A Study on Effects of Initial water Saturation in The Reservoir on well characteristics

by

Kua'SaNt'Ryuichiltot,ToshiakiTnNarcnandMichihiroFurupe'(Received)

Abstract

The elfects of initial water satlrration in two-phase geotherrnal reservoirs on well

characteristics have been evaluated. A vertical wellbore model of uniform diameter

coupled with a radial horizontal flow in reservoir of uniform thickness was employed'

Momentum eqr-ration for two-phase flow in a wellbore was nttmerically evaluated with

a methocl introduced by orkiszewski (1967). Energy equation in the wellbore was

assumed to be isenthalpy. Mass flow rate and pressi;re at a feed zone of the well was

calculated using a steady radial flow in a reservoir. The numerical calculation showed

that the lower initial water saturation in two-pirase reservoir leads to less total flow rate

at a given wellhead pressure. Holvever, the decrease in the initial water saturation

results in the increase in the maximum discharge pressure. The initial water saturation

does not affect the steam flow rate significantly for low wellhead pressure.

Keyworcls : Two-phase reservoir, Water saturation, Well characteristics, Wellbore flow

1. Introduction

ln the course of geothermal developrnent for power generation, understanding on well

performances and evaluation of well deliverability are of imporlant task for reservoir

management. This is because that predicting steam discharge rate from wells and evaluating

the effects of reservoir conditions on the steam flow rate provides valuable information for

designing a size of power plant. As a prodllction stage of the field begins, stable production of

steam from wells is often impeded by problems such as declines of reservoir pressure and

temperature.

A wellbore simulator is a powerful tool for evaluating well performances such as

pressure and temperature distributions in a well ancl well deliverability. Wellbore simulators

basically solve a set of mathematical equations that describe two-phase flows of steam and

water mixture in a wellbore. ThLrs, the simulators can be used for evalLrating the effects of

well specification such as diameter ancl depth on well performances' Furthermore, reservolr

conditions give effects on welI performances; thus, reservoir fluicl flow in the vicinity of wells

should be inclucled in the rveltbore simulator for providing reliable results'

l-here are several papers discussed or reviewecl about r'vellbore simttlator with their own

capabitity. Bjornsson and Bodvarsson (1987) developecl a mLrlti-feedzone wellbore simulator

that allows for calculations of downhole pressure, temperature and quality in wells with an

arbitrary number of feed zonest). [t can simulate fluid parameters in well during both

discharge and injection and evaluate internal flow and relative contributions of feedzones'

Another mgltipurpose wellbore simulator was developed by Hadgu and Freeston (1990)2)'

The simulator oan handle the flow with dissolved solids and gases and allow multifeed well

simulation when adeqr-rate information on feed zones is available' It also includes heat loss

and different geometry of wells with different pipe roughness' The two simulators above have

not included a flow in reservoirs in their model. Takahashi (1999) developed a wellbore

simulator that was coupled with a reservoir flow and can handle single and two-phase flows

in the reservoir3). He does not consider distributions of density and viscosity of two-phase

fluids in the vicinity of weil. Hadgu et ar. (1995) described the weilbore simulator (WFSA)

coupled with reservoir simulator (TOUGH)4). The cor-rpled simulator is capable, for example'

of determining pressure and mass flow rate verslls time and evah-rating power output scenarios'

The model of Hadgu et al. canhandle both initially the single and two-phase conditions in the

reservoir. The above tr,vo simulators presented the samples of deliverability curves as a tool

for evaluating well performances.

one of the pLlrposes of wellbore simulation is to evaluate well performances expressed by

its deliverability curve as discussed by several papers: Gudmundsson (1984)' Brennand and

watson (1987), Garget al. (2000)and Gutierrezet al' (2001)s)-8)' The last paper includedthe

reservoir model and discussed on computed deliverability cLlrve to evaluate the well

characteristics and pressure profiles irr wellbore. The main feature of the model is its use of

two-phase homogeneous flow model in wellbore. A fypical deliverabitity curve obtained from

field measurements of Well S-4 at the Sumikawa geothermal fielcl in Japan was presented by

Garg (1995)e). The similar curve was observed on Well 703 at Rotorua, New Zealand

(Brennand and watson, lg87). The obtained deliverability curves from measllrements were

fairly similar to those produced by above wellbore simulators.

The study on two-phase reservoir is imporlant because reservoirs of initially single liquid

phase and of high temperature have easily evolved to two-phase conditions as reservoir

pressure declines with development. The wellbore model above can be divided into two

groups: models including and not including reservoir flow' Although the former model can

handle two-phase conditions in reservoir, the effects of water satttration of reservoir on well

characteristics have not been discussed in detail. This paper, therefore, deals with the efflects

of initial water saturation in the fwo-phase reservoir on well characteristics using a wellbore

model that includes steady radial flow in a reservoir'

-2-

2. Governing Equations

The governing equations consist of those for wellbore that are cor-rpled with the fluid flow

in a reservoir. The schematic diagram of the reservoir anc.l wellbore rnodel is ilh-rstrated in Fig'

1.S

Fig. 1 Schematic diagram of reservoir-wellbore model'

Equations for reservoir

Basic equations for fluid flow in the reservoir are derived under assumptions as follows

(Itoi etal., 1988)ro)

1) reservoir is of radial symmetric and horizontal with a constant thickness

2) fluid flow obeys Darcy's law and is under steady state

3) no heat exchange between fluid and reservoir rock and the flow follows isenthalpic

process.

Continuity equations of mass and momentr"rm in the reservoir are expressed as:

I O(ru)

rark

Lt --

=Q

0p

(1)

(2)0r

mass flux density (kg/m2s),

--)-

vt

(m), a is the

$"

where r is the radial distance fr is the permeabilitY

(^'), r,is the total kinematic viscosity of two-phase fluid im2/s; andp is the pressure (Pa).

Mass flux density is the sum of steam and water flux densities (ll, and zr',) and written as

Here kn, and k,,are the relative permeability to water and steam (-), respectively, v,,andv,

are the kinematic viscosity of water and steam 1m2/s;, respectively.

Then, the total kinematic viscosity, r 1, Q?Ttbe defined as (Grant et al., 1982)11)

I kn,.k,,vt v* vs

From the definition of two-phase flr"rid enthalpy (Grant et al,, 1982), relative permeabilities

can be expressed using enthalpies ofsteam and water:

Ll=Lt\, +Lls

(kk^, kk,.,\ap=-l---{--l-

f. u,, v, ) ar(3)

(4)

k-".-=k,,,

vr(i,, -i)v,r(i-ir)

where i is the total enthalpy of fluid (J/kg), I'u is the

enthalpy of steam (J/kg).

Relative permeabilities can be expressed as a function

fractured media. In this study, the correlation between

X-curve that satisfies the following expression:

k,."+k,.r=1

then, v 1 can be expressed as:

I (i* -i,)vt v,r(i - i, ) +

Flow rate of two-phase mixture entering the

G = -uAl,;= 2m*

(5)

enthalpy of water (J/kg) and i is the

of water saturation in porous media or

relative permeabilities is expressed in

(7)

feed zone can be calculated by:

(6)

vr(ir,, - i)

wellbore at the

,( r ap\lv,0r)\ r /t

-4-

(8)

wittr boundary conditions written as:

t=t'v i p:pvb

f=f'e i P:PuWherer,uisthelvellboreradius (m),h isthethicknessof reservoir (m),A isthesLrrfacearea

of the feed zone (m2;. Subscript e denotes the outer boundary of reservoir. The pressure

gradient at r: r,,, in Eq. (8) is catculated lrom the solution of Eq. (i) under the boundary

conditions above. Because the total kinematic viscosity in Eq. (8) is a fr-rnction of either

pressure or temperature under saturated conditions, Eq. (8) is expressed in the integral fonn as

where R, is the normalized distance of r, (R,=ln(r"lr,,)). The integral part in Eq. (9) is

numerically evaluated with Simpson's method

Wellbore model

Basic equations for two-phase flow in the wellbore are derived under assumptions as

follows:

1) the fluid flows into the well from a single feed zone at well bottom

2) the well is vertical with uniform diameter

3) the fluid florv in the well is under steady state and isenthalpy

The basic equations r-rsed for two-phase flow in the wellbore consist of conservation of mass

and momentllm as follows:

lpe 0p

G = 2okhJ'*u \Rn

(e)

dM

--ud((t0)

(11)AP, -\aP" + aP, + Nr|=o

where M is the total mass flow rate (kg/s), I is the depth coordinate (m), AP, is the total

pressure drop (Pa), AP, is the pressure drop due to acceleration (Pa), APnis the pressure

drop dLre to potential (Pa) and APyis the pressure drop due to fi'iction (Pa).

Pressure losses evaluation

-5-

The classification of flow regimes for two-phase flow proposed by orkiszewski (1967)12),

as bubble, slug, transition and mist, is used for evaluating pressLlre losses. Each regime has its

or.vn formula for calculating void ratio. using this void ratio then the mixture density can be

determined. All pressLrre drop components require the mixture density to calculate their

values.

1) Pressure drop due to potential 1 A P n)

potential loss for vertical two-phase flow in a small length increment A L (m) is given as

where p n, is the mixture density (kgim3) and g is the gravitational acceleration (mls2)' a is

the void ratio (-) that is determined using the method by orkiszewski.

2) Pressure drop dure to acceleration ( A P,,)

Acceleration loss for the two-phase flow with the specific volume (vr,) between two adjacent

elements at points below and above of the length increment A L, ind\cated by sLrbscript 1 and

2 respectivelY, then

APn = ,, t -lu,,z)

where F is the cross sectional area of the well (m2)'

3) Pressure drop due to friction ( A Pf)

Friction loss is calculated using the following equation

AP1, = pr,.B.N

P,, = dPg +(t - a)p,

AP'=)''' M'^AI'r 2D p.,A'

t (s/D 5.7J)ll

SLos,ol n .W ))

roughness (-) and Re is Reynolds number (-)' Re is

where average kinematic viscosity and velocity are

(12)

(13)

( l4)(f)'t"

(ls)

where D is the well diameter (m). 2,,, is the friction factor for two-phase flow o and

evaluated using the equations proposed by Swamee & Jain (Lindeburg, lggD:t3)

0.2 5ln, = (16)

where t is the PiPe

two-phase flow flLrids

calcr-rlated for

evaluated for

-6-

respective fl olv regimes.

3. Calculation Procedures

3. 1 Input parameters for calculation

The parameters for reservoir and wellbore used in this study are summarized in Table L'

A reservoir temperature is given to be 2800c. T'he corresponding saturation pressure with this

temperature is 64 bar. To evaluate the effects of permeability thickness (/cft) on well

characteristics, two values are used, i.e. 3x l0-12m3 (3 darcy-m) and 5x 10-12m3 (5 darcy-rn).

The initialwater saturation (Srui) is given as 0.5 and 0.1' The roughness of inner surface of the

wellbore is taken as that of commercial steelpipe'

Table 1 Reservoir and lvellbore parameters.

Parameter Value

Reservoir

Temperature (f 280 cc)Pressure (P) 64 (bar)

Horizontal Extent (r") 50 (m)

Permeabil ity Thickness (&ft ) 3x1o-12,5x10-121m3;

Initial Water Saturation (Su'i) 0.5, 0.1 (-)

Wellbore

Diameter (D) 0.2 (m)

Length (Z) 1000 (m)

Roughness (e) 0.0000a6 (m)

3. 2 Calculation procedures

The first step of calculation is to read input data for both reservoir and wellbore. Then,

the feed zone pressllre, Pwb, is calculated using Eq. (9) under a specified mass flow rate' As

Pwb is implicitly expressed in the integration term of Eq. (9), the bi-section method was

adopted to find the value of Pwb that satisfies Eq. (9). The obtained Pwb together with the

mass flow rate are used as input data for wellbore calculation.

The wettbore calculation basically computes the total pressure drop in the wellbore that

depends on the flow regimes. A small pressure incrementAP (0'5 bar) is given, and a

corresponding length increment lZ is calculated. This procedure stafis from the feed zone at

pressure ptvb, and is repeated until the summation of N reaches to the total length of well'

Then, wellhead pressure, PtvH, is obtained. Fluid thermodynamic properties are calculated

with pRopATH (ito et a\.,1993)14) and are evaluated for average values using pressures at

both ends of the segment. Then, the pressure, enthalpy and dryness fraction at the wellhead

-1-

are finally attained.

4- Results and Discussion

4.1 Deliverability curves for different permeability thickness (kh)

Figure 2 shows deliverability curves for different permeability thickness, kh, with

parameters of initial water saturation of reservoir, Swi. Two cases of permeability thickness, 3

darcy-m and 5 clarcy-m, are considered. Swiare given to be 0.1 and 0.5. Deliverability aLlrves

represent relationship between mass flow rate versus wellhead pressure, which can be used to

evaluate well prodr,rctivity as well as well characteristics.

Tr= 280'CD=0.2mL=1000m

S v ts0.5

150150

s

100a

OJtt- 50,lt-oF

0

..100rF

OJt'- 50JFOF

r0 20 30 40 50 60

lVELLHEAD PRESSURE 6AT)

(a) kh: 5 darcy-m

010203040t{ EI,I-HEAD PRESSURE

50 60

6ar)

(b)kh:3darcy-m

Fig. 2 Effects of initial water saturation in reservoir (.Swi) on deliverability curves'

As shown in Fig. 2(a) for kh=5 darcy-m, total mass flow rate for Swi:0.5 gradually

decreases with an increase in wetlhead pressure, Prys,btl it quickly starts decreasing at about

ppvrr31 bar. After the curve reaches the maximum wellhead presslrre at about 37 bat, both the

wellhead pressure and the flow rate start decreasing. On the other hand, the curve for Swi:O.1

shows smallchanges in flow rate overawide range of wellhead pressllre. Small flowrates as

ptvs decrease in both curves may not be realized in practical sitr-rations as a flow in wellbore

become unstable. Relationship between wellhead pressure and total flow rate represents

similar curves regardless to the values of kh as shown in Fig. 2(a) and (b). The lower kh,

however, gives lower flow rate in general.

The water and steam flow rates versus wellhead pressure are illustrated in Fig. 3(a), (b)

for kh:5 darcy-m, and Fig. 4(a), (b) for kh:3 darcy-rn. The curves for water flow rate versus

wellhead pressure show sirnilar relationship as the case of the total flow rate. However, small

decline in the water flow rate can be seen as Ppyp decreases in both curves of Swl. On the

other hand, both curves of steam flow rate show the highest values at the lowest Pwu in a

pressllre range as shown in Fig. 3(b) and Fig. 4(b), then decreases linearly with an increase in

-8-

Tr = 280 "CD =0.2mL=1000m

S w ts0.5

S v tsO.1

Ptvrr up untit it reaches to the maximutn value' High values of steam flow rate atlow Pwu

result in a slight increase in total flolv rate in loi.v Pyplrange as shown in Figs. 2(a) and (b) in

spite of decreases in the water flow rate as Ppvs decrease'

40

330rdFEzoOJ

=10E]Fa

0

r50

p

r,rl00F&

OJ.^IL DU

rllF

0

kh = 5 D-mD =0,2mL =1000m

S v p0.5

Sv ts0.1

l0 20 30 40 50

lV ELLHEAD PRESSLTRE bar)(a) WHP vs Water Flow Rate

10 20 30 40 50

\VELLHEAD PRESSURE bAT)

(a) WHP vs Water Floiv Rate

Fig. 3 Well deliverability curves for different Swi (kh:S darcy-m)'

kh=5D-mD =0.2mL =1000m

S w i'0.1

S w ?0.5

l0 20 30 40 50

WELLHEAD PRESSURE 6AT)

(b) WFIP vs Steam Flow Rate

kh = 3 D-mD =0.2mL = 1000m

Swt 0.1

S w r0.5

l0 20 30 40 50

WELLHEAD PRESSURE bAd

(b) WFIP vs Steam Florv Rate

330LllF

ezoOJ

=r0tdFU)

150

r,r 100F

Jtr- 50

rilF

0

Fig.4WelldeliverabilitycLlrvesfordifferentSwi(kh:3darcy.m).

4.2 Behaviors of deliverability curves

Figure 2 shows that the total flow rates for lower initial water saturation,,Swi:0.1, are

always below those with higher Swi:0.5 for relatively high total flow rates. [n other words, as

the flow rate increases the deliverability curve for Swi:0.1 intersects tl'rat for Swl:0'5, then it

goes further below it. To explain this situation, the relationships between feed zone pressures

and total flow rates for different Swi are examined. All curves in Fig. 5 indicate that the

decrease in the feed zone pressures, PwD, results in the increase in the total flow rate

regardless of Su,i. At a given Pwb,the smaller the Srui is, the less total flow rate will be. It is

kh = 1l D-mll =0.2mL =1000m

S v'ts0.5

S w ts0.1

-9 -

obvious from Eq. (9) that the total flow rate depends primarily on two dependent variables;

feed zone pressure (Pvtb) and integrated total kinematic viscosity ( z,). Distribr-rtions of v,in

the vicinity of wett for different values of Slr,l are presented in Fig. 6. It can be seen thatthe v

, of ,Swi:O, 1 shows h igher values compared with that of Srvi:0.5 at a given mass flow rate (50

t/h). Mass flow rate from reservoir into wellbore depends on permeability thickness, total

kinematic viscosity and pressure gradient at wellbore radius as indicated by Eq' (9). This is

the reason why the decrease in ,Swi causes the decrease in the total flow rate at a given feed

zone pressure as shown in Fig. 5.

^70!r$3r65

f:lzR60aGl tc

<b0

:-45aCrl

Kso

.> 70

-ooc

rizR60uEb5B.

k50rd3ss(notilcE40o-

o

'oaAx^

F.)(nO^t) 'z(n

1

'tr:= 280 "C

D =0,2mL = 1000m

20 40 60 B0 100 120

TOTAL FLOW RATE (/h)

(a)kh:5darcY-m

Tr = 280'CD =0.2m

S l'F0.1

o 20 40 60 B0 100

TOTAL FLOlv RATE Vh)t20

(b) kh :3 darcy-m

Fig. 5 Pressure at feed zone vs total flow rate.

G=50t/hS w r0.l

kh = 5 D-m

Pr= B0 bar

s * tso.s Tr = 280 "c

RADUS (n )

Fig. 6 Viscosity distributions in the reservoir for diffurent Swl'

4.3 Characteristics of flowing pressures in wellbore

The pressure profiles in wellbores are presented in Fig. 7 for two different wellhead

pressures, Pryp, of 35.1 bar and 10,1 under the conditions of two values of initial water

saturations; Swi:0.5 (o, r) and 0.1 (o, r). Circles represent the high wellhead pressure

whereas squares represent the low lvellhead pressllre, The permeability thickness of 3

darcy-rn is used. The main features of the pressllre profrles due to the differences in Pwucdn

be found at the uppermost part of the wells. For the case of high Pevpl the pressure profile

t510

- 10-

decreases linearly up to the r,vellhead, white the case for low Ppys the pressure drops markedly

near wellhead. Simulated results are sllmmarized in Table 2.

PRESSURE bar)0204060

Fig. 7 PressLrre profiles for different Pwu and Swi.

Table 2 Calculated results on well pal'ameters.

Wellhead

Pressure

(bar)

Feedzone

Pressure

(bar)

Pressure DropInitialWater

Saturation

(-)

Mass

Flow

Rate

(tih)

ln

Wellbore

(ba0

in

Reservoir

(bar)

35.1 55.3 20.2 8.6s 0.5 42

35. 1 49.1 14.0 t4.9 0.1 35

10.1 4s.2 35.1 18.9 0.5 80

10.1 45.0 34.8 19.1 0.1 43

In the case of high Pwa (o, o) in Fig. 7,the effects of Su.,i are significant on the pressure

profiles. The decrease in Swi from 0.5 to 0.1 gives remarkable change in well bottom pressure

from 55.3 bar to 49.1 bar. On the other hand, for low PpyTl case the changes in Swi from 0.5 to

0.1 does not give any differences in pressure profiles. The pressure profiles are linear from the

bottom up to the depth of about 500 m and rapidly decrease as it approaches to the wellhead.

In terms of total mass flow rates, the decrease in Swi from 0.5 to 0. 1 leads to decrease in mass

flow rates by 15% for Pyyvl:35.1 bar and 52%o from for Pt'vu:l0.1ban

Figure 8 (a), (b), (c) and (d) show the components of pressure drop versus depth in

wellbore (potential, acceleration and friction) at given Pryp1and,Swi. In general, the potential

- 1l

component decreases as the fluid approaches to the wellhead while others tlvo increase. In the

case of high Pwaand Swi (35.1 ancl 0.5 respectively), as shown in Fig.8 (a), the pressure drop

is dominated by potential component fotlowed by friction and finally acceleration component.

In the case of high Peypl (:35.1) and low Srli (:0.1), as shown in Fig. 8 (b), the potential

component decreases significantly as the flLrid approaches to the wellhead while the friction

component increases markedly. The increase amolrnt in acceleration component is very small

and has a negligible value when fluid flows upwards. An important feature for high Pvucase

regardless of Slyi is the change in the pressure drop for each component follows the nearly

linear correlation from the bottom to the top. This is the reason why the pressure profiles in

wellbore for high Pwualso takes nearly linear correlation'

PRESSURE DROP (rPa)

02040

(a)

PRESSURE DR0P (<Pa)

0204060

PRESSURE DR0P kPa)02040

(b)

PRESSURE DR0P (<Pa)

0204060

(c) (d)

Fig. 8 Pressure drops components at depth for diffelent Pvll and Swi.

yr = 35.1 barSv i -- 0.5

--f- A ccebratbn--*- Fri:tbn

P rr = 35.L barSri - O.l

--I- P otentb I4-A ccebratbn{-Frbtbn

P yn = l0.l barSri = 0.5

--I- P otential

-O-Accebratbn-*- Frbtbn

I'

2.

a

Different fbatures are observed in Fig. 8 (c) and (d). For the low Pwn ease (:10.1 bar),

the pgtential component decreases as the fluid flolvs upwards while the acceleration and

friction cornponents increase. Regardless to ,Swi, the potential component reaches negligible

small at wellhead. The decrease rate in the pressure drop due to potential component to the

depth is larger for high Swi (=0.5). The increase rate for friction component to the depth is

also larger for high Swi (-0.5). In the case of low P1,yp1 case (:10.1 bar), acceleration

component increases markedly when flLrid approaches to the wellhead. The larger increase

rate is observed for high Swi (:0.5). This acceleration pressure drop profile at the uppermost

of the well may cause the pressure profiles decreases sharply as the fluid approaches to the

wellhead. This is because the velocity increases rapidly as it approaches to the wellhead'

l.

5. Conclusions

From the above analyses, the following conclusions have been drawn:

In the analysis of well characteristics for two-phase reservoiq the initial water saturation

in the reservoir plays an important role besides the reservoir pressure and temperature'

The smaller ilitial water saturation results in the less total flow rate at a given wellhead

pressure. Howeveq the decrease in the initial water saturation causes the increase in the

maximum discharge pressure.

The total flow rate stabilizes as the wellhead pressllre becomes smaller whereas the steam

flow rate linearly increases.

4. The correlation between total flow rate and pressure at feed zone shows in a non-linear

manner. At a given feed zone pressure, lower initial water saturations results in higher

mass flow rates.

5. At low wellhead pressures, the pressure profiles at the uppermost of the wellbore is

controlled by the pressure drop due to acceleration.

Acknowledgements

The first author gratefully acknowledges the financial support of the Ministry of Education,

Culture, Sports, Science and Technology, Government of Japan in the form of scholarship, so

the first author gets a chance to attend the master and doctor course at Dept. of Earth

Resources Engineering, Kyushu University.

References

1) Bjornsson, G. and Bodvarsson, G. S. (1987) A Multi-Feedzone Wellbore

Simulator.Geothermal Resources Council Trans., Vol.l I , 503-507.

-13-

Z) Hadgu, T. and Freeston, D. H, (1990) A Multipurpose Wellbore Simulator. Geothermal

Resources Council Trans, Vol. 14, 1279-1286'

3) Takahashi, M. (1999) Development of a Flow Simulator (WELCARD-V) Considering

Inflow Performance and Wetlbore Performance of Geothermal Well. Journal of the

Geothennal Research Society of Japan, Vol.2i ,341-352.

4) Hadgu, T. and Zimmerman, R. w. and Bodvarsson, G. s. (1995) Coupled

Reservoir-Weltbore Simulation of Geothermal Reservoir Behavior. Geothermics, Vol.24,

145-t66.

5) GudmLrndsson, J.S. (1984) Discharge Analysis Method for Two-Phase Geothermal Wells.

Geothermal Resources Council, Vol.8, 295-299.

6) Brennand, A.W and Watson, A. (1987) Use of ESDU Compilation of Two-Phase Flow

Correlations for The Prediction of Well Discharge Characteristics. Proceedings of 9th NZ

Geothermal Workshop, 65-68.

7) Garg, S. K., Combs, J. and Livesay, B. J. (2000) Optimizing Casing Designs of Slim

Holes for Maximum Discharge of Geothermal Fluids from Liquid Feedzones.

Proceedings World Geothermal Congress, Kyushu-Tohoku, Japan May 28 - June 10,

I 145-1 150.

8) Guiterrez, A. G., Paredes, G. E. and Ramirez,l. H. (2001) Study on The Flow Production

Characteristics of Deep Geothermal Wells. Geothermics, Vol.3l, 141-167.

9) Garg, S.K. and Combs, J. (1995) Production/lnjection Characteristics of Slim Holes and

Large-Diameter Wells at The Sumikawa Geothermal Field, Japan. Proceedings Twentieth

Workshop on Geotherrnal Reservoir Engineering, Stanford University, Stanford,

California, 3l-39.

l0) Itoi, R., Kakihara, Y., Fukuda, M. and Koga, A. (1988) Numerical Simulation of Well

Characteristics Coupled with Radial Flow in A Geothermal Reservoir. International

Symposium on Geothermal Energy, Kumamoto and Beppu, Japan, 201-204.

ll) Grant, M.A., Donaldson, I.G. and Bixley, P.F. (19S2) Geothermal Reservoir Engineering.

Academic Press, 369p.

12) Orkiszewski, J. (1967) Predicting Two-Phase Pressure Drops in Vertical Pipe. Journal of

Petroleum Technology, 829-83 8.

l3) Lindeburg, M.R. (lgg2) Engineering - Training Reference Manual 8d'' Edition.

Professional Publications, Inc,.

14) Ito, T. (1993) PROPATH: A prograrn package for thermodynamical properties of fluids,

version 8.1.

Nomenclature

A : Surface area of the feed zone (m2)

D : Welldiameter (m)

-14-

Fob

G

h

ii,

i,,

k

ku

k,,,

{ALMp

Pe

Psat

Pwb

APn

APr

APn

AP,

rR

fg

R"

Re

lnw

Rnu

Swi

LT

U7

Lls

Utv

Ynr

Vl

V2

a

Cross sectional area of the well (m2)

Gravitational acceleration (9.8 m2ls)

Mass velocity (kg/m2s)

Reservoir thickness (m)

Totalenthalpy of fluid (J/kg)

Enthalpy of steam (J/kg)

Enthalpy of water (J/kg)

Permeability (m2)

Relative permeability to steam ( - )

Relative permeability to water ( - )

Depth coordinate (m)

Length of small increment (m)

Totalmass flow (kg/s)

Pressure (Pa)

Pressure at outer boundarY (Pa)

Satr-rration pressure (Pa)

Pressure at feed zone (or well bottom) (Pa)

Pressure drop due to acceleration loss (Pa)

Pressure drop due to friction loss (Pa)

Pressure drop due to potential loss (Pa)

Total pressure drop (Pa)

Radial distance (m)

Normalized distance (:rlr*) ( - )Outer bor-rndary radius (m)

Normalized distance 1:r/r*) ( - )

Reynoldsnumber(-)

Well radius (m)

Normalized distance for n, ( - )

Initial water saturation ( - )

Mass flux density (kg/m2s)

Total fluid velociry (m/s)

Mass flux density of steam (kg/m2s)

Mass flux density of water (kg/m2s)

Specific volume of mixture lm3ltcg;

Specific volume of fluid at inlet of an interv al A P lmrlkg;

Specific volume of fluid at outlet of an interv al A P 1m3/t<g;

Voidratio(-)Pipe roughness (m)

- 15 -

tr,,'l/ s

/1

lw

Pm

Friction factor for mixture ( - )

Kinematic viscosity of steam 1m2is1

Total kinematic viscosity of two-phase fluid im2/s1

Kinematic viscosity of water 1m2/s;

Density of mixture (kg/m3)

-t6-