Twodimensional unsaturated flow through a circular inclusion

WATER RESOURCES RESEARCH, VOL. 38, NO. 7, 1113, 10.1029/2001WR001041, 2002

Two-dimensional unsaturated flow through a circular inclusion A. W. Warrick University of Arizona, Tucson, Arizona, USA

J. H. Knight Australian National University, Canberra, Australia Received 26 October 2001; revised 19 February 2002; accepted 22 February 2002; published 23 July 2002.

[1] Two-dimensional unsaturated flow is considered through a circular inclusion. The hydraulic conductivity is of the form Kiexp(ah) where the saturated conductivity Ki is different in the main flow regime and the inclusion, a is a constant in the entire flow domain, and h is the pressure head. The problem reduces to the Helmholtz equation as previously used by J. R. Philip and colleagues for solving problems with impermeable regions or cavities in an unsaturated regime. The pressure heads are continuous on the interface of the inclusion. The normal flow velocities at the interface are matched approximately using the analytical element method recently exploited for saturated domains in several studies. Flow enhancement and exclusion through the circular inclusions are dependent on the value of a and the radius of the cylinder but otherwise are similar to that for the saturated case. For example, for ratios of the inclusion to background saturated conductivity of 0.5 the flow is 0.74 of what it would be without the inclusion compared to 0.67 for the saturated case. This was calculated for a dimensionless radius (0.5 a multiplied by the physical radius) of 1. When the ratio of the inclusion saturated conductivity to the background is 5, the comparable value for the unsaturated case is 1.45 INDEX TERMS: 1866 Hydrology: Soil moisture; 1875 and for the saturated case is 1.67. Hydrology: Unsaturated zone; 1829 Hydrology: Groundwater hydrology; KEYWORDS: groundwater hydrology, irrigation, soil moisture, unsaturated zone, modeling

1. Introduction [2] The theory of water exclusion from or entry into subsurface inclusions has been presented by Philip et al. [1989] and other related papers. Possible applications include descriptions of flow in regions proximate to macropores, tunnels, cavities and around underground obstacles such as stones or structures. [3] A much simpler problem is the effects of inclusions within saturated flow regimes. For example, the two dimensional flow through a single circular object in infinite space compared to no inclusion is given by [Wheatcraft and Winterberg, 1985, equation (42)] F¼

2k1 1 þ k1

ð1Þ

where k1 is the ratio of the inclusion conductivity to the background conductivity. Analytical expressions also exist which relate effective conductivity for a system which includes spherical inclusions [Peck, 1983, p. 208]. [4] Barnes and Jankovic´ [1999] used the analytic element method to examine saturated, two-dimensional flow through large numbers (e.g., 10,000) of circular inclusions. They found appropriate forms of the complex potential which provided continuous normal velocities across all interfaces between the inclusions and an approximate form for the changes in potential across the interfaces. The locations, Copyright 2002 by the American Geophysical Union. 0043-1397/02/2001WR001041

sizes and relative conductivities of the inclusions were all arbitrary although no two were intersecting. The result was a detailed picture of the flow domain which is sufficient to enable particle tracking or whatever might be of interest. Jankovic´ and Barnes [1999] applied the same approach for large number of spheroidal inhomogeneities (in a threedimensional flow regime). [5] The objective of the present study is to apply the analytical element method of Barnes and Jankovic´ [1999] and Jankovic´ and Barnes [1999] to the unsaturated case for a single circular inclusion in a two-dimensional flow domain. The Gardner [1958] model for unsaturated flow is used, which results in a linear equation for the matric flux potential. Philip [1968] discusses the physical significance of the capillary length parameter a1 in the Gardner model. If it is assumed that the surrounding soil and the inclusion have the same value of the capillary length parameter but different saturated hydraulic conductivities, this reduces to the solution of the Helmholtz equation and, initially, follows the direction of Philip et al. [1989]. In fact, for an impermeable inclusion, the problem reduces to that of Philip et al. However, when the inclusion is permeable, it is necessary to resort to the analytical element method.

2. Theory [6] Consider the circular inclusion of radius r1 centered at the origin in Figure 1. The saturated hydraulic conductively is K1 and the unsaturated hydraulic conductivity is K = K1 exp(ah). The inclusion is embedded within a larger flow region described by K = K0exp(ah). The background flow is

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WARRICK AND KNIGHT: FLOW THROUGH A CIRCULAR INCLUSION

[9] The solution of H may be taken of a separable form [Moon and Spencer, 1961, p. 11]: H¼

1 X

Rn ðrÞYn ðyÞ

ð8Þ

n¼0

where r is the radial coordinate (r2 = x2 + z2) and y the polar angle with z = r cos y and x = r sin y. The functions Rn and Yn are given by Rn ðrÞ ¼ An Jn ðcrÞ þ Bn Yn ðcrÞ (9)

Yn ðyÞ ¼ Cn sin ðnyÞ þ Dn cos ðnyÞ

using Bessel functions Jn and Yn. The solution should be an even function of y for which Cn must be 0. As c is an imaginary number, consider as a trial solution H = H + for the outer (r > r1) and H = H for the inner (r < r1) region: H þ ðr; yÞ ¼

Figure 1. Flow geometry including circular inclusion.

1 X an Kn ð0:5arÞ cos ðnyÞ Kn ðsÞ n¼0

ð10Þ

1 X an In ð0:5arÞ cos ðnyÞ In ð s Þ n¼0

ð11Þ

H ðr; yÞ ¼

characterized by a pressure head h = h0 < 0 and vertical flow rate of K0 exp(ah) at large distances from the circle. [7] The flow rate everywhere is given by ~ J w ¼ Ki ðrf afrzÞ

ð2Þ

with the matric flux potential equal to Kif ¼

Ki exp ðahÞ a

ð3Þ

and with i = 0,1. The first part of equation (2) is flow due to pressure difference and the second part is due to gravity. For these definitions, Richards’ equation becomes r2 f a

@f ¼0 @z

ð4Þ

(Note that f does not contain either hydraulic conductivity term.) [8] Without loss of generality, we introduce the Helmholtz function H with

s ¼ 0:5ar1

ð5Þ

The value of f0 = (K0/a)exp(ah) is the limiting value of f at large distances from the inclusion; note the corresponding value of H is 0. Note that for background regions f is f0 and H is 0. As shown by Philip et al. [1989], H satisfies the Helmholtz equation r 2 H þ c2 H ¼ 0

ð6Þ

c2 ¼ 0:25a2

ð7Þ

ð12Þ

(For imaginary arguments of cr = 0.5iar, Bessel Functions Jn(cr) and Yn(cr) are constants multiplied by the modified Bessel functions Kn(0.5ar) and In(0.5ar). Further note that if the flow field were not symmetric around x = 0, a ‘‘sin’’ series would be needed as well as the ‘‘cosine’’ series. This condition will be satisfied by equations (10) and (11) for all choices of an). [10] Physically, the pressure head h and normal velocities should be continuous at every point along r = r1. Matching pressure head h requires H þ jr¼r1 ¼ H jr¼r1

[11] The requirement for continuity of the normal velocity at r = r1 is

H¼

f 1 expð0:5azÞ f0

where In and Kn are modified Bessel functions of first and second types, respectively and s is

@f K0 aðcos yÞf @r

@f ¼ K1 aðcos yÞf @r r1

ð13Þ r1

In terms of the Helmholtz function H, the continuity at r = r1 results in

@H þ 0:5að cos yÞ½H þ þ 2 exp ðs cos yÞ @r r1 (14) @H 0:5ð cos yÞ½H þ 2 exp ðs cos yÞ ¼ K1 @r r1 K0

where Note for K1 = 0, the last expression is equivalent to that of Philip et al. [1989, equation 32].


[12] Use of equations (9), (11), and (14) after dividing through by 0.5 a, leads to 1 X

an

n¼0

K0n ðsÞ In0 ðsÞ k1 cos y cos ðnyÞ cos y K n ðsÞ In ð s Þ

¼ 2ðk1 1Þð cos yÞ exp ðs cos yÞ

ð15Þ

where k1 = K1/K0. The coefficients an follow optimization of equation (15) for multiple choices of P and using a finite number of terms in the above series. In the solution of Barnes and Jankovic´ [1999], the continuity of the flux was satisfied exactly whereas the continuity of head was approximate. Here the situation is reversed and it is much simpler to match the head and approximate the flux term. [13] By equation (5) the value of f/fo is

f ao Ko ð0:5arÞ ¼ 1 þ expð0:5azÞ f0 Ko ðsÞ 1 X an Kn ð0:5arÞ cos ðnyÞ ; þ Kn ðsÞ n¼1 f ao Io ð0:5arÞ ¼ 1 þ expð0:5azÞ f0 Io ð s Þ 1 X an In ð0:5arÞ cos ðnyÞ ; þ In ð s Þ n¼1

r > r1

r < r1

ð16Þ

ð17Þ

3. Results and Examples [14] Equation (15) can be rewritten using Dwight [1961, equations 803.1 and 804.1] using N terms and evaluating at y = ym : N X

an fm;n cos ðnym Þ ¼ gm

ð18Þ

n¼0

with fm;o ¼ cos ðym Þ þ

K1 ðsÞ I1 ð s Þ k1 cos ðym Þ K0 ðsÞ I0 ð s Þ

ð19Þ

0:5½Kn1 ðsÞ þ Knþ1 ðsÞ k1 fm;n ¼ cos ðym Þ þ Kn ðsÞ 0:5½In1 ðsÞ þ Inþ1 ðsÞ ; n ¼ 1; 2; . . . ; N ð20Þ

cos ðym Þ I n ðsÞ gm ¼ 2ðk1 1Þ cos ðym Þ exp ð2s cos ym Þ

ð21Þ

Values of an were evaluated using the ‘‘linfit’’ function of Mathcad [Mathsoft, 1999] to ‘‘minimize the sum of the squares of errors.’’ The M + 1 values of ym were taken equally spaced over y = 0 to p. Results proved N = 5 to be generally sufficient provided s 1 although N = 10 was used in the following numerical examples. Larger N were needed for larger s, and N = 50 was used for computations with s = 5. Values of M used for matching the normal flux were taken generally to be 100 or more in lieu of comments by Barnes and Jankovic´ [1999] and because increasing M

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does not increase computational effort significantly. The average differences in magnitude of the approximations of the left side of equation (17) and the right side were generally on the order of 108 compared to absolute values of gm which were as large as 1 –10 for small s and several hundred for larger s. 3.1. Comparisons of F/Fo for Impermeable Circular Regions [15] Resulting f/fo were compared for s = 0.25, 1 and 5 using an impermeable circular region. This was in order to compare results offered by Philip et al. [1989] (Figure 2). Contours of f/fo are presented as Figure 2 with f/fo following from equation (5): f sz ¼ 1 H exp f0 r1

ð22Þ

The same patterns were captured for all three values of s. In each case, there is a maximum value of N at the top center of the interface, corresponding to a pressure buildup. At the lower center, there is a small value of f corresponding to the driest region obtained. Away from the interface, a value of f/fo = 1 is approached with the least disturbance of the original field for the small s = 0.25 and the largest disturbance (in terms of f) for s = 5. 3.2. Results for Contrasting Inner Conductivities [16] Contours and flow nets were prepared for an intermediate value of s = 1 and contrasting inner conductivities with k1 = 0, 0.5 and 5. Results for the dimensionless matric flux potentials are in Figure 3 (for ease of comparison, the results for k1 = 0 are repeated). Note that for k1 = 0.5, the contour values are less extreme with values closer to the background (f/fo = 1), even next to the inclusion. When the inclusion is more conductive than the surrounding (k1 = 5), the values of N tend to be less above the inclusion and more below, the opposite of the cases where k1 < 1. [17] The flow net calculations require calculation of hydraulic head and the stream function on a detailed grid. The hydraulic head was determined by first finding a dimensionless h from equation (3) in the form h h0 ¼ r1

1 f ‘n 2s f0

ð23Þ

A dimensionless hydraulic head was found by subtracting z/r1 from the above expression. The stream function follows by integrating a dimensionless form of the vertical velocity, starting at x = 0 and along the elevation of interest. The vertical velocity normalized by the background velocity is given by Jz = Ki@f/@z + Ki leading to Jz ki @ ðf=f0 Þ 2sðh h0 Þ þ ki exp ¼ @Z r1 K0 exp ðaho Þ 2s

ð24Þ

Where ki remains Ki/Ko and dimensionless Z is z/r. In order to evaluate @(f/fo)/@Z the following expression was used: @ ðf=f0 Þ @H @R @H @y ¼ s exp ðsZ ÞH þ esZ þ @Z @R @Z @y @Z

ð25Þ

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Figure 2. Contours of f/fo for impermeable inclusion and s of 0.5, 1, and 5. Contour values are from 0.25 to 4 in increments of 0.25.

Here R is the dimensionless radial coordinate r/r1 and from Figure 1, @R/@Z = Z/R and @y/@Z = Y/R2. Values of @H/@R and @H/@y for R > 1 are N @H þ sX an ½Kn1 ðsRÞ þ Knþ1 ðsRÞ cos ny ¼ 2 n¼0 K n ðsÞ @R

ð26Þ

N X @H þ nan Kn ðsRÞ sinðnyÞ ¼ Kn ðsÞ @y n¼1

ð27Þ

For inside of the inclusion, ‘‘K’’ is replaced by ‘‘I’’ throughout and the sign on @H /@R changed to give s/2 rather than s/2.

Figure 3. Contours of f/fo for k1 = 0, 0.5, and 5, all for s = 1.

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Figure 4. Flow nets for s of 0.5, 1, and 5, all for s = 1.

[18] The form of the stream function used was Y ¼ ½K0 expðaho Þ1 X

ZY

JZ dX 0

ð28Þ

0

with X = x/r1. The Y values were prepared by first finding the vertical velocity from equation (24) on a fine grid (50 100 for 0 < X < 2.5 and 2.5 < Z < 2.5). The integral of equation (28) was approximated by multiplying the average velocities at adjacent X nodes by their separation. When crossing the interface (at X 2 + Z 2 = 1), the average velocity was approximated by weighting according to the distance of each node to the interface. [19] Resulting flow nets are in Figure 4 with all plots for s = 1. The first result is for k1 = 0, again for the impermeable inner circle for which the streamlines can be compared to Philip et al. [1989] (Figure 2c). The streamline values along the upper limit (Z = 2.5) are equally spaced and nearly equal to X. The water moves downward and away from the impermeable region and then comes back below the obstruction. Where the streamlines became ‘‘close’’ to the inclusion, such as X = 1, Z = 0, the flow is accelerated. Conversely, when the streamlines diverge (such as near X = 0 and Z = 1), the flow is slowest. The hydraulic head contours are consistently orthogonal to the streamlines and flow goes from higher (top) to lower values. [20] The result with k1 = K1/K0 = 0.5 are considerably different than for k1 = 0 (compare Figures 4a and 4b). The streamlines are still deflected away from the less conductive region and return below the region; however, the pattern is less extreme. In fact, the pattern is much closer to the case for uniform flow (for which the flow net would be a rectangular mesh). There are breaks in slope of both the streamlines and hydraulic heads when crossing the interface. There is less flow through the lower permeable region than for the uniform case (about 70% of the uniform case) (see Table 1). [21] When the inner circle is at a higher conductivity, extra flow occurs within the inclusion (Figure 4c). For

k1 = 5, the flow through the inner cylinder is about 1.45 times that for the uniform case. For this situation, the flow lines converge to the cylinder from above and then diverge below, much like the saturated case considered by Wheatcraft and Winterberg [1985]. Again, there are discontinuities in the slopes of both the streamlines and hydraulic head contours at the interface. Streamline values at X = 1 and Z = 0 were used to define flow exclusion or enhancement through the internal cylinder. The value of dimensionless Y from equation (28) is a ratio of the flow through 0 to p at any elevation relative to the background flow. In Table 1, are the relationships for s = 0.25, 1 and 5. In each case, the values are less than 1 for k1 < 1 and greater than 1 for k1 > 1. [22] Values can also be computed for exclusion or enhancement of flow for saturated conditions using equation (1) [Wheatcraft and Winterberg, 1985] (see Table 1). For the less permeable case of k1 = 0.5, there is less flow through the inclusion for the unsaturated compared to the saturated case (e.g., 0.67 compared to 0.7). For k1 = 5, the ratio of 1.67 is higher than any of the unsaturated cases. For both relative permeability ratios, the tendency is for the effects on the saturated flow to be more extreme, i.e., more exclusion for the lower ratio and more enhancement for the higher ratio. Also, the saturated and unsaturated cases are in closer agreement for the smaller s values which would correspond to either a lower or a smaller radius.

Table 1. Flow Exclusion or Enhancement by Inclusionsa k1 = K1/K0

s = 0.5 s=1 s=5 Saturated

0

0.5

1

5

0 0 0 0

0.70 0.74 0.87 0.67

1 1 1 1

1.57 1.45 1.19 1.67

a Ratios are for the streamlines at x = r1, z = 0 compared to that for no inclusions.

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Figure 5. Dimensionless pressure head (h h0)/r1 for k1 = (a) 0, (b) 0.5, and (c) 5. Values of s are 0.5, 1, and 5. 3.3. Distributions of Pressure Along the Interface [23] Along the interface, the Helmholtz function H simplifies from equation (9) or equation (11) to H ð1; yÞ ¼

N X

an cos n y

ð29Þ

n¼0

A dimensionless pressure head s(h ho)/r is defined from equation (23). Plots of s(h ho)/r1 are given as a function of the polar angle y in Figure 5. The most extreme changes in pressure head are for the impermeable inclusion (k1 = 0), with dimensionless values of about 0.5 just below (y = 0) to a maximum of about 0.8 at the top (y = p) for s = 0.5. For k1 = 0.5, there are the same tendencies but the maximum and minimum values of h are less extreme. However, as noted previously on Figure 3, when the inclusion is at a higher conductivity than the surroundings (k1 = 5), there is a reversal of pressure changes; the maximum pressure head occurs below (y = 0) and a reduction of pressure occurs above (y = p). In this case, the reduction is as extreme as the impermeable case, but the minimum pressure occurs above rather than below (compare s = 0.5 for Figures 5a and 5c).

4. Summary and Conclusions [24] The analytical element method of Barnes and Jankovic´ [1999] and Jankovic´ and Barnes [1999] was found to be successful for unsaturated flow through circular inclusions. This leads to expressions for potential and flow fields for a permeable inclusion as well as an impermeable inclusion such as that considered by Philip et al. [1989]. Generally, computations are easier for smaller values of the dimensionless radius s = 0.5 as compared to s = 5. For example, 5 terms of the truncated Fourier series were sufficient for the smaller s values whereas 50 were used for s = 5. Flow exclusion and enhancement through the inclusion was less extreme than for the saturated case. This is due to a ‘‘compensation’’ of the conductivity reflected in a changing pressure head for regions near and within the

inclusion. Another interesting result is that for an inclusion of high conductivity, the pressure decrease is above and pressure increase below the discontinuities. This is the opposite of what happens for an inclusion with a conductivity lower than the background. [25] Successful extensions of the approach appear feasible. These include systems with multiple inclusions, with other shapes of inclusions and for three-dimensional domains. Also, the preciseness of the flow description should be advantageous for particle-tracking approaches to study solute dispersion. [26] Acknowledgments. This research was supported in part by Western Regional Project W-188.

References Barnes, R., and I. Jankovic´, Two-dimensional flow through large numbers of circular inhomogeneities, J. Hydrol., 226, 204 – 210, 1999. Dwight, H. B., Table of Integrals and Other Mathematical Data, 4th ed., Macmillan, New York, 1961. Gardner, W. R., Some steady state solutions of the unsaturated moisture flow equation with application to evaporation from a water table, Soil Sci., 85, 228 – 232, 1958. Jankovic´, I., and R. Barnes, Three-dimensional flow through large numbers of spheroidal inhomogeneities, J. Hydrol., 226, 224 – 233, 1999. Mathsoft, Mathcad user’s guide, Mathsoft, Inc., Cambridge, Mass., 1999. Moon, P., and D. E. Spencer, Field Theory Handbook, Springer-Verlag, New York, 1961. Peck, A. J., Field variability of soil physical properties, in Advances in Irrigation, vol. 2, edited by D. Hillel, pp. 189 – 221, Academic, San Diego, Calif., 1983. Philip, J. R., Steady infiltration from buried point sources and spherical cavities, Water Resour. Res., 4, 1039 – 1047, 1968. Philip, J. R., J. H. Knight, and R. T. Waechter, Unsaturated seepage and subterranean holes: Conspectus, and exclusion problem for circular cylindrical cavities, Water Resour. Res., 25, 16 – 28, 1989. Wheatcraft, S. W., and F. Winterberg, Steady state flow passing through a cylinder of permeability different from the surrounding medium, Water Resour. Res., 21, 1923 – 1929, 1985.

J. H. Knight, School of Mathematical Sciences, Australian National University, ACT 0200, Australia. ([email protected]) A. W. Warrick, 429 Shantz Building 38, University of Arizona, Tucson, AZ 85721, USA. ([email protected])