Theoretical and numerical analyses of convective instability in porous media with upward throughflow

*Correspondence to: C. Zhao, CSIRO Division of Exploration and Mining, P.O. Box 437, Nedlands, WA 6009, Australia

CCC 0363}9061/99/070629}18$17.50 Received 26 September 1997Copyright ( 1999 John Wiley & Sons, Ltd. Revised 22 May 1998

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS

Int. J. Numer. Anal. Meth. Geomech., 23, 629}646 (1999)

THEORETICAL AND NUMERICAL ANALYSES OFCONVECTIVE INSTABILITY IN POROUS MEDIA WITH

UPWARD THROUGHFLOW

CHONGBIN ZHAO*, B. E. HOBBS AND H. B. MUG HLHAUS

CSIRO Division of Exploration and Mining, P.O. Box 437, Nedlands, WA 6009, Australia

SUMMARY

Exact analytical solutions have been obtained for a hydrothermal system consisting of a horizontal porouslayer with upward through#ow. The boundary conditions considered are constant temperature, constantpressure at the top, and constant vertical temperature gradient, constant Darcy velocity at the bottom of thelayer. After deriving the exact analytical solutions, we examine the stability of the solutions using linearstability theory and the Galerkin method. It has been found that the exact solutions for such a hydrothermalsystem become unstable when the Rayleigh number of the system is equal to or greater than thecorresponding critical Rayleigh number. For small and moderate Peclet numbers (Pe)6), an increase inupward through#ow destabilizes the convective #ow in the horizontal layer. To con"rm these "ndings, the"nite element method with the progressive asymptotic approach procedure is used to compute theconvective cells in such a hydrothermal system. Copyright ( 1999 John Wiley & Sons, Ltd.

Key words: exact analytical solution; convective stability; horizontal layer; porous medium; through#ow;"nite element analysis

1. INTRODUCTION

Analytical solutions are very important for scienti"c and engineering problems.1 For example, ananalytical solution can be used as a powerful means to gain an understanding of the solutionscenarios under some extreme conditions for a given problem. In addition, an analytical solutionis often a useful or even in some circumstances an unique measure in the assessment andvalidation of any numerical method. However, most scienti"c and engineering problems aremathematically described by a set of partial di!erential equations. This makes it extremelydi$cult to obtain analytical solutions for such problems. As a result, to the best of the authors'knowledge, an analytical solution for a hydrothermal system consisting of a horizontal porouslayer with upward through#ow has not been available. Thus, one of the main purposes ofthis study is to derive analytical solutions for this problem with boundary conditions ofconstant temperature, constant pressure at the top of the layer and constant vertical temperaturegradient, constant Darcy velocity at the bottom of the layer. The governing partial di!erentialequations of the problem consist of a continuity equation, Darcy's equation and an energybalance equation.

As demonstrated in recent studies,2~4 convective pore-#uid #ow driven by a temperaturegradient can play a signi"cant role in mineralization and ore body formation in #uid-saturatedporous rock masses. It is necessary to study the stability of the trivial solution for the above-mentioned hydrothermal system because only a non-trivial solution results in temperaturegradient driven convective #ow in such a system. The resulting convective #ow can enhance themixing of species, and therefore promotes mass transport, chemical reactions and mineralization.In this sense, temperature gradient driven convective #ow acts as a very powerful catalyst formineralization in hydrothermal systems. However, the previous studies2~4 have only dealt withconvective instability problems in porous media without upward through#ow. The main di$-culty in dealing with convective instability in porous media with upward through-ow is that onehas to solve a complete fourth-order ordinary di!erential equation instead of solving anincomplete fourth-order (with the "rst-order and the third-order terms exactly being zero)ordinary di!erential equation as in the no upward through#ow case. This will lead to a formi-dable di$culty in mathematics and therefore, requires to develop a new solution methodology. Itis both the consideration of upward through#ow and the newly proposed solution method thatmake this study signi"cantly di!erent from the previous ones.2~4

If no vertical #ow goes initially through the layer, the trivial steady-state solutions fortemperature and pore-#uid #ow become unstable when the Rayleigh number of the layeris equal to or greater than the corresponding critical Rayleigh number. However, if verticalpore-#uid #ow occurs initially through the layer, driven by imposed #uid pressure gradientsor imposed mass #uxes, the trivial solutions may be stabilized for some boundary conditionsbut destabilized for other boundary conditions,5~7 compared with the no initial pore-#uid#ow situation. In the case of no initial through#ow, the trivial steady-state temperaturepro"le is caused by heat conduction only, and therefore is linear in the layer thicknessdirection. In the case of initial through#ow, heat is transferred by both conduction and ad-vection in the layer so that the trivial steady-state temperature pro"le is non-linear acrossthe thickness of the layer. Because of this signi"cant di!erence, theoretical research on temper-ature gradient driven pore-#uid convective instability in porous media with through#ow isvery limited. As mentioned above, the main di$culty in dealing with this kind of problemis that one must solve a complete fourth-order ordinary di!erential equation instead ofsolving an incomplete fourth-order (with the "rst-order and the third-order terms identicallyequal to zero) ordinary di!erential equation as in the no initial through#ow case. This, to a largeextent, has led to the lack of understanding of how the upward through#ow interacts with theconvective #ow in a porous medium. Besides, little, if any, numerical research has been carried outon this kind of problem. Thus, this paper will focus on both theoretical and numerical analyses oftemperature gradient driven pore-#uid convective instability in porous media with upwardthrough#ow.

This kind of convective instability problem has a very strong practical background in geo-physics and geoenvironmental engineering. For example, a horizontal layer of a #uid-saturatedporous medium may undergo constant temperature and overpressure at its top, whereas it mayundergo constant vertical temperature gradient and constant injection of mass #ux at its bottom.The overpressure can result from the presence of impermeable seals in geophysics, while it can beinduced by surface structures or human activities in geoenvironmental engineering. The verticaltemperature gradient may be generated by either geothermal sources in geophysics or buriedheat-generating waste in geoenvironmental engineering. The mass #ux may be generated bydehydration reactions or by compaction of underlying soft layers or reservoirs. From the

630 C. ZHAO, B. E. HOBBS AND H. B. MUG HLHAUS

Copyright ( 1999 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech., 23, 629}646 (1999)

Figure 1. Geometry and boundary conditions of a hydrothermal system

geomechanics point of view, the overpressure will lead to consolidation in the underlying softlayers so that the pore-#uid can be squeezed out of the layer. Compared with convective #owdriven by the temperature gradient only, the squeezed pore-#uid #ow may be strong at thebeginning of the soft layer consolidation, but vanishes at the end of the consolidation. In thissituation, it is necessary to study temperature gradient driven pore-#uid convective instability inporous media with decreasing upward through#ow. Clearly, the results from such a study aresigni"cant from the following two points of view: (1) If a decrease in upward through#owdestabilizes temperature gradient driven convective #ow, then the strongest temperature gradientdriven #ow takes place at the end of the underlying soft layer consolidation. This implies that ifthere is no temperature gradient driven convective #ow at the end of the consolidation, there isde"nitely no temperature gradient driven convective #ow during the consolidation. (2) If a de-crease in upward through#ow stabilizes the temperature gradient driven convective #ow, then thestrongest temperature gradient driven #ow occurs during the consolidation. This means thatthe existence of the temperature gradient driven convective #ow at the end of the consolidationcan guarantee the existence of the temperature gradient driven convective #ow during theconsolidation.

In view of the above, we begin with a statement of the problem, which is followed by thederivation and stability analysis of exact analytical solutions in Section 2. The exact analytical(trivial) solutions are then derived, followed by a stability analysis of the exact solutions usingconventional linear stability theory and the Galerkin method. In Section 3, the "nite elementmethod with the progressive asymptotic approach procedure2 is used to con"rm the convective#ow instability predicted by linear stability theory and the Galerkin method. Finally, someconclusions are given in Section 4.

2. THEORETICAL ANALYSIS

2.1. Statement of the problem

The hydrothermal system considered is a horizontal layer of in"nite length and thickness, H,with upward through#ow in a #uid-saturated porous medium. As shown in Figure 1, theboundary conditions are constant temperature and constant pressure at the top of the layer, andconstant vertical Darcy velocity and constant vertical temperature gradient at the bottom of the

CONVECTIVE INSTABILITY IN POROUS MEDIUM WITH UPWARD THROUGHFLOW 631


layer. This means that the layer is heated from below. The steady-state governing equations forthis hydrothermal system are expressed as [2]

ui, i"0 (1)

ui"

1

kK

ij(!p

,j#o

fgj) (2)

(o0cp)u

j¹

,j"(je

ij¹

,j),i

(3)

of"o

0[1!b (¹!¹

0)] (4)

jeij"/j

ij#(1!/)j4

ij(5)

where ui

is the Darcy velocity component in the xi

direction: p and ¹ are pressure andtemperature; o

0and ¹

0are the reference density of pore-#uid and the reference temperature of the

medium; k and cp

are the dynamic viscosity and speci"c heat of pore-#uid; jij

and j4ij

are thesecond-order conductivity tensors for the pore-#uid and solid matrix in the porous medium;/ and b are the porosity of the medium and the thermal volume expansion coe$cient ofpore-#uid; K

ijis the second-order permeability tensor of the medium and g

iis the gravity

acceleration component in the xidirection.

It is noted that in the "elds of geomechanics and geoscience, Darcy's law has been widely usedto describe the pore-#uid in #uid-saturated porous media.8,9 The justi"cation of using Darcy'slaw can be found in many classical literatures. For example, Nield and Bejan9 commented that&Darcy's law has been veri"ed by the results of many experiments. Theoretical backing for it hasbeen obtained in various ways, with the aid of either deterministic or statistical models.'Thus, it isreasonable to use Darcy's law to describe the pore-#uid #ow in this study. On the other hand,from the #uid dynamics point of view, Darcy's law, in essence, belongs to the momentumequation, rather than the constitutive equation. In other words, Darcy's law, which is used todescribe the pore-#uid #ow in a porous medium, is the analog of the Navier}Stokes equation,which is used to describe the #uid #ow in a pure #uid medium. Clearly, Darcy's equation (i.e.,equation (2)) is coupled with the energy equation (i.e., equation (3)) via both the pore-#uid velocityand the density, which is a function of temperature (see equation (4)). It is the coupling e!ectbetween Darcy's equation and the energy equation that plays an important role in the analysis ofconvective instability in a #uid-saturated porous medium.

The boundary conditions of the problem can be expressed as

¹"¹0, p"p

1(at x

2"H)

(6)u2"<

0, ¹

,2"¹0

,2"!q

0/j

e0(at x

2"0)

where ¹0, p

1, <

0, ¹2

,2, j0

e0and q

0are constants. Physically, q

0is the conductive thermal #ux and

je0

is a reference conductivity of the porous medium. Thus, q0

being constant implies that theconductive thermal #ux is constant at the bottom of the layer.

Note that if the porous medium of the layer is homogeneous and isotropic, equations (1)}(4)can be written in the following dimensionless form:

u*i, i"0 (7)

u*i"!p*

,i#Ra ¹* e

i(8)

u*j¹*

,j"¹*

,jj(9)



where e is a unit vector and e"e1i#e

2j for a two-dimensional problem. Other dimensionless

variables are de"ned as

x*i"

xi

H, ¹*"

je0

(¹!¹0)

q0H

, u*i"

Ho0cp

je0

ui

p*"K

)o0cp

kje0

(p!p0) , ¹*

,2"

je0q0

¹,2

Ra"(o

0cp)o

0gbK

)q0H2

kj2e0

(10)

where x*i

are the dimensionless coordinates; u*i

is the dimensionless Darcy velocity component inthe x

idirection; p* and ¹* are the dimensionless pressure and temperature; ¹*

,2is the dimension-

less vertical temperature gradient; K)

is a reference medium permeability coe$cient in thehorizontal direction; Ra is the modi"ed Rayleigh number expressed in terms of the conductivethermal #ux rather than in terms of the conventional temperature di!erence; H is the thickness ofthe layer and p

0is the static pore-#uid pressure.

The boundary conditions of the problem in equation (6) can also be written in the dimension-less form as

¹*"0, p*"p*1

(at x*2"1) (11)

u*2"Pe"

Ho0cp

je0

<0, ¹*

,2"!1 (at x*

2"0) (12)

where Pe is the Peclet number of the hydrothermal system.

2.2. Derivation of analytical solutions

For the hydrothermal system considered, the trivial solution for the horizontal Darcy pore-#uid velocity is zero.

u*1"0 (13)

Substituting equation (13) into equation (7) yields the following equation:

u*2,2

"0 (14)

It straightforwardly follows from equations (12) and (14) that

u*e"Pe (15)

This indicates that the vertical velocity, which is referred to as upward through#ow in theintroduction, is constant throughout the whole layer.

Substituting equations (13) and (15) into equation (9) yields the following equation:

Pe¹2,2"¹2

,ii. (16)



The solution for equation (16) can be expressed as

¹*"C1ePex*2#C

2(17)

where C1

and C2

are two independent constants. In order to determine C1

and C2

constantsuniquely, we have to use two thermal boundary conditions, one of which must be a temperatureboundary. Insertion of equations (17) into equations (11) and (12) yields

¹*"1

Pe(ePe!ePex*

2). (18)

It is noted from equations (8) and (13) that p* is a function of x*2

only. Hence,

Lp*

Lx*2

"

dp*

dx*2

"Ra¹*!Pe. (19)

Integrating equation (19) with respect to x*2

yields the following equation:

p*"Ra

Pe AePex*2!

1

PeePe x*

2B!Pex*2#C

3(20)

where C3

is a constant which can be determined from equation (11):

C3"p*

1#Pe!

Ra

Pe AePe!1

PeePeB . (21)

It follows that the "nal solution for the dimensionless pressure can be expressed as

p*"Ra

Pe AePex*2!

1

PeePe x*

2B!Pex*2#p*

1#Pe!

Ra

Pe AePe!1

PeePeB (22)

Up to now, we have obtained the exact solutions for the dimensionless Darcy velocities(equations (13) and (15)), dimensionless temperature (equation (18)) and dimensionless pressure(equation (22)) of the hydrothermal system.

2.3. Stability analysis of exact trivial solutions

Next, the stability of the above solution is investigated in a linear stability analysis. Supposingthe hydrothermal system is subjected to a small disturbance, the total solutions for the dimen-sionless velocities, temperature and pressure of the system can be expressed as

u*ti"u*

i#uL *

i, ¹*

t"¹*#¹K *, p*

t"p*#pL * (23a)

where uL *i, ¹K * and pL * are the corresponding perturbation solutions due to the small disturbance.

From the linear stability theory point of view, if and only if all these perturbation solutions arezero, then the exact solutions obtained in Section 2.2. are stable. This implies that the stability ofthe exact solutions for the hydrothermal system considered here can be judged by examining theexistence of the non-zero solutions for uL *

i, ¹K * and pL *.

Note that the small disturbance may be caused by a small tremor of the earth. From theclassical perturbation theory, we can introduce a small parameter, e, to express the consequence



of this small disturbance. For example, using this small parameter, it is possible to express theresulting perturbation velocity, temperature and pressure of a system in the following form:

uL *i"e(uL *(0)

i#euL *(1)

i#e2uL *(2)

i#2)

pL *"e(pL *(0)#epL *(1)#e2pL *(2)#2)

¹K *"e(¹K *(0)#e¹K *(1)#e2¹K *(2)#2) (23b)

Substituting equations (23a) and (23b) into equations (7)}(9), considering the linear perturbationterms only (i.e., e terms only) and then dropping the unnecessary superscripts, we obtain thefollowing eigenvalue problem:

uL *i, i"0 (24)

uL *i"!pL *

,i#Ra ¹K * e

i(25)

uL *j¹*

,j#Pe¹K *

,2"¹K *

,jj(26)

where

¹*,1"0 and ¹*

,2"!ePe x*2 (27)

The corresponding boundary conditions for the perturbation solutions are

¹K *"0, pL *"p*1

(at x*2"1) (28)

uL *2"0, ¹K *

,2"0 (at x*

2"0) (29)

Note that pL *"0 in equation (28) implies that LuL *2/Lx*

2"0 at x*

2"1.10 Inserting

uL *2"< (x*

2)e~*k

*1x

*1 , ¹K "h (x*

2)e~*k

*1x

*1, (30)

into equations (24)}(26) yields

<A(x*2)!(k*

1)2<(x*

2)"!(k*

1)2 Ra h (x*

2) (31)

hA (x*2)!Peh@(x*

2)!(k*

1)2h(x*

2)"!<(x*

2)ePe x*

2 (32)

where k*1

is the dimensionless wave number in the x*1

direction:

k*1"k

1H (33)

k1being the wave number in the x

1direction. Using equation (30), equations (28) and (29) can also

be rewritten as

h"0, < @"0 (at x*2"1) (34)

<"0, h@"0 (at x*2"0) (35)

Substitution of equation (31) into equation (32) yields the following equation:

<(IV)(x*2)!Pe<@@@ (x*

2)!2(k*

1)2<A(x*

2)#Pe(k*

1)2<@ (x*

2)#(k*

1)2[(k*

1)2!RaePe x*

2]< (x*2)"0

(36)



The analytical solution of equation (36) is troublesome because it contains all order derivatives.Instead we solve equations (31) and (32) approximately through single member Galerkin ansatze.The quality of approximation is excellent for Peclet number less than one, as will be shownsubsequently by comparison with numerical solutions.

With <(x*2)"A<M (x*

2) and h (x*

2)"BhM (x*

2), equations (31) and (32) can be rewritten in the

Galerkin form as follows:

P1

0

[A<M A (x*2)!(k*

1)2A<M (x*

2)#(k*

1)2RaBhM (x*

2)]<M (x*

2) dx*

2"0 (37)

P1

0

[BhM A(x*2)!PeBhM @ (x*

2)!(k*

1)2BhM (x*

2)#A<M (x*

2)ePex*

2]hM (x*2) dx*

2"0 (38)

where A and B are independent constants;<M (x*2) and hM (x*

2) are trial functions for<(x*

2) and h (x*

2).

They must be chosen in such a way that all boundary conditions of the problem are satis"edidentically. From equations (37) and (38), it follows that

CC

11C

21

C12

C22DG

A

BH"G0

0H (39)

where

C11"P

1

0

[<M A(x*2)!(k*

1)2<M (x*

2)]<M (x*

2) dx*

2

C12"Ra(k*

1)2 P

1

0

hM (x*2)<M (x*

2) dx*

2

C21"P

1

0

ePex*2<M (x*

2)hM (x*

2) dx*

2

C22"P

1

0

[hM A (x*2)!PehM @ (x*

2)!(k*

1)2hM (x*

2)]hM (x*

2) dx*

2(40)

Clearly, the condition, under which equation (39) has a non-zero solution, is

C11

C22!C

12C

21"0 (41)

In theory, any function, which satis"es the boundary conditions of the problem considered, canbe chosen as the candidate of the trial function. However, in practice, for the purpose of avoidingany unnecessary di$culty in mathematics, it is favourable to use the polynomial function as thetrial function because many preliminary functions can be expressed as the combination ofpolynomial functions using the Taylor expansion. For the hydrothermal system considered, wehave tested various candidate trial functions (see Appendix). The best result (minimum error inhorizontal wave number at Pe"0) was obtained with

<M (x*2)"1

2(x*

2)2!x*

2

hM (x*2)"1

5[(x*

2)5!1] (42)



Substituting equation (42) into equation (40) yields the following equations:

C11

"!C1

3#

2

15(k*

1)2D

C12"

17

336Ra(k*

1)2

C21"

1

Pe8[ePe(1

2Pe6!2Pe5#3Pe4#12Pe3!108Pe2#306Pe!504)

#(504#144Pe#15Pe5#1

5Pe6)]

C22

"!C1

9#

1

33(k*

1)2!

1

50PeD (43)

From equations (41) and (43), the critical Rayleigh number, for which temperature drivenconvective #ow may occur, can be expressed as

Ra#3*5*#!-

"

366C11

C22

17(k*1)2C

21

(44)

Since Ra#3*5*#!-

is a function of (k*1)2, its minimum value is obtained from

LRa#3*5*#!-

L[(k*1)2]

"0 (45)

Substituting equations (43) and (44) into equation (45) leads to the following condition, underwhich Ra

#3*5*#!-has a minimum value for a given Peclet number:

k*1"A

55

6 B1@4

A1!9

50PeB

1@4(46)

Obviously, our approximate solution is only valid for

Pe(50/9 (47)

The above restriction is a consequence of the speci"c trial function we have used in our linearstability analysis and convective solutions may exist (and indeed exist, see Section 3) forPe*50/9.

Figure 2 shows the variation of the minimum critical Rayleigh number (Ra*#3*5*#!-

) with theupward through#ow, which is represented by Pe . In the range considered, Ra*

#3*5*#!-decreases as

Pe increases, i.e., temperature gradient driven convective #ow is destabilized by an increase inupward through#ow. In other words, a decrease in the upward through#ow stabilizes temper-ature gradient driven convective #ow.

3. NUMERICAL ANALYSIS

To con"rm our "nding that an increase in upward through#ow destabilizes temperature gradientdriven convective #ow in the hydrothermal system considered, we complement our theoretical



Figure 2. Variation of predicted minimum critical Rayleigh number with the magnitude of through#ow

analysis by a "nite element study.11 Numerical analyses are carried out with the progressiveasymptotic approach procedure. The method was thoroughly discussed and validated in a pre-vious study.2 A brief summary of the "nite element formulation is given below.

CQ

0

!B

E(U)D GU

TH"GF

GH (48)

where Q, B and E are global property matrices of the system; U and T are global nodal velocityand temperature vectors of the system; F and G are global nodal load vectors of the system. Theyare assembled by the following element property matrices and vectors:

Q%"M1 %#1

eA% (M%

1)~1(C%)T

M1 %"CM%

0

0

M%D , U%"GU%

1U%

2H , F%"G

F%1

F%2H

B%"GB%1

B%2H , A%"G

A%1

A%2H , C%"G

C%1

C%2H

M%1"P

A

wwT dA, E%"D%i(u*

i)#L%

i,

G%"!PS

q*u dS (49)



where

A%i"P

A

u,iwT dA, B%

i"P

A

u Ra uTeidA, C%

i"P

A

wuT,i

dA,

D%i(u*

i)"P

A

uu*iuT,i

dA, L%i"P

A

u,iuT,i

dA, M%"PA

uuT dA,

F%i"P

S

r*i

u dS, r*i"

K)o0cp

kje0

ri, q*"

q

q0

(50)

where U%1

and U%2

are the dimensionless nodal velocity vectors of the element in the x1

and x2

directions, respectively; A%i, B%

i, C%

i, E% and M% are the property matrices of the element;

F%i

and G% are the dimensionless nodal load vectors due to the dimensionless stress andheat #ux on the boundary of the element; u is the shape function vector for the temperatureand velocity components of the element; w is the shape function vector for the pressure of theelement; r

iand q are the stress and heat #ux on the boundary of the element; A and S are

the area and boundary length of the element; e is a penalty parameter since the penalty"nite element approach11 is used to eliminate the pressure variable in the process of derivingequation (48).

From the "nite element computational point of view, the problem domain must be "nite in size.This is in contradiction with the hydrothermal system considered here because it is a horizontallayer of in"nite length. However, consideration of the periodic nature of the solutions (seeequation (30)) for high Rayleigh numbers makes it possible to overcome this di$culty. Theoret-ically, it is appropriate to place two vertical boundaries of the "nite element computationaldomain at the cell boundaries, where both the horizontal velocity and the normal thermal #ux arezero. Since there are two cells (a clockwise cell and an anticlockwise cell) in a periodic cycle of thesolutions, the length of a cell in the horizontal direction is equal to half the wavelength of thesystem. This indicates that the minimum length of the "nite element computational domainshould equal half the wavelength of the system. From equation (33), such a minimum length canbe deduced and expressed as

¸*"¸/H"n/k*1

(51)

where ¸ is half the wavelength of the system; ¸* is the minimum dimensionless length of the "niteelement computation domain and can be estimated using equations (51) and (46). Figure 3 showsthe computation domain, which is discretized into 2304 four-node quadrilateral element with2401 nodes in total.

Figure 4 shows the comparison of the analytical results with the numerical results. It needs tobe pointed out that the analytical results are calculated using equations (43), (44) and (46), whichare based on linear stability theory and the Galerkin method. The numerical results are obtainedin the following manner. For a given Peclet number, ¸* is estimated using equations (46) and (51)and a "nite element model is established. Note that Pe"6, ¸* is negative. In this case, ¸* isestimated using Pe"5, instead of Pe"6. By varying the Rayleigh numbers, the same modelis repeatedly used to calculate the convection cell until the minimum Rayleigh number, below



Figure 3. Finite element mesh for the hydrothermal system

Figure 4. Comparison of analytically predicted results with numerically calculated results

which temperature gradient driven #ow does not take place, is determined. It is obvious fromFigure 4 that the numerical results agree well with the analytical ones. Both the numerical andanalytical results demonstrate that the minimum critical Rayleigh number decreases as the Pecletnumber increases.



Figure 5. Dimensionless total velocity distribution due to di!erent upward through#ow

Figures 5 and 6 show the dimensionless total velocity and perturbation velocity distributiondue to di!erent upward through#ow conditions. The perturbation velocity is calculated by takingthe trivial velocity from the total velocity. Figure 5 shows that the Peclet number Pe hasa signi"cant e!ect on the pattern of total convective #ow in the hydrothermal system. Since theperturbation velocity is induced by the temperature gradient only, the corresponding convectioncells are clearly exhibited in Figure 6.



Figure 6. Dimensionless perturbation velocity distribution due to di!erent upward through#ow

Figure 7 shows the dimensionless temperature distribution in the hydrothermal system due tonon-uniform upward through#ow. If temperature gradient driven convective #ow does not occur,the isotherms are horizontal lines. However, once temperature gradient driven convective #owtakes place, the isotherms become curved lines. Since the distance between two neighbouringisotherms represents the magnitude of the temperature gradient, Figure 7 shows that the



Figure 7. Dimensionless temperature distribution due to di!erent upward through#ow

temperature gradient at the top of the layer becomes much larger than that at the bottom of thelayer. Generally, the temperature gradient at the top of the layer increases as the Peclet numberPe increases, as is indicated by equation (18).

To investigate the e!ect of temperature gradient driven convective #ow on ore body formationand mineralization in permeable rock masses, the dimensionless Rock Alternation Index(RAI)8 has been calculated and shown in Figure 8 for three di!erent upward through#ow cases.According to Phillips,8 the RAI is de"ned as the dot product of the pore-#uid velocity



Figure 8. Dimensionless RAI distribution due to di!erent upward through#ow

and temperature gradient. Physically, the down temperature pore-#uid #ow most likelyresults in minerals precipitation, whereas the up temperature pore-#uid #ow mostlikely results in minerals dissolution in a hydrothermal system. This implies that for tem-perature gradient driven mineralization, the minimum negative value of the rock alterationindex indicates the most probable region of mineral precipitation. Without temperaturegradient driven convective #ow, the isopleth of the rock alteration index is horizontal.



Figure 8 shows that the most probable region of mineral precipitation is remarkably localizedand concentrated.

4. CONCLUSIONS

For the hydrothermal system considered in this paper, the exact analytical solutions indicatethat the trivial horizontal and vertical Darcy velocities are zero (equation (13)) and constant(equation 15)), respectively, throughout the whole layer. The trivial temperature pro"le is ofa non-linear nature along the layer thickness direction and is given by equation (18).

From linear stability theory and the Galerkin method, it has been demonstrated thatthe hydrothermal system becomes unstable when the Rayleigh number is equal to or greaterthan the minimum critical Rayleigh number of the system. To represent the hydrothermalsystem considered here, the modi"ed Rayleigh number used is expressed as a functionof the injected conductive thermal #ux at the bottom of the layer. In addition, it has alsobeen demonstrated that the minimum critical Rayleigh number only depends on the upwardthrough#ow (Pe) within the range considered in this study. This implies that the onset oftemperature gradient driven convective #ow in the hydrothermal system considered is onlydependent on the bottom temperature gradient, upward through#ow and the hydro-thermal parameters of the system, but independent of the initial pressure distributionwithin the layer. The reason for this is due to the fact that the convective #ow is drivenby the temperature gradient, rather than by the initial pressure gradient in the hydrothermalsystem.

Both theoretical and numerical analyses have indicated that for a small or moderate Pe (up toPe)6), an increase in upward through#ow destabilizes the temperature gradient driven #ow inthe hydrothermal system considered. This implies that within the range of Pe)6, the occurrenceof temperature gradient driven convective #ow becomes easier as the upward through#owbecomes stronger, for a layer of given thickness.

It is concluded from the related numerical results that not only can temperature gradientdriven convective #ow occur in systems with upward through#ow, but also it potentially plays animportant role in the process of ore body formation and mineralization in the hydrothermalsystem considered.

ACKNOWLEDGEMENTS

The authors are very grateful to the anonymous referees for the valuable comments on an earlydraft of this paper.

APPENDIX

In the case of Pe"0, the analytical solution to the dimensionless horizontal wave number incorrespondence with the minimum critical Rayleigh number for the hydrothermal system con-sidered10 is equal to 1)75. Thus, error in the dimensionless horizontal wave number resulted fromdi!erent groups of trial functions can be compared with this analytical solution, as shown inTable I below.



Table I. Comparisons of di!erent trial functions (Pe"0)

Group Trial functions Dimensionless horizontal Relativewave number error (%)

1 <M (x*2)"1

2(x*

2)!x*

2, hM (x*

2)"1

2[(x*

2)2!1] k*

1"2SA

5

2B 9)71

2 <M (x*2)"2

3(x*

2)3@2!x*

2, hM (x*

2)"2

3[(x*

2)3@2!1] k*

1"4SA

105

16 B 8)57

3 <M (x*2)"1

2(x*

2)2!x*

2, hM (x*

2)"2

3[(x*

2)3@2!1] k*

1"4SA

75

12B 9)71

4 <M (x*2)"2

3(x*

2)3@2!x*

2, hM (x*

2)"1

2[(x*

2)2!1] k*

1"4SA

105

16 B 8)57

5 <M (x*2)"2

3(x*

2)3@2!x*

2, hM (x*

2)"4

5[(x*

2)5@4!1] k*

1"2SA

63

24B 7)43

6 <M (x*2)"1

3(x*

2)3!x*

2, hM (x*

2)"1

2[(x*

2)2!1] k*

1"4SA

2625

482 B 12)6

7 <M (x*2)"1

2(x*

2)2!x*

2, hM (x*

2)"1

3[(x*

2)3!1] k*

1"4J7 6)86

8 <M (x*2)"1

2(x*

2)2!x*

2, hM (x*

2)"1

4[(x*

2)4!1] k*

1"4SA

225

28 B 4)00

9 <M (x*2)"1

2(x*

2)2!x*

2, hM (x*

2)"1

5[(x*

2)5!1] k*

1"4SA

55

6 B 0)57

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459}463 (1989).8. O. M. Phillips, Flow and Reactions in Permeable Rocks, Cambridge University Press, Cambridge, 1991.9. D. A. Nield and A. Bejan, Convection in Porous Media, Springer, New York, 1992.

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Theoretical and numerical analyses of convective instability in porous media with upward throughflow