M ixtures T he Effect of NaCl S alt on M ethane Solubility T h e Effect of O th er G ases on M ethane S olubility in W a te r A nd in B rine Solutions C H 4 / C 2H 6 /H 20 /N a C l System C H 4 / C 0 2/ H 20 /N a C l System S u m m a r y... Chapter 6. Conslusions And Recom m endations V

11 Page References 160 Appendices A Equation of State Developed by Nakamura et a l B Orientation-Averaged Molecular Interaction C Potential Energy for W ater Molecule D Computer Program V ita vi

12 List of Tables Page T able 2-1 T ab le 2-2 A vailable published d a ta on m ethane solubility in w ater an d w ater-n ac l solution..22 A com parison of solubilities of m eth an e in w ater calculated by H aas, B lount, and C oco-johnson s correlations w ith published d a ta T ab le 3-1 C lassification of su bscript num ber used in th is study and com puter program T ab le 3-2 Interm olecular potential energy used in th is stu d y T ab le 4-1 C om parison of various co rrelations of liquid w ater d en sity w ith ste a m tab les T ab le 4-2 C om parison of A p predicted by equation (4-21) w ith Rogers an d P itz e r s experim ental density d a ta for N ac l-h 20 solution T ab le 4-3 Sensitivity stu d y to show the effect of various physical p aram eters on predicted m eth an e solubilities and p artial m olar volum es T ab le 4-4 N um erical values for physical p aram eters as reported in th e lite ra tu re T ab le 4-5 Individual F itted <7h 2o an d for various d a ta sets along w ith its corresponding p a rtia l m olar volum e of m eth an e T ab le 4-6 P a rtia l F test of p aram eter fittin g of C H 4 / H 20 system T ab le 4-7 C om parison of RM SD of present m odel w ith B lo u n t s polynom ial in NaCl brine solution... I l l T ab le 5-1 C om parison of the m agnitu d e of each term s in equation 3-56 for m eth an e-w ater system T ab le 5-2 C o n trib u tio n of various term s in equation T ab le 5-3 C om parison of V q jj an d /?q jj predicted by th is stu d y w ith reported values in C H 4 / H 20 system T ab le 5-4 C alcu lated vapor and liquid phase m eth a n e fractions in C H 4 / H 20 system

13 T ab le 5-5 C alculated salting-out coefficient based on our m o d e l T ab le 5-6 R eported values for salting-out coefficient o f C H 4-N ac l p a ir T ab le 5-7 E xperim ental gas solubility d a ta of C 0 2, C 2H6, and C H 4 in w ater T ab le 5-8 C alculated vapor and liquid phase therm odynam ic properties for C 2H 6 /C H 4 / H 20 /N a C l system a t 423 K, 1530 a tm, and 1.83 m T ab le 5-9 R atio of undissociated C 0 2 to the to tal am o u n t of C 0 2 in w ater a t 423 K and a t different ^ 7 a / l l j. ^ values T ab le 5-10 C alculated vapor and liquid phase therm odynam ic properties for C 0 2 /C H 4 / I I 20 /N a C l system a t 423 K, 874 a tm, and 0.89 m T ab le A -l C om parison of predictions of fugacity coefficients of SRK an d N a k am u ra s equation of sta te in C H 4 /H 20 system v iii

14 List of Figures Page Figure 1-1 L ennard-jones p o ten tial Figure 1-2 H ard sphere p o tential Figure 1-3 Square well potential Figure 1-4 T h e split of p o tential according to B arker-h enderson Figure 1-5 T h e split of potential according to W eeks-c handler-a ndersen Figure 3-1 S chem atic representation of the system under consideration Figure 3-2 E x trap o latio n of liquid sa tu ra tio n pressure into hypothetical liquid region Figure 3-3 V ariation of H enry's law co n stant w ith com position in a tern ary system Figure 4-1 C om parison of A p of our correlation (equation 4-20) w ith experim ental d a ta of Rogers and P itzer Figure 4-2 Flow chart of program to com pute m eth an e solubility given system specifications and physical p aram eters of com ponents F igure 4-3 In itial com parison of th e discrepancies betw een predicted m e th an e solubility and experim ental d a ta of C ulberson and M ck etta F igure 4-4 T h e difference of p redicted m eth a n e solubility in pure w a ter using various correlation of w ater density Figure 4-5 M inim um m eth an e solubility in pure w ater predicted by our m odel w ithout adju stin g the a p aram eter of w ater from th e initial values chosen from the literatu re Figure 4-6 T h e effect of a linear variatio n of <?h2o w ith tem p eratu re on predicted m eth an e solubility in pure w ater Figure 4-7 F ittin g of irh i) 0 and *n p o tential w ith experim ental solubility d a t a Figure 4-8 F ittin g of in L-J p o ten tial w ith experim ental solubility d a ta using co nstant value ix

15 Figure 5-1 M inim um m ethane solubility in pure w ater around 344 K predicted by our m odel and the experim ental d a ta of C ulberson an d M c K e tta Figure 5-2 M axim um apparent ideal solution fugacity of m eth an e predicted by our m odel around 360 K a t various pressure F igure 5-3 C om parison of m eth an e so lu b ility in pure w ater predicted by different correlations w ith experim ental d a ta of Price and Sultanov Figure 5-4 C om parison of m eth an e solubility in pure w ater predicted by B lo unt, Coco, and our m odel w ith experim ental d a ta of Price and O Sullivan Figure 5-5 C om p arison o f m eth an e solubility in w ater p redicted by B lou nt, Coco, and th is study w ith experim ental d a ta of Sultanov and M ck etta Figure 5-6 Fugacity coefficient of pure m ethane from C hoi s stu d y and th a t in m ethane-w ater m ix tu re from SRIv equation o f sta te in th is stu d y over m oderate presures and tem p eratu res F igure 5-7 Salting effect on m eth ane solubility a t 600 a tm and m oderate tem p eratu re Figure 5-8 Salting effect on m eth ane solubility predicted by B lount, Coco, and th is stu d y F igure 5-9 C om parison of m eth an e solubility in low an d high sa lt co n te n t brine solutions predicted by various correlations w ith B lo u n t s d a ta Figure 5-10 C om parison of m eth ane solubility in 0.89m brine solution a t low tem p eratu re and low pressure conditions Figure 5-11 C om parison of m eth an e solubility a t low tem p eratu res and pressures in 0.89m brine solution F igure 5-12 T he existance of a m inim um salting out coefficient over the tem p eratu re range and the v ariatio n of Ks w ith pressure a t m = l based on our m odel F igure 5-13 P redicted p artial m olar volum e of m ethane a t various salt concentration Figure 5-14 T he effect of C 2H6 011 th e m ethane solubility in brine solution a t different dissolved gas ra tio

16 Figure 5-15 T h e ra tio of m eth an e to eth a n e in the vapo r phase predicted by our m odel a t B lo u n t s experim ental condition of 1530 atm, 423 K, and m olality of F igure 5-16 P red icted m eth an e solubility an d to ta l gas dissolved in 1.82m N ac l solution a t tw o different pressure a n d various C 0 2 percentage in th e dissolved gas Figure 5-17 C om parison of predicted m ethane and to ta l gas dissolved in m N ac l soluition w ith B lo u n t s experim ental d a ta a t various C 0 2 percentage in the dissolved gas F igure 5-18 T h e ratio of C H 4 to C 0 2 in the vapor and phase predicted by our m odel a t B lo u n t s experim ental condition of 874 atm, 423 K, and a salt content of 0.89m xi

17 Abstract T he ability to predict m ethane solubility in underground reservoirs is essential to eventual exploitation of geotherm al-geopressured reservoirs as an altern ate energy source. In th is study, second-order Leonard-B arker-iienderson p ertu rb atio n theory was applied to brine solutions containing II2O, C H 4, C 2II6, C 0 2, an d N acl to develop a fundam entally-based th erm odynam ic approach to handle m ore effectively th e broad tem p eratu re and pressure ranges encountered in underground reservoirs, and to include the effect of th e presence of various salts and o th er gases on m eth an e solubility. T he apparen t ideal solution fugacity, f ', for each solute gas is proposed and developed th ro u g h p ertu rb atio n theory as a direct m easure of th e net forces acting upon the solute gas from all the species in the system. Since f l depends on the brine com position, there is no need to use a H enry s law constant evaluated a t infinite dilution along w ith an activ ity coefficient th a t varies w ith the solute gas m ole fraction. Inst' id, th e pro d u ct, (x^ f, ), gives the fugacity of each solute gas in the equations representing phase equilibrium. B oth th e p a rtia l m olar volum e of solute gas and the isotherm al com pressibility of its p a rtia l m olar volum e were self- generated from the p ertu rb atio n theory approach to predict the gas solubility a t high pressure, a n d they b o th agreed well w ith rep o rted values. E ssentially all published experim ental d a ta on m eth an e solubility were utilized to test the relationships developed in this work. T h e ranges of conditions covered by th is stu d y are: tem p eratu res from 298 to 589 K, pressures from 10 to 2000 a tm, an d salinities from 0 to 5 m. T he p aram eters used in this stu d y were all based upon reported literatu re values. Because of th e extrem e sensitivity of calculated m eth an e solubilities to the values of a p aram eter in the L ennard-jones p o ten tial, it was found necessary to determ in e th e best values for these xii

18 p aram eters for each com ponent to insure a m inim um least-squares global fit of th e solubility d a ta. In ad d itio n, it was found th a t allow ing the a p aram eter for w ater to vary w ith tem p eratu re significantly im proved th e overall global fit to the d a ta. x iii

19 List of Symbols A A c A aa aj a_ _ a D D 0 d ej fj H elm holtz free energy configurational H elm holtz free energy configurational H elm holtz free energy of reference system m olecular activ ity ionic diam eter activ ity of cation activ ity of anion D ielectric constant of salt solution D ielectric co n stan t of w ater diam eter ionic charge fugacity o f com pon ent i f? fugacity of com ponent i a t reference state ap p a re n t ideal solution fugacity of solute gas i gc gj p artial m olar G ibbs energy for cavity form ation p a rtia l m olar G ibbs energy for interaction g rad ial d istrib u tio n function of reference system h P lan ck s co n stant H? H enry s co n stan t of com ponent i in pure w ater Ho i H enry s co n stan t of solute 2 in pure solvent 1 H, H enry s law co n stan t of com ponent i in the solution K K j B oltzm ann constant saltin g coefficient xiv

20 m ^ m a m 5 m^_ m Nj N 0 P Pj PJ P p Its P ' Q Q c Qj R r S T t U (r) U (r) Stoichiom etric concentration m olecular concentration salt m olality cation m olality anion m olality num ber of m olecules of species i in m ix tu re A vogadro s co n stant pressure p a rtia l pressure of com ponent i sa tu ra tio n pressure of com ponent i a t tem p eratu re T p ertu rb atio n p a rt of pressure equation hard sphere equation of sta te p a rtitio n function co n figurational p a rtitio n function quadrupole m em ent gas constant in term o lecu lar distance salin ity in g ram s per liter tem p eratu re tem p eratu re in degree F ah ren h eit to ta l potential energy reference system p o ten tia l energy U ^(r) pertu rb ed potential energy V Vj Vs volum e liquid m olar volum e of com ponent i liquid m olar volum e of salt XV

21 Vf p a rtia l m olar volum e of com ponent i a t infinite dilution V,- pair potential energy \ j pair potential energy of reference system Vfj X,- X 0 Y f z Zc pertu rb ed pair potential energy liquid phase m ole fraction o f com ponent i solute m ole fraction in pure w ater vapor phase m ole fraction o f com ponent i com pressibility factor co n figurational integral OO co nfigurational in teg ral of reference system xvi

22 G reek Sym bol a a,- p e rtu rb a tio n param eter, m easuring inverse steepness of repulsive potential polarizabilitv of com ponent i 0 B oltzm ann factor (1 /K T ) /?,: iso th erm al com pressibility of p a rtia l m olar volum e of solute gas i 0 O com p ressibility of pu re w ater activ ity coefficient of com ponent i 7 _j_ m ean ionic a c tiv ity coefficient e energy param eter in L ennard-jones potential 9 polar angle A p ertu rb atio n p aram eter, m easuring stren g th of a ttra c tiv e p o tential p t chem ical poten tial of com ponent i hs / /,' chem ical potential of a hard sphere fluid lis r Pi reduced chem ical p o ten tial for a h ard sphere fluid chem ical potential of an ideal gas Pi dipole m o m en t of com ponent i n reduced density p solution density Pi n um ber of i m olecule per u n it volum e p* density of pure w ater A p cr <j) u density increase due to the presence of salt distance param eter in th e L ennard-jones p o tential fugacity coefficient acentric factor xvii

23 Chapter 1 Introduction 1-1 T h e G copressured E nergy R esource in T h e U.S. T he shock of the energy crisis in the 70 s focussed considerable a tte n tio n in the U.S. on the developm ent of altern ativ e energy resources for the future. It was recognized by the U.S. D epartm ent of Energy (D O E ) th a t the various altern ate energy resources in th e U.S. need to be identified and characterized p rior to the tim e, which surely m ust come, w hen the oil and gas resources of the world are depleted. O ne energy source th a t was selected for stu d y by D O E is the group of geopressured-geotherm al reservoirs th a t are found in the U nited S tates in the north ern G ulf of M exico basin, m ainly along the T exas and L ouisiana G u lf C oasts, a t depths of G000 ft to ft. As a result of D O E-sponsored effort, the locations of these reservoirs are now generally know n, and som e inform ation about th eir tem p eratu res, pressures, and salinities has been obtained from logs of oil and gas wells previously drilled in the im m ediate vicinity of each geopressured reservoir. In addition, som e special test wells were drilled a t D O E expense during the late 70 s and early 80 s to gain m ore inform ation ( including m ethane content ) about a few of the high-potential reservoirs, b u t well logs of previously drilled wells rem ain the p rim ary source of inform ation for estim atin g th e n atu re and am o u n t of this energy resource. T h e energy contained in the geopressured reservoirs is classified as : (A) th erm al energy from the hot brine. (B) m echanical energy from the high fluid pressure of the reservoirs, and (C ) com bustive fuel energy from the n atu ral gas, m ostly m ethane, dissolved in the u n dergroun d brines (G regory, 1981). Early estim ates for th e to ta l in-place resource were up to 1

24 quads ( 1 quad = 1.0 xlo 15 B tu ) of low level ( F) th erm al energy and 60,000 quads of dissolved m ethane. ( T h e estim ated available m echanical energy was several orders of m agnitu d e less than these tw o.) T he econom ically recoverable resources were estim ated to be as m uch as 300 quads of therm al energy and 800 quads of m eth an e (W esthusing, 1981). Because of previous success in the exploiation of high tem p eratu re geotherm al wells in the far w estern U.S., initial interest focused prim arily on the heat th a t could be ex tracted from the hot brine an d converted to electrical energy th ro u g h tu rb in es a t th e surface. In 1974, 30.4% of the energy consum ed in the U nited S tates cam e from n atu ral gas (C am pbell, 1977). As a result of both increasing dem and and the de-regulating N atu ral Gas Policy Act of 1978 by Congress, the price of deregulated n atu ral gas soon rose by ten fold above pre-1973 levels to the $ 7-8 /1 000-SCF range. As a result, econom ic interest in geopressured resources shifted from the low-level therm al energy of the reservoirs to the fuel energy of the dissolved m ethane in the brine. A t th a t tim e, techno-econom ic studies of specific h ig h -potential reservoirs, such as those m ade by Johnson et al. (1980), indicated th a t n atu ral gas prices m u st equal or exceed the 8S/1000-SC F level for the geopressured energy resource to becom e econom ically viable. Since 1983, conservation efforts in itiated in th e 70 s, coupled with a w orld-w ide energy glut has forced the price of both oil and n a tu ra l gas back down to m ore m oderate levels ( /barrel and S I.5-2.0/1000 SC F ), alth o u g h these prices are still about tw ice the price levels th a t existed prior to Hence, te n ta tiv e projects for exploitation of the geopressured m eth an e have been shelved by in d u stry u n til such tim e th a t gas prices again reach econom ically a ttra c tiv e levels. T he m eth an e entrain ed in geopressured reservoirs rem ains a vast b u t dilute resource th a t continues to be unexploited because drilling and production costs are high and because m any u ncertainties are associated w ith its exploitation and com m ercialization.

25 1-2 DOE-Sponsored Research on Geopressured Energy and Methane Solubility D uring the late 70 s and early 80 s, an extensive research effort was sponsored by D O E to o b tain m ore reliable assessm ents of the geological, engineering, envirom ental, legal, and social aspects of developing the geopressured resource in order to provide a d a ta base for th a t future tim e when this resource becom es econom ically viable. A m ong the m any related sub-projects carried out under D O E sponsorship was one to stu d y the solubility of m ethane in brine a t conditions equivalent to the underground pressure, tem p eratu re, and salinity ranges of know n reservoirs. A q u a n tita tiv e description of vapor-liquid equilibria pertin en t to reservoir conditions is very im p o rtan t for estim atin g the m ethane conten t of a prospective reservoir. A. E. Johnson. Jr.. who served as th e director of a m ulti-disciplinary, D O E-sponsored project at th a t tim e, undertook as one of the sub-tasks of the D O E project a study of the thenavailable m eth an e solubility d a ta. Coco and Johnson (1981) developed a com puter subroutine called SO L U T E to correlate the available solubility d a ta based on a fundam entally sound th erm odynam ic approach. SO L U T E can be used to predict the solubility of m ethane in a brine a t given salinity, tem p eratu re, and pressure conditions. It therefore provided the LSU teclm o-econom ic com puter m odel of a geopressured, geotherm al reservoir w ith the capability of calculating the m axim um (satu rated) am o u n ts of m ethane which could be dissolved in the brine solution of th e reservoir, given the salin ity of the brine and its te m p e ra tu re a n d pressure. T his work was reasonably successful in correlating the then-available d a ta, but it revealed th a t there rem ained a strong need to im prove the m odel used to correlate the d a ta by extending the m odel s fu n d am en tal basis. T he correlation w hich was developed does not consider the presence of C 0 2 in the dissolved gas, which is the m ajo r secondary gas com ponent in geopressured brines, ranging from tw o to twelve m ole percent. Som e prelim inary d ata from reco m b in atio n experim ents run by IG T (In stitu te of G as Technology) on gas from

26 geopressured test wells had suggested th a t the effect of C 0 2 on to ta l gas dissolved is both su b stan tial and highly nonlinear. Some later D O E-sponsored experim ental results by B lount et. al. (1982) indicated th a t the solubility of m ethane is enhanced by an extrem ely low concentration of C 0 2 b u t is reduced a t higher concentrations of C 0 2. It also revealed the am o u n t of C 0 2 dissolved in geopressured brines can be m uch greater th a n originally suspected. T he presence of higher-m olecular-w eight hydrocarbons, m ainly ethane, m ay also play an im p o rta n t role in a com prehensive study of m ethane solubility. B lount et al. (1982) reported th a t a t low concentrations, ethane salted m ethane into solution, while above 6 to 8 mole percent eth an e of the dissolved gas in solution, m ethane was strongly salted out by the ethane. T his is a significant d epartu re from th a t observed by A m irajatari and C am pbell (1972) who found th a t solubility of the binary m ethane-ethane m ixture is greater th a n th e solubility of the pure com ponent at. the sam e tem p eratu re and pressure. In addition, geopressured brines contain various salts other th a n sodium chloride, prim arily calcium salts. T h e previous tre a tm e n t of Coco and Johnson (1981) correlated d a ta of brine containing only sodium chloride salt an d dissolved gases contain in g only m ethane.

27 1-3 Goal of This Research and Nature of The Problem In this research, our goal was to (1) expand the m eth ane solubility correlation to include additional gases and additional salts, (2 ) develop an even m ore fundam ental therm odynam ic approach, com pared to the classical solution therm odynam ics applied by Coco and Johnson, and (3) extend the global n atu re of the correlation over a wider range of experim ental conditions, while a t the sam e tim e im proving the accuracy ( sum of squares error) betw een the predicted values an d experim ental d a ta. T he coexistence in aqueous solution of strong electrolytes such as N a d together w ith dissolved gases such as C II4, C 0 2, and C 2H0 creates a com plex interm olecular force field th a t affects the solubility of m ethane, which for extrem ely dilu te gas concentrations can be expressed in term s of its H enry s law co n stan t in the liquid phase and its fugacity coefficient in the equilibrium gas phase. Therefore, the success of any m eth an e solubility prediction m ethod is highly dependent on developing an effective and fundam entally correct m odel to account for the variation of the H enry's law constant of m ethane w ith tem p eratu re, pressure, and both the electrolyte and dissolved gas com position of the brine. T he solubility prediction is also, of course, dependent upon the gas phase fugacity coefficient of m ethane, which in th is work was calculated from th e SRK eq u atio n of state.

28 6 1-4 C orrelation of H enry s Law C o n stant E xtensive research has already been done to correlate H enry s law constants of various gases in binary non-electrolyte solutions. H ayduk and Laudie (1973) p lo tted straig h t lines for lu ll vs. 1 /T for a num ber of gases in a single solvent and found th a t they all intersected a t a single value of 1nH c a t the solvent critical tem p eratu re T c. Therefore, a single m easurem ent of InH can be com bined w ith lnh c t.o o b tain values a t other tem p eratu re. E xtensions of th is concept have been m ade to m ore com plex solvents, b u t it is not clear how it can apply to a system in w hich II goes through a m axim um w ith tem perature. Y orizone and M ivano (1978) developed from a th ree-p aram eter corresponding state theory a generalized correlation for predicting H enry s law constant. B ut this work was lim ited to non-polar m ixtures. Benson and K rause (1976) studied thoroughly the solubilities of sim ple gases in w ater and developed an em pirical equation valid from 0 to 50 C. Cysewski and P rausnitz (1976), based on A lder s perturbed hard sphere equation of state, developed a sem i-em pirical correlation of H enry s law constant over a wide tem p eratu re range, yet the accuracy of the prediction was not very high. O th er a tte m p ts include the work of N ak ah ara and Ilira ta (1969) for hydrogen-hydrocarbon m ixtures an d those of Brelvi (1980) and S agara and Saito (1975) for v ariations on the form ulae of solubility param eters in regular solution for non-polar system s. In co n trast to th e relatively extensive studies th a t have been conducted on H enry s law co n stan ts of gases in binary non-electrolyte solutions, little a tte n tio n has been given to th a t of gases in electrolyte solutions. E arly approaches correlated the salting effect through the sem i-em pirical Setchenow equation (1889). W ith recent im provem ents in our understan d in g of solution theories, typified by scaled particle theory and statistical m echanical p ertu rb atio n theory, a m ore fundam ental approach to this subject can be atte m p te d. In the following section, th is topic is developed m ore fully.

29 1-5 Sctchenow Equation A well know n phenom enon is th a t when salt is added to a carbonated beverage, the dissolved C 0 2 gas bubbles out from the liquid solution. T his phenom enon has long been recognized and is term ed the saltin g -o u t effect. M athem atically, the solubility of a non- electrolvte gas in a salt solution was first related by the sem i-em pirical Setchenow equation (1889) : log X " = K S 111 s (1-1) w here, for a specified tem p eratu re and pressure of the system, X is th e solubility of the dissolved gas in an aqueouus salt solution- of concentration m s, X D is its solubility in pure w ater, and K s is the salting coefficient. In its original application, the Setchnow equation was applied to aqueous salt solutions in equilibrium w ith a gas phase a t m oderate tem p eratu res and pressures, so th a t the w ater content of the gas phase a t equilibrium is essentially zero. T he Setchnow equation is valid prim arily a t low salt concentrations, and K s has a specific value for each p a ir of sa lt and dissolved non-electrolyte gas. T he effect of salt on gas solubility is actually a com plex phenom enon. In som e cases, adding salt enhances the gas solubility (salting-in), while in other cases th e effect is the opposite (salting-out). C learly, a positive value of K s in equation (1-1) corresponds to a saltin g -o u t effect, whereas, a negative Ks value indicates salting-in. It is easily recognized th a t the ratio of solubilities expressed in the Setchenow equation actu ally reflects the effect of the salt on the fugacity, f,, of the dissolved gas, which a t equilibrium w ith a given gas phase m ust be th e sam e as for the pure w ater case. Since ff = II{ X ; for th e gas in a d ilu te bin ary

30 8 solution, equation ( 1-1) can be rew ritten as log = K s m s ( 1-2) W here II is H enry s co n stan t of th e solute gas in pure w ater, and H m is H enry s co n stan t in the electrolyte solution. V arious a tte m p ts were m ade to predict th e salting coefficient K s. Section 1-6 discusses m ore on this. 1-6 E lectro static T heory T h e electrostatic theory w as proposed originally by Debye and M ca ulay (1925). It tre a ts the solvent as a continuous dielectric. T he am o u n t of work necessary to discharge the ions in pure solvent of dielectric constant D 0 and recharging th em in a solution of dielectric constant D containing the non-electrolyte is related to the saltin g coefficient I \ s through equations (1-3) and (1-4) : tv &_Np, s xlooo K T D 0 A- a D = D 0 ( 1-0 n) (1-4) w here N 0 is the A vogadro s num ber D is th e dielectric co n stan t of th e solution D 0 is th e dielectric constant of w ater n is th e num ber of m olecules of non-electrolyte solute per cubic centim eter ci is ionic charge

31 a,- i/j is ionic diam eter is the num ber of ions of type i per m ole of electrolyte Species which lower the dielectric co n stan t should be salted o u t by all electrolytes. T his theory predicts th e rig h t order of m ag n itu d e of K, values. However, the predicted K s values vary very little w ith the n atu re of the electrolyte (M asterton, 1970), an d the theory fails entirely to predict a shift from salting-out to salting-in of a nonelectrolyte gas in different electrolyte solutions, such as 7 -B utyrolactone in N ab r and N al solutions (Long, 1952). T h is theory was later refined by Long and M cd evit (1952) to relate the sign of Ivs to the influence of salt on th e solvent stru c tu re by equation (1-5): w here CO CO K _ V,- (Vs ~ Vs ),. Ks 2.3R T /30 V s CO V,- CO V s is th e liquid m olar volum e of salt is th e p a rtia l m o lar volum e of th e non-electrolyte at infinite dilu tio n is th e p artial m olar volum e of th e electrolyte a t infinite dilution f30 is th e com pressibility of pure w ater In th is theory, the Vs value is difficult to evalute, and predictions are usually of incorrect order of m ag n itu d e (M asterton, 1969). T his theory gives relative values of K 4 for different salts w ith th e sam e solute which fall in the correct order. However, the absolute value of Iv5 calculated are in poor agreem ent w ith exp erim en t (M asterto n, 1970). C onw ay, Desbiyers, and S m ith (1964) took into account th e dielectric sa tu ra tio n effect. Each ion is assum ed to have a p rim ary hydratio n shell which contains n w ater m olecules. T hese w ater m olecules are so tied-up w ith the ion, therefore, th a t they lose their av ailab ility to fu rth e r dissolve oth er solute m olecules. O u tside of th is shell, w ater m olecules rem ain effective

32 and the dielectric co n stan t is assum ed to be th a t of the pure w ater. G ood agreem ent was usually found betw een predictions and actu al experim ents for dilute electrolyte solutions, but not a t all a t higher concentrations of electrolyte, since negative solubilities are predicted (T iepel, 1973). For solutions containing tw o salts, the salting effect can t be explained by a single saltin g coefficient nor by a sim ple p roduct of tw o coefficients. T hough there were some m ixing rules proposed by G ordon, and T horne (1967), no theoretical basis to su p p o rt them was given. In the sense th a t salting-out is an effect which increases the activ ity of the dissolved gas in solution, while salting-in decreases it, the salt effect results from th e com bined effect of all interm olecular forces upon the gas molecules. Presum ably, th e m ore detailed and accurate the consideration of these forces becomes, the m ore com prehensive the m odel can be. B ut the cost of additional detail is th a t there are m ore electrostatic p aram eters for w hich values are needed. T h is is th e p rim ary w eakness of all th e electrostatic dependent theories.

33 Scaled Particle Theory T he scaled particle theory of Reiss, Frisch, H elpfand, and Lebowitz (1959, 1960) yields an approxim ation of the reversible work required to introduce a spherical particle of solute i into a solvent. P iero tti (1963,1965) considered the solution process to consist of tw o steps : (A) the creation of a cavity in th e solvent of a suitable size to accom m odate the solute m olecule, and (B) the introduction into the cavity of a solute m olecule w hich in teracts w ith the solvent. H enry s co n stan t is given by : i + a ( i - 6) R T R T + R T 1 0j w here Y x is the m olar volum e of the solvent, g c is the p a rtia l m olar G ibbs energy for cavity $ form ation, gj is the p artial m olar G ibbs energy for interaction, and H, 1 ^ is the H enry s constant of solute 2 in pure solvent 1 at satu ratio n pressure of com ponent 1 a t tem p eratu re T. T h e G ibbs energy, g c, for cavity form ation is a know n function of th e tem p eratu re, the m olar volum e of the solvent, and the hard sphere d iam eters o f the solvent an d solute m olecules respectively. T his theory was later extended by Shoor and G ubbins (1969) to o b tain an equation for the solubility of a non-electrolyte in an aqueous salt solution. T he saltin g out effect was explained as due to changes in the cavity work term. Such changes arise prim arily from nonpolar solute-ion interactions, and not from the ionic charges them selves. M asterton and Lee (1970) applied Shoor and G u b b in s m odel to get a general expression for the salting- out coefficient in term s of the ap p aren t m olar volum e of the salt, and the d iam eters and polarizabilities of cation, anion, and non-electrolyte gas applicable to any salt-nonelectrolyte gas pair. Scaled particle theory suffers from the fact th a t it is only tractab le for a hard sphere fluid. T h e basic assu m p tio n th a t m olecules possess h a rd cores som etim es leads to predicted

34 heats of solution th a t are too high (Shoor, 1969), and for large gas solute m olecules, salting-out coefficients calculated based on this theory are in poor agreem ent w ith experim ent (M asterton, 1970). Schulze and P rausnitz (1981) applied scaled particle theory to correlate phase equilibrium d a ta of aqueous system s over a wide range of tem p eratu re (0-300 C ). A greem ent w ith experim ental d a ta was achieved by allowing a slight tem p eratu re dependence of the m ixing rule on the d iam eter of individual molecules. However, th is work is lim ited to low pressure system s where H enry's law holds. Therefore, as we shall discuss in our param eter fittin g procedure in chap 4, the p aram eters they obtained to fit their solubility d a ta could p redict unreasonable results for o th er therm odynam ic properties. T his m ay explain why the size p aram eter they obtained for carbon m onoxide (0.395 nm ) is m uch bigger th a n th a t for carbon dioxide (0.332 nm ). Choi (1982) com pared the correlation of Schulze and P rausnitz w ith his experim ental d a ta for both m ethane and nitrogen in pure w ater a t high pressure and found very poor agreem ent. Scaled particle theory is an im provem ent over electrostatic theory in the sense th a t it can include the m olecular forces in to consideration. However, the tre a tm e n t is not general enough and an even m ore detailed approach is needed.

35 1-8 Perturbation Theory It is a well known principle th a t when a physical problem can not be exactly represented m ath em atically, one can often m ake progress by a series of successive approxim ations. P ertu rb atio n theory is one way of doing this w ithin the context of solution theory. In the last three decades, num erous efforts have been m ade to apply p ertu rb atio n theory to the u n derstanding of the liquid state. P ertu rb atio n theory for the H elm holtz free energy provides us a m ethod of relating the therm odynam ic properties of a real system to those of a som ew hat ideal reference system through T a y lo r s expansion of the p artitio n function. L onguet-iiiggins (1951) was the first one to expand the free energy of a liquid m ix tu re about th a t of an ideal solution. His m ethod can only be applied to near-ideal solution. Later Zwanzig (1954) showed how to relate the properties of a high tem p eratu re non-polar gas w ith a L ennard-jones potential (Figure 1-1) to those of a rigid sphere fluid (Figure 1-2). None of the above developm ents got m uch a tte n tio n because it was assum ed th a t the p ertu rb ed potential has to be very sm all to o btain m eaningful results. T he a rt of applying p ertu rb atio n theory consists of choosing a reference system w ith tw o p rim ary concerns : (a) it is sim ple enough to handle and (b) it contains as m uch as possible of the physically im p o rta n t p a rts of the real system. U n fo rtu n ately, desiderate (a) and (b) are generally in co m p atib le. T he first generally successful approach was th a t due to B arker an d H enderson (1967 a). T hey recognized th a t the failures of previous a tte m p ts a t low tem p eratu re o f Sm ith and Alder (1959), Frisch et al. (1966), and M cquarrie and K atz (1966) were due either to the lack of a satisfactory tre a tm e n t of the softness of the repulsive forces, w ith consequent extrem e sensitivity to the choice of hard sphere diam eter, or to an u n fo rtu n ate choice of separation into unpertu rb ed and p erturbing potentials. Barker and H enderson used a hard sphere system and chose the hard sphere d ia m e te r of th e reference fluids in order to m inim ize the effect of the

36 14 Figure 1-1 Lennard- Jones potential U(r) Figure 1-2 Hard sphere potential U(r) 0 <7 I \ R cr- r Figure 1-3 Square well potential

37 15 repulsion. T h a t is, they used the freedom of choosing the d iam eter to throw as m uch as possible the effects of the repulsion into the zeroth order expansion term. Barker and Henderson were able to obtain excellent, results even for pure liquids by changing the expansion procedures for both square well potential (Figure 1-3) ( B arker, et al a) and Lennard- Jones poten tial (B arker, et al b). Successful extension of their work to liquid m ixture was m ad e by L eonard, H enderson, a n d B arker (1970). R ealizing th e effect of th e repulsive forces in determ ining th e stru ctu re and therm odynam ics of sim ple fluids. W eeks, C handler, and A ndersen (W -C -A ) proposed a new first order p ertu rb atio n approach in T he prim ary difference between th eir approach and the B arker H enderson approach is the way in which the interm olecular p o tential is divided into an unpertu rb ed and a perturbed p a rt, as shown in Figure 1-4. Instead of separating the potential into positive and negative p arts as in B arker-h enderson theory, Figure 1-5, they sep arated the potential into a p art containing all the repulsive forces where < 0, and a p a rt containing all the a ttra c tiv e forces where > 0. T he separation can be described as follow : U (r) = U (r) + U p (r) (1-7) U (r) = U (r) + c r < i'o ( 1-8) 0 r > ro (1-9) r < r0 ( 1-10) U(r) r > r0 ( 1-11) W here U (r) is the real potential; U is the reference potential; UP is the perturbed potential;

38 u(0 l<t Figure 1-5 Figute 1-4 The split of potential according to Barker-Ilenderson The split of potential according to Weeks-Chandler-Andersen CTi

39 and rq is the location of the low est p o tential energy. Use of the W CA reference system is considered m ore realistic th an a hard sphere system (M cq uarrie, 1973), and the p ertu rb atio n series was found to converge m ore rapidly a t high density (V erlet, et al. 1972). However, since there are no purely repulsive fluids in natu re, the properties of the reference fluid m u st be determ ined from a theoretical calculation. T his som ew hat offsets th e ad v an tag e it has over the B-H p e rtu rb a tio n theory. Lee and Levesque (1973), has extended W -C -A th eo ry to m ixtures. Since the developm ent of the B-H and W -C -A m ethods, m uch work has been done on p ertu rb atio n theory of spherical an d nonspherical m olecules through the use of m odels based on a m ore sophiscated potential or a m ore suitable reference system. G ubbins (1973), G ubbins, Shing, and Street (1983), G ubbins and Tw u (1978 a,b ), and M ansoori and Haile (1983) have w ritten general reviews on these recent developm ents. M ost of the research on this subject has been a t too m icroscopic a level to have m uch engineering application to our system, as we shall see in chapter 3 th a t the evaluation of H elm holtz free energy requires th e knowledge of radial d istribution function and reference system which are not usually available except for very sim ple system. T he evaluation of chem ical potential and hence th e H enry s law constant need even m ore work. As a result, use of p ertu rb atio n theory to predict H enry s co n stan t have been lim ited to very sim ple system s which allow som e sim plifications to be m ade. Neff and M cq uarrie (1973) derived a equation for H enry s co n stan t using the result of Leonard, H enderson, and B arker s (1970) first, order p ertu rb atio n theory. G ood agreem ent was obtained for Neon in A rgon. Uno et al. (1975) included the second order p ertu rb atio n term and applied it to 22 system s of binary m ixtures. T hey found th a t a set of effective K ih ara p o tential p aram eters m ust be found to perform the calculation w ith reasonable accuracy. G oldm an (1977) applied the W CA p ertu rb atio n theory to sim ple binary m ixtures w ith success. All these works were concerned w ith sim ple binary fluids. For our system, containing polar w ater m olecules and electrolytes as well as gaseous com ponents, it seem ed infeasible to consider

40 including the orientation effect and other latest advancem ents in p ertu rb atio n theory a t this stage of p ertu rb atio n theory developm ent. It was hoped th a t these m inute details can be included into a m acroscopic adju stm ent. T h e work of Tiepel and G ubbins (1972 a,b) on the application of p ertu rb atio n theory to the prediction of H enry s law constants in electrolyte solutions appeared to be very prom ising for the developm ent of a global m acroscopic m odel for our specific system of interest. Tiepel and G ubbins (1973) also proposed a m odel based on the second order p ertu rb atio n theory of Leonard, H enderson, and B arker (1970) for the therm odynam ic properties of gases dissolved in an electrolyte solution. T h e final equations closely resem ble those of scaled particle theory, b u t the hard sphere diam eters are tem p eratu re dependent. In th is approach, the H elm holtz free energy of a fluid is expressed as a T ay lo r series expansion about th a t of a fluid m ixture of h ard spheres of different d iam eters w ith respect to a and A, which m easure the inverse steepness of the repulsive potential and the depth of the a ttra c tio n p art of the poten tial, respectively. T h e equations of p ertu rb atio n theory can be m ade to yield those of scaled particle theory as a special case. A d istin ct ad v an tag e is th a t num erical values for m ost of the fu n d am en tal p aram eters needed are readily available from the literatu re or can be correlated from solubility experim ents w ith th e system. A lthough this m odel showed the m ost prom ise over the others, it h ad not yet been tested for com plex system s of the ty p e we were going to deal w ith.

41 1-9 Summary T his research is aim ed a t the problem originally sponsored by D O E : predicting how m uch m eth an e (and other gases) is dissolved a t equilibrium, know ing th e tem p eratu re and pressure of the system, the electrolyte content (salinity) of the gas-free aqueous phase, and the com position (relative m ole ratios) of the dissolved gases in th e geopressured reservoir. In approaching the solution of this problem, we outlined the original objectives of this research as follows : (a) to develop a m odel for predicting H enry s law co n stan t and its associated activity coefficient for each dissolved gas specie using p ertu rb atio n theory to account for the various interm o lecu alr in terac tio n s in th e liquid phase. (b) to incorporate all available experim ental d a ta a t various conditions and use param eter- fittin g techniques as needed to o b tain a global m odel for predicting m eth ane solubility. (c) to gain insight into the various effects of tem p eratu re, pressure, salinity, and other gas com ponents on m eth an e solubility. In ch apter 1, we have introduced th e purpose and relevant background o f this research. V arious aspects and approaches to this general ty p e of problem were review ed, including the Setchnow equation, electrostatic theory, scaled particle theory, and p ertu rb atio n theory. C h apter 2 reviews specifically the sub-system consisting of pure m eth ane gas dissolved in pure w ater or in brine solutions. E xperim ental d a ta on this system are presented. V arious previous correlatio n s are discussed. T h e work o f Coco an d Joh n so n (1981), w hich was the