MULTIPHASE, MULTICOMPONENT COMPRESSIBILITY IN GEOTHERMAL RESERVOIR ENGINEERING. L. Macias-Chapa and H. J. Ramey, Jr.

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1 PROCEEDNGS. Twelfth Workshop on Geotherml Reservoir Engineering Stnford University, Stnford, Cliforni. Jnury 2022, 1987 SGPTR109 MULTPHASE, MULTCOMPONENT COMPRESSBLTY N GEOTHERMAL RESERVOR ENGNEERNG L. McisChp nd H. J. Rmey, Jr. The University of Tuls, Tuls, OK Stnford University, Stnford, CA ABSTRACT Coefficients of compressibilities below the bubble point were computed with thermodynmic model for single nd multicomponent systems. Results showed coefficients of compressibility below the bubble point lrger thn the gs coefficient of compressibility t the sme conditions. Twophse compressibilities computed in the conventionl wy re underestimted nd my led to errors in reserve estimtion nd well test nlysis. NTRODUCTON The expnsion properties of twophse liquidvpor system re different from the corresponding properties of the individul phses. t is common error to ssume tht the expnsion properties of twophse system re volumeiric vergge of the liquid nd gs properties. Mrtin derived n expression relting three phse flow (oil, gs, nd wter) to n equivlent single phse flow. This expression defined Totl System Compressibility tht ws volumetric verge of individul phse compressibilities nd considered solution of gs in liquids. This expression is widely used in well test nlysis nd reservoir simultion. Although this pproch ppers resonble for the segregted flow of the three phses, the true thermodynmic pth for mny expnsion processes requires different verging conditions. Work by Grnt nd Sorey3 indicted tht pprent system compressibility for twophse systems in porous medium could be much lrger thn gs compressibility for geotherml systems. Thus the objective of this pper ws to study twophse compressibilities for single nd multtcomponent systems. n order to perform the study, the chnge in volume for the totl mixture with respect to pressure ws computed with thermodynmic model for flsh system. The thermodynmic model included of mss nd energy blnce, with pproprite thermodynmic reltionships for enthlpy nd equilibrium rtios utilizing the viril eqution of stte. The totl system effective compressibility for multiphse systems for different production modes ws lso computed in this study. The production modes included either gs production, or production ccording to reltive permebilitysturtion reltionships. Results from these clcultions provide informtion on the multiphse coefficient of compressibility nd the totl system effective compressibility useful for the interprettion of well test nlysis nd other reservoir clcultions. Theory nd pertinent literture concerning the coefficient of compressibility nd totl system compressibility will be considered in the next section. THEORY AND DEFNTONS The common kinds of compressibilities considered re: coefficient of isotherml compressiiblity, coefficient of dibtic compressibility, totl system compressibility, nd twophse pprent compressibility. A brief description of ech follows. Coefficient of sotherml Compressibility The isotherml compressibility is point function, nd cn be clculted from the slope of n isotherm of pressure versus specific volume curve, or from differentition of n eqution of stte, nd is defined s: c (l/v)(dv/dp)t... (1) For the coexisting twophse compressibility (gs nt liquid), it cn be shown from pv digrm, Fig. 1, tht the inverse of the slope of n isotherm for the twophse region will be lrger thn the corresponding reciprocl slope of either the gs or the liquid regions. We now turn to considertion of dibtic compressibility..adibtic Compress i b il i t y Mesuring the chnge in temperture nd volume for given smll pressure chnge in reversible dibtic process provides enough informtion to clculte the dibtic 119

2 compressibility, which is given by: cs = (l/v)(dv/dp)h... (2) 5 Keiffer in study of the velocity of sound in liquidgs mixtures, clculted sonic velocities for wterir nd wterstem mixtures tht were smller thn the sonic velo city of the gs phse. Sonic velocity cn be relted to dibtic compressibility by the expression: us = (cs.v)o.5...(3) From Eq. 3, it is pprent tht smll sonic velocity us corresponds to lrge compressibility. The existence of gs or vpor bubbles in liquid reduces the speed of sound in he liquid. This phenomenon ws explined by suggesting tht twophse system hs the effective density of the liquid but the compressibility of gs. This phenomenon is considered to be of importnce in systems with the presence of bubbles from process like gs leving solution. t ws suggested tht sonic velocity mesurements cn be used to me ure compressibility B of gsliquid mixture. Totl System Compressibility Perrine6 presented n empiricl extension of singlephse pressure buildup methods to multtphse flow situtions. He suggested tht improper use of singlephse buildup nlysis for multiphse flow could led to errors in the estimtion of sttic formtion pressure, permebility nd well condition. A theoreticl foundtion for Perrine's suggestion ws estblished by Mrtin.' t ws found tht under certin conditions of smll sturtion nd pressure grdients, the equtions for multiphse fluid flow my be combined into n eqution for effective singlephse flow. ter some mthemticl 3 mnipultions, Mrtin developed to the following definition of totl system isother ml compressibility: sg ( 1 /Bg ) ( Bg/ p) 1... (4) 7 Perrine6, Mrtin', nd lter Rmey, showed tht for multiphse buildup nlysis, the prmeter corresponding to isotherml compressibility in the dimensionless time group should refer to the totl system compressibility of oil, gs, wter, nd reservoir rock, including chnges of solubility of gs in liquid phses. TwoPhse Apprent Compressibility Grnt nd Sorey' considered the fluid volume chnge nd het evolved from rock in phse chnge process with gs production to give n pproximtion of the dibtic twophse pprent compressibility.3 An exmple presented by Grnt nd Sorey showed twophse pprent compressibility tht ws 30 times lrger thn the gs compressibility t the sme conditions. From the proceding, it ppers tht mny fluid thermodynmic fctors ffect multtphse system compressibility. n view of the importnce of this fctor in petroleum reservoir engineering, the min objective of this study ws to consider methods for n investigtion of multiphse compressibility. We now consider the method used in this study. METHOD OF SOLUTON The method of solution to determine the coefficient of compressibility nd the totl system compressibility for twophse, single or multicomponent system will be discussed in the following section. Coefficient of Compressibility Compressibil ty computtions were considered for either singlecomponent or multicomponent system with specified conditions of temperture, pressure, composition, nd frctionl vporiztion for flshing process llowing n increse in volume with fixed decrese in pressure. To obtin the chnge in totl volume with respect to pressure for single nd multicomponent systems, the following procedure ws used: where : c = (l/vmix)(~~mix/~p)t or... (5) VporLiquid Equilibrium clcultions for given system of m components t specified pressure, composition nd temperture were performed with thermodynmic flsh model utilizing the viril eqution of stte, which is pproprite for polr compounds t pressures below 1440 psi ( kp). Further detils re given by McisChp. Totl System Apprent Compressibility n order to clculte the totl system pprent compressibility in similr fshion 3 to tht clculted by Grnt nd Sgrey, the thermodynmic model ws modified to perform constnt volume flsh, fter smll pressure drop within given rockfluid system. Below, or t the bubble point, there is phse chnge in the system, llowing production from fixed volume reservoir. The thermodynmic model ws modified to include porous medium contribution to the enthlpy blnce. Two modes of production were llowed: gs production only nd pro 120

3 duction of both liquid nd gs ccording to reltive permebilitysturtion reltionship. The totl system pprent compressibility is: where AV rod. corresponds to the initil fluid vofume fter the flsh minus the fluid volume remining fter production. The Vpore term represents the fixed pore volume. Figures 2 nd 3 given schemtic representtion of the two production modes considered in ths study. RESULTS Results re presented for ech system studied. First, the coefficient of compressibility is considered, then the production controlled systems. Coefficients of compressibility below the bubble point were clculted for fluid systems rnging from geotherml fluids to hydrocrbon systems. The systems include pure wter, wtercrbon dioxide, severl multicomponent hydrocrbon systems, nd hydrocrbonwter systems. The systems contining pure wter nd wtercrbon dioxide were treted s dibtic. The hydrocrbon nd the hydrocrbonwter systems were considered to be isotherml. Discussion of isotherml systems re given in Refs. 9 nd 10. H20 nd H~OCOZ systemsfig. 4 shows the dibtic coefficient of compressibility of wterstem t n initil pressure of 1440 psi ( kp) together with the coefficient of compressibility of stem t the sme conditions computed with the viril eqution of stte. The dibtic coefficient of compressibility for the twophses is lrger thn the coefficient of compressibility for the stem t the sme conditions, een for smll stem qulities. Similr behvior ws observed for runs mde t initil sturtion pressures of 576 psi ( kp) nd psi ( kp), Fig. 5 nd 6. These results re in greement with thermodynfmtc theory (pv digrm) nd with studies of sonic velocities in liquidgs 1 mixtures. An explntion of this phenomenon is tht liquidgs system hs nerly the density of the liquid but the compressibility of the gs, specilly when the gs quntity is smll. The effect of qu ity is more 4 noticeble t low pressures. n order to study simple twocomponent system, crbon di xide ws dded to wter in 9 the liquid phse. Contmintion of single component system cuses reduction in the coefficient of twophse compressibility. Results for the dibtic coefficients of compressibility for the H20CO2 systems showed the sme generl behvior s the H20 system: The twophse dibtic coefficient of compressibility ws lrger thn the gs (H20CO2) coefficient of compressibility. A comprison of onecomponent, twophse compressibility nd twocomponent, twophse compressibility with gs compressibility, Fig. 7, shows tht the twophse compressibility of singlecomponent system is lrger thn the twophse compressibility of twocomponent system, nd lso lrger thn the gs compressibility for the sme conditions. Totl System Apprent Compressibility ws computed for pure wter system t different sturtion pressures nd rock porosities in n dibtic process for two modes of production: gs production (stem), nd production from geotherml system wherein both wter nd stem re producted s multiphse flow reltive permebility reltionships would dictte. Both modes of production re discussed in the following. Gs Production After pressure drop, some of the initil liquid vporizes. The mount of gs remining in the system fills the volume tht ws occupied by the vporized liquid. The rest of the gs is produced. Singlecomponent wter systems t different initil pressures were studied. These systems include rock component evident through the enthlpy blnce. Figure 8 presents results for system of sturted wter t n initil pressure of 576 psi ( kp) in 10% porosity rock. The computed pprent compressibility is shown versus pressure. As the system depleted, the pprent compressibility incresed. The vlues of pprent compressibility obtined in this process greed with the vlues obtined by Grnt nd S~rey.~ The twophse pprent compressibility ws mgnitudes lrger thn the gs (stem) compressibility t the sme conditions. Multiphse Production For this production mode, liquid nd gs were produced in proportion to reltive permebilities s determined from n verge sturtion. Figure 9 represents system of sturted wter t n initil pressure of 576 psi ( kp) in 25% porosity rock. Apprent compressibility is initilly lrge becuse of chnge in mss due to liquid production nd liquid chnging to gs. Apprent compressibility then decreses becuse gs sturtion increses sufficiently to llow gs to be produced. Apprent compressibility ws much lrger thn the gs (stem) compressibility t the sme condit ions throughout the twophse reservoir condition. From the results shown in Figs. 8 nd 9 for different modes of production, it cn be seen tht the pprent compressibility depends strongly on ths mnner in which reservoir is produced. 121

4 CONCLUSONS A thermodynmic model hs been used to compute coefficients of compressibility nd pprent compressibility for both isotherml nd dibtic conditions. For the cses studied, it ws observed tht the compressibility of twophse system ws lwys lrger thn the compressibilty of the corresponding gseous phse t the sme conditions. The model used cn yield informtion bout coefficients of compressibility nd pprent compressibility for well test nlysis nd simultion of reservoirs below the bubble point. Other equtions of stte pproprite for different fluids cn be used in the model. n ddition pprent compressibility could be lrger thn the coefficient of dibtic twophse compressibility, or the compressibility of the gseous phse t the sme conditions, nd is lso ffected by the wy fluids re removed from given reservoir LiquidLiquid Equilibri, PrenticeHll, McisChp, L. : "Multiphse Multicomponent Compressibility in Petroleum Reservoir Engineering," Ph.D. Disserttion, Petroleum Engineering Dept., Stnford University, Mcis, L., H. J., Rmey, Jr.: "Multiphse Multicomponent Compressibility in Petroleum Reservoir Engineering," SPE 15538, pper presented t the 6lst Annul SPE Fll Meeting, New Orlens, Oct. 58, NOMENCLATURE B C formtion volume f ctor res B/std B; res m 9 3 /st m coef icient of compressibility psi{; ke' ACKNOWLEDGEMENTS Finncil support ws provided by Stnford University, NTEVEP, nd the Deprtment of Energy. REFERENCES McWillim, D. nd Duggins, R. K.: "Speed of Sound in Bubbly Liquids," Proc. nstn. Mech. Engrs., 184(3C), 102, Mrtin, J. C.: "Simplified Equtions of Flow in Gs Drive Reservoirs nd the Theoreticl Foundtions of Multiphse Pressure Buildup Anlysis,' Trns A, 216, , Grnt, M. A. nd Sorey, M. L.: "The Compressibility nd Hydrulic Diffusivity of WterStem Flow," Wter Resources Reserch, 153, , H P S T V 'mix 'pore enthlpy BTU/lb; J/Kg pressure psi ; kp sturtion, frction of pore volume temperture OF; OC sonic velocity ft/s; mls sp5cific yolume or totl volume ft /lb; m /Kg mi ture v lume ft 4 /lb; m 9 /Kg pose vglume ft ; m 4. Sge, B. H. nd Lcey, W. N.: Volumetric nd Phse Behvior of Hydrocrbons, Gulf Publishing Co., "prod chnge in volume due to produc tiy 3 ft ; m 5. Keiffer, S. W.: "Sound Speed in LiquidGs Mixtures : WterAir nd WterStem," Journl of Geophysicl Reserch, 82, 20, X Subscripts qulity, mss of gslmss totl lb/lb; Kg/Kg 6. Perrine, R. L,: "Anlysis of Pressure Buildup Curves," Drilling nd Production Prctice, 482, g P i pprent gs 7.. Rmey, H. J., Jr.: "Rpid Nethods of Estimting Reservoir Compressibilities,1 JPT (Apr. 1964), Prusnitz, J. M., Anderson, T. F., Grens, E. A., Eckert, C. A., Hsieh, R., O'Connel, J. P.: Computer Clcultions for Multicomponent VporLiquid nd 1 = liquid 0 = oi 1 S = dibtic W i wter 122

5 Volume. cu Wlb z ' >. t 2 m v) 0.01 v) W 0 u g z w u LL ll P NTAL = kp T NTAL = 311.C, x GAS COMPRESSBLTY TWOPHASE COMPRESSBLTY _ Fig. 1 Pressure vs specific volume, two component system nheptne npentne, from Sge B. H., nd Lcey W.N4 t Vp Prod. " 0.1 P NTAL 4000 kp T NTAL = 252'C vp V) vporc Vpore Llq (L) lk z w 0 V A S TWOPHASE LL Fig. 2 Gs production mode PRESSURE (kp) Fig. 5 Semilog grph of coefficient of dibtic compressibility vs. pressure, pure wter system. VPOl F vp (V) Lg (L) Note:?=*. LP kl >. t m =! v) v) W U % w u ll LL W 0 0 TWO ~ PHASE =. = GAS one component ( HfJ) P NTAL 930 kp0 T NTAL * 177%..., Fig. 3 Multiphse production mode (ccording to reltive permebilitysturtion reltionship) 123

6 . f ">".rnl Pressure borl Fig. 7 Coefficient of dibtic compressibility vs. pressure, pure wter nd H20C02 systems. " P NTAL 4000 kp0 T NTAL = 252.C POROSTY 10% V w 0.1 TWOPHASE One component ( HzO) 0 z PRESSURE (kp) 41 O Fig. 8 Apprent compressibility vs. pressure for stem production from 10 % porosity rock initilly contining hot wter. v. i.\ P NTAL * 4000 kp0 T NTAL. 252'C POROSTY 25 X one component (Hz $,,,, PRESSURE (kpo) O Fig. 9 Apprent compressibility vs. pressure for multiphse production from 25% porosity rock initilly contining hot wter. 124

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