linearized moisture flow with loss at the soil surface1

5
Linearized Moisture Flow with Loss at the Soil Surface 1 D. O. LOMEN AND A. W. WARRICK 2 ABSTRACT The general solution of a linearized moisture flow equation is derived for point and line sources whether buried or on the soil surface. The surface flux is taken to be proportional to the matric flux potential, which is consistent with the physical situation of small evaporative losses for dry soil and larger losses for wet soil. The fraction of water lost at the surface turns out to be identical for point and line sources and for the steady-state case is [ml(2 + m)] exp(-orf) where m is the surface flux proportionality constant, d the depth of source, and a a constant from the assumed unsaturated hydraulic conductivity function. This fraction is plotted as a function of time for a fixed source depth. Also given are curves depicting the total water loss as a function of source depth. A generalized solution is derived for any type of source located in a horizontal plane. Additional Index Words: high-frequency irrigation, matric flux potential, soil water, evaporation. Lomen, D. O., and A. W. Warrick. 1978. Linearized moisture flow with loss at the soil surface. Soil Sci. Soc. Am. J. 42:396-400. R ECENT ADVANCES IN SOLVING the moisture flow equa- tion for unsaturated soil have been enhanced by a linearization technique. The corresponding results are es- pecially attractive because the geometries involved are those used in trickle and subsurface irrigation. By taking an unsaturated hydraulic conductivity of the form [1] the moisture flow equation for an isotropic soil medium can be expressed as d0 a0 d<f> dt [2] where </> is a matric flux potential (Note: 0 as used here is not to be confused with the matric potential) defined by [3] 0= K(h)dh = K/a, 6 is the volumetric water content, h is the pressure head, / is time, V 2 is the Laplace operator, and thez-axis is chosen to be positive in the downward direction. Values of K 0 and a are constant. For steady-state conditions, Eq. [2] is linear with no further assumptions. Solutions for such steady-state cases are included or referenced in Philip (1969), Raats (1972), and Zachmann and Thomas (1973). In addition, Philip and Forrester (1975) give results for a hydraulic 'Arizona Agric. Exp. Stn. Paper no. 2767. Support was from Western Regional Project W-128 and funds provided by the U. S. Dep. of the Interior, Office of Water Resour. Res., as authorized under the Water Resour. Res. Act of 1964, Projects B-045-ARIZ and B-064-ARIZ. Received 18 July 1977. Approved 13 Feb. 1978. "Professor, Department of Mathematics; and Professor, Department of Soils, Water and Engineering, respectively, The Univ. of Ariz., Tucson, AZ 85721. conductivity dependent upon pressure head and depth as K = K 0 exp (ah + a/3z) with ft a constant. Solutions of the time-dependent Eq. [2] have been obtained for point, line, disc, and strip sources [Warrick (1974), Lomen and Warrick (1974), and Warrick and Lomen (1976)] by making the additional assumption that dO/d<f> be a constant (a/k). In this paper we develop a solution of Eq. [2] for a point or line source located at depth d below the soil surface. The boundary condition at the soil surface (z = 0)is for a vertical component of velocity proportional to the matric flux potential (or equivalently proportional to the hydraulic conductivity). Written in terms of 0 this is v,= -(d<f>/dz) + a<t>= -(am/ 2)0, z = 0, [4] with m a constant. In the papers mentioned previously an infinite medium or an impermeable upper boundary (m = 0) was assumed. Equation [4] approximates evaporation from the surface by giving large losses for wet conditions and small losses for dry conditions. As 0 is exponentially related to the pressure head h, then 0 is linearly related to the relative humidity of the soil vapor. A linear relationship of evaporation rate to vapor pressure of the surface has been useful for bare soil, at least under isothermal con- ditions (Staple, 1974, Eq. [3]). The generalized solution is + exp(-2D)0 B (X,y,Z + D,r)-2(l + m) exp (roD) exp [(2 + m)(Z- Tj)]0 fl (X,r,Tj,7Vi7, [5] I" ' J Z + D where 0 B is the solution of a "buried" line or point source given in Table 1. The dimensionless coordinates are defined by X = ax / 2, Y = ay / 2, Z = a z / 2, and T = a la / 4, [6] and the dimensionless depth isD = ad/2. The proof of Eq. [5] is tedious and presented in Appendix A. The result is similar to that of Raats (1972), but is more general in that a surface flux (Eq. [4]) and time dependence are allowed. For m = 0 and for <f> B without time-dependence, Eq. [5] reduces to Raats' results. The rate of surface loss can be calculated by Eq. [4] once 0 is known. Solutions for multiple sources and laterally confined systems may be found by adding individual point or line sources as previously shown by Warrick (1974) and Lomen and Warrick (1974). If we define the total amount lost at the surface as q tvap , we find for the point source To Jo $(R,0,T)RdR, <}> from a point source. [7] 396

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Linearized Moisture Flow with Loss at the Soil Surface1

D. O. LOMEN AND A. W. WARRICK2

ABSTRACTThe general solution of a linearized moisture flow equation is

derived for point and line sources whether buried or on the soilsurface. The surface flux is taken to be proportional to the matric fluxpotential, which is consistent with the physical situation of smallevaporative losses for dry soil and larger losses for wet soil. Thefraction of water lost at the surface turns out to be identical for pointand line sources and for the steady-state case is [ml(2 + m)] exp(-orf)where m is the surface flux proportionality constant, d the depth ofsource, and a a constant from the assumed unsaturated hydraulicconductivity function. This fraction is plotted as a function of time fora fixed source depth. Also given are curves depicting the total waterloss as a function of source depth. A generalized solution is derived forany type of source located in a horizontal plane.

Additional Index Words: high-frequency irrigation, matric fluxpotential, soil water, evaporation.Lomen, D. O., and A. W. Warrick. 1978. Linearized moisture flow withloss at the soil surface. Soil Sci. Soc. Am. J. 42:396-400.

RECENT ADVANCES IN SOLVING the moisture flow equa-tion for unsaturated soil have been enhanced by a

linearization technique. The corresponding results are es-pecially attractive because the geometries involved arethose used in trickle and subsurface irrigation. By taking anunsaturated hydraulic conductivity of the form

[1]

the moisture flow equation for an isotropic soil medium canbe expressed as

d0 a0d<f> dt

[2]

where </> is a matric flux potential (Note: 0 as used here isnot to be confused with the matric potential) defined by

[3]0= K(h)dh = K/a,

6 is the volumetric water content, h is the pressure head, / istime, V2 is the Laplace operator, and thez-axis is chosen tobe positive in the downward direction. Values of K0 and aare constant. For steady-state conditions, Eq. [2] is linearwith no further assumptions. Solutions for such steady-statecases are included or referenced in Philip (1969), Raats(1972), and Zachmann and Thomas (1973). In addition,Philip and Forrester (1975) give results for a hydraulic

'Arizona Agric. Exp. Stn. Paper no. 2767. Support was from WesternRegional Project W-128 and funds provided by the U. S. Dep. of theInterior, Office of Water Resour. Res., as authorized under the WaterResour. Res. Act of 1964, Projects B-045-ARIZ and B-064-ARIZ.Received 18 July 1977. Approved 13 Feb. 1978.

"Professor, Department of Mathematics; and Professor, Department ofSoils, Water and Engineering, respectively, The Univ. of Ariz., Tucson,AZ 85721.

conductivity dependent upon pressure head and depth as K= K0 exp (ah + a/3z) with ft a constant. Solutions of thetime-dependent Eq. [2] have been obtained for point, line,disc, and strip sources [Warrick (1974), Lomen andWarrick (1974), and Warrick and Lomen (1976)] bymaking the additional assumption that dO/d<f> be a constant(a/k).

In this paper we develop a solution of Eq. [2] for a pointor line source located at depth d below the soil surface. Theboundary condition at the soil surface (z = 0)is for avertical component of velocity proportional to the matricflux potential (or equivalently proportional to the hydraulicconductivity). Written in terms of 0 this is

v,= -(d<f>/dz) + a<t>= -(am/ 2)0, z = 0, [4]

with m a constant. In the papers mentioned previously aninfinite medium or an impermeable upper boundary (m =0) was assumed. Equation [4] approximates evaporationfrom the surface by giving large losses for wet conditionsand small losses for dry conditions. As 0 is exponentiallyrelated to the pressure head h, then 0 is linearly related tothe relative humidity of the soil vapor. A linear relationshipof evaporation rate to vapor pressure of the surface hasbeen useful for bare soil, at least under isothermal con-ditions (Staple, 1974, Eq. [3]).

The generalized solution is

+ exp(-2D)0B(X,y,Z + D,r)-2(l + m) exp (roD)

exp [(2 + m)(Z- Tj)]0fl(X,r,Tj,7Vi7, [5]I" 'J Z + D

where 0B is the solution of a "buried" line or point sourcegiven in Table 1. The dimensionless coordinates aredefined by

X = ax / 2, Y = ay / 2, Z = az / 2, and T = a la / 4, [6]

and the dimensionless depth isD = ad/2. The proof of Eq.[5] is tedious and presented in Appendix A. The result issimilar to that of Raats (1972), but is more general in that asurface flux (Eq. [4]) and time dependence are allowed. Form = 0 and for <f>B without time-dependence, Eq. [5]reduces to Raats' results. The rate of surface loss can becalculated by Eq. [4] once 0 is known. Solutions formultiple sources and laterally confined systems may befound by adding individual point or line sources aspreviously shown by Warrick (1974) and Lomen andWarrick (1974). If we define the total amount lost at thesurface as qtvap, we find for the point source

To

Jo$(R,0,T)RdR,

<}> from a point source. [7]

396

LOMEN & WARRICK: LINEARIZED MOISTURE FLOW WITH LOSS AT SOIL SURFACE

Table 1—Solutions for line and point sources

397

Time dependent Steady-stateBuried1. Point

(q units LVT)

to(R,Z,T)= '

+ e'p erfc

2. Line(q units L

<t>B(X,Z,T) = -if- Tr4ir J 0

Generalized

1. Point

</> = <t>B(X,Z-D,T)+exp(-2D)<j>B(X,Z+D,T)+F(R,Z,n(<t>B from 1 above)

F(R,Z,T) = -

2ir

is a modified Bessel function of the first kind)

. 0 9"

O7T

f T £-'J o

2. Line

qd+m)-2(1+ m)exp(mD+(2 + m)Z)

x

(4>B from 2 above)

The results of the integration for a step input of q starting attime zero give

_ mq<7evap 2 m

+ exp(-2D)erfc(D/2VT-Vf) _ 2(1 + m)m + 2 m(m + 2)

exp[/nD + m(m + 2)7] erfc [D / 2VT + (m + l)Vr]|.J[8]

The integration details, while quite lengthy, are straightfor-ward and use the following results borrowed in part fromEq. 7.4.36 of Abramowitz and Stegun (1964):

I erf (b = [exp (aX) erf (b + cX)

- exp(-ab/c + a2/4c2) erf (b + cX - a/2c)] [9]

Jexp(±V/;2 + Z2)erfc[-±v^+ ^R2 + Z*/2\/T]RdR

o V/?2 + Z2

= ± exp(-r) erfc (Z/2VF)

exp(±Z) erfc (Z/2\/f± [10]

For a line source, similar calculations lead to the result that<7evap is also given by Eq. [8]. The appropriate units forqrevapand q are L3/T for the point source and L2lT for the linesource (see Table 1). For the steady-state case Eq. [8]reduces to simply

= [mq / (m + 2)] exp(-ad). [11]

Equation [11] is not only valid for point and line sources,but is true for any well-defined distribution which is on ahorizontal plane. Intuitively, because it is a linear problem,any distribution approximated by a finite number of pointor line sources would also have this form. A proof of thisstatement is given in Appendix B.

For the important case of d = 0, we note that qevap ISfinite and in fact less than q even though the flux rate itselfgiven by Eq. [4] is undefined at the source. For $ greaterthan Kg/<x (corresponding to h = 0), Eq. [4] is not

398

IJO

O.I

001

0.001

OjOOOl

SOIL SCI. SOC. AM. J . , VOL. 42, 1978

0.01

10.0

Iad

Fig. 1—Total fraction of applied water lost at the surface, (the steady-state case.)

applicable. One choice for the case when d = 0, would beto define an input rate equal to q minus the excessevaporation calculated by Eq. [4] near the source. Forexample, we could calculate the distance away from thesource where h = 0 and assume the correct flux inside theregion was the maximum value allowed. The solutionwould be ignored very close to the source, but would besatisfactory a short distance away.

CALCULATIONSFigure 1 is a plot ofqnaf/q as a function of ad for steady-state

conditions. Physically qenp/q is the fraction of applied water lostat the surface. Values of m = 0.01, 0.1, 1, and 10 were used. Asd or a increases the amount of surface loss decreases. As mincreases the surface loss increases. An appropriate value of mmay be estimated by setting a realistic upward surface flux for awet soil Es (cm/day) equal to the negative of the surface flux fromEq. [4] f o r k = 0 resulting in

Es = I 2. [12]

Thus, if Es was 5 mm/day and K0 was 2.5 cm/hour then m wouldbe 0.017. If the evaporative demand were higher, m wouldincrease.

In Fig. 2, values ofqnap/q are plotted as a function of time form = 0.02 and ford = 1, 10, 50, and 100 cm. The values of a andk were taken as 0.01 cm"1 and 100 cm/hour, respectively. As /increases so does q^-df/q until the steady-state value given by Eq.[11] is reached. As d increases there is less surface loss and inaddition the steady-state results are approached only at largertimes.

0005 -

o.ooi

0 2 4 6t(hours)

Fig. 2—Fraction of applied water lost at the surface with m = 0.02.

DISCUSSION

The generalized solution given in Eq. [5] is valid for aburied water source of any horizontal shape and seemsappropriate for analyzing loss of water at the soil surfacedue to evaporation. Certainly, the complex process of soilevaporation is not accounted for in detail, but neverthelessthe local atmospheric and soil conditions can be taken intoaccount, in part, by the appropriate choice of the parameterm. The simple form of the solution should prove valuablefor checking numerical schemes adaptable to more com-plicated expressions of surface loss.

APPENDIX A

Derivation of the Generalized Solution Listed as Eq. [5]

In terms of dimensionless coordinates, Eq. [2] becomes

dr dx2 ay2 az2 az'

while the boundary condition at the surface is

az

[Al]

[A2]

First we note that for any function g which has two continuousderivatives, if <$>B satisfies Eq. [Al] so will

fJzg(Z -

This can be shown by substituting the integral expression into Eq.[Al] and integrating by parts. The function

<£ = <t>B(X,Y,Z - D,T) + exp(-2D)<j>B(X,Y,Z + D,T)

+ I g(Z-r,)^B(X,Y,-n,T)df, [A3]J Z + D

will, by our previous remarks, satisfy Eq. [Al] with <f>B thesolution of Eq. [Al] for a source buried in an infinite medium. Thefunction g is now chosen so the boundary condition at the surface,Eq. [A2], is satisfied. Straightforward calculations give

+ exp(-2D)[-^'B(X,Y,D,T) + (2 + m)<t>B(X,Y,D,T)]

LOMEN & WARRICK: LINEARIZED MOISTURE FLOW WITH LOSS AT SOIL SURFACE 399

r- APPENDIX B+ [-*'(-»») + ( 2 + «)g(-i7)]«MX,r,7j,r)di»

•> o Proof that Eq. [11] is Valid for an Arbitrary+ g(-D)<t>B(X,Y,D,T) = Q. [A4] Horizontal Source

T,, ,« . , , ,.cf .... ... . .. ., Assume we have a source located on a horizontal plane at depthThe pnme o n . f a denotes differentiation with respect to the d ̂ &QUK& can ̂ ̂ ^ & finjte number rf p^/^ lines £ a

ir argument. II continuously distributed source. However, we require it either bewithin a finite lateral region or, in the case of infinite sources, that

g(Z - 7)) = A expf(2 + m)(Z - rj)] [A5] there are symmetric vertical planes. The input q over theappropriate region A (either for all space or within the symmetric

the integral in Eq. [A4] vanishes identically and we choose A so vertical planes) is related to the integrated surface flux qenp bythe remainder of the expression is zero. Before we can show thisis possible for general horizontal sources we must first consider „ = „ + ijm I I Vz<ix:dy [Bl]the solution for a point source. The solution of Eq. [Al] for a point evap

z_» J Jsource of strength q buried in an infinite medium is obtained from , ,Carslaw and Jaeger (1959, p. 261) q = 9evap + lim a^dxdy [B2]

2-»oo J J

$BP = jg^s/2 J £ 32 exp[-(X2 + K2)/4f] sulce as z grows large the flux is given by the limiting0 [A6] conductivity A' = a<f>. By use of Eq. [5] and [6], we find

fexp[-(Z - 2£)2/4f]df = /(X,y,£)exp[-(Z-2f)2/4f]df f f rrr1 ? V V h m j j a<l>dXdy = (4/a){ <t>B(X,Y,°°)dXdY [B3]

. , 2~"° A Awithoa + exp(-2D) I [ (j>B(X,Y,°°)dXdY - 2(1 + m) exp(mD)/(x,r,0=-£li|-3«exP[-(x« + y*)/4f]. ; j j w , , ;

The second form of Eq. [A6] is convenient for showing that the x J J ]™ jz+D exPt(2 + «) (z ~ *)) <^a (^.y. 1))^^^^]

boundary condition at the soil surface, Eq. [A2] is satisfied. A ~*Differentiation shows that

The first two integrals on the right-hand side are each equal to aqld<t>BP _ _ CT (Z - 2^) r_/7 _ <jf\2 iAf-\ t 4 since (f>B is the solution for a source "buried" in a medium of~dZ~ ~ ~ J0 f(X'Y'*> 2| expL~(Z - 2f) /4§Jdf infinite vertical dimensions for which all of the loss must occur

[A7] below the source depth. Similarly, the final multiple integral is= - l T f ( X , Y , { ) (-) exp[-(Z - 2f)2/4f]^ + </>BP

£asily eVa'Uated after substituting « = ^ - (Z + ^) g^ingJo Uf/

„.. . , c r A , n j r A _ , . . .. c Mm a^dwfy = 9[l + exp(-2D)-2(l + m)cxp(-2D)/(2 + m)].It the expressions from Eq. [A6] and [AT] are substituted into Eq. *^oo J J[A4], it reduces to [B4]

-2(1 + m)<t>Bp(X,Y, -D,T) - g(-D)<t>BP(X,Y,D,T) = 0. [A8] Substitution of this last quantity into Eq. [B2] and solving for^evapleads to

Bychoosing gevap = 9-?t l+exp(-2Z>)A=-2( l+m)exp[m£>] [A9] - 2(1 + m)exp(-2D)/(2 + m)] [B5]

the boundary condition at the soil surface is satisfied for a point which simplifies to Eq. [11].source buried at Z = D and

4> = (t>B(X,Y,Z - D,T) + exp(-2D)<t>s(X,Y,Z + D,T)-2(1+ m) exp(m£>)

f exp[(2 + m)(Z-i,)].MX,y,T,,7Vii [5]J Z + D

is the appropriate form of the solution for a point source. Thesolutions of Eq. [Al] for buried horizontal sources in the shape oflines, strips, discs, rings, and rings of finite width, as summarizedin Lomen and Warrick (1976), are all obtained from the pointsource solution by integrating the point source solution, given byEq. [A6], over the appropriate domain in anX — Y plane. Sincethe operations in showing that the point source solution, Eq. [A6],satisfied the surface boundary condition did not involve X and Y,any expression obtained from Eq. [A6] by integration in anX — Yplane will also satisfy the surface boundary condition, Eq. [A2].Thus Eq. [5] is the solution for a buried source of arbitraryhorizontal shape.

400 SOIL sci. soc. AM. J., VOL. 42, 1978