SIMULATED ANNEALING – ADVANCES, APPLICATIONS AND HYBRIDIZATIONS pdf

Denis Jos6 Schiozer Simultaneous of Reservoir and Surface Facilities Simultaneous Simu1atio:n of Reservoir and Surface Facilities Denis Jos6 Schiozer March 1994 Reservoir Simulation Industrial Affiliates Program (SUPRI-B) SIMULTANEOUS SIMULATION OF RESERVOIR AND SURFACE FACILITIES B Y Denis Josi: Schiozer March 1994 @ Copyright 1994 b Y Denis Josk Schiozer 11 Abstract The main objective of this work is to develop efficient techniques to couple solutions for performance of reservoir and surface facilities, which is a fundamental step for the development of management routines and for optiinization of total system periormance Use of conventional techniques for full-field models can be very computation time intensive and, therefore, such models are either rarely used or they are simplified by handling surface facilities explicitly For this reason, a model with domain decornposition where the degree of implicitness and the timestep size in each domain are related to its own characteristics was developed The reservoir is represented by a Black-Oil simulator with three important features: local grid refinement, domain decomposition, and local timestep in each domain The combination of these three features allows for irnproved predictions over conventional techniques, with no significant increase of the computation time The model for surface facilities is treated as a network problem where nodes, which are defined as wells or groups, are connected by different units which can be wellbores, pipeline segments, or chokes Steady-state multip’hase flow correlations arc used to calculate pressure drop through these units The interaction between reservoirs and surface facilities can have a great influence on the performance of the whole system, and this interaction is greatly affected by the use of chokes in surface facilities Existing choke model:; are evaluated and their effect on the performance of the whole system is studied and discussed iv Different approaches have been investigated to couple production systems to reservoirs Advantages and disadvantages of each method, as well as acceleration techniques, where domain decomposition is used to improve the performance of implicit methods, are presented The efficiency of each method is investigated under different conditions and some examples are presented With these acceleration techniques, full-field simulation can be achieved more effectively and accurately than is possible with conventional techniques The efficiency of this new approach increases for large systems and is naturally suited for parallel machines V Acknowledgements I would like to express my deep gratitude and sincere appreciation t o Dr Khalid Aziz for his support, encouragement, and guidance throughout the course of this study The appreciation is extended to Dr Roland Horne and Dr Thomas A Hewett for participating on the reading and examination committees, and to Dr Nathan Meehan for participating on the examination committee I also wish to thank the faculty, staff, and colleagues of the Petroleum Engineering Department for their contributions to this work Special thanks to Les Dye, Hikari Fujii, Raphael Guzman, K.T Lim, Evandro Nacul, Cesar Palagi, Deniz SumnuDindoruk, Santosh Verma, Chick Wattenbarger, Terry Wong, and Christina Del Villar I am grateful to UNICAMP and FAPESP for their financial support Reservoir Simulation research at Stanford University is supported by the SUPRI-B Industrial Affiliates Program Finally, I would like to express my eternal gratitude to my family for their love, encouragement, and understanding I dedicate this work to my very precious wife * Ad riana vi APPENDIX A PROGRAM DESCRIPTION 102 shift Evaluate increment (E) 158 for numeriacal derivatives 103 simul Solves each local timestep for each subdomain 104 skr Calculates phase relative permeabilities and their derivatives 105 space Calculates distance between two adjacent grid points, and between a grid point and its boundaries 106 srho Calculates phase densities and their derivatives 107 srsgo Calculates gas solubility and its derivatives 108 surcon Checks group constraints and sets control for each subdomain 109 surder Evaluates derivatives of the pressure drop with respect to bottom-hole pressure, and reservoir block pressure and saturations 110 surft Calculates gas-oil surface tension using the Baker and Swerdloff [lo] correlation 111 surmodel Calculates pressure drops for each surface facilities domain 112 svis Calculates phase viscosities and their derivatives 113 switfi Switches from fully implicit to IMPES formulation after first iteration 114 sx Calculates mass fractions and their derivatives 115 tabbin Linear interpolation 116 testdat Checks for consistency of some of the input data 117 tfile Reads data and prepares file to read subroutine 118 transm Calculates transmissibility terms and their derivatives 119 tratab Generates transformation tables €or local and global numbering 120 travers Calculates pressure drop in wellbores APPENDIX A PROGRAM DESCRIPTION 159 121 tstep Timestep control 122 update Updates unknowns after Newtonian iterations 123 velocit Calculates fluid properties 124 visc Determines gas viscosity using Dempsey [29] correlation; estimates oil viscosity using the correlations of Beggs and Robinson [14], and Vasquez and Beggs [76]; water viscosity is estimated from an empirical correlation 125 webopr Calculates bottom-hole pressure for each layer 126 welcon Evaluates well connections and geometrical factors 127 welpro Calculates fluid properties at wells 128 wgcon Gets connections between groups and wells 129 writco Output for connections and geometric factors 130 writin Writes input data 131 zfachy Calculates the compressibility factor using the Hall and Yarborough [8Q] correlation for curve fitting the Standing-Katz reduced pressure-reduced temperature z-factor chart Appendix B Mult iphase Flow Correlations This appendix contains a brief review of multiphase flow in pipes The correlations used in this work are very common in the petroleum literature and therefore, they will not be described here Different correlations were implemented just to test the model for various conditions No attempt was made to evaluate or to compare these correlations One new correlation was developed as a modification of existing correlations in order to ensure smoothness over the entire range of operating condit tions A smooth correlation is necessary when the whole system is treated implicitly, as explained in Chapter B.1 Principles of Multiphase Flow in Pipes The basic idea behind multiphase correlations used to calculate pressure drop in tubings and pipelines is that pressure gradient can be obtained if all energy changes in the system can be predicted A general pressure gradient equation can be obtained, starting with the general energy balance equation for a steady state system, i e , where U’is the internal energy, p is the energy V of expansion or compression, mu2 is the kinetic energy, mgZ is the potential energy, q’ is the heat energy added to the 160 APPENDIX B MULTIPHASE FLOW CORRELATIONS 161 system, W: is the work done in the the system, and and are two generic points of the system The differential form of the energy per unit mass can be obtained by dividing the above equation by rn, yielding The internal energy term can be substituted by using the thermodynamic relation where T is temperature and S is entropy Substituting Eq B.3 into Eq B.2 gives According to the Clausius inequality, for an irreversible process, where Lw represents the losses due to irreversibilities, such izs friction Substituting Eq B.6 into Eq B.4 and considering a pipe inclined at an angle to the horizontal (where dZ = dlsin$ ), and multiplying this equation by p / d L , APPENDIX B MULT1PHAS.E FLOW CORRELATIONS The final expression can be written as "=($) +(-$) +($), dL el ace P.9) f where (2) = psin$, (B.10) gc e6 pvdv (B.ll) acc and (B.123 where the subscripts el, ucc, and f refer to pressure gradients due to potential en+ ergy (or elevation), kinetic energy (or acceleration), and viscous shear (or friction) respectively Single-phase Flow For single-phase flow, the pressure gradient is normally represented as in Eq B.9 where the term due to friction is given by (u; = f PV2 -2g,d (B.13) where f is the Moody friction factor which is a function of Reynolds number and pipe roughness APPENDIX B MULTIPHASE FLOW CORRELATIONS 163 Two-phase Flow The analysis of the pressure gradient equation is much more complicated when multiphase flow occurs The phases may travel at different velocities, and they may form different flow patterns In such cases, the pressure gradient equation must be modified The flow is treated as two-phase flow where water and oil are combined into a liquid phase The pressure gradient equation can be modified for two-phase flow and written as (B.14) where total fluid density, or the slip density (p,) is used in the elevation component, and different values of f f , p f , uf, pact, and vaccare defined for different correlations Basically, for the elevation component, an important part is the correct determination of liquid holdup ( H L ) which is used in the definition of pm For the friction and in the determinacomponent, the methods differ widely in the definition of t ~ f tion of the friction factor ( f f ) Little attention has been given to the acceleration component in the literature and, at times, it is even ignored in some flow patterns One important consideration in the determination of these parameters is the flow pattern which must be predict for different sets of conditiclns A common problem is that many methods have different models for different flow patterns which causes the correlation to be discontinuous at the interface between two different flow patterns For such cases, convergence problems may occur in the entire simulator, and if discontinuities are large, convergence may not be achieved Calculation Procedure The pressure drop calculation normally consists of an iterative procedure where the conduit is divided in pressure or length increments arid the pressure gradient is evaluated at average conditions Basically, for each segment, the first step is the APPENDIX B MULTIPHASE FLOW CORRELATIONS 164 flow pattern determination followed by holdup and then presijure gradient calculation The accuracy of the pressure drop determination can be improved by increasing the number of segments, but the computer time also increases, especially when such calculations are performed many times, inside loops of other iterative procedures For more details about the calculation procedure and fluid properties determination, see Beggs [13] B.2 Vertical Flow The correlations used for vertical flow are presented in Table B.l Table B.l: Multiphase correlations for vertical flow Correlation Poettmann and Carpenter [61] no slip, no flow regime Hagedorn and Brown [42] slip, flow regime Orkiszewski [53] slip, flow regime Aziz, Govier and Fogarasi [7] B.3 Horizontal and Inclined Flow The correlations used for horizontal and included flow ase : Poettmann and Carpenter [61], Beggs and Brill [18], Dukler et d [30], Eaton et d [31], PanhandleFlanigan [36], or a combination of these correlations (see 4ppendix A for possible combinations available in the program) B.4 Discontinuities In Chapter 4, it was observed that most of the existing correlation are not smooth over the entire production conditions The most common reasons for discontinuities in the models are presented next APPENDIX B MULT1PHAS.G FLOW CORRELATIONS 165 Flow regime consideration To consider different flow regimes can improve the accuracy of multiphase flow correlations in pipes but it can make them discontinuous if different models are used for each flow regime Chokemodel Discontinuities in choke models can occur in the transition between critical and subcritical flow Iterative procedures Any iterative procedure in the surface facilities model has to used with caution because a large tolerance may cause discontinuities in the model while a small tolerance may increase significantly the total computation time Fluid properties calculation Many of the existing fluid properties correlation not have a smooth transition when one phase appears or disappears Friction factor calculation Moody Friction factor is not continuous in the transition from laminar to turbulent flow B.5 A Smooth Correlation In order to test the coupling of surface facilities and reservoirs with a smooth correlation, a new correlation was introduced in this work The correlation is similar to the Poettmann and Carpenter [61] correlation, with no flow regime consideration Fluid properties and friction factor subroutines were modified to avoid discontinuities Another modification was made in the viscosity definition which was proposed by Beattie and Whalley [la] to account for different flow regimes They have proposed a hybrid definition of viscosity, APPENDIX B MULTIPHASE FLOW CORRELATIONS 166 (B.15) which replaces viscosity definitions for bubble and annular flow The homogeneous void fraction P= ( p ) is given by PLX PLX + pg(l - x) (B.16) * where x is the mass fraction of gas This correlation was not evaluated or compared against other correlations or against any existing data The purpose of this correlation is only to test the implicit method described in Section 6.4 and to solve the production facilities model as described in Section 4.4.2 Appendix C Example Problems C.1 Example The first example is the SPE Comparative Problem Number [51] with one modification: the well constraint is the well-head pressure instead of the oil rate as in the original problem The system chara.cteristics are presented in Tables C.l, C.2, and C.3 The producer is completed in the bottom layer of it corner gridblock (1,1,3) and the injector is completed in the top layer of the corner gridblock opposite to the producer ( l O , l O , l ) C.2 Example The second example is a three-dimensional case with connected surface facilities where the system constraint is the separator pressure (Fig C.1) Surface facilities characteristics are presented in Table C.4 The reservoir characteristics are the same as in Example with the following modifications: the grid is 19 x 19 x 3, injector at the center grid.block (400 MMscf/d), and four producers at the corner gridblocks controlled by the surface facilities 167 APPENDIX C EXAMPLE PROBLEMS 168 Table C.l: Reservoir data for Example B Y B W CT hbottom hop kmed Lz Pi Pb Qinj Swi SWT Pitd std Pw Pitd std Pw P psia 14.7 264.7 514.7 1014.7 2014.7 2514.7 3014.7 4014.7 5014.7 9014.7 0.167 bbl/SCF 1.041 bbl/STB 3x psiia-l 50 ft 20 ft 50 md 10000 ft 4800 p i a 4014.8 p i a 100 MMscf/d 0.12 % 0.12 % 0.008 cp 0.31 cp 0.0647 lbm/ft3 62.238 lbm/ft3 1.062 bbl/STB 1.3687 x ft x IO-6 pia-' 30 ft 200 md 500 md 10000 ft L4.7 psia 1000 psia 0.00 % 0.02 % 0.30 % 1.04 cp 0.30 % 46.244 lbm/ft3 10 x 10 x Table C.2: Fluid properties for Examplle -_ B W B O Pw P O Pg BY bbl/STB bbl/STB bbl/scf C P -C -P 1.062 0.166 666 1.0410 1.040 0.080 1.150 O 12093 1.0403 0.975 0.00916 0.31 1.207 0.006274 1.0395 0.910 0.011'2 0.31 0.003 197 1.0380 0.830 O 1410 0.31 1.295 0.00 I 614 1.0350 0.695 0.0 18'9 0.31 1.435 1.500 0.00 I 294 1.0335 0.641 0.020'8 0.31 1.565 0.00 I 080 1.0320 0.594 0.022'5 0.31 1.695 0.000811 1.0290 0.510 0.0268!3 0.31 0.449 0.030'9 0.31 1.827 0.0 0 649 2.357 0.04710 0.31 0.0 0 386 1.0130 0.203 - I & 90.5 180.0 371.0 636.0 775.0 930.0 1270.0 1618.0 2984.0 169 APPENDIX C EXAMPLE PROBLEMS Table C.3: Production facilities data for Example [ Connecting I abs roughness I incl I did.=\ hh-I BH bottom-hole w3 r;;7 Reservoir w4 Figure C l : Example APPENDIX C EXAMPLE PROBLEMS 170 Table C.4: Surface facilities data for Example Connecting node node incl BH WH 90 10 Gl/G2 abs roughness (in> WH 0.0008 Gl/G2 0.0008 G3 0.0008 node ("> I 30600 10600* Oe5 1.0 :mperature (\ O F I) BH WH G C.3 bot tom-hole well- head surface group 2100 1100 60 Example The third example includes Local grid refinement around wells to improve accu+ racy in the near-well region, and to shown the advantages of domain decomposition The system is schematically represented in Fig C.2 Surfface facilities characteristics are presented in Table C.6 Most of reservoir characteristics are the same as in Example with the following modifications: two-dimensional case, two producers at opposite corner gridblocks, refinement' in the well regions, and three different cases are described in Table C.5 'The refined region is one coarse block for the injector, and it occupies the same region in the producer subdomains, ie., one coarse block in Case A , four coarse blocks in Case B , and sixteen coarse blocks in Case C Each coarse block has a x refinement Cartesian refinement APPENDIX C EXAMPLE PROBLEMS 171 w1 Figure C.2: Example Table C.5: Reservoir data for Example Case A B C grid - refined coarse blocks each producer injector x 19 x 19 39 x 39 16 total number of gidblocks 2346 I 172 APPENDIX C EXAMPLE PROBLEMS Table C.6: Surface facilities data for Exarnple node Temperai ure ( 200 100 60 ... or in different processors Several authors (Wasserman [79], Killough and Wheeler [43], Brand and Heinemann [16], and Ewing and Lazarov [34]) have developed domain decomposition techniques at the... surface facilities and reservoir models A description of other topic$ related to reservoir simulation can be found in Aziz and Settari [8], Mattax and Dalton [46], Peaceman [55], and Crichlow [26]... this study The appreciation is extended to Dr Roland Horne and Dr Thomas A Hewett for participating on the reading and examination committees, and to Dr Nathan Meehan for participating on the