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# RESERVOIR FLUIDS PETE 310

Texas A&M

GPA 3.72

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This 20 page Class Notes was uploaded by Dr. Christopher O'Hara on Wednesday October 21, 2015. The Class Notes belongs to PETE 310 at Texas A&M University taught by Staff in Fall. Since its upload, it has received 39 views. For similar materials see /class/225886/pete-310-texas-a-m-university in Petroleum Engineering at Texas A&M University.

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Date Created: 10/21/15

PETE 310 Lectures 34 to 35 e 1 Cubic Equations of State Last Lectures Instructional Objectives El Know the data needed in the EOS to evaluate fluid prope ies Know how to use the EOS for single and for multicomponent systems Evaluate the volume density or zfactor roots from a cubic equation of state for U I Gas phase when two phases exist I Liqu d Phase when two phases exist I Single phase when only one phase exists Equations of State EOS El Single Component Systems Equations of State EOS are mathematica relations between pressure P temperature T and mo ar vo ume 0 El Multicomponent Systems For multicomponent mixtures in addition to P T amp the overall molar composition and a set of mixing rules are needed Uses of Equations of State EOS Evaluation of gas Injection processes miscible and immiscible El Evaluation of properties of a reservoir oil liquid coexisting with a gas cap gas El Simulation of volatile and gas condensate production through constant volume depletion evaluations Recombination tests using separator oil 5 El l Many more Equations of State EOS El One of the most used EOS is the Peng Robinson E05 1975 This is a three parameter corresponding states model aoc VVbbV b re p atfr Equations of State EOS El PengRobinson EOS is a threeparameter corresponding states model I Critical Temperature TE l Critical Pressure PE l Acentric factor n PV Phase Behavior Pressure volume behavior indicating isotherms for J Pr lz pure component 2 runs mm mm Equations of State EOS El The critical point conditions are used to determine the EOS parameters e 6V a 6213 0 KaVZJz Equations of State EOS El Solving these two equations simultaneously for the Pang Robinson EOS provides 2 RT aQa PC andeQb PC 5 Equations of State EOS Where Qa 045724 and Qb 007780 with oc1m1 Jfz m 037464 15422603 02699032 EOS for a Pure Component EOS for a Pure Component n MaxWeH equa area rme I n Q m X m 1 A m 3 U Q g a E ower than T the vapor pressure rs tound when A A 2 El Equatons otState cannot a H be quadratrc po vnorma s A k n Lowest root rs hqu d mo ar N gt V vomne argest root 5 gas 1 24m mo ar voume t El dd eroothasno r phvs ca srgnrncance mun Equations of State EOS El Phase equilibrium for a single component at a given temperature can be graphically determined by selecting the saturation pressure such that the areas above and below the loop are equal these are known as the van der Waals loops Equations of State EOS El PR equation can be expressed as a cubic polynomial in V density or Z 23BilZZ 2 A733 721927 A 7 mp ABiBziBE0 RT 1 with B bi RT Equations of State EOS El When working with mixtures am and b are evaluated using a set of mixing rules El The most common mixing rules are I Quadratic for a I Linear for b Quadratic MR for a NC NC 05 aamzllzllxlxjalajalaj l klj I J El where ku39s are the binary interaction parameters and by definition kij kji kii 0 Linear MR for b NC bm 239191 i1 Example El For a three component mixture Nc 3 the attraction a and the repulsion constant b are given by 0 2 01 sz 17 km 2W Whoever 17 kn Um 0101 an 1 7190 xi dial X azaz ew b x b Jrgb xb Equations of State EOS El The constants a and b are evaluated l Overall compositions z with i 1 2Nc l Liquid compositions x with i 1 2Nc l Vapor compositions y with i 1 2Nc Equations of State EOS El The cubic expression for a mixture is then evaluated using 2 Bm RT RT Analytical Solution of Cubic Equations El The cubic EOS can be arranged into a polynomial and be solved analytically as follows Z3B 1ZZ A 3BZ ZBZ AB B2 B30 Analytical Solution of Cubic Equations D Let39s write the polynomial in the following way X3 a1x2 82Xa3 0 Note X could be either the molar volume or the density or the zfactor Analytical Solution of Cubic Equations El When the equation is expressed in terms of the 2 factor the coefficients a1 to a3 are a1 B 1 2 a2 A 3B 2B 2 3 a3 AB B B Procedure to Evaluate the Roots of a Cubic Equation Analytically El Let 7 3aZ alz 9611612 7 27a3 7 26113 54 S RQ3 RZ T R7Q3 RZ R Procedure to Evaluate the Roots of a Cubic Equation Analytically El The solutions are 1 ST 9 3611 x2 ST ali S T 1 1 1 S T 39 3 S T 9g 2 3611 21 Procedure to Evaluate the Roots of a Cubic Equation Analytically El If a1 a2 and a3 are real always here The discriminant is D Q3 R2 Then I One root is real and two complex conjugate ifD gt I All roots are real and at least two are equal ifD I All roots are real and unequal ifD lt 0 Procedure to Evaluate the Roots of a Cubic Equation Analytically 421P cos 16 flu 3 3 1 a 1 IfDlt0gt xz2 chos 61207 al Where x32 l7 cos 624097 a1 c056i 1in Procedure to Evaluate the Roots of a Cubic Equation Analytically x1 x2 x3 a1 xlx2 xzx3 x3x1 a2 x1x2x3 a3 where X1 X2 and X3 are the three roots Procedure to Evaluate the Roots of a Cubic Equation Analytically I The range of solutions useful for engineers are those for positive volumes and pressures we are not concerned about imaginary numbers Solutions of a Cubic Polynomial P I i We are only an interested in the first quadrant V7b V D Vb 10 Solutions of a Cubic Polynomial El httpwwwunikoelndemathnat hr ml contains Fortran codes to solve the roots of polynomials up to fifth degree Web Site to download Fortran Source Codes to solve polynomials up to fifth degree Mllxrnvllnu 1w mm nilla llurlleuml mm qillllum EOS for a Pure Component Parameters needed to solve EOS Ei Tc Pc acentric factor for some equations ie Peng Robinson El Compositions when dealing with mixtures El For a single component I Specify P and T gt demrmine Vm l Specify T and Vm gt determine P Tartaglia the solver of cubic equations nttp WWW um winFm 3971 WWW Cubic Equation Solver El Only to check your results El You will not be able to use it in the exam if needed El Special bonus HW will be invalid if using this code you MUST provide evidence of work El Write your own code Excel is OK Twophase VLE El The phase equilibria equations are expressed in terms of the equilibrium ratios the K values Dew Point Calculations El Equilibrium is always stated as Al Av xi iP yi i P I 1 2 3 Nc El with the following material balance constrains NC NC Zx 1 Zy 1 Zz1 11 1 K Dew Point Calculations El At the dew point A A xi i 21 i123Nc xiKt Zi Dew Point Calculations El Rearranging we obtain the Dew Point objective function Bubble Point Equilibrium Calculations El For a Bubble point Flash Equilibrium Calculations El Flash calculations are the work horse of any compositional reservoir simulation package El The objective is to find the fv in a VL mixture at a specified T and P such that N 21Ki 1 i11fvKi1 Evaluation of Fugacity Coefficients and Kvalues from an El The general expression to evaluate the fugacity coefficient for component is P RT In if PX EFF o P 741m Evaluation of Fugacity Coefficients and Kvalues from an E The final expression to evaluate the fugacity coefficient of component i in the vapor phase using an EOS is El A similar expression replacing v by is used for the liquid Equations of State are not perfect El EOS provide self consistent fluid properties I Density 0 amp g trends are correctly predicted with pressure temperature and compositions and all derived properties I Same phase equilibrium model for gas and liquid phases material balance consistency Equations of State are not perfect El However predicted fluid property values may differ substantially from da El EOS are routinely calibrated to selected amp limited experimenta da a After calibration EOS predictions beyond ran e of data can be used with confidence El El EOS are extensively used in reservoir simulation What is EOS calibration El Minimization of squared differences between experimental and predicted fluid properties Ndala d I d I Z 9 a 9 2 mm 7 El These Properties g include I Densities saturation pressures l Relative amounts of gas and Iquid phases I Compositionsetc What is EOS calibration u Accumpiisned by changing Within certain iimits mm E05 parameters a Minur adjustments 1 tn 2 ur binary interactiun parameters ku can 20 tn 30 a Different pruperties uftne c fractiun affect iiquid drupuut and densities These pruperties inciude Muiecuiar Weight uncertainty is 7 1n Speci c gravity Criticai pruperti 25 and scentr c femurs wn ch are nigniy dependent an curreiat ans 7 Cannnt be Easiiy measured and nut usuaiiy dune Pre and post calibration predictions from an EOS Pre and post calibration predictions from an EOS Pre and post calibration predictions from an EOS Pre and post calibration predictions from an EOS Special Homework Bonus III Determine the equilibrium ratio of C1 from multiple flash calculations using S Select a m39xture a suitable pressure temperature ra I Discuss the trends how does kgchange with fixed P I Discuss the trends how does kochange wi P a a f39 ed 7 p r I rovide well documenmd graphs Special Homework Bonus El Compare the equilibrium ratio of C1 at 4000 psia and at 200 F with that of the convergence pressure chart using I A mixture of C1 and C2 I A mixture of C1 and C4 I A mixture of C1 and C8 I Discuss the results obtained and provide overlapped plots Calibrate one of EOS s in SOPE to the bubble point data reported by Standings in the following table El Special Homework Bonus m mm unmw smm nun xix ulx lk lan In or Tm mum M mu Special Homework Bonus I Mole fraction of C1 Dew point pressure I Bubble oint pressurpe A Z factors of mixture gas and liquid AMolar volumes of mixture gas amp liquid El All at T 160 F not shown here DIquot 185559305 RP 009013 1507 200 V 400 01001 1 593 19 Special Homework Bonus El Select one EOS El Select the best kij de RK SRK PR that matches the or Cubic4G bubble point El Select one bubble Pressure point pressure for El Compare the one composition of values of met ane experimental vs El Plot pb predicted Pred39Cted m ar V5 inary volumes interaction parameter selected You should be obtaining a plot like this one a uhhle mm Pressure c cl Mixture 1n39l cu 1 v 16quot r Experimental p is 339 psia You CANNOT use this same composition in Your homework Vrtssut psi k Cuhc ac p n 14 n x n x vnw This is the end we survived

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