S. Irmay

— \951d. Soil Sci. 84:257. — 1957 e. Soil Sci. 84:329. — 1957/ Australian J. Physics 10:29. — 1959. Soil Sci. 88:91. — 1960a. Australian J. Physics 13:1. — 19606. Australian J. Physics 13:13. — 1960c. Soil Sci. 89:132. — 1960a\ Soil Sci. 89:353. — 1964. / . Chem. Physics 41:911. — 1965. J. Atm. Sci. 22:196.

Solutions of the non-linear diffusion equation with a gravity term in hydrology

S. Irmay Israel Institute of Technology, Haifa, and University of California, Berkeley

A B S T R A C T : The partial differential equation ( P D E ) of unsaturated flow of liquids in porous media, especially of water in stable soils, is discussed: steady and unsteady flows, with or without gravity, in one-, two-, and three-dimensional cases, in cartesian and cylindrical coordinates. The initial and boundary conditions are described, and the hodograph sphere (circle) method of saturated flow is extended to unsaturated media. The inverse P D E are also developed.

Various auxiliary functions are used: effective concentration C , hydraulic conductivity K, capillary potential y>, and diffusivity D , but the equations become m u c h simplified when expressed in terms of the diffusivity potential F, when they often become linear in steady flow. It is shown that K1!3 is linear in C , and y> is linear in log C in a wide range.

A number of steady flow solutions is studied analytically and graphically, either transforming the P D E into an ordinary differential equation ( O D E ) , finding a first integral or the general solution. The cases treated include: horizontal parallel, radial and two-dimensional flows; vertical flows and flows in a vertical plane and axi-symmetric. In downflow two cases are possible: from higher to lower C , and conversely.

In unsteady flows the P D E are transformed into an O D E by the Boltzmann transformation, by a linear wave-like transformation, by similarity transformations, by separation of variables and by other methods. A m o n g the new solutions the effect of a rising or falling aquifer is described, and unsteady linear inclined flow, evaporation and infiltration effects.

The method outlined above can be used in solving similar diffusion equations in other fields.

R É S U M É : O n discute l'équation aux dérivées partielles du mouvement non permanent des liquides dans les milieux poreux et spécialement de l'eau dans les sols stables : mouvement permanent ou non, avec ou sans gravité, pour des cas à une-, deux- ou trois dimensions, en coordonnées cartésiennes ou cylindriques. Les conditions initiales et celles aux limites sont décrites et la méthode de la sphère (cercle) hodographe des sols saturés est étendue à des milieux non saturés. L'équation aux dérivées partielles inverse est aussi développée.

Diverses fonctions auxiliaires sont utilisées : concentration effective C , conductivité hydraulique K, potentiel capillaire y> et la diffusivité D , mais les équations deviennent beaucoup plus simplifiées quand elles sont exprimées à l'aide du potentiel de diffusivité F, car elles deviennent souvent

478

Solutions of the non-linear diffusion equation with a gravity term in hydrology

linéaires en mouvement permanent. O n montre que K1/3 est linéaire en C et que y> est linéaire en log C dans un très vaste domaine.

Tout un n o m b r e de solutions pour le mouvement permanent sont étudiées analytiquement et graphiquement en transformant l'équation aux dérivées partielles en équation différentielle ordinaire, en trouvant une première intégrale ou la solution générale. Les cas traités comprennent: le mouvement horizontal parallèle, le mouvement radial et à deux-dimensions, des mouvements verticaux et ceux dans un plan vertical et présentant une symétrie axiale. Dans le mouvement descendant, deux cas sont possibles; d'une valeur plus élevée de C à une valeur plus faible et inversement.

Dans le cas du mouvement non permanent, les équations aux dérivées partielles sont trans­formées en équations différentielles ordinaires par la transformation de Boltzmann, par une transformation linéaire ondulatoire, par des transformations de similitude, par séparation des variables et par d'autres méthodes. Parmi les solutions nouvelles, l'effet d'une nappe ascendante ou descendante est décrit ainsi que le mouvement non permanent linéaire incliné et les effets de l'évaporation et de l'infiltration.

La méthode exposée ci-dessus peut être utilisée pour la solution d'équations de diffusion analogues dans d'autres domaines.

I. INTRODUCTION

Steady flow of a viscous, homogeneous, isotropic, chemically and physically inactive constant-density liquid through an isotropic, geometrically stable, wetted and saturated porous medium with interconnected pores, in the absence of other immiscible liquids, suspended solid particles and compressible gas bubbles, at low Reynolds numbers, obeys the experimental Darcy law:

q = KJ = -KVq> q> = z + p¡y (1.1)

where z is the vertical elevation. Experimental and theoretical evidence seems to show that the hydraulic conductivity

K increases with porosity n according to the Kozeny-Carman formula, (Carman, 1956), as modified by Irmay (1954, 1964, 1966):

Koc( f i -n 0 ) 3 / ( l -n ) 2 (1.2)

n 0 is the irreducible (stagnant, ineffective) porosity, due to stagnant water at the surface of the soil grains and in the pore corners. In general n0> 0.1 n, exceeding 0.5 n in very fine-textured soils.

Richards (1931) extended Darcy law to unsaturated flow. The moisture content m a y be expressed by the dimensionless volume concentration in water c, degree of saturation s, effective volume concentration C or effective degree of saturation S:

n0 < c ^ n 0 < C = c — H 0 < n —n0

n0jn < s = c/n ^ 1 0 < S = (c — n0)/(n — n0) < 1

The pressure head ply is replaced by the capillary potential \¡/ as measured by the pressure head of water in a porous tensiometer cup (or plate) in equilibrium with water in the unsaturated soil. Richards assumed K = K(i¡/), so that (1.1) becomes:

-q = A ' ( ^ ) V ( ^ + z) q> = ijj + z (1.4)

D u e to hysteresis \¡/(c) and K(\¡i) are not single-valued, but depend upon the previous moisture history—whether the soil is draining or wetting—and upon former points of reversal from draining to wetting and conversely (scanning curves) (Morrow and Harris, 1965; Topp and Miller, 1965).

Gardner (1946) assumed K = K(c), so that (1.4) becomes:

-q = K(c) VO/r + z) = D(c) Vc + K(c) Vz (1.5)

(1.3)

479

S. Irmay

with

D(c) = K(c) V{c) (1.6)

D{c) is the unsaturated water diffusivity in the soil. Experiments (Topp and Muller, 1965) seem to show that K{c) exhibits little hysteresis

and is single-valued. Childs and George (1948, 1950), Klute (1952) and Irmay (1956) applied continuity

considerations and obtained the non-linear parabolic diffusion equation in c(x, v, z, t) with gravity term:

c, = -divq = d iv [KV0/f + z)] = d iv (DVc + K V z ) (1.7)

F e w exact solutions are known, mostly in homogeneous soils and steady-state hori­zontal or vertical flows. The unsteady state horizontal solutions in parallel flow (Irmay, 1959; Philip, 1957) or radial flow (Singh, 1965) are based on Boltzmann's transformation (Boltzmann, 1894; Crank, 1956). In vertical flow a linear wave-like transformation can be used (Irmay, 1956). The method of separation of variables has been used in a similar equation (unsteady aquifer flows) by Boussinesq (1904). Pirverdian (1960) and Irmay (1968) use a generalized form of Boltzmann's transformation. A m e s (1965) surveys general methods for solving non-linear or parabolic equations.

II. A U X I L I A R Y F U N C T I O N S

Most experiments seem to justify for the monotonously increasing single-valued func­tion K(c) the form:

K = k(c-n0)m = kCm (m Ss 3) (2.1)

Irmay (1954, 1964) has shown that theoretically m = 3. This can be confirmed plotting K* versus c—a straight line is obtained intersecting the c-axis at c = n0 (Wyckoff and Botset, 1936, for two-fluid systems; Topp and Miller, 1965, for glass beads and water), (fig. 1). Then:

\3 \3 _ / , / -3 _ / , c3 K = k(c-n0y = kC5 = k^S* = fct

n — n.

k^kin-n»)3 K'(Q = 3kC2; C = (K/k)i

(2.2)

kx is the saturated hydraulic conductivity. This linear K$(c) relationship does not seem to hold near complete saturation.

N o such simple relationship is k n o w n concerning \¡/{c), except that it increases m o n o ­tonously on all its branches: the main draining and wetting branches ( M D , M W ) and the draining and wetting scanning curves (SD, S W ) . The function \¡/{c) increases monoton­ously only on S D curves ; on the others it decreases from c = n0 to some intermediate value of c, approximately as a linear function of log (c-n0), then increases. Near saturation (c = n), i// is not exactly determined, as it depends on the amount of entrapped air; \¡/(c)->-co. Near c = n0, i/ '̂(c)->oo. K(\j/) increases monotonously, with K'(ij/)-*0 as c-*n. The diffusivity D(c) of (1.6) does not vary so regularly, as it is the product of K(c) by i¡/'(c) which first decreases, then increases. D{c) increases at larger c. Near the dry end (c = n0), by (2.2) isf—>0, so D - > 0 . Near saturation (c = «), as i¡/'(c)-*co, D-*co.

480

Solutions of the non-linear diffusion equation with a gravity term in hydrology

A useful auxiliary function is the diffusivity potential F (Kirchhoff, 1894; Carslaw and Jaeger, 1953; Crank, 1956; A m e s , 1965), defined up to a constant:

F(c) = I D{c')dc' or F(i/>) = I KW)dij/'

F'(c) = D F'W) = K

(2.3)

F I G U R E 1. K versus C/n (based on Topp-Miller's data (1965))

c0(or \j/0) are arbitrary, e.g., n0 or n. A s D , K> 0, F(c) and F(\p) increase monotonously, but are affected by hysteresis. It seems that F is finite even near saturation (c = «). c, ij/, F, K, D can be expressed as functions of each other (table 1).

T A B L E 1. The auxiliary functions

Dependent variable Auxiliary functions

c(x, y, z, t)

y>(x,y, z, t)

F(x, y,z,t)

K(x,y,z,t)

D(x,y,z, t)

K{c), y(c), F(c), D(c)

c(i), K(ip), F(v>), D(y»

c(F), K{F), y>(F), D(F)

c(K), y>(K), F(K), D(K) c(D), K(D), y>(D), F(D)

Usually c or \j/ are chosen as the dependent variable, but F simplifies the computations. In what follows w e shall assume the auxiliary functions to increase monotonously and

481

S. Irmay

to be single-valued, which is compatible with hysteresis, provided the system is only draining or wetting.

Crank (1956) lists some suggested forms of D(C), which give analytic solutions in parallel horizontal flow. Other forms are added in Appendix I.

N o . 3 is alluring in its simplicity, as it renders the steady-state equation of flow linear in F. The 1^(0 curve has then the simple form:

I/Í = £¡o + «i l o g C = ao + ai log ( c - n 0 ) = b 0 + «i log S (2.4)

Fig. 2 shows that this is, e.g., true for 0.1 < cjn < 0.8 on the main draining curve with « 0 = 0 . 1 0 M ; and on the main wetting curve for 0.2 < c¡n< 0.6 with n0 = 0 .19n(Topp and Miller, 1965).

1 U

0.8 0.6

0.4

0.2

0.1 0 08

0.06

0.04

0.02

0.01

---

-

1

/

/

1

1

f /

/

i

; /

/MD

v\,

/ /

¥

i

/ M W

1

---

_

--

-

-70 -60 -50, -40 -30 -20 l|/ (cm) *

c - no \ F I G U R E 2. y> versus log ( J (based on Topp-Miller'j dala (1965))

III. DIFFERENTIAL E Q U A T I O N S

In the variables C, ij/, F, K, D , Darcy's flow equation (1.4), (1.5) becomes:

-q = D(C)VC + K(C)Vz = K(ij/) ( V ^ + V z ) = V F + /s.'(F)Vz = = F' (K) V K + KVz = F' (D) V£) + K (D) Vz (3.1)

Introducing into (1.7) the differential equation of unsaturated flow becomes:

C, = K ' ( C ) C + div[£>(C)VC] (3.2)

C(iP)yt = K'(«/')</'z + d.iv[K(lA)Vl/0 (3.3)

C (F)Ft = K'(F)FZ + V 2F (3.4)

C (K) Kt = Kz + div IF' (K) V K ] (3.5)

C'(D)D, = K ' ( D ) £ z + div[F(£>)VD] (3.6)

V 2 F i s the laplacian of F, defined in cartesian (x, y, z) and cylindrical (r, 0, z) coordinates by:

V 2 F = F „ + F „ + F„ = - (rFr)r + F„ + -2Fe

r r (3.7)

482

Solutions of the non-linear diffusion equation with a gravity term in hydroloby

A n equivalent form of (3.2), e.g., is:

C, = K'(C)CZ + D(C)V2C + D'(C)(VC)2 (3.2a)

(VC)2 = Cl + C) + C2 = C2 + C2 + — C 2 (3.8) r

In one-dimensional parallel vertical flow eq. (3.2) to (3.4) become:

C , = K'(C)C:+ID(C)CJZ = K'(C)CZ + D(C)CZ: + D'(C)C2 (3.9)

C'Wtfr, = X ' W ^ + E ^ W - A . L = K'W)ij,z + K(iP)il,zz + K'(il,)il,2z (3.10)

C{F)Ft = K'(F)FZ + FZZ (3.11)

etc. In parallel horizontal flow:

C, = [D(C)CJ, = D ( O C „ + D ' ( O C Í (3.12)

C'(iA)iA, = [^(<A)^L = KW)iPxx + K'(il,)ij,2x (3.13)

C\F)Ft = Fxx (3.14) etc.

All the above equations of unsteady-state flow are non-linear second order partial differential equations (PDE) of the parabolic type. In order to get a sinlge solution the problem must be well posed. This means that appropriate boundary conditions (PC) and initial conditions (IC) have to be specified. Usually one specifies the values of C,\j/,q= - ( D V C + ^ V z ) or q}, For their derivatives.

The simplest form is where F is the dependent variable (3.4), (3.11), (3.14). In what follows F will be chosen, and occasionally C . Then C'(F) — \ / D .

In other branches of physics D = D 0 = const, and K = kC, so that K'(C) = k = const. Then many solutions are known, as (3.2) becomes:

Ct = kCz + D0V2C (3.15)

Carslaw-Jaeger (1953), Crank (1956), Irmay (1947) applied it to infiltration to and evoporation from aquifers.

In steady flow q, C, F and the auxiliary functions depend only upon space (x, y, z). The P D E are n o w of the elliptic type, which require only B C . Eq. (3.4), e.g., becomes:

K'(F)FZ + V2F = 0 (3.16)

In horizontal flow in the (x, v)-plane, we get the linear Laplace equation:

V2F = Fxx + Fyy = - (rFX + - Fee = 0; F = F(x, y) or F = F(r, d) (3.17) r r

In plane flow in the vertical (jc,z)-plane:

Fxx + Fzz + K'(F)FZ = 0 ; F = F(x, z) (3.18)

In one-dimensional vertical flow, we get an ordinary differential equation ( O D E ) of the second order:

F" + K'(F)F' = 0 ; F = F(z) (3.19)

483

S. Jrmay

W h e n there is no flow, q = 0. By (1.4):

i[f + z = const; *l/2~ll/i ~ z\~zi (3.20)

In a tensiometer at equilibrium with soil moisture, w e get the hydrostatic law:

- + z = const; Pl~Pl = Z l - z 2 (3.21)

The anology between (3.20) and (3.21) should not lead to the conclusion that the capillary potential \¡/ is equivalent to the pressure-head ply (Irmay, 1951, 1956).

Some authors (Miller, 1956; A m e s , 1965) suggested a useful transformation of (3.9): the dependent variable C(z, t), function of the independent variables (z, f), is replaced by the dependent variable z(C, t), function of the independent variables (C, t). Eq. (3.9) of unsteady vertical flow becomes (Appendix II):

D{C)zcc-D'(C)zc = z¿[zt + KXQ¡] (3.22)

Applying the procedure to F(z, t) and replacing it by z(F, / ) w e get:

zFF = z2FlC(F)zt + K'(F)-] (3.23)

In horizontal parallel flow, with F(x, t) replaced by x(F, t):

xFF = C'(F)xFxt (3.24)

The advantage is that C'{F),K'(F) are functions of the independent variable F. Appendix II gives the extension of (3.23) to flow in space.

Another transformation consists in replacing (/, x, y, z) by (9, £, r¡, ç):

e = t Ç = x r¡ = y Ç = z + bt (3.25)

Eqs. (3.2) and (3.4) become:

C„ = div [ D ( C ) V C ] + [K'(C)-fc]C { (3.26)

Fe = FK + F„ + Fa + lK'(F)-bC'(F)] Fç (3.27)

In a limited range Ci^C^C2, K(C) = kC3 can be approximated by a straight line:

K(C) = bC + b0 (3.28)

Then (3.26) and (3.27) are transformed into the diffusion equation without the gravit terms:

Ce = div[D(c)VC]] F, = V 2 f f

IV. B O U N D A R Y A N D INITIAL CONDITIONS

In steady flow the P D E is of the elliptic type. The solution of a well-posed problem requires that adequate boundary conditions (BC)be given on the boundaries. In unsteady flow the P D E is of the hyperbolic type, which requires besides the B C on the boundaries, also appropriate initial conditions (IC) in the initial flow domain. There are different types of boundaries and B C .

484

Solutions of the non-linear diffusion equation with a grauity term in hydrology

A . IMPERVIOUS BOUNDARY

Along it the flux vector q is tangent to the boundary.

B . WATER BOUNDARY

The pressure head ply is the same on both sides, and in isotropic soils q is normal to the boundary. In completely saturated soils (c = n), \j/ is to be replaced by ply, so that (p =z + +p/y is the same on both sides. If the water body is stationary, q> = const, along the boundary. If there remains some entrapped air, the soil is unsaturated (c = c0 < n). The B C is then: c = c0; \j/(c0) =p/y.

C . FREE SURFACE

In saturated flow a free surface can be defined by the phreatic surface, i.e., the isobaric surface of atmospheric pressure:

P = Pa <P = z + Pjy (4-1)

This requires that capillarity be neglected. As such a surface is an unknown of the problem, an additional B C is supplied, taking the hydrodynamic (material) derivative of (4.1):

Dp/Dt = pt + pxV!+ py V2 + pz V3 = 0 (4.2)

P C i > ^2» ^3) ¡s t n e effective velocity related to q(qlt q2, #3) by:

q = V\cf-Ci\ (4.3)

c¡ is the concentration just outside the front,

Cf is the moisture just inside the front (Irmay, 1968).

In unsaturated flow there are no free surfaces. W h e n moisture moves downwards by infiltration or seepage and upwards by surface evaporation, the flow domain has a front separating the still unaffected soil (c = c¡) from the flow domain (c^c¡). The B C there can be:

c = c¡ or C = C¡ = c¡-n0 (4.4)

This B C m a y lead to a conceptual difficulty. If the soil is initially air-dry, i.e.:

c = n0 or C = 0 (4.5)

B y (2.1) K->0 and presumably also D - * 0 . In order to have a finite flux, eq. (3.1) requires that V C and Vi/r-»oo. C and xjj vary practically almost jump-like. In our case (C = 0) and beyond the front flow occurs mainly in form of vapor.

A s the form of the front is an unknown of the problem, w e m a y proceed as in (4.2). The hydrodynamic derivative of (4.4) is:

DC/Dt = Ct + C^ + CyVi + C^ = C, + VC-V = 0 (4.6)

Introducing the effective flux:

q = F ( C - Q ) (4.7)

(C-C¡)Ct + VC-q = 0 (4.8)

485

S. Irmay

By (3.1) the B C becomes:

(C-C¡)Ct = DÇ7C)2 + KCZ = K[f(c)(VC)2 + CJ ( c - g f X V F f u ' F ,

(4.9)

In steady flow, the front is a streamline, along which the B C are:

CJF)2 + C'(F)F: = 0 = F2+F2+F2 + KFZ

(VC)2 + C2/^'(C) = 0 = C2x + C2

y + C¡ + CJ<l/'(C) (4.10)

These B C are non-linear, so in general front problems are not easy to solve. O n e simplification though is the fact that the front is in general an iso-F or iso-C surface.

Eq. (4.10) show that C 2 < 0 and Fz < 0 at the front, i.e., the moisture is always higher under the front than above. A front can exist in general only along the upper half of the flow domain. In the lower half the flow must either extend to infinity or reach an aquifer or the physical limit of the porous matrix. Eq. (4.10) show that the hodograph method of saturated flow (Polubarinova, 1968; Irmay, 1966) can be used also in unsaturated flow. At the front C=Ch F= F¡, K = const = K(C¡), í/\p'(c) = const = 2b. Then (4.10) is rewritten:

F2 + F2y+(Fz + KI2)2 = (K/2) 2

C2x + C2 + (Cz + b)2 = b2 (4.11)

The vector O P = V F describes a sphere of radius K/2 and center M at (0; 0 ; — K/2), tangent to the horizontal plane F . :

By (3.1):

: 0. Its projections on 3 Cartesian axes are (Fx, F F,) .

-VF-KVz (4.12)

Plotting the vector O L = K vertically downward (fig. 3), then a vector L P ' parallel to O P , we find that the resultant vector O P ' represents q. The locus of the ends of all vectors q on the front lies on the sphere (4.11). q is orthogonal to V F : the iso-F line is a streamline.

V F /

/ <

P>^v

0

T 5

,M

VF

P' 1 A. 2

1 K 2

\

3*

F I G U R E 3. The hodograph

q must go down: in vertical downflow it reaches its m a x i m u m absolute value (g3 = K); horizontally there can be no flow in steady-state—such a point is a stagnation point. In between the horizontal component passes through a m a x i m u m value (ql = K/2) at 45°. Sometimes B C consists in prescribing the normal or vertical component of q along the boundary: the rate of infiltration through or evaporation from the soil surface. This B C has to be used when passing from the saturated to unsaturated soil.

486

Solutions of the non-linear diffusion equation with a gravity term in hydrology

V. STEADY-STATE SOLUTIONS

In steady flowj, C, F, etc. do not vary in time. Then the auxiliary functions are unique, though they m a y depend on the history of wetting.

In horizontal flow the P D E is the Laplace equation (3.17). Its solutions have been thoroughly studied ( M o o n and Spencer, 1961; Muskat, 1937; Polubarinova, 1962; Irmay, 1968).

In one-dimensional horizontal parallel flow eqs. (3.17), (3.1) give by integration:

Fxx = 0; -qx = D(QCX = Fx = const; qL = q

q.(x-x0) = F(C0)-F(Q; F = D(C)dC -l> (5.1)

A graphical solution is shown (fig. 4) by plotting F(C). Here C = C0 or F = F(C0) = F0

at x = x0, is a B C . A s an example consider horizontal flow from a linear water source at x = 0, (C = C , , F= F¡) towards an outlet into air at x = x0 = L (C = C0, F = F0). The rate of flow q is given by:

<? = F.-Fo HCJ-F&o)

(5.2)

F I G U R E 4. Horizontal parallel flow

O f the 4 variables (q, L, Ct, C 0 ) only 3 can be arbitrarily prescribed, e.g., ¿ , C 0 , C \ or L, q, Cx. If w e increase q while ct = n (saturation), then the downstream end dries out (c < «0) : moisture flows in form of vapor.

In vertical upward flow, e.g., evaporation from an aquifer, by integration of (3.1):

•q3 = K(F) + FZ = const < 0; q3 = q > 0

dF i.(z-z0) = G(F0)-G(F); F = < F

í + K(F)lq

The graphical solution of fig. 4 can be used, replacing F(C) by G(C).

(5.3)

487

S. Irmay

W h e n q is very small {q< K), the moisture distribution approaches the static case of (3.20):

G{F) ~ q dF/K = q \ d\j/ = qtjj; z — z0~ij/0-ij/ (5.4)

W h e n q is very large {q> K), G~F, the moisture distribution approaches that of horizontal flow (5.1).

In vertical downward flow, e.g., infiltration into soil, by (3.1):

-q3 = K{C) + DC: = K(F) + FZ = const = q > 0 (5.5)

In general the rate of infiltration q < K(n), where K(n) is saturated hydraulic conducti­vity. Therefore there exists a value C = Cs or F= Fs for which q = K(CS). Integrating (5.5), two cases m a y be considered.

1) C > C s ; K(C) > q

q.(z-z0) = N(F0)-N(F); N{F) = àF

K(F)lq~l J

(5.6)

This occurs in infiltration to a water table at z = 0. It is interesting to note that here flow occurs from lower to higher moisture. This happens whenever the downward gravity flux exceeds the upward diffusion flux: K> D\CZ\.

At the soil surface C> Cs (fig. 5a). W h e n q<K,v/e get the static distribution: z - z 0 ~

2) C < C s ; K(C) < q

q.(z-z0) = M (F) - M (f0) ; M (F) •ÍT

dF •K(F)lq

(5.7)

This occurs in infiltration towards a pressure-plate. At the upper limit C< Cs (fig. 56). W h e n q> K, M~F, and w e get the distribution of horizontal flow.

F I G U R E 5. Vertical downflow

Steady flow in the vertical (x, z)-plane obeys (3.18):

F„ + F„ + K'(F)FZ = 0

A special solution is obtained making the linear transformation

F = f(v) v = z — mx

(5.8)

(5.9)

488

Solutionq of the non-linear diffusion equation with a gravity term in hydrology

Then the PDE (5.8) becomes an O D E :

(l + m2)f" + K'(f)f = 0

whose solution is:

v = z — mx = A0 — (l + m2) df A!+K(f) (5-10)

This is parallel flow normal to the planes v = z—mx = const of slope m , between two parallel inclined boundaries:

v = z — mx = v j ; v = z — mx — v2

The B C are: F = f(Vl) = f (C , ) ; F = f(v2) = F(C2).

This gives A0, At. Eq. (5.8) becomes linear F in case 3 (appendix I), where:

K = 2cnF; K'(F) = 2a

Fxx + Fzz + 2aFz = 0 (5.11)

Eq. (5.11) is the so-called telegraph equation. M a n y of its solutions can be obtained by superposition (Moon and Spencer, 1961).

As spatial flow is much more complicated, let us find the flows of iso-Flines, which are also iso-C lines, obtained by vertical displacement of the surface at the soil, (F = F0). The iso-F lines are then of the form:

z=f(x) + cp(F) (5.12)

It is convenient to apply von Mises' transformation (App. II. 2). Then (5.11) becomes after separation of variables:

^—11 = ^- = const = -b (5.12a) 1 + / ' 2 <P'2

Integrating, and introducing a number of integration constants:

a2 = (1 +2a¡b)IA ; A, b > 0 (5.13)

f{x) = f0±AH-1\(a2 + exp2bx)i + a-log\l - — 2a j] (5.14a) L 2 I (a2 + exp2bx)* + aJJ

or

fix) = / 0 ± ^ - ' L 2 - e x p 2 ^ + glog{l - y - » P y II (5.146) L 2 (. (a2-exp2foc)* + aJJ

(p(F) = b-1log(F-a0) + (Po (F>a0) (5.15a) or

(p(F) = b-1log(a0-F) + <p0 (F<a0) (5.156)

In (5.14a), for x-+— oo the F-lines have straight asymptotes. For ;c->-+oo they rise (or descend) exponentially. In (5.146), for x-y — oo the F-lines have the same asymptote, but

489

S. lrmay

but for larger x their slope decreases (increases) until it vanishes along the vertical * = ¿>-1loga.

Along an F-line the vertical flux is constant. The horizontal flux varies like the slope of the .F-line:

q3 = -K-Fz = -2a.F-ljzF = -F(b + 2a) +ba0)

<7i = -Fx = -zJzF = -bf'(x)(F-a0) (5.16)

W e have: q3 > 0 (evaporation) for F < ba0l(b + 2<x)

q3 < 0 (infiltration) for F > ba0¡(b + 2a)

q3 = 0 (horizontal flow only) for F = ba0j(b + 2a)

(5.17)

All these possibilities m a y occur at different F-lines.

In case (5.15a), zF = <p'(F)>0: the moisture increases upwards. In the case (5.15¿), zF < 0: the moisture decreases upwards. A n in vertical down-flow (5.6), flow m a y occur from lower to higher moisture. W e can also consider the case b < 0.

A similar solution exists in axisymmetric flow. Putting:

z=f(r) + <p(F) (5.18)

and applying the axi-symmetric equivalent of (App. II 2), w e get after separation of variables:

i*-r-rir = <!r_=_b 5]8a) l + r 2 <P12

W e obtain (p(F) in the form (5.15). The O D E for f(r) is not easily integrable.

VI. UNSTEADY-STATE FLOWS

In unsteady flow eqs. (3.2) to (3.6) do not suffice in general to determine a unique solution. Even if the problem is mathematically well-posed with adequate B C and IC, there remains the hysteresis of the auxiliary functions, with the exception of K(C). In wetting ( C ( > 0 ) , ij/(C) follows the wetting curve. In draining ( C , < 0 ) , \j/(C) follows the draining curve. W h e n some parts are draining while others are wetting, \¡/{C) follows draining and wetting scanning curves. In what follows w e shall neglect hysteresis by limiting the investigation to soils which are either only wetting by infiltration or draining by evaporation.

There exists no general method of solution. The known solutions in one-dimensional parallel or radial horizontal flow are mainly based on Boltzmann's transformation of the P D E in C(x, t) into an ordinary differential equation ( O D E ) in the new variable v = *//* ; on a linear transformation, giving an O D E in the new variable v = x + ut (Irmay, 1956); and on separation of variables (Boussinesq). S o m e other transformations and general solutions wil be mentioned briefly.

VII. BOLTZMANN'S TRANSFORMATION

In horizontal parallel flow solutions of (3.14) which depend on a single variable v.

v = x/t* F(x, t) = f(v) (7.1)

490

Solutions of the non-linear diffusion equation with a gravity term in hydrology

Eq. (3.14) becomes an O D E in v.

2f"(v) + vC'(f)f'(v) = 0 (7.2)

v = 0 corresponds tox = 0ori-»oo;i>->oo corresponds to x-* oo or / = 0. Eq. (7.1) is equivalent to x = t^cpiF), which transforms the P D E (3.24) into the O D E

in v = (p (F): 2q>" = C'(F)cp<p'2 (7.3)

For x > 0, q> > 0 and q> " > 0. Crank (1956) gives a number of exact solutions. Other new solutions are obtained

assuming the function C{F) and solving (7.3) for q>(F). E.g. :

1) C(F) = b(F-a0)m-<$

2)

3)

q>(F) = [2(l+m)/i»m(»i-l)]*(F-flo)" (""1) /2

<p(oo) = 0(m > 1); cp(co) -» oo(m < 1)

C ( F ) = fll-6(F-fl0)-"

<p(F) = [2(w-l)/fc/n(»i + l)]*(F-flo)("+1)/2

<p(oo)-»oo or C(oo) -» ax(m > 0)

C ( F ) = Û ! — ft exp(-ü-F)

q>(F) = (2/ftfl)* exp(aF/2)

<p(oo)->oo or C(oo)-> a!(a > 0)

(7.4)

(7.5)

(7.6)

Other new solution can be obtained assuming the form of <p(F) with a number of arbitrary coefficients (a,, a2, a3, ...). Then C(F) can be computed from (7.3) by integra­tion:

C(F) = at+2 (cp'l(p<p'2)dF (7.7)

The coefficients (a1, a2, -..) can be determined by the method of least squares, so as to approximate C(F) to observed data.

Another method is to start from a function based on the shape of the C{x, t) or Fix, t) curve, and try an adequate function F = f(v) with arbitrary coefficients (a2,a3, . . . ) . C(F) can then be computed from (7.2) by integration:

C(F) = al-2\(f"/v)dv (7.8)

The coefficients (a,, «2» •••) c a n De found so as to approximate C(F) to observed data. A modified Boltzmann's transformation is (Irmay, 1956):

F(x,t)=f{v); v = (x + x0)¡(t + t0)* (t>-t0) (7.9)

W e obtain again all former results.

Another possibility is to assume

F(x, t) = f(v); v = (x + xo)/(fo-0* (< < t0) (7.10)

491

S. Irmay

The O D E corresponding to (7.2), (7.3) are:

2 / » = vf'(v) C'(f) 1

2(p"(F) + C'(F)cp(F)cp'2(F) = 0\

(7.11)

W e can use again the former methods. W h e n we can approximate C(F) by a straight line of C'(F) = 4b, the solution of

(7.2) has the form of the error function:

(7.12) F = f{v) = a0 + a exp( — bv2)dv

It can be shown that (7.9) is the only solution of the types:

F = fíxcp(t)l F = fitcpixj], F = fla(t) + xb(t)l (b'(í) * 0

In radial flow a similar transformation to Boltzmann's gives:

»=r/( í + í0)*; F(r,t)=f(v)

Eq. (7.2) is replaced by:

2f(v)+f'(v)[2lü + vC'(fy] = 0

Eq. (7.3) is replaced by:

r/(t + t0f = v = <p(F); Icpcp" = q>'2 (<p2 + 2) C (F)

(7.13)

(7.14)

(7.15)

One can either assume C(F) and solve forf(v) or <p(F); or assume f(v) and (p(F) with appropriate, yet indeterminate coefficients; then compute C{F).

VIII. LINEAR T R A N S F O R M A T I O N S

Consider horizontal flow with solutions of the type:

F=f(x-ut); C'(F)F, = FX (8.1)

The general integral is:

x — ut = d / / [ a 0 - « C ( / ) ] + a i ; q3 = - / ' = uC{f)-a0 (8.2)

This case corresponds to a wave travelling at speed u in the direction +x; an observer moving at that speed will observe the same value of F. This occurs, e.g., whenever the water source moves at speed u.

In vertical flow there exist similar wave solutions:

F = f(z + ut); C(F)F, = FZZ+'K (F)FZ

The general integral is:

-{z + ut) df

K(f) + q3

d/ K(f)-uC(f)-a0

+ const = (p(f)

-q3 = f' + K = uC(f) + a0

(8.3)

(8.4)

492

Solutions of the non-linear diffusion equation with a gravity term in hydrology

u> 0 (« < 0) corresponds to an observer moving upwards (downwards) at speed |« | , e.g., when the water table rises (drops) uniformly.

In upflow: q3 > 0, / ' < 0, the moisture decreases with y. In downflow: q3 < 0 and w e have to consider two cases:

1) \q3\ < K(f), the results are the same as above.

2) \q3\ > K(f), eq. (8.4) is to be replaced by:

iC{f)-K(f) Z + Ut= Í ^ r- (8"5)

H e r e / ' > 0 , the moisture increases with z in spite of downflow; everywhere C < Cs, when Cs is defined by:

| i 3 ( Q | = K(CS)

Within a limited range (C t < C ^ C 2 ) it is sometimes possible to choose u so that approximately:

a0 + uC(f)-K(f) ~ const = b0l

F = b0 -(z + uí) j

A given function v = z + ut = <p(F), e.^. one obtained experimentally w h e n the water table rises or drops at a uniform rate, allows us to determine the form of C(F) from (8.4), for it is compatible with the B C at the water table ( F = const for z + ut = const):

a0 + uC(F)-kC\F) = W ( F ) (8.7)

a0 is determined by the I C and B C ; C(F) is the root of a cubic equation. T h e w a v e solution can be extended to an observer moving at constant speed « in a

direction v defined by its three direction cosines (a, /?, y):

(8.8) F = f(v); v = ctx + Py + yz-ut; a2 + /5 2 +y 2 = ll

C'(F)F, = FXX + Fyy + Fzz + K\F)FZ j

The solution is:

-v = J dfi(yK + uC-a0) + const (8.9)

The specific flux in direction v is:

qv = yK + uC-a0 = - / ' ( » ) (8.10)

Here also two cases are possible, as above, depending upon whether <7vsg;0. This solution corresponds to oblique flow.

Another case of flow in the vertical (*, z)—plane obtained assuming a uniformly rising (falling) water table:

F=f(t,x,v); v = z + ut; C'(F)Ft = FXX + FZZ + K'(F)F2 (8.11)

The solution is given by:

/ x x + / „ + / u [ ^ ' ( / ) - « C ' ( / ) ] = 0 (8.12)

Here the P D E of unsteady flow (8.11) has been reduced to that of steady flow (5.8) with K'(F) replaced by K'{F)-uC'{F).

493

S. Irmay

IX. SIMILARITY TRANSFORMATIONS

In horizontal parallel flow, find similarity solutions of (3.14) of the form:

F(x, t) = ff(v); v = xtm; C'(F)F, = Fxx (9.1)

Assuming C'(F) = aF~", we get nfi + 2 m + 1 = 0 and the O D E in/(¿>):

a-lf"r = nf+mvf' (9.2)

The case n=0, m = — \ gives the Boltzmann transformation (7.1). For n= — l//i, m = 0, w e can integrate:

F = r^fix); / " / " - 1 = na; x = ± ¡df/t^-f^+b) (9.3)

In the general case n can be chosen arbitrarily. The transformation:

F(x, t) = x"f(v): v = xf

gives essentially the same results, for F = r~mnf(v). The transformation:

F(x, t) = e"f{v); v = xe"' (9.4)

gives P = — a/i/2 and O D E in f(v), with arbitrary a:

{*la)rf»=f-invf (9.5)

In vertical flow w e can apply to P D E (3.11) the similarity transformation (9.1):

C(F)F, = K'(F)F1 + F„; F(z,t) = t"f(v); v = ztm (9.6)

Assuming:

C ' (F ) = aF"""; K'{F) = bF"\ i.e. v = 2-3/ i by (2.2), (9.7)

w e get the O D E :

rr+bfi"+*r = <j+vvr)aivi F(z, i) = ¿'"Azi"*); n = \ln\ m = v/n

(9.8)

fi and v can be adapted to experimental data. In the special case: v = 0, ft = 2/3, w e have m = 0, n = 3/2, F(z, / ) =/(z) /3 / 2 , and the

O D E : f" + bf = 3afi/3/2, which gives a first integral. The similarity transformation (9.4) gives here:

F(z,t) = enlf(v); v = ze""; f4/5f" + bf2'5f' = anif~^vA (9.9)

n can be chosen arbitrarily, but one has to assume: /i = 4/5, v = —2/5. In all the above transformations x, z, t can be replaced by x + x0, z+z0, t + t0 or

t0 — t (in the last case the term in F, changes its sign). Most of the above O D E cannot be integrated analytically, but numerically or graphi­

cally only.

494

Solutions of the non-linear diffusion equation with a gravity term in hydrology

In radial horizontal flow:

C'{F)Ft = Frr + r-lFr- C ' ( F ) = ^ " " 1 ( 9 1 0 )

F(r, I) = »"/(«); v = rf" j

W e find that m = -(1 +n¡i)¡2, and the O D E is:

f,(vf+f') = a(nf+mvf')

n can be chosen arbitrarily. For n = 0, m = - ^ w e get (7.14). In axi-symetric flow in a vertical meridian plane:

C'(F)F, = Fzz + Frr + rlFr + K'(F)Fz; C (F) = aF~"; K'(F) = bFv (9.11)

F(r, z, t) = t"f(v, w); v = rtm; w = zf"

W e find /J = m = -v/(2v + /i); n = -l/(2v + /i)

fl(n/+o/„ + w / J = if„ + v-lfv+fm + brfH).r (9-12)

This is a P D E in / O , ¡¿%) simpler than the original P D E .

X. SEPARATION OF VARIABLES

Solutions can also be obtained by separation of variables. In horizontal parallel flow we get the similarity solution:

C(F)F, = F„; F = X(x) T(t)

aT-x-HT, = x„xi-v = ±p. F = Qax2

+alX + ao)l(t0+i)

for C'(F) = aF_1(/i= 1,); while for C'(F) = aF ~ "(ju =£ 1 ) we have the solution:

7\0 = [ KUtnt

i In X — Xr¡ "T"

dX

q0-2mX2-"l(2-n)

Another solution is:

T(t) = jxm(f( »o-0J

x = x0 ± dZ

a0 + 2 m A r 2 ~ 7 ( 2 - / i )

For a0 = 0, we get the Boltzmann solution. Another solution is obtained starting from (3.12) and assuming:

C, = [D(C)CJ»; D(C) = aCn + bCm; C = X(x) T(t)

W e have: T'¡T"+i = a(XnX')'IX + b(XmX')IX • Tm~n

Assuming further (A""A")' = 0, w e get a Boltzmann solution:

X(x) = [(m+l)a(x 0 + x)] l / ( 1 + m ) ; n = 2(l + m)l

H O =[2a(2 + m)(l + m)a2m/<1+m)(íü + 0 ] " 1 / 2 < 1 + m ) |

(10.1)

(10.2)

(10.3)

(10.4)

(10.5)

(10.6)

(10.7)

S. Irmay

In vertical flow we get a similar solution:

C, = ID{C)C2 + K(C)~\:; D(C) = aC" + bCm; K(C) = kC3 (10.8)

C = Z{z) T(t); T'IT3 = aT"~2-(Z"Z')'/Z + bTm-2(ZmZ')IZ + k(Z3)'IZ (10.9)

Assuming again:

(ZmZ'y = 0; Z(z) = [0n + l)a(zo±z)]1/(1+m),

and

m = 1, n = 4,

Z(z) = [(m + l)a(z0±z)]1/(1+"\

eq. (10.9) becomes:

T ' = ±a(2/c + 3a r 2 )T 3

This equation is easily integrable. Another solution of (10.8), (10.9) is obtained assuming:

D(C) = aC2; n = 2 ; C = Z(z) T(t)

T'/T3 = [a(Z2Z') ' + 3 /cZ 2 Z' ] /Z = const = ± a

T(t) = 2a(t0Tt)-*; adC'/dC = « C 1 / 3 / C -fc; C = Z3(z)

(10.10)

(10.11)

The last equation is easily computed numerically or graphically. In axi-symmetric flow separation of variables is possible in steady flow in a meridian

(r, z)—plane, and in unsteady flow in a horizontal plane:

C (F)Ft = r ~l (rFr)r; C'(F) = a F " " ; F(r,/) = R(r) T(t)

aT'¡T" + l = Rfl'1(rR')'/r= +a

r(i) = [a/a/i(fo±0]1/''; R(r) = cT0r21»; a¡¡ = ± a ^ 2 / 4

(10.12)

Using inverse coordinates it is possible to find some special vertical flow solutions of the type:

z = f(F) T(t), e.g. z = a0lF; C'(F) = aF~i; K'(F) = ¿F* (10.13)

or of the type:

z=f(F) 7(0 + 0(0 with T(0 = ío-í; 0(0 = «o iog(í-ío) (10.14)

XI. SURFACE EVAPORATION A N D INFILTRATION

W h e n there is evaporation at a constant rate e(dim. L T ~ ' ) from the surface of a hori­zontal flow, the P D E (3.9) becomes in parallel flow:

ct + qx = E; q = -Fx; c, = Fxx + e; C'(F)Ft = Fxx + s (11.1)

In steady flow, with £ = e0 constant,

Fxx = -«oí F(x) = F0 + q0x-iex2; q(x) = q0-e0x (11.2)

496

Solutions of the non-linear diffusion equation with a gravity term in hydrology

The iso-F surfaces are parabolas and the flux decreases linearly from q0 at x = o to 0 at x = qjs0.

If e depends on x:

s = e(x); F(x) = F0 + qox - e(jc')dx' (11.3)

If 8 depends on the moisture content:

dF E = s(F); x = x 0 +

j0±(b0-2\ *

(11.4)

e(F)dFy

In radial flow, the P D E becomes:

C(F)Ft = r-l(rFr)r + e (11.5)

In steady radial flow, the solution is for e = e0 constant:

F(r) = F0 + a logr-ie0r2; q(r) = a\r-\z0r (11.6)

The flux decreases with r until it vanishes at r = (2a/e0)* In the case of infiltration at rate i, replace e by — i. In radial steady flow the flux starts

from source (q~a/r0, if r0 small), decreases to a m i n i m u m (2ae0)* at r = (2a/e0)*, then increases almost linearly with r.

In unsteady flow P D E (11.1) is transformed into an O D E by the linear transformation:

F = j\v); v = x + ut; uC'(f)f =f"+e0 (11.7)

A first integral gives: / ' = a0-e0u + uC(f) (11.8)

This equation has an analytical solution for certain forms of C'{F), e.g. C(F) = aF~^ otherwise numerical or graphical methods can be used.

APPENDIX I

ANALYTIC FORMS OF THE AUXILIARY FUNCTIONS

No. Source

1 Fujita (1952) D = - ^ 2 _ p = P^ log—^— ; 1—aC a 1—aC

C = - [ l - exp( -aF /D 0 ) ] a

^ - F = ^[21og(l-aC) + — (1-aC)2 a 1-aC

2 Irmay (1956) D = — F = -[2log(l-aC) + --(1-aC)]

Irmay D = 3aC2 F = aC3; C = a~1/3 F1'3; K = - F a

\¡i = a0 + (3a¡k)logC

497

S. Irmay

4 Irmay D = maCm~l F = aCm; C = a~l/m Fl/m;

K = /c6j"3/mF3/m

i¡/ = a0 + Cm~3 ma¡k(m-3){m # 3 )

F{C) is computed by (2.3); C{F) by inversion; ^(F) by (2.2); i/r(C) by integration of

i¡i\C) =D(QIK(C).

APPENDIX II

V O N MISES' TRANSFORMATION

Introducing into (3.9)

F = / 0 > y, z. 0 = fix, y, z(F, x, y, t), f]

applying successively the operations

d ô d d ô2 d1 ô2 ô2 d2

ex' dy' dt' OF' ôx2' dy2' dF2' ôxôF' ôydF

w e get:

(HI)

F = - — • F = - — • F = - ii • F = — -*( ï * x ) * y » a z

Zp Zp Zp Zp

z 1 Fzz = ~ ~ T ! f» = — ( 2 Z / Z X Z X F - Z F ^ - Z F F Z X

2 ) zF zF

and a similar expression for Fyy. Introducing into (3.4) we get:

z2plK'(F) + C'(F)zt-(zxx + zyf)l + 2zp(zxzxF + zyzyp) = zFF(l+z2x+z2) (H.2)

The B C (4.10) becomes:

z2+z2y + l + KZp = 0 (11.3)

REFERENCES

A M E S , W . F. 1965. Nonlinear partial differential equations in Engineering. Academic Press, N e w York, 511 p.

B O L T Z M A N N , L . 1894. Zur Integration der Diffusionsgleichung bei variabeln Diffusions-koeffizienten. Ann. Physik., 53, 3: 959-964.

BOUSSINESQ, J. 1904. Recherches théoriques sur l'écoulement des nappes d'eau infiltrées dans le sol et sur le débit des sources. Compl. J. Math., 10, 4: 363-394.

C A R S L A W , H . S. and J. C . J A E G E R . 1959. Conduction of heat in solids. 2nd ed. Oxford Univ. Press, London, 510 p.

C H I L D S , E . C . and N . C O L L I S - G E O R G E . 1948. Soil geometry and soil-water equilibria. Disc. Faraday Soc, 3: 78-85.

— 1950. The permeability of porous materials. Proc. Roy. Soc. 201 A : 392-405. C R A N K , J. 1956. The mathematics of diffusion. Oxford Univ. Press, London, 347 p. FUJITA, Hiroshi. 1952. Textile Research J. 757- , 823- . G A R D N E R , W . and J. A . W I D T S O E . 1946. Infiltration. Trans. Am. Geophys. Union. 27: 126-128.

498

Solutions of the non-linear diffusion equation with a gravity term in hydrology

G A R D N E R , W . R . 1957. Some steady-state solutions of the unsaturated moisture flow equation

with application to evaporation from a water table. Soil Sci. 85: 228-232.

I R M A Y , S. 1947. Le mouvement des eaux dans le sol. Revue Universelle des Mines, Liège. 3,

4: 129-139.

— 1951. The effect of air bubbles on the law of hydrostatic pressures in unsaturated soils.Proc,

Gen. Meeting, Brussels, U . G . G . I . , ¡ASH, 2: 179-183.

— 1954. O n the hydraulic conductivity of unsaturated soils. Trans. Amer. Geophys. Union,

35, 3: 463-467.

— 1956. Extension of Darcy law to unsteady unsaturated flow through porous media. Proc.

Symp. Darcy, U . G . G . I . , J ASH, 41: 57-66.

— 1958. O n the theoretical derivation of Darcy and Forchheimer formulas. Trans. Amer.

Geophys. Union, 39, 4: 702-707.

— 1964. Theoretical models of flow through porous media. R I L E M Symp. on the Transfer of

Water in Porous Media, Paris. Bull. R I L E M , 29: 37-43)

— with D . Z A S L A V S K I J. B E A R , ,1968. Physical principles of percolation and seepage. Unesco,

(Paris, 465 p.).

K L U T E , A . 1952. Some theoretical aspects of the flow of water in unsaturated soils. Soil Sci.,

Soc. Amer., 16: 144-148.

— 1952. A numerical method for solving the flow equation for water in unsaturated materials.

Soil Sci. 73: 105-116.

M I L L E R , E . E . 1956. Physical theory for capillary flow phenomena. J. Appl. Phys., 27, 4: 324-332.

M O O N , P. and D . E . S P E N C E R . 1961. Field theory for engineers. Van Nostrand, Princeton, N.J . ,

530 p.

M U S K A T , M . 1937. The flow of homogeneous fluids through porous media. M c G r a w Hill, N e w York

and Edwards, A n n Arbor, Mich. (1946), 763 p.

PHILIP, J .R. 1957. Theory of infiltration. Soil Sci. 83: 345-357; 435-448; 84: 163-178; 85 (1958)

278-286.

P O L U B A R I N O V A - K O C H I N A , Ya . 1962. Theory of groundwater motion. Princeton Univ. Press, 613 p.

P I R V E R D I A N , A . M . 1960. O n self-similar motions in the theory of the non stationary filtration of a

gas in a porous medium. Prikl. Mat. Mekh. (in Russian), 24: 558.

R I C H A R D S , L . A . 1931. Capillary conductivity of liquids through porous mediums. Ph. D.Thesis,

Cornell Univ.; Physics, 1: (1931), 318-333.

S I N G H , R . 1965. Unsteady and unsaturated flow in soils in two dimensions. Techn. Rep. N o . 54,

Stanford Univ., Dept. Civ. Eng. 129 p.

S T O K E S , R . H . 1952. Trans. Faraday Soc, 48: 887- .

T O P P , G . C . and E . E . M I L L E R , 1965. Hysteretic conductivity; calculation versus measurement.

Agronomy Abstracts, 1964 Meeting of A S A ; and private communication.

W A G N E R , C.J. 1950. / . Chem. Phys., 18: 1229- .

W Y C K O F F , R . D . and H . G . B O T S E T , 1936. The flow of gas-liquid mixtures through unconsolidated

sands. Physics. 7: 325-345.

Y O U N G S , E . G . 1957. Moisture profiles during vertical infiltration. Soil Sci., 84, 4: 283-290.

499