LBL-36849 UC-250 Lawrence Berkeley Laboratory 1/67531/metadc712514/...I LBL-36849 Lawrence Berkeley Laboratory UC-250 1 2 UNIVERSITY OF CALIFORNIA /- ’# P ‘3 EARTH SCIENCES DIVISION

I

LBL-36849 UC-250

1 Lawrence Berkeley Laboratory UNIVERSITY OF CALIFORNIA 2

/- ’ #

P

‘3

EARTH SCIENCES DIVISION

To be presented at the World Geothermal Congress, Florence, Italy, May 18-3 1, 1995, and to be published in the Proceedings

Numerical Simulation of Water Injection into Vapor-Dominated Reservoirs

K. Pruess

January 1995

Prepared for the US. Department of Energy under Contract Number DE-AC03-76SFO0098

DISCLAIMER

This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency Thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

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i

January 1995

LBL-36849 UC-250

Numerical Simulation of Water Injection into Vapor-Dominated Reservoirs

Karsten Pruess

Earth Sciences Division Lawrence Berkeley Laboratory

University of California Berkeley, California 94720

This work was supported by the Assistant Secretary for Energy Efficiency and Renewable Energy, Geothermal Division, of the U.S. Department of Energy under Contract No. DE-ACO3-76SFOOO98.

Pruess

NUMERICAL SlMuLATION OF WATER INJECTION INTO VAPOR-DOMINATEI) RESERVOIRS

KrnsknPzucss

EarthScicnccsDivisiiLawrcnceBerkeleyLaboratory Univasity of CdZonk, B&q, CA 94720

Key words: Reinjection, Ocysas sttam field, coupled fluid and heat nswS, deaf simulation techniques. '

including the gravitational instability of water over steam (pruess. 1991b). and viscous instabilities at the water-vapor interface

ABSTRACT

Water injection into vapordominated &trvoirs is a means of (Fiagtrald et al., 1994). condensate disposal, as well as a reservoir management tool for enhancing energy recovery and reservoir life. We review different Different conceptualizations have been used in thc mathematical approaches to modeling the complex fluid and bat flow processes modeling of water injection into vapor zones. Early work generally during injection into vapordominated systems, Vapor pressure simplified the reservoir as a homogeneous rous continuum. and lowering, grid orientation effects. and physical dspcrsion of focussed on one-dimwiond horizontal (O'Sullivan and injection plumes from reservoir heterogeneity arc important Pruess, 1980; Schroeder et d.,_1982; Pruess et al.. 1987). Two- considerations for a realistic modeling of injection effects. An dimensional flows including gravity effects and fracture-matrix example of detailed three-dimensional njection interactions wen modeled by Calon et al. (1986). These authors experiments at The Gcysers is given. found that injection plumes tend to slump downward and that

temp- and phase fronts bccome very broad in fractured- porous medm In tht vicinity of the injection point two-phase mnes

1. INTRODUCIlON with low and pressure devdop. while tcmpcratures and pres-= dtcptr and more distant regions of the plume.

Extensive steam production from thc vapordominated at Stcam is gcwated by e hotter portio? of injection p f u s and is Larderello. Italy. and The Geysers. California, has caused a decline consumed in cooler ngrons. giving me to a very efficient heat of reservoir pnssuns and well flow rates, and has led to an transfer mechanism known as "heat pipe." in which liquid is underutilization of installed electric generating capacities, These flowing away from the injection point while Vapor is flowing reservoirs arc beginning to run out of fluid, while heat rescrvcs in towards i' (Won et al.. 1986; PNeSs and Enedy. 1993). Coarse- place are still enormous. gnd studm wen performed by several authors in an effort to

determine reservoir-scale effects of water injection into vapor- ly water-short dominated systems (Shook and Faulder. 1991; Lai and Bodvanson.

. -

of Energy's software betn enhanced with a dient solven. making l0,oOO grid blocks or

CAL PROCESSES

PI - P y = P,&) < o r (1)

* Energy Science and Technology Sofrwore Center, P.0, Box 1020. Oak Ridge, TN37831. .

Pmess is a function of liquid saturation Si and is tamed the suction pnssun. Pnre Vapor pressure above a liquid bdd by capillary or adsorptive forces is reduced in comparison to saturated vapor pnssurc Pa above the flat surfact of a bulk liquid. The duction is exprissed in terms of a vapor pnssun lowaing factor f = PJprat which is given by Kelvin's equation

Here. M, is the molecular weight of Water, pi is liquid phase density, R is the universal gas constant, and temperature T is measured in OC. f depends chiefly on suction prrssun. which in tum is primarily a function of liquid Saturation. Si. At typical vapor- dominated conditions of T = 240 O C the suction pnssurcs rtquked for 1%, 10 46. and 20 % vapor pressure lowcring (i.c., f equal to

-430 bars. Thus. sigplficant ducclon m vapor pnssun w d OCCUT only for very strong suction pnssuns.

In a bulk two-phase mixture of liquid and vapor, vapor p u n depends solely on temperature. while inside porous media the dependence on liquid saturation can become very strong, and can significantly affect vapor pressure rcsponsc to injection. To demonstrate the effects, we consider a fluid-dcpleted matrix block of T = 240 OC. with vapor at a pressure of Pv = 10 bars. At a porosity of 5 %, the block can hold appbximately 40.7 kgm3 of water at full saturation. Suction p u n dationsbips for rcscrvQir rocks at The Geysen and Lardcnllo arc not tly available. We use data obtained by Peters et al. (1984) =-le of tightly welded tuff. designated G4. This has a permeability of 1.9 microdarcies. comparable to unfractund rocks from vapor- dominated systems, so that the suction px!ssurc nlationships may be similar. The TOUGH2 simulator IS used to detcmune the

block in a series of inamental steps. After each injection step the water is assumed to k uniformly distributed throughout the block. Results for the dependence of vapor pressure on mass of injected water arc shown in Fig. 1.

0.99.0.90. and O.SO).are. e ? ' e l y , -19.4 bm, -203 and

pnssure r r s p o w as water of 20 o c ttmperahve is injcued into the

0.0 0.1 1 10

Injected Mass (kglrn?

Figure I. Vapor pressure lowaing (WL) effects in a ZQD-

~ h m vapor pressure lowering is dcglected, the injtacd water is initially completely vaporized, causing vapor pressure to rise snd temperahlre to decline. After injection of about 0.65 kg/d, vapor pressure reaches the saturation pressure Pmt(T). and the block makes a transition to two-phasc conditions. Subsequently vapor pressure is controlled by tempuatm. and both decline upon further injection. When vapor pressure lowcring is taken into eccount. the behavior is quite different. There is less vaporization initially because some of the injected water is adsorbed. Vapor pressure increases during injection arc controlled by increasing liquid saturation and weakening suction and VPL effects according to Eq. (2).

3 3 Grid Orientation Effects

Numerical simulation of injection is subject to grid orientation effects, i.e.. simulation results depend not only on finite diffmnce grid spacing but also on the orientation of the grid nlative to the vertical (Pruess, 1991b). This is demonstrated by modeling injection into the system shown in Fig. 2, which represents a vertical section through a depleted vapor zone. Using "parallel" and "diagonal" grids Fig. 3) nsults in dramatically different predictions

dimcnsionalmanix blocksubjecttowaterinjection.

for injtctioa p~umes @ii. 4). Fig. 5 shows that more consistent (less grid-dcpendent) results can be obtained by using a higher order differencing method ("9-poinc Forsyh and Wasow. 1960).

..L s11-U

Figure 2. Vutical section model for smdy of grid orientation cffm (from Prucss. 1991b).

Tbe grid orientation effect ariscs from crro~s introduced by the finite difference approximation of the gravity flow ttmt Numerical aispeniOn is gumally anisotWpic and depcnds on the orientation of the computational grid rcl?tive to the vertical. 8s well as on the finite diffmnce approxh~~Don uscd (see Table 1).

in finite disercnce grids of square blocks mth side lengtb b (from Pruea, 1991b).

Tpble L HOrizOntal pnd V d d nmeri+dirpersivities ch,

parallel 5-point

The strong grid orientation observed in the parallel 5-pint grid arises from an interplay between gravitational instability and the extremely anisotropic numerical dispersion. For 9-point differencing. as well as for the diagonal 5-point grid, numerical dispersion IS nearly isotropic. so thar grid orientation effects arc reduced. Note that results with less grid orientation arc not ncccssady "bencr, they still contain numerical dispersion effects but avoid obvious inconsistencies simply because these effects are more nearly isotropic.

In order to diminirh the sensitivity to space dis.ci&ation effects and attain a &tic description of the behavior of in'tction plumes, it is necessary to explicitly represent the p h y s d dispersion of liquid plumes from medium heterogeneities (see Mow).

33 Phase Dispersion

Water injection in fractured vapor-dominated reservoin is dominated by gravity effects. which tend to pull the injection plume downwards. However. "sIraight" downward flow is only possible wbcn appropriate permeability is a d a b l e in the verticfzl e o n . . Water flowing downward in sub-vertical fractures IS llkely to encounter low-permeability obstacles. such as asperity contacts bctwccn fraaun walls, or fracture terminations. Water will pond atop the obstacles and be diverted sideways, until other predominantly vcrtical pathways arc reached Fig. 6).

- W e have developed an ap roach that seeks to account for heterogeneity-derived phase &penion by a suitable extension of conventional multiphase flow theory (pruess. 1994). A continuum approach to phase dispersion is formulated in analogy to Fickian

.2

h e s s

rows 1 2 3 4 5 6 7 8 9 1011

. _ ~

1 2 3 4 5 6 7 8 rows

1

2

3

4

5

6

XBL 911-98

XBL 911-97

Figure 3. schematic a f ” p d w and “diagonal“ grias uscd for mdcling injection in ZD vatical section (Rues, 1991b).

Length (m) 0 50 100 150 200

xu. 9110-1129

wgurc 5. simulated injection after 717 days for 9-point ~ ~ g ( R u c s s .

phase dispersion is compared in Figs. 8 and 9. As expected, phase dispersion enhances the lateral and diminishes the vertical migration of injected fluid An obvious imptiation that m g k t of phasedispersive effects may underestimate the potentid for water breakthrough at neighboring produdon wells. Reservoir pre.mre distributions may also be strongly affected. A more detailed discussion is given by fruess (1994).

Reccnt injection experiments ptrfomed by Northern California Power Agency (NCPA) in the Southeast Geysers have shown dramatic pantrns of interference with produaion (Enedy et al.. 1991; hues and Enedy, 1993). During 1990 water was injected into W d d c d 4-2 for periods Of from One to several Weeks at rates of 200-600 gpm (approximately 12-36 kgls). A nearby

3

h e s s

, Q

Figure 6. Schematic of liquid plume descent in a hemogcnwus medium. Impuzucable obstacles arc shown by dark shading (frwa Pmess. 1994).

0

-250

6 0 0

-750

1000 0 50 100 150 200

Radial Distance (m)

Figure 7. Gridding for 2-D R-Z injection problem (Raess, 1994).

production well. Q-6. responded to injection with rapid m n g ratc declines. When injection was stopped production not only

pattern could k repeated over many injection cycles, and (over- )recovery of production was stronger for longer periods of injection

ECOV& but O V ~ ~ - ~ C C O V C ~ ~ ~ . AS shown in Fig. 10 the intafcrcnce

shut-in.

The NCPA test has yielded unique field data on injection- production interference. Replicating these effects would k a were test for the capabilities of numerical simulation models. We have developed a model that attempts to captun in detail the nsuroir conditions and processes deemed responsible for the peculiar observed behavior (Pruess and Enedy. 1993). The strength md rapidity of interference between Q-2 and Q6 suggest that both wells intersect the same frachnts or fracture zones. Accordingly. our simulation model contains a vertical fractun coupled to a large background reservoir (Fig. 11). Heat transftr from the wall rock to the fracture was included. as wen effects of finite wall rock permeability. An "effective continuum" treatment was employed for the fractund-porous background rcsenroir. Our modd involves fully threc-dimtnsional fluid and heat flow, and simultaneously nsolves processes on scales from centimeters to hundreds of meters.

Typical results of our TOUGH2 simulations arc shown in Fig. 12. Prior to start of injection the production well is placed on deliverability. Production is simulated for a five-year period to obtain reasonably stabilized ram. When subsequently injection is started. production rate is seen to decline through a combination of temperature. pressure and relative permeability effects. When injection is terminated production rates not only recover but over- mover. This behavior agrees with the field observations. although no attempt was madc to match them in quantitative &taiL

Radial Distance (m)

F i 8. Injetxion plume (liquid saturation contours) after 692 days, no phase aispasion (Raess, 1994).

- 1 0 0 0 ~ . . . . . I . . . I . . . . I . 1 0 50 100 150 200

Radial Dhnce (m)

Figure 9. As Fig. 8. but uansversc aispcrsivity of 10 m (Ruess, 1994).

The main results from this study can k summarized as follows (for a more detailed discussion see Pruess and Enedy. 1993). (i) Current numtrical modding techniques arc capable of simulating the highly non-linear fluid flow and heat transfer plocesses during injection in considerable detail. even including the complications of flow in highly pumtable fractures. (ii) The most significant reservoir processes during iqjection include gravity-driven downward migration of injected water. local heat exchange between the injection plume and e m 0 2 rock. capillary imbibition of injected water into matrix rock, vapor condensation in cooler portions of the plume. and W i g in the hotter podoh. (iii) In'cction is subject to heat transfer limitations. Cooler portions ofl injection plumes consume large amounts of Icservoir steam. while hotter portions contribute additional steam. (iv) From the standpoint of reservoir management, injection should not be concentrated in a few wells operating at large rates. Better pressure support is achieved by distributing injection among many wells with modest rates. well Wow their capacity for accepting fluids.

5. DISCUSSION AND CONUUSIONS

Aftcr considerable uncertainty md controversy in the 1970s and 80s. the essential role of watcr injection in long-term management and enhanced energy recovery of vapordominated systems is now well aecogniztd at The Geysers and Larderello. Optimization of water injection .and avoidance of detrimental effects remain challenging tasks for the rcscrvoir engineer. cuntntly available simdation techniques give a comprehensive description of the coupled fluid and heat flow proccssts during injection. and arc capable of dcaiing with the complexity of "res' field problems. Recent developments attempt to better represent reservoir heterogeneities. to increase the size of problems that can be handled. and to make capabilities for treating large three- dimensional problems available on "small" computers, such as PCs (A~~ULUZ et aL. 1994). Tpesc advances make numerical'simulation a powcrful tool for injectaon design.

In practical applications. the impact of water injection on nearby production wells is probably dominated by reservoir heterogeneity on a local scale. Detailed forecasting of injection effects appears feasible "in principle," but is limited in practice by our ability to actually characterize reservoir heterogeneity in suufficient detail.

4

Pruess

Figure 10. Injection and production data from wclls 4-2 and Q-6, southcast Geystn (from PNess and Encdy, 1993).

I

€1 8 7

I

c I

I t I

140 300 SOD

12, I 1

Figure 12. Simulated production before, during, and after injection (RUCSS and Encdy, 1993).

B e d , R, C. more, 0. Cqpetti, R. -ti and F. D'Amore (1985). A Three-Year Recharge Test by Reinjection in the Central Area of Ladecello Field Analysis of Production Data. In: ProceeaYngs, I985 International Symposium on Geothennal

OIIS 9. part U. pp. 293-298.

ACfCNOWLEDG

The author is indebted to Dn. Emilio Antuntz and Stefan Fmterle for a critical review of the manuscript and the suggestion Of improvements. This work was supported by the Assistant Sccrctary for Energy Efficiency and Renewabte Energy, Geothermal Division, of the U.S. Lkparrmcnt of Energy under Contract No. DE- AC03-76SF00098. Stanford, California, pp. 161-168.

Calore, C.. K. -ess. and R. Cclati (1986). Modeling Studies of Cold Water Injection into Nuid-Depleted. Vapor-Dormnated Geothermal Reservoirs. in: Proceedings, 11th Workshop Geothennal Reservoir Engineering, Stanford University.

5

RUCSS

Wefsen. N. E. and A. B. C. Aoduson (1943). Thtrmodynamics of

Enedy. K. L. (1992). The Role of Dedine Curve Analys at thc Geysers. In: Special Report No. 17, Monograph on the Geysers Geothermal Field, C. Stone. (ed.). Geothermal ResoUrcts Council. Davis, CA, 1992, pp. 197-203.

Enedy, S., K. Enedy and J. Maney (1991). RescrvoKResponse to Injection in the Southeast Geysers. In: Proceedings. Sirtrmth Workshop on Geothermal Reservoir Engineering. Stanford University, Stanford, CA, January 1991,pp. 75-82.

Finsterle. S. and K. Pruess (1994). Estimating ~wo-Phase Hydraulic Properties by Inverse Modeling. In: Proceedings, Fvth Infernational High-Level Radioactive Waste Management Conference. Las Vegas, NV, May 1994. pp. 2160-2167.

Soil MO~S~UGS. H i l g d . 15 (2). pp. 31-298.

Fitzgerald, S.D.. A.W. Woods. and M. Shook (1994). Viscous Fingers in Superheated Geothermal System, resented at Nineteenth Annual Workshop on Geothermaf Reservoir Enginccxing. Stanford University. Stanfod, CA, January 1994.

Forsythe. G. E. and W. R. Wasow (1960). Finite-Difference Methods for Partial Dflermtial E+aiom, John Wiley & Sons, h.. New York, London.

Giovannoni. A.. G. Allepini. G. Cappetti and R Cclati (1981). First Results of a Remjection Experiment at L a r d d o . In. Proceedings, Seventh Workshop Geothemal Reservoir Engineering. Stanford University. Stanford, CA, December 1981.

Goyal, K. P. and W. T. Box (1992). Injection Recovery Based on Production Data in Unit 13 and Unit 16 Areas of The Ocysers Field. In: Proceedings. Seventeenth Workshop on Geothermal Reservoir Engineering, Stanford University. Stanford, CA,

Goyal, K. P. and W. T. Box (1990). Reservoir Response to Production: Castle Rock Springs Am, East Geysers, califoaaia, U.S.A. In: Proceedings, Fifteenth Workshop on Geothermal Reservoir Engineering. Stanford University. Stanford, CA,

Herkelrath, W. N.. A. F. Moench and C. F. O'Neal 11 (1983). Laboratory Investigations of Steam Flow in a Porous Medium, WaterResour. Res., 19 (4). pp. 931-931.

Hsieh. C. H. and H. J. Ramey, Jr. (1981). Vapor Pressure Lowering in Geoahennol Systems, paper SPE-9926, presented at the 1981 Society of Petroleum Engmeers Califorma Regional Meeting. Bakersfield, CA, March 1981.

International Formulation Committee (1967). A Fomuiation of thc Thennodynamic Properties of Ordinary Water Substance, IFC Secntariat, wisseldorf. Oumany.

Lai. C. €I. and G. S. Bodvanson (1991). Numerical Studies of Cold Water Injection inw Vapor-Dominated G e o t h e d System, paper SPE-21788. resented at the Society of Petroleum Engineus Western 8ational Meeting. Long Beach, CA, Msrch 1991.

Narasimhan. T. N. and P. A. Withmpoon (1976). An rotcgratcd Finite Difference Method for Analyzing Fluid Flow in Porous Media. WaterResour. Res., Vol. 12(1),pp. 57-64..

p ~ . 11-83.

Jan~ary 1992, pp. 103-109.

January 1990, p ~ . 103-1 1 2

Appendix: TOUGH2 Equations and Methods.

TOUGH2 is a general-purpose simulator for nonisothumal flows of NK fluid components distributed among NPH phases. For the case of a single fluid component (water) in two coexisting ph? (liquid. vapor). thc mass and energy balance equations for an arbitrary subdomain V, bounded by the surface ra can be written in the following form.

dt "n rn "0

O ' S u l l i ~ . M. J. and K. h a s (1980). &y& of Injection Tcsting of Gcothumal Resuvoirs. In: Trans-, Geothennol

Persoff, P. and K. Prutss (1993): Flow Visualization and Relative Permeability Measurements in Rough-Walled Fractures. In: Proceedings, Fourth Annual High-Lcvel Radioactive Waste Management Internattom1 Conference, Las Vegas. Nevada.

Grange Park, Ill.

Persoff. P.. K. Pnress and L. Myer (1991). Two-Ph&e Flow Visualization and Relative Permeability Measurement in Transparent Replicas of Rough-Walled Rock Fractures. In: Proceedings, Sixteenth Workshop on Geothermal Reservoir Engineering, Stanford University, Stanford, CA, January 1991.

Petus. R R.. E A. Klaveacr. L J. Hall, S. C. Blair. P. R. Heller and G.W. Gee (1984). Fracture and Matrix Hydrologic Characteristics of TMueous Maferialsfrom Yucca Mountclin. Nye C o q , Nevada. Report SAND84-1471. Sandia National

Rutss, K. (1991a). TOUGH2 - A General Purpose Numerical Simulator for Muhiphase Fluid and Heat Flow . Repon NO. LBL-29400, Lawrcnce Berkeley Labontory, Berkeley. CA.

Capillary Pressun Effects in the Simulation of Wazer InJcchon lnto Deplekd Vapor Zones.

Prutss. K. (1994). Liquid-Phase Dispersion During Injection info Vapor-Dominated Reservoirs. Lawrence Berkeley Laboratory Report LBL-35059. presented at Nin.mth Annual Workshop on Geothermal Reservoir Engineenng. Stanford,University. Stanford, CA, January 1994.

Prucss, K. and S. M y (1993). NLmrcricol Modcling of Injectwn Experiments at The Geysers. Lawrence Berkeley Laboratory Report LBL-33423. presented at Eighteenth Workshop on Geothermal Reservoir Engineering, Stanford University. Stanford, CA, January 2628,1993.

Prueu. K. and M. O'Sullivan (1992). Effects of Capillarity and Vapor Adsorption in the Depletion of Vapor-Dominated G e o t h d Rcsenroin. L: Proceedings, Seventeenth Workshop on Geothermal Reservoir Engineering, Stanford University.

PNeSs, K. C. Wore, R. Celati and Y. S. Wu (1987). .An Analytical Soluhon for Heat Transfer at a Boiling Front Movmg Through a Porou~ Medium. Int. J. of H a ! and Mars Tran$er. Vol. 30 (12).

RcJourcu c o d , 4, pp. 401-404.

April 1993, Vol. 2. pp. 2033-2041, NWlW Society. La

pp. 203-210.

' Laboratones, A l b m t . NM.

Rutss. K (1991b). Grid Mcnt?tio! L

G e ~ t h e m ~ k ~ , Vol. 20 (516). pp. 253-277.

Stanford, CA, January 1992, pp. 165-174.

pp. 2595-2602.

Prutss. K. and G. S. Bodvanson (1984). Thermal Effects of Reinjection in Geothermal Reservoirs with Major Vertical FraaunS. J. Pet. Tech., Vol. 36 (10). pp. 1567-1578.

Schrocdcr, R. C.. M. J. O'Sullivan, K Prucss. R. Celati e d C. Ruffilli (1982). Reinjection Stumes of Vapor-Dommated

Shook, M. and D. D. Fauldcr (1991). Analysis of Reinjection Strategies for The Geysers. In: Proceeakgs. Sixteenth Workshop Geothermal Reservoir Engineerin , Stanford University. Stanford, CA, January 1991. pp. 97-Id

systems. Geothennicr. 11 (2). pp. 93-12Q.

Here M is the "accumulation turn". qresenting mass or internal energy per unit reservoir volume. F represents flux terms, and q sinks and sources (wells). The accumulation terms for mass (m) and heat (h) arc given by, respectively.

Mln = O(SlPI+SVP") (A.3

Mb = ~ ( S l P I ~ I +Svpvuv)+(l-4)PRCRT (A3)

I

6

6

Pruess

For numerical solution. the continuum equations (A.l) arc discretized in space and timc. Space discretization is made with the "Integral Finite Diffyence" method (IFD; Narasimhan and Witherspn. 1976). Tius method permits imgularly shaped grid blocks in 1.2. and 3 dimensions. It includes double porosity. dual permeability. and multiple interacting continua (MINC) formulations for fractured-porous media as special cases. For. grid systems of regular blocks referred to a fmed global coordinate system, the IFD reduces to conventional finite differences. Time is discretized filly implicitly as a first-order (backward) finite difference.

Discretization results in a system of coupled non-linear algebraic equations. These arc cast in residual form and solved simultaneously by means of Newton-Rapluon iteration. Iteration is continued until all residuals arc reduced blow a user-specified convergence tolerance. The bear equations arising at each iteration step arc solved either by direct matrix methods. or by means of preconditioned conjugate gradients.

Hen $ is porosity. S is saturation, p is dcnsity. u is internal energy. C is specific heat, and T is temperature. The subscripts 1. v, and R denote liquid, vapor, and rock. respectively. The . m s f h . F is a sum over the fluxes in liquid and vapor phases. wluh arc wnmn as a multiphase version of Darcy's law, as follows (B = 1. v).

k denotes the permeability tensor. 14 is relative permeability. p is viscosity, Pp is the pressure in phase 8. and g is acceleration of gravity. Heat flux contains conductive and convcctive components:

Fh = -KVT+(hlF1+hvFv) (As)

with K the thermal conductivity of the rock-fluid mixhue. and h the specific enthalpy. Thermophysical propties of water substance arc calculated, within experimental accuracy. from steam table e uations given by the International Formulation Committee (IFC. 1167).

7