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Some properties of flows in unsaturated soils with an exponential dependence of the hydraulic conductivity upon the pressure head
Journal of Hydrology 14 (1971) 129-138; © North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
S O M E P R O P E R T I E S OF F L O W S I N U N S A T U R A T E D S O I L S W I T H AN E X P O N E N T I A L D E P E N D E N C E OF T H E H Y D R A U L I C C O N D U C T I V I T Y U P O N T H E P R E S S U R E HEAD P. A. C. RAATS The Johns Hopkins University *
Abstract: A systematic development of the consequences of an exponential dependence of
the hydraulic conductivity upon the pressure head is presented. Alternative expressions for the flux are discussed in detail. For steady flows, partial differential equations in terms of the matric flux potential, the pressure head, and the total head are derived. For steady, plane and axially symmetric flows, partial differential equations for the stream function are given. A theoretical basis for the construction of viscous flow analogs for steady, plane and axially symmetricflows is also presented. Introduction
It was shown by Gardner 1) that if the hydraulic conductivity decreases exponentially as a function of the pressure head, the partial differential equation for steady flows is linear in terms of a certain function. This function may appropriately be called the matric flux potential2). The first detailed example of an analytical solution of this linear equation was given by Philip 3) in his discussion of steady infiltration from buried point sources and spherical cavities. Immediately following, Wooding 4) analyzed the difficult problem of steady infiltration from a circular pond. Philip 5) also briefly discussed the solution for a single, horizontal line source in an infinite region. Recently I have shown that for plane and axially symmetric flows the stream functions also satisfy linear partial differential equations z, 6, 7). In fact, for plane flows the matric flux potential and the stream function satisfy the same linear partial differential equation. The equation for the plane stream function was used as a basis in a study of steady infiltration from line sources and furrows 2). The solution of this problem provides a quite complete description of the flow in terms of the distributions of the stream function, the components of the flux, the matric flux potential, the pressure head, and the total head. If the water content is * Present address: USDA-ARS-SWCRD, Department of Soil Science, University of Wisconsin, Madison, Wisc. 53706 USA. 129
130
P.A.C.
RAATS
given as a function of the pressure head, its distribution will of course also be known. The distribution of the pressure head and the total head for infiltration from buried point sources and cavities (thus completing Philip's analysis) and a complete description of infiltration from surface point sources and basins has been found7). On the basis of this latter solution, Philip s) has constructed a superposition theorem, which gives the solution for any surface source. Subsequently I have generalized Philip's superposition theorem to an arbitrary distribution of sources located at arbitrary distances below the soil surface 9). This paper presents a systematic development of the consequences of the exponential dependence of the hydraulic conductivity upon the pressure head. Of particular interest are the alternative representations of the flux and the simple flow equation in terms of the total head. In the last section a theoretical basis for the construction of viscous flow analogs for the plane and axially symmetric flows is presented.
The matrix flux potential Let 0 be the volumetric water content and v be the velocity of the water. Assuming the fluid to be incompressible and the flow to be steady, the balance of mass may be written as: V.(0v) = 0.
(1)
Let z be the vertical direction, positive downward. The volumetric flux Ov is assumed to be given by:
Ov = - kVH,
(2)
= - - k[h] Vh + k[h] Vz,
(3)
= - D [0] V0 + k [0] Vz,
(4)
=-
(5)
V~o + k [~o] v ~ ,
where k is the hydraulic conductivity, H is the total head, which is the sum of the pressure head h and the gravitational head - z (i.e., H = h - z ) , D=k(dh/dO) is the diffusivity, and q) is the matric flux potential defined by: h
0
: [" kfk] dh = [" O[0] dO,
(6)
mr/
ho
0o
where ho and 0o are reference values, and 0o = 0 [ho]. It is assumed that the steady flow is established through monotonic changes in water content, so that hysteresis need not be considered. The hydraulic conductivity k is assumed to be a unique function of the water content 0, which in turn is
PROPERTIES OF FLOWS IN UNSATURATED SOILS
131
assumed to be a unique function of the pressure head h. It is understood that D is undefined whenever d h / d O ---, ~ . Substitution of(5) into (1) gives: v%
-
ak [m] ~z
(7)
In general, Eq. (7) is nonlinear. The flux, when k is an exponential function of h
At this point it is convenient to assume that k and h are related by an expression of the form 1-5, 7-10) k = k o e~h.
(8)
The parameters ko and ~ are characteristics of the porous medium. The parameter ko represents the hydraulic conductivity at h = 0, i.e., the hydraulic conductivity of the saturated porous medium. The k-weighted mean pressure head between h = 0 and h ~ - ~ , which Bouwer (ref. 11 and references given there) has called the critical pressure head, is simply equal to -1/~1°). The parameter 1/e may be regarded as a characteristic length of the partially saturated porous medium. According to Philip3), a typical value of ~ is 0.01 cm -~ and the range 0.05 to 0.02 cm -1 seems likely to cover most applications. Most of the data presented by Rijtema 12) fall in this range. Taking ho -+ - m, and introducing (8) into (6) gives: k = ~¢p.
(9)
e ~'h = ~ o / k o .
(10)
e~n = ~0 e -~z . ko
(11)
Substitution of (9) into (8) gives:
Multiplying (10) by e -~z gives:
Using Eqs. (8)-(11), the flux Ov can be expressed in many forms. Introducing (9) into (5) gives: Ov = - V~p + ~q~Vz.
(12)
According to (12) the flux is linear in the matric flux potential and, in view of (10), also linear in e ~h. Introducing (9) into (2) gives: Ov
= - aq~VH.
(13)
In vector analysis it is well known that a n y vector field ¢ can be represented
132
P . A . C . RAATS
as the sum of a lamellar field Vf and a complex-lamellar field mVn (cf. Ericksen 13): c = Vf + mVn, (14) where the scalar fields f, m, and n are the so-called Monge potentials. However, the form of (12) is special in that two of the Monge potentials are scalar multiples of one another and the third Monge potential is just the coordinate z. Any vector field represented by (12) is also represented by (13) with H = ( 1 / ~ ) In t p - z . In other words, any vector field represented by (12) is in effect a complex-lamellar field. [Equations of the form (12), in which a flux 0v appears as the sum of a diffusive component - V t p and a drift component cupVz, occur in many other applications, The drift component may arise from the movement of a source of some entity with respect to the medium in which the entity diffuses, or from the action of an external force field such as the gravitational field. The fact that all such flows are complexlamelIar appears not to have been noted before.] According to (13) the vector field Ov is perpendicular to a family of iso-H surfaces whose distance function is -cup. The representation (13) is not unique. In fact there are infinitely many representations of Ov exhibiting the complex-lamellar nature. One such representation is Ov = _ __k°e~Ve~n .
(15)
According to (15) the vector field 0v is perpendicular to a family of iso-H surfaces whose distance function is - ( k o / ~ ) e ~'z. It is remarkable that, for what in essence is the same family of surfaces, in (13) the distance is a function of ~o and of the constant ~, while in (15) the distance is a function of z and the constants c~and ko. The results in this section are, of course, not resrticted to steady flows.
Flow equations, when k is an exponential function of h The alternative expressions for Ov naturally lead to different partial differential equations describing the flow. Introducing (12) in (1) or (9) in (7) gives ~-5, 7-9): V2q~ = ~ - - . Oz
(16)
Since ko and c~ are constants, it follows from (10) and (16) that the partial differential equation: V2e~h = c~
t~e~h
8z
.
e ~h
satisfies
(17)
PROPERTIES
OF FLOWS IN LrNSATURATED
SOILS
133
Certain solutions of Eq. (16) subject to boundary conditions in terms of ~o have been given by Philip 3), Raats 7), and Wooding4). Using Eq. (17), the same problems could of course be formulated in terms ofe ~h. Substituting (15) into (1) gives: t~e~u V 2 e ~// = - - ~ - -
(18)
Oz This equation is new. Note that there is a minus sign on the right hand side of (18). In some problems the boundary conditions are simplest, when expressed in terms of H. Equation (18) may be used as the basis for the study of a whole new class of boundary value problems. Plane and axially symmetric flows
For flows plane with respect to the y-direction, Eq. (1) reduces to: t30u t30w +- - =0.
~x
(19)
#z
Equation (19) suggests that there exists a stream function ~kpsuch that: Ou = -
~P ~z '
Ow =
~x
(20)
.
From the results in the previous section it is obvious that there are many different forms for the dynamic expressions for Ou and Ow. For example, from (15) and (20) it follows that ko_e~z ~3_e~n=_c~p, ~3x Oz
k 0e "z-d e~n _ c~ 3z
~bp ax
(21)
These equations may be regarded as analogs of the Cauchy-Riemann conditions. Elimination of e~n from (21) gives the partial differential equation for ~p 2): t325r' 02~b° ~¢P (22) a x z + ~ZzZ = ~ a z
In some problems the boundary conditions are simplest when expressed in terms of ~kp. For example, in an earlier paper 2) (22) was used to analyze steady infiltration from line sources and furrows. The reductions of Eqs. (16), (17) and (18) to plane problems are obvious. For axisymmetric flows, Eq. (1) reduces to : r O--rrOu, + d z Ow
= 0.
(23)
134
P.A.C. RAATS
Equation (23) suggests that there exists a Stokes' stream function ~,~ defined as (Stokes 14):
1 O~b. Ou, =-2~z~r Oz'
Ow-
1 O~p~
(24)
2nr Or
From (15) and (24) it follows that ko ~t
0 e"u 8r
1 O~a 2nr Oz '
ko e~z 8 e~u c~ Oz
1 0I/]a 2~r Or
(25)
Eliminating e ~n from (25) gives 7):
020a 0r 2
I O@a 02ffIa r
01//a + O~--Ct 0Z Or
(26)
Note that the left hand side of (26) has a minus sign in the second term and, thus, is not the Laplace operator for axially symmetric flow (cf. Zaslavsky and Kirkham15,16), Raats7). The reduction of Eqs. (16), (17), and (18) to axisymmetric problems are obvious.
Viscous flow analogs Preston 17) considered the steady slow motion of an incompressible Navier-Stokes fluid between two rigid plates which are a small distance d[x, y] apart and where the variation of d with x and/or y is slow. One boundary is taken as fiat and lying in the x, y-plane. Preston assumes that: (a) the velocity of the fluid perpendicular to the x, y-plane is negligibly small, (b) the acceleration is negligibly small, (c) the only viscous terms that need to be considered are those arising from the variation in the z-direction of the components of the velocity in the x- and y-directions. Considering the general case with the flat wall undergoing an arbitrary rigid motion in the x,y-plane, Preston derives partial differential equations for the stream function ~ and the fluid pressure p. When the walls are at rest with respect to each other, these equations reduce to:
a20 a20 3/adaO
+ 0y
a0
= 0,
(27)
Op + 0y 0 = 0. Ox~ + ay - - 2 + d Ox ax
(28)
~x~ + Oy~ -- a oX ax 02p 02p 3 rod Op
0d y)
Special cases result from choosing particular forms for the function d [ x , y ] . For example, if d--constant Eqs. (27) and (28) describe plane
PROPERTIES OF FLOWS IN UNSATURATED SOILS
135
potential flow. In this special case, and in this special case only, ~ and p satisfy the same partial differential equations. The analogy between potential flow and steady viscous flow in films of uniform thickness was first utilized in experiments by Hele-Shaw 18) and demonstrated theoretically by Stokes 14). Apparently, the Hele-Shaw model was first used in connection with flow in porous media by Dachler 19). Setting: y = z,
(29)
d = do exp (~z/3). reduces (27) to (22), the equation for the stream function when the flow is plane, while Eq. (28) for the fluid pressure reduces to: c~2p 02p ~ X 2 71- (~Z 2 - -
c3p ~ OZ
(30)
Comparison of (30) and (18) shows that p in (30) is the analog of e ~n for plane flow. In other words the lines along which the pressure p is constant in the viscous flow model correspond to lines along which the total head H is constant in the partially saturated porous medium. At Preston's suggestion, Horlock 2o) used the special case represented by (29) as the basis for a viscous flow analogy for the stream function in a particular problem in magnetohydrodynamics. Neither Horlock20) or Preston 17) offered an interpretation of the fluid pressure p. Horlock constructed an apparatus in which the distance d=do exp(O.742z/3). In other words, the viscous flow in Horlock's apparatus is analogous to steady flow in a partially saturated soil with e=0.742 cm-1. This value is an order of magnitude larger than the typical values indicated earlier. However, the flow pattern in the viscous flow model depends on the product of e and some characteristic length L representing the macroscopic dimensions in the x,y-plane. By choosing e large, the dimensions of the cell may be kept small. As an example, consider steady infiltration from an array of equally spaced line sources or furrows at the surface of a semi-infinite soil profile. Let L be the half distance between the line sources. The solution of this problem and calculations for e L = 1 and 5 were given before 2). To model these calculated results, the half distance between the sources in Horlock's apparatus should be 1.35 cm and 6.75 cm respectively. Figure 1, which was taken from Horlock's paper, shows the flow pattern for upward flow around a cylinder. For e L = 0 the flow on the upstream and downstream sides would be symmetric. In the photograph the flow is clearly asymmetric.
136
P.A.C.
RAATS
Fig. 1. Viscousflow model for upward flow around a cylinder (from Horlock2°). Preston 17) has pointed out that with x = r,
(31)
d = do r~ ,
Eqs. (27) and (28) reduce to the governing equations of axially symmetric potential flow. (~eqen et a121) used this analogy in a study of flow to horizontal collector wells completely penetrating the horizontal impervious base of an isotropic, homogeneous porous medium. By combining (29) and (31) one obtains a viscous flow analog for steady, axially symmetric flow in a partially saturated soil: y = z, d = do r~ exp (~z/3).
(32)
PROPERTIES OF FLOWS IN UNSATURATED SOILS
137
Such an a n a l o g could be used to m o d e l the flows studied by Philip3), R a a t s 7, 9), a n d W o o d i n g 4 ) . Just as for p l a n e flow, the p r e s s u r e p is the a n a l o g o f e ~H. The viscous flow a n a l o g y for steady p l a n e a n d axially s y m m e t r i c flows in p a r t i a l l y s a t u r a t e d soils with a n e x p o n e n t i a l d e p e n d e n c e o f k u p o n h opens m a n y possibilities. It m a y be used to study the influence o f realistic t o p o g r a p h i c features u p o n steady infiltration from furrows. It offers a convenient basis for studying steady u p w a r d a n d d o w n w a r d flow a r o u n d i m p e r m e a b l e objects with p l a n e or axial symmetry, which c o u l d lead to some insight into the effect o f the heterogeneous n a t u r e o f soils. Also, by extending (29) or (32) to include further d e p e n d e n c e o f d u p o n p o s i t i o n one could m o d e l certain steady flows in layered soils.
Acknowledgements I wish to t h a n k Mr. F. W. Blaisdell o f the St. A n t o n y Falls H y d r a u l i c Lab. for bringing to m y a t t e n t i o n the p a p e r by (~e~en et a121) a n d Prof. J. H. H o r l o c k o f C a m b r i d g e University for his p e r m i s s i o n to r e p r o d u c e Fig. 1. This research was s u p p o r t e d by a G r a n t f r o m the N a t i o n a l Science F o u n d a t i o n to the Johns H o p k i n s University.
References 1) W. R. Gardner, Some steady-state solutions of the unsaturated moisture flow equation with application to evaporation from a water table. Soil Sci. 85 (1958) 228-232 2) P. A. C. Raats, Steady infiltration from line sources and furrows. Soil Sci. Soc. Amer. Proc. 34 (1970) 709-714 3) J. R. Philip, Steady infiltration from buried point sources and spherical cavities. Water Resources Res. 4 (! 968) 1039-1047 4) R. A. Wooding, Steady infiltration from a shallow circular pond. Water Resources Res. 4 (1968) 1259-1273 5) J. R. Philip, Theory of infiltration. In : Ven Te Chow, ed., Advances in Hydroscience (Academic Press, New York, 1969) 305 p., vol. 5, pp. 215-296 6) P. A. C. Raats, Addendum to "Steady infiltration from line sources and furrows". Soil Sci. Soc. Amer. Proc. 35 (1971) 161 7) P. A. C. Raats, Steady infiltration from point sources, cavities and basins. Soil Sci. Soc. Amer. Proc. (1971) (submitted for publication) 8) J. R. Philip, General theorem on steady infiltration from surface sources, with application to point and line sources. Soil Sci. Soc. Amer. Proc. (1971) (submitted for publication) 9) P. A. C. Raats, Steady infiltration from sources at arbitrary depth. Soil Sci. Soc. Amer. Proc. (1971) (submitted for publication) 10) P. A. C. Raats and W. R. Gardner, A comparison of some empirical relationships
138
11) 12)
13)
14)
15) 16) 17)
18)
19) 20) 21)
P . A . C . RAATS between pressure head and hydraulic conductivity, and some observations on radially symmetric flow. Water Resources Res. (1971) (accepted for publication) H. Bouwer, Theory of seepage from open channels. In: Ven Te Chow, ed., Advances in Hydroscience (Academic Press, New York 1969) 305 p., vol. 5, pp. 121-172 P. E. Rijtema, An analysis of actual evapotranspiration. Agr. Res. Reports no 659, 107 p. Centre for Agricultural Publications and Documentation, Wageningen, The Netherlands (1965) J. L. Ericksen, Tensor Fields (Appendix to The Classical Field Theories by C. Truesdell and R. A. Toupin). In: Handbuch der Physik (S. Flugge, Ed.) Vol. IlI/1 (1960) p. 794-858 G. G. Stokes, Mathematical proof of the identity of the stream lines obtained by means of a viscous film with those of a perfect fluid moving in two dimensions. Report of the British Association (1898). [Also: Mathematical and physical papers 5 (1905) 278-282.] D. Zaslavsky and D. Kirkham, The streamline function for axially symmetric groundwater movement. Soil Sci. Soc. Amer. Proc. 28 (1964) 156-169 D. Zaslavsky and D. Kirkham, Streamline function for potential flow in axial symmetry. Amer. J. Phys. 9 (1965) 677-679 J. H. Preston, Extensions of the Hele-Shaw analogy. Proceedings of the Second Australian Conference on Hydraulics and Fluid Mechanics (1966) p. A1-A23 and D 1 H. S. Hele-Shaw, Investigation of the nature of surface resistance of water and of stream-line motion under certain experimental conditions. (Second paper) (Institution of Naval Architects, London, 1898) 26 pp. and 11 plates R. Dachler, Grundwasserstromung (Julius Springer, Vienna, 1936) 141 pp. J. H. Horlock, Some two-dimensional magneto-fluid-dynamic flows at low magnetic Reynolds number. J. Fluid Mech. 6 (1963) 17-32 K. Ce~en, E. Omay and A. Si~iner, The study of collector wells by means of viscous flow analogy. 13th Congress of the International Association for Hydraulic Research. Proceedings - Vol. 4 (Subject D) (1969) 1-! 0
S O M E P R O P E R T I E S OF F L O W S I N U N S A T U R A T E D S O I L S W I T H AN E X P O N E N T I A L D E P E N D E N C E OF T H E H Y D R A U L I C C O N D U C T I V I T Y U P O N T H E P R E S S U R E HEAD P. A. C. RAATS The Johns Hopkins University *
Abstract: A systematic development of the consequences of an exponential dependence of
the hydraulic conductivity upon the pressure head is presented. Alternative expressions for the flux are discussed in detail. For steady flows, partial differential equations in terms of the matric flux potential, the pressure head, and the total head are derived. For steady, plane and axially symmetric flows, partial differential equations for the stream function are given. A theoretical basis for the construction of viscous flow analogs for steady, plane and axially symmetricflows is also presented. Introduction
It was shown by Gardner 1) that if the hydraulic conductivity decreases exponentially as a function of the pressure head, the partial differential equation for steady flows is linear in terms of a certain function. This function may appropriately be called the matric flux potential2). The first detailed example of an analytical solution of this linear equation was given by Philip 3) in his discussion of steady infiltration from buried point sources and spherical cavities. Immediately following, Wooding 4) analyzed the difficult problem of steady infiltration from a circular pond. Philip 5) also briefly discussed the solution for a single, horizontal line source in an infinite region. Recently I have shown that for plane and axially symmetric flows the stream functions also satisfy linear partial differential equations z, 6, 7). In fact, for plane flows the matric flux potential and the stream function satisfy the same linear partial differential equation. The equation for the plane stream function was used as a basis in a study of steady infiltration from line sources and furrows 2). The solution of this problem provides a quite complete description of the flow in terms of the distributions of the stream function, the components of the flux, the matric flux potential, the pressure head, and the total head. If the water content is * Present address: USDA-ARS-SWCRD, Department of Soil Science, University of Wisconsin, Madison, Wisc. 53706 USA. 129
130
P.A.C.
RAATS
given as a function of the pressure head, its distribution will of course also be known. The distribution of the pressure head and the total head for infiltration from buried point sources and cavities (thus completing Philip's analysis) and a complete description of infiltration from surface point sources and basins has been found7). On the basis of this latter solution, Philip s) has constructed a superposition theorem, which gives the solution for any surface source. Subsequently I have generalized Philip's superposition theorem to an arbitrary distribution of sources located at arbitrary distances below the soil surface 9). This paper presents a systematic development of the consequences of the exponential dependence of the hydraulic conductivity upon the pressure head. Of particular interest are the alternative representations of the flux and the simple flow equation in terms of the total head. In the last section a theoretical basis for the construction of viscous flow analogs for the plane and axially symmetric flows is presented.
The matrix flux potential Let 0 be the volumetric water content and v be the velocity of the water. Assuming the fluid to be incompressible and the flow to be steady, the balance of mass may be written as: V.(0v) = 0.
(1)
Let z be the vertical direction, positive downward. The volumetric flux Ov is assumed to be given by:
Ov = - kVH,
(2)
= - - k[h] Vh + k[h] Vz,
(3)
= - D [0] V0 + k [0] Vz,
(4)
=-
(5)
V~o + k [~o] v ~ ,
where k is the hydraulic conductivity, H is the total head, which is the sum of the pressure head h and the gravitational head - z (i.e., H = h - z ) , D=k(dh/dO) is the diffusivity, and q) is the matric flux potential defined by: h
0
: [" kfk] dh = [" O[0] dO,
(6)
mr/
ho
0o
where ho and 0o are reference values, and 0o = 0 [ho]. It is assumed that the steady flow is established through monotonic changes in water content, so that hysteresis need not be considered. The hydraulic conductivity k is assumed to be a unique function of the water content 0, which in turn is
PROPERTIES OF FLOWS IN UNSATURATED SOILS
131
assumed to be a unique function of the pressure head h. It is understood that D is undefined whenever d h / d O ---, ~ . Substitution of(5) into (1) gives: v%
-
ak [m] ~z
(7)
In general, Eq. (7) is nonlinear. The flux, when k is an exponential function of h
At this point it is convenient to assume that k and h are related by an expression of the form 1-5, 7-10) k = k o e~h.
(8)
The parameters ko and ~ are characteristics of the porous medium. The parameter ko represents the hydraulic conductivity at h = 0, i.e., the hydraulic conductivity of the saturated porous medium. The k-weighted mean pressure head between h = 0 and h ~ - ~ , which Bouwer (ref. 11 and references given there) has called the critical pressure head, is simply equal to -1/~1°). The parameter 1/e may be regarded as a characteristic length of the partially saturated porous medium. According to Philip3), a typical value of ~ is 0.01 cm -~ and the range 0.05 to 0.02 cm -1 seems likely to cover most applications. Most of the data presented by Rijtema 12) fall in this range. Taking ho -+ - m, and introducing (8) into (6) gives: k = ~¢p.
(9)
e ~'h = ~ o / k o .
(10)
e~n = ~0 e -~z . ko
(11)
Substitution of (9) into (8) gives:
Multiplying (10) by e -~z gives:
Using Eqs. (8)-(11), the flux Ov can be expressed in many forms. Introducing (9) into (5) gives: Ov = - V~p + ~q~Vz.
(12)
According to (12) the flux is linear in the matric flux potential and, in view of (10), also linear in e ~h. Introducing (9) into (2) gives: Ov
= - aq~VH.
(13)
In vector analysis it is well known that a n y vector field ¢ can be represented
132
P . A . C . RAATS
as the sum of a lamellar field Vf and a complex-lamellar field mVn (cf. Ericksen 13): c = Vf + mVn, (14) where the scalar fields f, m, and n are the so-called Monge potentials. However, the form of (12) is special in that two of the Monge potentials are scalar multiples of one another and the third Monge potential is just the coordinate z. Any vector field represented by (12) is also represented by (13) with H = ( 1 / ~ ) In t p - z . In other words, any vector field represented by (12) is in effect a complex-lamellar field. [Equations of the form (12), in which a flux 0v appears as the sum of a diffusive component - V t p and a drift component cupVz, occur in many other applications, The drift component may arise from the movement of a source of some entity with respect to the medium in which the entity diffuses, or from the action of an external force field such as the gravitational field. The fact that all such flows are complexlamelIar appears not to have been noted before.] According to (13) the vector field Ov is perpendicular to a family of iso-H surfaces whose distance function is -cup. The representation (13) is not unique. In fact there are infinitely many representations of Ov exhibiting the complex-lamellar nature. One such representation is Ov = _ __k°e~Ve~n .
(15)
According to (15) the vector field 0v is perpendicular to a family of iso-H surfaces whose distance function is - ( k o / ~ ) e ~'z. It is remarkable that, for what in essence is the same family of surfaces, in (13) the distance is a function of ~o and of the constant ~, while in (15) the distance is a function of z and the constants c~and ko. The results in this section are, of course, not resrticted to steady flows.
Flow equations, when k is an exponential function of h The alternative expressions for Ov naturally lead to different partial differential equations describing the flow. Introducing (12) in (1) or (9) in (7) gives ~-5, 7-9): V2q~ = ~ - - . Oz
(16)
Since ko and c~ are constants, it follows from (10) and (16) that the partial differential equation: V2e~h = c~
t~e~h
8z
.
e ~h
satisfies
(17)
PROPERTIES
OF FLOWS IN LrNSATURATED
SOILS
133
Certain solutions of Eq. (16) subject to boundary conditions in terms of ~o have been given by Philip 3), Raats 7), and Wooding4). Using Eq. (17), the same problems could of course be formulated in terms ofe ~h. Substituting (15) into (1) gives: t~e~u V 2 e ~// = - - ~ - -
(18)
Oz This equation is new. Note that there is a minus sign on the right hand side of (18). In some problems the boundary conditions are simplest, when expressed in terms of H. Equation (18) may be used as the basis for the study of a whole new class of boundary value problems. Plane and axially symmetric flows
For flows plane with respect to the y-direction, Eq. (1) reduces to: t30u t30w +- - =0.
~x
(19)
#z
Equation (19) suggests that there exists a stream function ~kpsuch that: Ou = -
~P ~z '
Ow =
~x
(20)
.
From the results in the previous section it is obvious that there are many different forms for the dynamic expressions for Ou and Ow. For example, from (15) and (20) it follows that ko_e~z ~3_e~n=_c~p, ~3x Oz
k 0e "z-d e~n _ c~ 3z
~bp ax
(21)
These equations may be regarded as analogs of the Cauchy-Riemann conditions. Elimination of e~n from (21) gives the partial differential equation for ~p 2): t325r' 02~b° ~¢P (22) a x z + ~ZzZ = ~ a z
In some problems the boundary conditions are simplest when expressed in terms of ~kp. For example, in an earlier paper 2) (22) was used to analyze steady infiltration from line sources and furrows. The reductions of Eqs. (16), (17) and (18) to plane problems are obvious. For axisymmetric flows, Eq. (1) reduces to : r O--rrOu, + d z Ow
= 0.
(23)
134
P.A.C. RAATS
Equation (23) suggests that there exists a Stokes' stream function ~,~ defined as (Stokes 14):
1 O~b. Ou, =-2~z~r Oz'
Ow-
1 O~p~
(24)
2nr Or
From (15) and (24) it follows that ko ~t
0 e"u 8r
1 O~a 2nr Oz '
ko e~z 8 e~u c~ Oz
1 0I/]a 2~r Or
(25)
Eliminating e ~n from (25) gives 7):
020a 0r 2
I O@a 02ffIa r
01//a + O~--Ct 0Z Or
(26)
Note that the left hand side of (26) has a minus sign in the second term and, thus, is not the Laplace operator for axially symmetric flow (cf. Zaslavsky and Kirkham15,16), Raats7). The reduction of Eqs. (16), (17), and (18) to axisymmetric problems are obvious.
Viscous flow analogs Preston 17) considered the steady slow motion of an incompressible Navier-Stokes fluid between two rigid plates which are a small distance d[x, y] apart and where the variation of d with x and/or y is slow. One boundary is taken as fiat and lying in the x, y-plane. Preston assumes that: (a) the velocity of the fluid perpendicular to the x, y-plane is negligibly small, (b) the acceleration is negligibly small, (c) the only viscous terms that need to be considered are those arising from the variation in the z-direction of the components of the velocity in the x- and y-directions. Considering the general case with the flat wall undergoing an arbitrary rigid motion in the x,y-plane, Preston derives partial differential equations for the stream function ~ and the fluid pressure p. When the walls are at rest with respect to each other, these equations reduce to:
a20 a20 3/adaO
+ 0y
a0
= 0,
(27)
Op + 0y 0 = 0. Ox~ + ay - - 2 + d Ox ax
(28)
~x~ + Oy~ -- a oX ax 02p 02p 3 rod Op
0d y)
Special cases result from choosing particular forms for the function d [ x , y ] . For example, if d--constant Eqs. (27) and (28) describe plane
PROPERTIES OF FLOWS IN UNSATURATED SOILS
135
potential flow. In this special case, and in this special case only, ~ and p satisfy the same partial differential equations. The analogy between potential flow and steady viscous flow in films of uniform thickness was first utilized in experiments by Hele-Shaw 18) and demonstrated theoretically by Stokes 14). Apparently, the Hele-Shaw model was first used in connection with flow in porous media by Dachler 19). Setting: y = z,
(29)
d = do exp (~z/3). reduces (27) to (22), the equation for the stream function when the flow is plane, while Eq. (28) for the fluid pressure reduces to: c~2p 02p ~ X 2 71- (~Z 2 - -
c3p ~ OZ
(30)
Comparison of (30) and (18) shows that p in (30) is the analog of e ~n for plane flow. In other words the lines along which the pressure p is constant in the viscous flow model correspond to lines along which the total head H is constant in the partially saturated porous medium. At Preston's suggestion, Horlock 2o) used the special case represented by (29) as the basis for a viscous flow analogy for the stream function in a particular problem in magnetohydrodynamics. Neither Horlock20) or Preston 17) offered an interpretation of the fluid pressure p. Horlock constructed an apparatus in which the distance d=do exp(O.742z/3). In other words, the viscous flow in Horlock's apparatus is analogous to steady flow in a partially saturated soil with e=0.742 cm-1. This value is an order of magnitude larger than the typical values indicated earlier. However, the flow pattern in the viscous flow model depends on the product of e and some characteristic length L representing the macroscopic dimensions in the x,y-plane. By choosing e large, the dimensions of the cell may be kept small. As an example, consider steady infiltration from an array of equally spaced line sources or furrows at the surface of a semi-infinite soil profile. Let L be the half distance between the line sources. The solution of this problem and calculations for e L = 1 and 5 were given before 2). To model these calculated results, the half distance between the sources in Horlock's apparatus should be 1.35 cm and 6.75 cm respectively. Figure 1, which was taken from Horlock's paper, shows the flow pattern for upward flow around a cylinder. For e L = 0 the flow on the upstream and downstream sides would be symmetric. In the photograph the flow is clearly asymmetric.
136
P.A.C.
RAATS
Fig. 1. Viscousflow model for upward flow around a cylinder (from Horlock2°). Preston 17) has pointed out that with x = r,
(31)
d = do r~ ,
Eqs. (27) and (28) reduce to the governing equations of axially symmetric potential flow. (~eqen et a121) used this analogy in a study of flow to horizontal collector wells completely penetrating the horizontal impervious base of an isotropic, homogeneous porous medium. By combining (29) and (31) one obtains a viscous flow analog for steady, axially symmetric flow in a partially saturated soil: y = z, d = do r~ exp (~z/3).
(32)
PROPERTIES OF FLOWS IN UNSATURATED SOILS
137
Such an a n a l o g could be used to m o d e l the flows studied by Philip3), R a a t s 7, 9), a n d W o o d i n g 4 ) . Just as for p l a n e flow, the p r e s s u r e p is the a n a l o g o f e ~H. The viscous flow a n a l o g y for steady p l a n e a n d axially s y m m e t r i c flows in p a r t i a l l y s a t u r a t e d soils with a n e x p o n e n t i a l d e p e n d e n c e o f k u p o n h opens m a n y possibilities. It m a y be used to study the influence o f realistic t o p o g r a p h i c features u p o n steady infiltration from furrows. It offers a convenient basis for studying steady u p w a r d a n d d o w n w a r d flow a r o u n d i m p e r m e a b l e objects with p l a n e or axial symmetry, which c o u l d lead to some insight into the effect o f the heterogeneous n a t u r e o f soils. Also, by extending (29) or (32) to include further d e p e n d e n c e o f d u p o n p o s i t i o n one could m o d e l certain steady flows in layered soils.
Acknowledgements I wish to t h a n k Mr. F. W. Blaisdell o f the St. A n t o n y Falls H y d r a u l i c Lab. for bringing to m y a t t e n t i o n the p a p e r by (~e~en et a121) a n d Prof. J. H. H o r l o c k o f C a m b r i d g e University for his p e r m i s s i o n to r e p r o d u c e Fig. 1. This research was s u p p o r t e d by a G r a n t f r o m the N a t i o n a l Science F o u n d a t i o n to the Johns H o p k i n s University.
References 1) W. R. Gardner, Some steady-state solutions of the unsaturated moisture flow equation with application to evaporation from a water table. Soil Sci. 85 (1958) 228-232 2) P. A. C. Raats, Steady infiltration from line sources and furrows. Soil Sci. Soc. Amer. Proc. 34 (1970) 709-714 3) J. R. Philip, Steady infiltration from buried point sources and spherical cavities. Water Resources Res. 4 (! 968) 1039-1047 4) R. A. Wooding, Steady infiltration from a shallow circular pond. Water Resources Res. 4 (1968) 1259-1273 5) J. R. Philip, Theory of infiltration. In : Ven Te Chow, ed., Advances in Hydroscience (Academic Press, New York, 1969) 305 p., vol. 5, pp. 215-296 6) P. A. C. Raats, Addendum to "Steady infiltration from line sources and furrows". Soil Sci. Soc. Amer. Proc. 35 (1971) 161 7) P. A. C. Raats, Steady infiltration from point sources, cavities and basins. Soil Sci. Soc. Amer. Proc. (1971) (submitted for publication) 8) J. R. Philip, General theorem on steady infiltration from surface sources, with application to point and line sources. Soil Sci. Soc. Amer. Proc. (1971) (submitted for publication) 9) P. A. C. Raats, Steady infiltration from sources at arbitrary depth. Soil Sci. Soc. Amer. Proc. (1971) (submitted for publication) 10) P. A. C. Raats and W. R. Gardner, A comparison of some empirical relationships
138
11) 12)
13)
14)
15) 16) 17)
18)
19) 20) 21)
P . A . C . RAATS between pressure head and hydraulic conductivity, and some observations on radially symmetric flow. Water Resources Res. (1971) (accepted for publication) H. Bouwer, Theory of seepage from open channels. In: Ven Te Chow, ed., Advances in Hydroscience (Academic Press, New York 1969) 305 p., vol. 5, pp. 121-172 P. E. Rijtema, An analysis of actual evapotranspiration. Agr. Res. Reports no 659, 107 p. Centre for Agricultural Publications and Documentation, Wageningen, The Netherlands (1965) J. L. Ericksen, Tensor Fields (Appendix to The Classical Field Theories by C. Truesdell and R. A. Toupin). In: Handbuch der Physik (S. Flugge, Ed.) Vol. IlI/1 (1960) p. 794-858 G. G. Stokes, Mathematical proof of the identity of the stream lines obtained by means of a viscous film with those of a perfect fluid moving in two dimensions. Report of the British Association (1898). [Also: Mathematical and physical papers 5 (1905) 278-282.] D. Zaslavsky and D. Kirkham, The streamline function for axially symmetric groundwater movement. Soil Sci. Soc. Amer. Proc. 28 (1964) 156-169 D. Zaslavsky and D. Kirkham, Streamline function for potential flow in axial symmetry. Amer. J. Phys. 9 (1965) 677-679 J. H. Preston, Extensions of the Hele-Shaw analogy. Proceedings of the Second Australian Conference on Hydraulics and Fluid Mechanics (1966) p. A1-A23 and D 1 H. S. Hele-Shaw, Investigation of the nature of surface resistance of water and of stream-line motion under certain experimental conditions. (Second paper) (Institution of Naval Architects, London, 1898) 26 pp. and 11 plates R. Dachler, Grundwasserstromung (Julius Springer, Vienna, 1936) 141 pp. J. H. Horlock, Some two-dimensional magneto-fluid-dynamic flows at low magnetic Reynolds number. J. Fluid Mech. 6 (1963) 17-32 K. Ce~en, E. Omay and A. Si~iner, The study of collector wells by means of viscous flow analogy. 13th Congress of the International Association for Hydraulic Research. Proceedings - Vol. 4 (Subject D) (1969) 1-! 0