PROCEEDINGS SECOND WORKSHOP

GEOTHERMAL RESERVOIR ENGINEERING December 1-3,1976

SGP-TR-20-34

-- --- -- - -------- ---- - -- -- ------- -- - -- ----

PROCEEDINGSSECOND WORKSHOP

GEOTHERMAL RESERVOIR ENGINEERINGDecember 1-3, 1976

'Conducted under Grant No. NSF-AER-72-03490 supported by the RANN program of the National Science Foundationand Contract No. E043-326-PA-50 from the Geothermal Energy Division, Energy Research and DevelopmentAdministration.

BUOYANCY INDUCED BOUNDARY LAYER FLOWS IN GEOTHERMAL RESERVOIRS

*Ping ChengDepartment of Petroleum Engineering

Stanford University, Stanford, CA. 94305

Most of the theoretical study on heat and mass transfer ingeothermal reservoirs has been based on numerical method. Recent­ly at the 1975 NSF Workshop on Geothermal Reservoir Engineering,Cheng (1) presented a number of analytical solutions based onboundary layer approximations which are valid for porous mediaat high Rayleigh numbers. According to various estimates theRayleigh number for the Wairakei geothermal field in New Zealandis in the range of 1000-5000, which is typical for a viable geo­thermal field consisting of a highly permeable formation and aheat source at sufficiently high temperature.

The basic assumption of the boundary layer theory is thatheat convective heat transfer takes place in a thin porous layeradjacent to heated or cooled surfaces. Indeed, numerical solutionssuggest that temperature and velocity boundary layers do exist inporous media at high Rayleigh numbers (2). It is worth mentioningthat the large velocity gradient existing near the heated or cooledsurfaces is not due to viscosity but is induced by the buoyancyeffects. The present paper is a summary of the work that we havedone on the analytical solutions of heat and mass transfer in aporous medium based on the boundary layer approximations sincethe 1975 ~orkshop.

Similarity Solutions to Boundary Layer Equations

Free Convection about a Vertical Impermeable Surface with UniformHeat Flux

The solution for free convection about a vertical impermeablesurface with wallAtemperature being.a power function of distance,i.e., T

W= T + Ax for x> 0 is given by Cheng and Minkowycz (3).

The constant heat flux solution can be obtained by a simple trans­formation of variables and by setting A = 1/3 in Ref. 3. The

-'-~Visiting Professor. On sabbatical leave from Department ofMechanical Engineering~ University of Hawaii, Honolulu, Hawaii96822.

-236-

expressions for local Nusselt number, the mean Nusselt number,and thermal boundary layer thickness are

1

Nu 1 [~a"; ] 3 =x x

1

Nu / [Ra'" ] 3 =L L

0.7723,

1.029,

(1)

(2 )

1

<5 T = 4. 81 [Ra,'; ] 3",x x

hx oxwith Nux - -k = k (T~-T-:r' NUL

Rat - PmgSKqL2/~ak, and TW-Too

(3)

qL

K(TW-T )LCXl

-t1 (TW-Too ) dx, where

the density of the fluid at infinity; g the gravitationalacceleration;~ and B the viscosity and the thermal expansion co­efficient of the fluid; K the permeabil ity of the porous medium;L the length of the plate; q the surface heat flux; a and K

the equivalent thermal diffusivity and the thermal conductivityof the saturated porous medium. The equivalence of Eqs. (1) through(3) and the corresponding expressions given by Cheng & Minkowycz (3)was shown recently by Cheng (4).

Free Convection about a Horizontal Impermeable Surface with UniformHeat Flux

The constant heat flux solution for a horizontal impermeablesurface can be obtained by a simple transformation of variablesand by setting A = 1/2 in the solution given by Cheng and Chang (5).The expressions for local Nusselt number, the mean Nusselt number,and thermal boundary layer thickness for the present problem are

1

Nu I[Ra*14 = 0.8588,x x

1

NUr! [RatJ"4 = 1.288,

-237-

(4)

(5 )

(6)

Mixed Convection from a Vertical Isothermal Impermeable Surface

( 7)

(8)

(9 )

(10)

(11)

(12)

6.31 .. 1

[Grx/Re:J 2

[-6' (0)]

l[-¢(Q)]'

-238-

[Re Pr]1/2 •x

2[-8' (0)]

x

Nux =-----1

[Re Pr]2x

1'2

NU/[Re.Pr] =0.444x x

The problem of mixed convection from a vertical impermeablesurface with a step increase in wall temperature (i.e., TW= Too + Afor x~O), embedded in a porous medium is considered by Cheng (6).The expressions for local Nusselt number, average Nusselt number,and thermal boundary layer thickness are

Nux---

[Re Pr]l/2x

The expressions for local Nusselt number, average Nusseltnumber, and thermal boundary layer thickness are given by (4,7)

U x00 vwith Re = and Pr = where U is the velocity outside

x v a 00 1 1

the boundary layer. The values of NUx/[RexPr]~ and [RexPr]~6T/x

as a function of Gr IRe are shown in Figs. 1 and 2, where thecorresponding value~ fo~ free convection about a vertical isothermalplate as given by Cheng and Minkowycz (3) can be rewritten as

Mixed Convection from a Horizontal Impermeable Surface with UniformSurface Heat Flux

3= 2[ -<pCOTT' (13 )

(14)

, tThe val~es of Nu /[Re Prl~ and [Re PrJ ~/x as a function

of Ra*/(Re Pr) are giv~n inxRef. 4, wher~ the asymtotes for freeconve~tionxabout a horizontal plate with uniform heat flux aregiven by Eqs. (4) and (6), which can be rewritten as

1"2Nu /[Re PrJ

x x

12 4"= O. 8588 [ Ra ~': / (Re pro) ] ,

x x(15)

1 1

[Re pr]28T/x=4.0/[Ra~/(Repr)2]4".x x x

(16)

The Effect of Uro on Heat Transfer Rate and the Size of Hot-Water Zone

To gain some feeling on the order of magnitude of variousphysical quantities in a geothermal reservoir, consider a verticalimpermeable surface at 215°C is embedded in an aquifer at 15°C.If there is a pressure gradient in the aquifer such that ground­water is flowing vertically upward, the values of heat transferrate and the size of the hot water zone can be determined fromFigs. 1 and 2. The results of the computations for Uoo varyingfrom 0.01 cm/hr to 10 cm/hr are plotted in Figs. 3 and 4 whereit is shown that the total heat transfer rate for a vertical sur­face, 1 km by 1 km, increases from 20 MW to 110 MW, whi Ie theboundary layer thickness at x = 1 km decreases from 130 m to20 m.

Val idity of Boundary Layer Approximations

The validity of the boundary layer approximations can beaccessed by a comparison of results obtained by similarity solu­tions to that of numerical solutions of exact partial differentialequations, or to experimental data. For free convection' in aporous medium between parallel vertical plates separated by.adistance H, the correlation equation given by Bories and Combarnous(8) is

-239-

References

( 18)

(17)

(19)Nu ' = O. 8 8 8 ( RaH

) 0 • 5 ( !i) 0 • 5H L ,

1. Cheng, P., "Numerical & Analytical Study on Heat and MassTransfer in Volcanic Island Geothermal Reservoirs." Pro­ceedings of 1975 NSF Workshop on Geothermal ReservoirEngineering,pp. 219-224 (1976).

-240-

Concluding Remarks

which can be rewritten as

where L is the length of the plate, NU H = h~ and RaH3= pgBK(T'p(TL) HhlCX. Eq. (17) isSval id for Ra H from 102 to 10 and forH/l between 0.05 and 0.15. On the other hand, the heat transferrate as obtained from boundary layer approximations for an iso­thermal vertical plate (3) is

2. Cheng, P., Yeung, K.C., & Lau, K.H., I'Numerical Solutionsfor Steady Free Convection in Island Geothermal Reservoirs."To appear in the Proceedings of 1975 International Seminaron Future Energy Production--Heat & Mass Transfer Problems.Hemisphere Publishing Corporation.

Eqs. (17) and (19) for H/L = 0.05 and 0.15 are plotted inFig. 5 for comparison. It is shown that they are in good agree­ment, especially at high Rayleigh numbers where the boundary layerapproximations are val id.

As in the classical convective heat transfer theory,boundary layer approximations in porous layer flows can result inanalytical solutions. Mathematically, the approximations are thefirst-order terms of an asymptotic expansion which is valid forhigh Rayleigh numbers. Comparison with experimental data andnumerical solutions show that the approximations are also accurateat moderate values of Rayleigh numbers. For problems with lowRayleigh numbers where boundary layer is thick, higher-orderapproximations must be used (9).

3. Cheng, P. and Minkowycz, W.J., "Free Convection about aVertical Flat Plate Embedded in a Porous Medium withAppl ication to Heat Transfer from a Dike." Accepted forpublication by J. Geophysical Research.

4. Cheng, P., "Constant Surface Heat Flux Solutions for PorousLayer Flows." Submitted for publication to Letters inHeat & Mass Transfer.

5. Cheng, P. and Chang, I-Dee, "Buoyancy Induced Flows in aSaturated Porous Medium Adjacent to Impermeable HorizontalSurfaces." Int. J. Heat Mass Transfer, .!.1, 1267 (1976).

6. Cheng, P., "Combined Free & Forced Boundary Layer Flowsabout Inclined Surfaces in a Porous Medium. '1 Acceptedfor publ ication by Int. J. Heat Mass Transfer.

7. Cheng, P., "Similarity Solutions for Mixed Convection fromHorizontal Impermeable Surfaces in Saturated Porous Media."Accepted for publication by Int. J. Heat Mass Transfer.

8. Bories, S.A. and Combarnous, M.A., "Natural Convection ina Sloping Porous Layer," J.F.M., 57, 63-79 (2973).

9. Cheng, P. and Chang, I-Dee, "Higher-Order Approximationsfor Convective Heat Transfer in Porous Layers" (in progress).

-241-

FIGURE 1. HEAT TRANSFER RESULTS FOR MIXED CONVECTION FROM AN ISOTHERMAL VERTICAL PLATE.

10,-----------------------------,

IN.l:­N

I

- - - Free Convection Asymptote

5 --- Forced Convection Asymptote4

3

2

1.0

0.5['---'-'-'-­0.4

0.3

0.2

O.ll-__-J...-_..I...--I..---L-....Io.-L-L.....L..L.__--t_--J.._l.-J.-J..-.I-oA-L....I-__--L-_.-l--J--J-...L-...L-.oIo.-J-J

0.1 0.2 0.3 0.4 0.5 1.0 2 3 4 5 10 20 30 40 50 100

Grx/Rex.

10,.-,---------------------------,~

t

_ - - - Free Convection Asymptote

_._.- Forced Convection Asymptote

54

.-._.~.-.-._.-.--.---.--.....

3 1r,2 f-

1.0

~0.5 f-

I

0.4 10.3 r

L0.2

10.1 . 11--1.1_.1..1-1.-1 ...l.1-J-.!..1.J-1 1 J-_L.--l--...L-..J-I--J-1 .l..l.l..l__-1.1_--..1..1_J.-J-l.-ll--..l....l.!~II0./ 0.2 0.3 0.40.5 1.0 2 3 4 5 10 20 304050 100

Grx/ Rex

)(

"-I-

toC\I.........,.---,

\0-

r 0-N ><J::-W (l)

I

0::~

Fig. 2 Boundary Layer Thickness for Mixed Convection From An Isothermal Vertical Plate

FIGURE 3. THE EFFECT OF Uoo ON TOTAL HEAT TRANSFER RATE FOR MIXED CONVECTION FROMAN ISOTHERMAL VERTICAL HEATED SURFACE IN A GEOTHERMAL RESERVOIR.

1000

500~K = 10-

12m2

3; f!. = 106 9 1m 3

2co

I f3 = 1.8 x10-41°C400t...

Q.)

3001

fL = 0.27g/sec-m0+-

0a = 6.3 xI0- 7 m2/sec0:::

200r\0- T - Tco = 200°C(1)I '+- I W

IN (f) I Area = 1000111 xlOOOnl.j::-.j::- c

100 1-

I

0\0-r- f- ------ Free convection asymptote /'~

1-040- I -- - - Forced convection asymptote .,.

1-

~0 I

Q.) 50 ~I 40 L '"

30 1- '/0 I

.,.0+-

------------------~0 20r-

I0 L-_--L~___l_L_LJ_l.J....l__ _L_J_l._J__L_.L.J_L.l__ __L_ _l__l_..l._...L......I.._J"..J_.J

.01 .02 03 .05 .10 .2.3.4 .5 1.0 2 3 4 5 10

Ua:> crn/hr

102 3 4 51.0.2 .3.4.5.10

Free convection asymptoteForced convection asymptote

.02.03 .05

I0 '---_---L._--l.---J---'-...l.-..J.---I--J-l-.-__-1.----L._.l-..J---'-~__l.._I...___J.__...I.___'__.....4_"__'__'__'_'

.01

--------------~

100r- "'~ ~

50 1= -~40[30,

I -------20 1-

I

1000-12 21= K - 10 m-

500f PoO

- 106 g/ m3-

400 f3 - 1.8 x 10-4/ o C-

300 fL - 0.27 g/sec-m- -

a - 6.3xlO-7 m 2/sec-200 T - ToO = 200°Cw

E·...

E000or-

--I 0N-l:"" ..--.\nI f-

ro--

Uco ern /h r

Fig. 4 The Effect of Uen on the Boundary Layer Thickness (Size of Hot Water Zone) forMixed Convection from a Vertical Isothermal Heated Surface in a Geothermal Reservoir

FIGURE 5. COMPARISON OF THEORY AND EXPERIMENT FOR VERTICAL POROUS LAYERS.

roor----------------------------,

, ,

--- Theoretical (Cheng 8 Minkowycz 1975)

50 ----- Experimental (Bories a Combarnous 1973)40 -

30r20

NU H r,Iv

1:[ /./;;~~~34~ /' /'r ~ /'2~ ,?-' ,/'

I ,//1

I ~I__-l-_..l--J--l-J--,l.,,!-.lor...l.t....!__-.1-_-,---,--,-,

10 10 2

INJ:­(J'\I