criterion for instability of steady-state unsaturated flows

Transport in Porous Media 25: 313-334, 1996. 313 (~) 1996 Kluwer Academic Publishers. Printed in the Netherlands.

Criterion for Instability of Steady-State Unsaturated Flows

VIVEK KAPOOR School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0355, U.S.A. e-mail: [email protected]

(Received: 13 December 1995; in final form: 3 July 1996)

Abstract. The stability of steady-state solutions to the unsaturated flow equation is examined. Condi- tions under which infinitesimal disturbances are amplified are determined by linear stability analysis. Uniform suction head profiles are shown to be linearly stable to three-dimensional disturbances. The stability of nonuniform suction head profiles to planar (:el - :e2) disturbances is examined. When the steady-state suction head solution (tg) increases with depth, :ca, (dfft/dx3 > 0), a condition for the amplification of infinitesimal planar disturbances is identified as

daK(tg) \ dffJ ] > d~ ~ K(~)

where K(ff t) is the hydraulic conductivity versus suction head characteristic of the porous medium. The same condition applies when d~/dx3 < -1. Therefore when the rate of change of the slope of the K - gJ characteristic curves is larger than the squared slope divided by K, even small disturbances can be amplified exponentially. The smallest wavelength of unstable planar perturbations is shown to be inversely related to the coarseness of the soil. Conditions under which the instability criterion is met are delineated for some commonly employed K - 9 curves.

Key words: unsaturated flow, instabilities, fingering, linear stability analysis, cutoff wavelength, instability criterion, nonlinearity, spatial-temporal complexity, exponential growth, moisture profiles.

1. Introduct ion

The hydraulic conductivity, K , of a variably saturated porous medium is a decreas- ing function of the suction head, ~, and an increasing function of the moisture content, 0 (Figure 1). Consequently, during infiltration into an initially dry soil, the rate of propagation of moisture or suction head fronts at any point will depend on the moisture content and suction head, and their gradient. This nonlinearity can result in an amplification of variations of the suction head or moisture content. The ensuing differential movement has been experimentally observed to often result inf ingers, which are regions of large moisture content and specific discharge (e.g., Miller and Gardner, 1962; Peck, 1965; Hill and Parlange, 1972; Diment and Watson, 1985; Glass et al., 1989(b); Glass et al., 1990; Baker and Hillel, 1990; Babel et al., 1995). In a f ingered f low, water may be transmitted through the vadose

314 VIVEK KAPOOR

K

Porosity -

0

(o,0) (o,o) v/

Figure l. Hydraulic conductivity characteristics for unsaturated flow. K (units: L/T) denotes the hydraulic conductivity. The suction head ~b = -P/P9 (units: L), where p is the pressure relative to the air pressure, which is assumed to be constant, p is the density of water, and 9 is the acceleration due to gravity. Themoisture content is denoted by 0 (units: L3/L3).

zone relatively rapidly, over a small fraction of the porous media. The transport of moisture out of these fingers, driven by gradients in the suction head, can inhibit them. The competing influences of variable advection and diffusion of moisture and how they determine the stability of steady-state unsaturated flow solutions is examined in this paper.

The capacity of the vadose zone to filter out some contaminants and thereby protect the underlying aquifer can be diminished due to fingering, thereby increasing the susceptibility of the saturated zone to contaminants. The highly nonuniform spatial distribution of moisture content can shorten the time-scales governing the movement of water from the vadose zone to the aquifers. Hence, the practical importance of understanding the circumstances under which fingered flow may occur or not, and describing the fingered flow. Of course, the heterogeneity of natural porous media will influence the flow description, in addition to instability mechanisms. The stability of the heterogeneous flow solutions to disturbances also needs to be understood. In this paper, however, the focus is on analyzing the instabilities of the steady-state flow descriptions in homogeneous media.

The topic of instability of unsaturated flows was studied early by soil scientists. From the body of experimental work referred to in the introductory paragraph, three common features emerge:

(1) In uniform coarse sands, the smaller the initial moisture content (i.e., the larger the initial suction head), the more likely and pronounced is the fingering.

(2) Coarse soils are more prone to developing pronounced thin fingers than fine ones.

(3) The presence of a layer of a fine porous medium overlying a coarse one causes a wetting front to finger, and the greater the contrast, the more pronounced is the fingering.

The porous continuum formulation of unsaturated flow requires experimentally determining the hydraulic conductivity-suction head-moisture content (K - ~b - 0)

CRITERION FOR INSTABILITY OF STEADY-STATE UNSATURATED FLOWS 315

characteristics (Figure 1), which are the source of strong nonlinearity in the unsaturated flow problem. Due to this nonlinearity, understanding the experimentally observed features theoretically is a difficult task. The developments in the method- ologies of numerical simulation of unsaturated flow are not easily employable to study fingering because of the difficulty in distinguishing between instability due to numerical artifacts and actual amplification of disturbances based on the physics inherent to the problem of unsaturated flow. To surely avoid the numerical instability artifacts small grid P6clet numbers will be needed. Therefore, studying instabilities by direct numerical simulation will be computationally intensive. Consequently, most of the insights gained into fingering have been through simplified analysis, and their application to explain experiments. A brief discussion of some of the existing approaches to theoretically understand the fingering problem is presented next.

Diment et al. (1982) formulated the eigenvalue problem for stability analysis of the unsaturated flow problem, similar to the formulation made here. However in a subsequent numerical solution to that problem, Diment and Watson (1983) did not predict any condition for linear instability and found all the flows they examined to be stable against small perturbations.

Earlier, Raats (1973) argued that wetting fronts would become unstable if the velocity increased with depth, using the Green and Ampt flow model. Philip (1975) and White et al. (1976) also employed a Green-Ampt infiltration model to examine the stability of a wetting front. Parlange and Hill (1976) have examined the stability of a wetting front, assuming instability induced fingers to be fully saturated.

Hillel and Baker (1988) identified a relatively simple criterion for predicting fingering and the wetted fraction in a horizontally layered system, in which a clay overlays a coarser medium. According to Hillel and Baker, if the value of the suction head at which the coarse layer admits water into it, ~be, is small enough so that the hydraulic conductivity of the coarse layer at that suction head,/s (~be), is larger than the vertical specific discharge through the clay, fingering will occur. They postulated that the wetted fraction is simply the specific discharge through the clay divided by K(~e) . The experiments conducted by Baker and Hillel (1990), in which ~be was experimentally measured, were in remarkable agreement with their simple 'flux discrepancy' criterion. An analysis to predict ~be and its experimentally observed variation with the initial moisture content of the coarse layer remains to be done to fully realize the predictive capability of Hillel and Baker's 'flux discrepancy' criterion. This criterion is based on considering the vertical gravitational flux of moisture and does not take into account the mechanism of lateral diffusion of moisture out of the fingers.

A different approach was taken by Glass et al. (1989(a)), who examined the fingering phenomenon based on combining a dimensional analysis of Richards equation and the Miller and Miller (1956) scaling theory. Glass et al. (1989(a)) developed new relationships between finger width, velocity, and initial moisture content. These relationships involved some parameters and functions that needed

316 VIVEK KAPOOR

to be experimentally determined (Glass et al., 1989(b)). A theoretical prediction of these functions, based solely on independently inferred K - ~b - 0 characteristic curves (0 being the moisture content) needs to be undertaken to further the predictive capability of the insights gained through Glass et al. 's analysis.

Insofar as nonlinearities of the unsaturated flow phenomenon can give rise to complex spatial distributions of the transported quantity, the moisture content, the unsaturated flow fingering problem has similarities with the viscous fingering problem in porous media (see Homsy, 1987, for a review). The viscous fingering problem has been studied extensively by petroleum and chemical engineers because of it's application in enhanced oil recovery and the performance of packed bed reactors. In addition to stability analysis of that problem, direct numerical simulations of viscous fingering in homogeneous and heterogeneous porous media have also been performed and critically compared to observations (e.g., Zimmerman and Homsy, 1992; Tchelepi et al., 1993).

The motivation of this theoretical study is developing an explicit criterion for instability of steady-state unsaturated solutions, based on the empirical K - ~b - 0 characteristics. Linear stability analysis of steady-state unsaturated flow solutions is undertaken to delineate such a criterion. Analytical results on sufficient conditions for the amplification of infinitesimal disturbances are presented. Based on the analysis, the previous experimental observations can be qualitatively interpreted. The explicit criterion for instability, developed here, is illustrated for different representations of the unsaturated hydraulic conductivity function. The analysis is employed to predict how the instability mechanisms would manifest in different types of soils (sand, loams and clays). The spatial scale of the smallest unstable wavelength is identified. The results found here on the stability of steady-state solutions complement the previous results on the stability of wetting fronts, as under the conditions that the steady-state solution is unstable, the classical solution to the unsaturated flow equation cannot be expected to be realized, even at large time. The limitations and strengths of linear stability analysis need to be underlined at the outset. The strong results of linear stability analysis are in identifying conditions under which small perturbations are surely going to be exponentially amplified. However, even under the conditions that the transport system is linearly stable, amplification of finite amplitude disturbances can take place. Such subcritical bifurcations have not been examined in this work.

2. Formulation

2.1. UNSATURATED FLOW EQUATION

Denoting the specific discharge by qi, the suction head by ~b, the hydraulic conductivity by K(~b), and the volumetric moisture content by 0, the empirical Darcy's law for unsaturated media

qi = + (1) ux i


(o,o

(o~

3'

Suction head ~(x~)

X 1

x3

x3

Figure 2. The stability of the steady-state suction head solution 9 (x3) to the unsuturated flow equation is examined by identifying the conditions under which perturbations to ~(x3) are exponentially amplified.

and conservation of mass statement for the constant density case

O0 Oqi 0-7 + ~ = 0, (2)

yield a nonlinear advection-diffusion equation for the suction head

c(~,) ~_~t + b(#))(~@/+ ~i3 ) 0~b 02~b ~ - K (~)ox ibx~ - o, (3a)

c(~b) dO dK(~b) - - > 0 , b(~b) - (3b)

d~b d~b

Summation over repeated indices is implied in (2) and (3a). A specification of the K - ~b - 0 curves, determined from experiments, completes this widely employed continuum model of unsaturated flow (Richards, 1931). In (3), it is assumed that the air in the pore-spaces is much more mobile than water (and gets out of its way), thereby assuming that the need for separately analyzing its movement doesn't arise. For the purpose of performing a stability analysis, it is further assumed that the K ~b 0 curves are nonhysteretic.

The nonlinear transport Equation (3) admits a steady-state solution '.II, which, for the configuration examined in this study (Figure 2), varies in x3 only. On setting the time derivative equal to zero in (3), it follows that the steady-state solution, ',9, is characterized by

d2~ b(~) ( d~ ~2 d~ ] dx~ -- K(~) k.~x3J + ~x3 " (4)

318 VIVEK KAPOOR

The nonlinear boundary value problem (4) can be solved analytically for some special K(tg) functions (Gardner, 1958; Bear, 1972), or numerically, for more general cases9 The purpose of this paper is to examine the stability of the steady- state solution ~. This study is carried out in multidimensional space (Figure 2). The key result of this work is the identification of conditions under which the steady-state solution is unstable to small disturbances, based on the K - ~b - 0 relationship. The expressions derived in this work are represented in terms of the steady-state solution, ~, and its derivative d~/dx3.

2.2. LINEARIZED PERTURBATION EQUATION

It is stressed that ga (lower case) quite generally denotes the suction head, and (capital case) denotes the steady-state solution (governed by Equation (4)), which varies in :c3 alone. We are interested in the evolution of ~b = ~ + ~b', where ~b' is the perturbation that the steady-state solution is subjected to. Does the nonlinear transport system amplify or diminish small perturbations, ~b', that t9 is subjected to? To answer that question, the nonlinear parameters of the transport Equation (3) can be represented by Taylor series expansions of the nonlinear functions around their value at the steady-state suction head @. For example,

s b(t9 + r = ~ ~b'n dnb(~b) [

,~=o n! d~b '~ ~, (5)

Substituting the first two terms of such expansions in (3a), and using the condition on q~ for it to be a valid steady-state solution to (3a) (i.e., O~/Ot = 0, and (4)), the linearized evolution equation for ~b ~ is

' [ d~] 0~b' -K (~) 02~ b' 1+257 3 b77.3 0x ox -/3r (6a)

b2( ) db(ff*) [ /3=- L -I- kxdx3J J "

(6b)

Summation over the repeated index i is implied in (6a). The differential equation for the steady-state solution (4) was utilized in evaluating/3 in the form presented in (rb).

The units of the parameter/3 are (LT) -1. By controlling the sink/source term in the perturbation equation, the parameter/3 importantly controls the stability of the steady-state solution, which is the main result of this work.

2.3. EIGENVALUE PROBLEM

As the coefficients of the ~b ~ Equation (6) vary in :c3 alone, and no flux boundary conditions are assumed at the vertical boundaries in the zl and x2 directions, ~b p


may be expanded in the form

\ L1 / cos \ L2 . ] '

hi, k2 = 4-3 . . . . (7)

Any spatial distribution of disturbance (consistent with the boundary conditions) can be represented as the superposition of the representation in (7). Therefore the growth of any of the Fourier modes (7), will render the steady-state solution, 9, linearly unstable to infinitesimal disturbances.

Substituting (7) in (6) gives

d2f df dx~ - A I I + A2 dx---3' (8a)

[(~lTr~ 2 " (~27r~ 2] (3(~I/) A1- K (~ + [ \ L1 / + \ Lz / J +sK(tg)' (8b)

dln C(r [2d ] Aa - d~ ] dx3 + 1 . (8c)

Equation (8) is the general eigenvalue problem, that needs to be solved to determine the stability of steady-state solutions to the unsaturated flow Equation (3). The assumptions made so far are:

(1) Nonhysteretic characteristic functions; (2) Infinitely mobile air phase; (3) Steady-state solution a function of x3 alone.

The growth constant s is an eigenvalue, and depends on the nondimensional wavenumbers kl and k2 and the steady-state solution ~. When s is positive, small perturbations will undergo exponential amplification.

For the case of constant ~ the eigenvalue problem can be solved (Section 2.4). Unfortunately it is not possible to solve the general eigenvalue problem analytically, even if 9 (x3) is known. In formulating a numerical solution to determine the growth constant, s, the eigenvalues of an asymmetric matrix will be required. For the case of planar (Zl -- X2) perturbations, considerable simplification is afforded and the results are presented in Section 3. Numerical solution to the general eigenvalue problem is not undertaken in this work. For an overview of eigenvalue problems that arise in hydrodynamic stability analysis, the reader is referred to Drazin and Reid (1981).

2.4. UNIFORM SUCTION PROFILES ARE NOT LINEARLY UNSTABLE

First let us consider the case in which the suction head is uniform (due to the boundary conditions 9(0) = ~(L3) = ~) , and f = 0 at x3 = 0 and L. That

320 VIVEK KAPOOR

such a uniform suction head profile is stable to small perturbations can be inferred directly from the perturbation Equation (6), or from the eigenvalue problem (8).

First let us consider the suction head perturbation Equation (6). The parameter fl is zero if d~/dx3 = 0, and the other coefficients in the equation are constants. The suction head perturbations r are therefore governed by a constant coefficient advection-diffusion equation

.... ,0r 00~ 3 02r ' + - = o (9)

and therefore cannot be amplified. This can be shown by formulating the equation for the spatially integrated squared perturbations of the suction head

(r =- f /a / r dxl dxadx3, (lO)

where the domain f~ is shown in Figure 2. It follows from (9) and the boundary conditions that

2K(%) /or or dt C(~u) \ Oxi Oxi "

(11)

A sum over the repeated index i is implied in (11). As the right-hand side of this equation is never positive, (r cannot increase with time. From the inequality

k, Oxi] / ) (Li/u) 2' (12)

it follows from (11) that (r will undergo an exponential decay and that

dln(r 2K(~u) [ ( _~) -2 ( _~) - : ( _~) -2] dt ~< ~(~,~) + + . (13)

Inequality (12) is a variant of Poincar6's inequality, and is sometimes also called the Wirtinger inequality (Hardy et al., 1952). Kapoor and Kitanidis (1995) employed this inequality to find an upper bound on the concentration fluctuations in the case of advection-local dispersion in heterogeneous porous media, that demonstrated the singular role of local dispersion in controlling dilution and the irregularity of concentration distributions.

Now consider the eigenvalue problem (8) for the constant suction head case. The constant parameters are

AI = \ L1/ + \ L: / ] + s~(~) ' A2- dlnK(~)d~ ~ o (14)


As these coefficients are constants, the eigenvalue problem can be solved without further approximation. The characteristic equation to find a solution (of the form e mz3) to the eigenvalue problem is m 2 = A1 + mA2, which yields m = (A2 4-

v/A 2 + 4A1)/2. For the perturbations to be zero at x3 = O, L3,A 2 + 4A1 < 0. Substituting the expressions for A1 and A2 into this condition implies

4 + \ La J J +4sK( tg~) dlnK(~) ]

d~ ~ 05)

Satisfying this condition requires s to be negative, which renders the constant suction head linearly stable, as previously shown by Diment et al. (1982).

The uniform suction case is linearly stable- small perturbations will not undergo an exponential amplification. It needs to be emphasized that this does not rule out amplification of finite amplitude perturbations. The notion of linear stability is a weak one. However linear instability is a strong notion insofar as it implies exponential amplification of small disturbances. The next step is to study the stability of nonuniform steady-state solutions ~(x3) to identify conditions under which infinitesimal perturbations are amplified.

3. Simplified Analysis of Planar Perturbations When the characteristic length scale of the perturbations is smaller in the horizontal (Xl - x2) plane than the x3 direction, considerable simplification is afforded to the linear stability problem, by simply dropping the terms involving the spatial derivatives of the perturbations in x3. This simplification amounts to studying the stability problem in two-dimensional (zl - x2 plane), with the x3 dependence appearing through the parameters of the two-dimensional problem. Physically, this amounts to considering whether the perturbations will grow or decay, subject to the source term on the right-hand side of the perturbation Equation (6), and the diffusion of the perturbations in the horizofltal (xl - . ' /72) plane. The steady-state solution obeys (4) and the boundary conditions imposed on it. As experimental observations (referred to in the first paragraph) have often shown narrow and long fingers due to instability, it is worthwhile to study the stability of the steady-state solution to those types of disturbances. An explicit condition on the hydraulic conductivity-suction characteristic, for linear instability, results from this simplification. The eigenvalue problem for this case is Al f = 0, which yields, A1 = 0, as the condition on the growth constant s. It follows that

s= \L1 J + \L2 / (16)

322 VIVEK KAPOOR

3.1. CONDITION FOR LINEAR INSTABILITY

If

[ \L1 J -~- \ L2 J J ' (17)

s will be positive and infinitesimal planar disturbances will be exponentially amplified. Expanding/3 (based on (6b) and (3b)) the sufficient condition (17) for the amplification of planar disturbances is

d~2 dff,' ) d~ 2 (kl~r)2 (k21r,~2] .

On the left-hand side of (18) are the parameters intrinsic to the porous media and the gradient of the steady-state solution, which is determined by the boundary conditions too. On the right-hand side are the dimensions of the domain, and the wavenumbers of the disturbances.

3.2. SMALLEST UNSTABLE WAVELENGTH

The growth constant versus wavenumber relationship (16) is shown in Figure 3. The cutoff wavenumbers (i.e., the wavenumbers at which s becomes zero) are given by

\ L1 J + \-L-222 J J cutoff - K~ffa) (units" L-2). (19)

The cutoff wavenumbers (k I / L I , k2/L,2) establish the size of the smallest wavelength of the perturbations that will be amplified by the transport system

Smallest unstable wavelength = 7 r ~ ~) . (20)

Planar disturbances with wavelengths smaller than that given in (20) are not amplified because the diffusive attenuation of those disturbances occurs at a rate faster than their amplification due to the dependence of hydraulic conductivity on suction head and the suction head gradient, which determine/3. The compact expressions (18), (19), and (20) will be employed later to compare the instability of steady- state flows in different types of soils. These expressions are pertinent to the stability of a steady-state solution to disturbances with spatial scales much smaller in the horizontal plane, compared to the vertical direction.

CRITERION FOR INSTABILITY OF STEADY STATE UNSATURATED FLOWS 323

r~

o

o

0

s : + -

~ ~ , ~ , , , , I , , , , I , ~1 I , , ,

t(12"'(~kllr" "l- - - -~ k2~l ~ 4 J~/K( t I~

Figure 3. Growth constant, s, as a function of perturbation wavenumber (for/3 > 0).

d2K(~) k ,~/ | [ d',I~ d/d'-!/"/2 ] d0(~, . /3 = d~ 2 . . . . . K(RJ) / Ldx3 + ~-~x3] ] ; c(~)-- dg~ ' 0 : moisture content.

J

3.3. CONDITION FOR LINEAR INSTABILITY

For a large domain problem (klrr/L1 ~ O, k2~r/L2 --+ 0), it follows from (18) that the positivity of/3 is the sufficient condition for linear instability

d2/-s \ d~ J d~ d~ 2 /3 ~-~ - d~ 2 K(~I/) " ~ -1-- ~ > O. (21)

For finite domains this remains a necessary condition for linear instability. If/3 is negative, the growth constant, s, is always negative - under these condition exponential amplification of infinitesimal planar disturbances does not occur. However if/3 > 0 then the growth constant s may become positive and an exponential amplification of small disturbances will occur. That the sign of/5 importantly controls the stability of the perturbations is evident from the linearized perturbation Equation (6) which has a source term equal to/3~b'.

324 V1VEK KAPOOR

For the purpose of delineating conditions under which/3 is positive, we will write it as

fl = X[J + j2]

Due to gravity and Due to suction head gradient suctionhead gradient J. (independent of gravity). (22)

(

d2K( ) | and J - d* (23) = d~ 2 K(~-----) ~ dx3"

The parameter X is an intrinsic property of the porous medium, as it is determined by the function K(~). It is useful to recognize the origin of the different terms in the definition of/3 (22). If the effect of gravity were neglected, the first term in the squared brackets would not be there. When J > 0, then both the terms in the square brackets act to destabilize the solution when X > 0, and both the terms stabilize the solution when X < 0. The effects of gravity have only an additive role in determining the linear stability of the steady-state solutions. Instabilities can occur without gravity.

3.4. STABILITY CHARACTERISTIC FOR DIFFERENT TYPES OF SOILS

Interpreting previous experiments will require knowing the dimensions of the domain, and boundary conditions. However, an intrinsic measure of the potential of steady-state solutions to be unstable in a soil is the value of/3. When it is positive the necessary conditions for instability of infinitesimal planar disturbances are met, and when it is negative, the steady-state solution does not exhibit linear instabilities. In this section the nature of/3 is explored for different K(qJ) functions. In Table I are listed the evaluations of X for different K(~) functions. An example calculation for the van Genuchten (1980) hydraulic conductivity function, which has four parameters, and is capable of representing a wide variety of soils, is also presented. The focus is on interpreting the general implications of the instability criterion for different K(qJ) functions, and making general inferences of the potential of instability of the steady-state solution in different types of soils.

Case 1 J > 0

It is generally perceived that instabilities occur when the suction head is increasing with depth. Most of the experimental documentation of fingering has been made


K( q9 Ksat

7 = Ksa, e

X Linearly stable to planar perturbations

i i i i i t i T i i i i i i I I l l ~ ' , , , 11 t I I r

(0,0) Suction head

Figure 4. For the Gardner hydraulic conductivity characteristic K(~) =/~-~t e -'~'~ (Model 1 in Table I), X = d~K(ffs)/d 'I'2 - (dK(~)/d~)2/K(~) = 0 for any 't~ >__ O. Therefore small planar disturbances will not be exponentially amplified.

for this case. For a base state solution having an increasing value of suction head with z3, the condition for amplification of planar disturbances is simply X > 0 (from (21) and (22)). This condition reads as follows: if the rate of change of the slope of the hydraulic conductivity is larger than the squared slope divided by the hydraulic conductivity, then planar disturbances will be amplified. Therefore the slope and the curvature of the hydraulic conductivity function both play an important role in controlling its stability characteristics. In subjecting an initially dry soil to high moisture at the top end, the smaller the initial moisture content, the larger the value of J, and therefore greater the potential amplification of planar disturbances (Equation (16)), when

From the point of view of stability, the exponential model (Model 1 in Table I, which is also referred to as the Gardner model) has a special characteristic insofar as the rate of change of its slope is precisely equal to the squared slope divided by its hydraulic conductivity (therefore X = 0). The exponential model is linearly stable to planar disturbances for all values of the suction head (Figure 4). However, the simple exponential model is generally not able to describe the empirically observed hydraulic conductivity function over a large range of suction heads. It is mostly employed for its sim- plicity, and because some explicit steady-state analytical solutions for it can be found.

tad

to

Table

I. H

ydra

ulic

con

du

ctiv

ity st

abili

ty ch

arac

teri

stic

s (a

)

K(~

) x

- -

-

d2

K(~

))

\~

//

d~

2 K

(~)

Con

dit

ion

s for

x>

O

x

0

/]a'

tI t-(n

+2

) a

a/*

~ 2t

~-2

[ M

odel

3. b

+ ~

-----7

(b

+ ff

-'~*)

3 1

[ (O

~'I/)

n--i ] 2

Ks

~t

1 (1

+ (

oe~

)'~

) m

(b)

Mod

el 4.

(1

+ (

~')

~)m

/:

imp

ossi

ble

im

pos

sib

le

g' >

~p

im

pos

sib

le

(#.7

-7:. 1)

b ]

~*'

> (

/.t -

1)

b ,Ir

a

0

0


K(70 gsat

- r i i i I i

K(70= a \ - b+ T"

i i l l l l l i l 1 i , , , 1

(0,0) Suct ion head

7'= [(#-l)b]

g~ Z> 0

P , , , , I , , , ,

Figure 5. For the hydraulic conductivity characteristic K(~) = a/(b + ~) (Model 3 in Table I), X > 0 for 'I' > [(# - l)b] l/~, and vice-versa. When X > 0, planar disturbance can be exponentially amplified when d~/dx3 > 0, and when d'It/dx3 < -1. For 0 > d~/dx3 > - 1, X < 0 is the necessary condition for linear instability of planar disturbances.

Hillel (1980) presents a list of different K(tg) functions - among them are models 2 and 3 of Table I. Model 2 is a power law. Its stability characteristics are quite different from the exponential model. In Model 2 the condition X > O is met over all the possible values of suction head, hence planar disturbances will undergo exponential amplification. For Model 3 the condition X > 0 is met only at sufficiently large suction heads (Table I and Figure 5). For the widely used van Genuchten (1980) model (Model 4 in Table I), the expression for X is lengthy and, therefore not presented in the table. The van Genuchten model is similar to Model 3 insofar as the condition X > 0 is met only at sufficiently large suction heads. In Figure 6 is shown an example calculation for the van Genuchten model.

Model 3 (Table I) will be used to compare the stability characteristics of different soils. It is generally observed that the parameter # is larger for coarser soils (Bouwer, 1964, 1978; Hillel, 1980; etc.) Table I lists the condition for instability as 9 > [(# - 1) b]UU. In Figure 7 is shown that the condition X > 0 is met at greater suction heads for finer soils, for the parameters employed here (after Bouwer, 1964, 1978). It is predicted that clays would require greater suction head for instabilities to occur, compared to loams and sands. This prediction is in general agreement with the experimental observations that coarse soils are more susceptible to instabilities.

328 VIVEK KAPOOR

0.8

K(q') 0.6

Ksat 0.4

0.2

~r = 21 cm

xO

(a~e)"-'

(1 +

o~= 0.03, n=2, m=0.5

0

0 10 20 30 40 50 60

Suction head t/.t (cm)

Figure 6. Example calculation for the van Genuchten (1980) hydraulic condictivity function K(~) (Model 4 in Table I). For any if' > ~I'c~, X > 0 and vice versa. When X > 0, planar disturbance can be exponentially amplified when d~/dz3 > 0, and when d~/dz3 dff'/dz3 > -1, X < 0 is the necessary condition for linear instability of planar disturbances.

This prediction holds even in the absence of gravity effects, although gravity would make the instability more pronounced.

For the three different soils dealt with in the previous paragraph, we now evaluate the cutoff wavelength (Equation (20)), as a function of suction head and its gradient. The physical significance of this length scale is as follows: small perturbations to the steady-state solution, with characteristic spatial length-scales smaller than the cutoff wavelength, will not be amplified because the transverse diffusion of moisture occurs rapidly enough over such lengthscales to counter the rate of amplification. Of course if the domain length is smaller than this length- scale then it can be concluded that the steady-state solution is linearly stable to perturbations. Model 3 is employed to evaluate the smallest unstable wavelength 7rv/K-/fl. Figure 8 shows how this wavelength varies with suction head for the three different types of soils considered here. Left of the vertical line X < 0 and the condition of linear instability to planar disturbances is not met. The value of X (and therefore/3) has a distinct maximum, which causes the smallest unstable


1000

100

K(7') 10

(cm/day)

0.1

0.01 0

:am

...................... } ......... ~ ~

i i i , I i , I I I i I 1 I I I i , I

50 100 150 200 Suction head ~ (cm)

Figure 7. Comparison of the necessary condition for linear instability for sand, loam, and clay. The hydraulic conductivity is represented as K (~) = a/(b + ~u) (Model 3 in Table I) and the parameters (with suction head expressed in cm) are taken to be (Bouwer, 1964):

Sand: a = 5 109, b =- 107; /z = 5, Ksat = 500crrdday. Loam: a = 5 x 106,b = 105; # = 3, Ksat = 50crn/day. Clay: a=5 / z=2, gsat= lcrn/day.

To the right of the vertical line on each curve, X > 0, and to the left of the vertical line X < 0. For d',It/dz3 > 0, planar perturbations can be linearly unstable when X > 0. Therefore when d@/dx3 > 0, sands can exhibit instability at smaller suction heads than loams, and loams can exhibit instabilities at smaller suction heads than clays.

wavelength to have a distinct minimum, in Figures 8a, b, c. For large suction heads it follows from the expressions in the Table I that

"~/) ) [(# - l )b ] l / /~

Smallest unstable wavelength = 7r~/r~-~ V~ ~r~ .--+ (24)

~/# ( j ~_ j2 ) "

As the parameter # is larger for coarser soils and smaller for finer soils, the analysis predicts that the size of the narrowest fingers resulting from instabilities will be inversely related to the coarseness of the soil. This has been observed in experiments. A physical interpretation of this is that the effects of gravity are more important in coarse soils, and the capillary action that will smear the fingers is less rapid in coarse soils. This argument is not entirely correct as the analysis predicts

330 VIVEK KAPOOR

(a) 1000

lO0

~ 10

1

0 .1 - -

10

(b) 104

1000

100

lO

~ 0.1 10

(c) 104

1000 o=

~: loo

. . . . . . t . . . . . . . ~ J=O.1

~ J=l J=lO ~ J=lO0

, , , r J , I _ , , t , , ,~ I00 1000

Suchon head t/J (cm)

/ J=0.1

~ J=l ~,~ ~ / J=lO

/ J=lO0

i ~ r J i l l

100 Suction head k u (cm)

~ J=0.1

~ J=l

i i i E i _

1000

~ J=10 10

,-~ ~ J=lO0

10 100 Suction head tart (ella)

1000

Figure 8. Smallest unstable wavelength as a function of suction head and its gradient for model 3 in (Table I). For the suction head vales to the left of the vertical line, the conditions for amplification of planar disturbances are not met when d~/dx3 > 0. The smallest wavelength is nonmonotonic in suction head. Therefore in an experiment, linear instabilities may be manifest in a certain range of the suction head, and not observed at too high or low values of the suction head. (a) Sand (Hydraulic conductivity model parameters in Figure 7 caption); (b) Loam (Hydraulic conductivity model parameters in Figure 7 caption); (c) Clay (Hydraulic conductivity model parameters in Figure 7 caption).


this to occur even without the influence of gravity. In fact for J >> 1, gravity has an unimportant role in the analysis made here. This should not be considered surprising as gravity effects and capillary effects are competing influences, with the latter being dominant in the situation of sharp gradients. To make a sharper quantitative assessment of this prediction, experiments, with careful measurements of the hydraulic conductivity functions K(~) , need to be performed.

The evaluations of the smallest unstable wavelength (24) (Figures 8a, b, c) show that is varies nonmonotonically with the suction head, with a distinct minimum. Therefore, it is predicted that in a laboratory investigation of unsaturated flows, fingering may be observed at some intermediate values of the suction head and not observed at very low or very large values of the suction head.

Case 2 J < -1

Notwithstanding the general belief that instabilities occur for the case that the suction head increases with z3, the analysis performed here shows that planar disturbances will be amplified when J < -1, under the same condition, X > 0, that they will be amplified when J > 0. For a large value of ]J[, in fact, as mentioned above, the sign of J is of little consequence in determining the condition, or rate of linear amplification of planar disturbances. It is reasonable that the action of gravity is not significant, if the suction head gradients are large. Therefore, instabilities in unsaturated flow should not be considered to be primarily gravity driven.

Case3-1 < J < O

This represents the situation in which the two different terms in/3 (Equation (22)) have opposing roles. Under this condition the condition for amplification of planar disturbances is X < 0. This condition is the opposite of the conditions under which instabilities occur if J > 0, or if J < -1 . Therefore as long as the mean suction head gradient is nonzero, it is possible to have linear instabilities due to planar disturbances, depending on the nature of the porous medium.

4. Summary

The objective of this work was to examine the stability of steady-state solutions to the unsaturated flow equation. The methodology developed to do that, the inferences made, and limitations of the analysis are briefly summarized here.

4.1. METHODOLOGY

(1) The eigenvalue problem for assessing the linear stability of steady-state unsaturated solutions was formulated. Linear stability of the one-dimensionally varying steady-state solution, to multidimensional disturbances, for general

332 VIVEK KAPOOR

hydraulic conductivity-moisture content-suction head characteristics, can be assessed from the eigenvalue problem.

(2) The stability of uniform suction head profiles to multidimensional perturbations was examined.

(3) For nonuniform suction head profiles, an explicit criterion for the amplification of planar disturbances was derived.

(4) An expression for the smallest unstable wavelength was derived for planar disturbances.

4.2. INFERENCES ABOUT INSTABILITIES

(1) Constant suction head flows are linearly stable to multidimensional perturbations.

(2) An increase in the suction head in the direction of gravity is not a sufficient or necessary condition for amplification of planar disturbances.

(3) Gravity has an additive role, along with the nonlinear diffusion effects, in determining instabilities of the steady-state solution. The steady-state solution can be linearly unstable even in the absence of gravity.

(4) When the suction head increases in the direction of gravity (i.e., positive suction head gradient), planar disturbances can be exponentially amplified when the rate of change of the slope of hydraulic conductivity versus suction head (K - ffJ) function is greater than the squared slope divided by the hydraulic conductivity, i.e., when d2K(~) /d~ 2 > (dK(qd)/d~)2/K(qd).

(5) For positive suction head gradients, the Gardner soil (with an exponential K(g') function) does not yield linear instabilities. Other models exhibit potential for amplification of small planar perturbations. In some commonly employed K - 9 representations, the instability criterion is met at sufficiently large suction head values. The Gardner model is special in terms of its stability characteristics, insofar as the second derivative of the hydraulic conductivity with respect to the suction head is precisely equal the squared first derivative divided by the hydraulic conductivity, at all suction values.

(6) The minimum suction head at which planar disturbances are linearly unstable was found for different types of materials. It was predicted that that minimum suction head will be inversely related to the coarseness of the material.

(7) It was shown that the smallest wavelength of unstable planar disturbances is inversely related to the material coarseness, with or without the influence of gravity.

(8) The smallest unstable wavelength was found to vary non monotonically with the suction head, and had a distinct minimum. Therefore it is predicted that linear instability induced fingering may be observed over an intermediate range of suction head values, and not observed at too high or too low suction head values.


4.3. LIMITATIONS AND OPEN PROBELMS

(1) This paper was limited to finding analytically tractable results on the stability of steady-state unsaturated flow solutions.

(2) The inherent limitations of linear stability analysis need to be kept in mind in judging the import of the criterion delineated here. When the criterion for amplification of small disturbances is met, instabilities are sure to occur. How- ever, instabilities can even occur when the criterion is not met, by nonlinear amplification of finite amplitude perturbations. Such subcritical instabilities were not explored here.

(3) The solution to the general eigenvalue problem needs to be performed to relax the assumption of planar perturbations made here in examining instability in the presence of gradients in the steady-state solution.

(4) The analysis performed here was limited to examining the stability of steady- state solutions. How disturbances influence a transient solution needs to be examined. The nature of the transient solution may better help determine the types of disturbances amplified in real flows.

(5) The stability of the unsaturated flow solution in heterogeneous media needs to be examined. For instance, layering has been previously shown to importantly influence stability. The stability of steady-state solutions in layered media, which have also been studied extensively to determine effective properties (e.g., Zaslavsky, 1986; Yeh and Harvey, 1990), need to be explored. A numerical solution to the eigenvalue problem will provide a framework for examining the influence of layering on instabilities, and the relevance of the instability criterion delineated in this work for layered media.

(6) Finally, the significance of the new instability criterion needs to explored by direct numerical simulations and experiments with carefully measured K ~ 0curves.

Acknowledgement

Comments of three anonymous reviewers are gratefully acknowledged.

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criterion for instability of steady-state unsaturated flows

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