A note on groundwater flow along a hillslope

Transcription

1 WATER RESOURCES RESEARCH, VOL. 40,, oi: /2003wr002438, 2004 A note on grounwater flow along a hillslope Eoaro Daly Dipartimento i Iraulica Trasporti e Infrastrutture Civili, Politecnico i Torino, Torino, Italy Amilcare Porporato Department of Civil an Environmental Engineering, Duke University, Durham, North Carolina, USA Receive 28 June 2003; revise 2 October 2003; accepte 22 October 2003; publishe 6 January [1] Particular cases of grounwater flow along a hillslope are stuie using the grounwater hyraulic theory (e.g., Dupuit approximation) escribe by the Boussinesq equation. Analytical solutions are foun as simple transformations of known similarity solutions. In particular, the expressions for phreatic surface level an volume flux an the corresponing loop-rating curve for the flow of a grounwater moun as well as the travelling wave solutions of the Boussinesq equation are escribe in etail. INDEX TERMS: 1829 Hyrology: Grounwater hyrology; 1899 Hyrology: General or miscellaneous; 3220 Mathematical Geophysics: Nonlinear ynamics; KEYWORDS: Boussinesq equation, Dupuit approximation, similarity solution Citation: Daly, E., an A. Porporato (2004), A note on grounwater flow along a hillslope, Water Resour. Res., 40,, oi: /2003wr Introuction [2] Unconfine grounwater flow in a sloping aquifer may be moele using the nonlinear Boussinesq equation, base on the classical Dupuit approximation. Solutions of such an equation are of interest in grounwater catchment hyrology [Troch et al., 1993; Sanfor et al., 1993; Szilagyi an Parlange, 1998; Mizimura, 2002], coastal grounwater hyraulics [e.g., Li et al., 2000a, 2000b], an several other nonlinear iffusion problems [e.g., Peletier, 1971; Aronson, 1986; Gratton an Minotti, 1990; Diez et al., 1992]. Although exact analytical solutions of the Boussinesq equation on a sloping be o not seem to have been given before, many approximate solutions have been propose in the literature [e.g., Lockington et al., 2000; Telyakovskiy et al., 2002], among which the solutions of the linearize equation [e.g., Brutsaert, 1994] an of the kinematic wave equation [e.g., Beven, 1981; Fan an Bras, 1998; Troch et al., 2002] must also be quote. [3] This paper presents some analytical solutions of the Boussinesq equation escribing the grounwater flow along a hillslope. By means of a simple travelling wave coorinate transformation, the Boussinesq equation is written as if the flow occurre on a horizontal impermeable be, for which exact similarity solutions are well known [e.g., Polubarinova- Kochina, 1962; Aravin an Numerov, 1965; Bear, 1972; Gratton an Minotti, 1990]. The practical applicability of the approach is somewhat limite by the fact that the same transformation must also apply to the initial an bounary conitions; however, the analysis may be useful to clarify some mathematical an physical aspects of the problem an to furnish benchmarks for the valiation of numerical simulations. 2. Moel Description [4] The water flow within a shallow saturate zone of a soil mantle overlying an impermeable berock of slope q can be stuie following the classical approach of Dupuit [e.g., Polubarinova-Kochina, 1962; Bear, 1972], which neglects the curvature of the flow streamlines, which are assume to be parallel to the impermeable be. As iscusse by Dagan [1967], such an approximation is equivalent to the shallow-water approximation in open channel flows [e.g., Stoker, 1957]. [5] For isotropic an homogeneous soils, the continuity equation an the equation of motion are cos q þ g þ 1 ¼ sin q ð2þ where t is time, x is the spatial coorinate, n is the average soil porosity, h is the water level, Q is the time volume flux per unit with, i.e., Q = hu, u is the seepage velocity, g is the gravity constant, sinq is the be slope, an j is the friction slope. Neglecting the inertial terms an assuming j = u/k s, where K s is the average soil hyraulic conuctivity, the equation of motion becomes u ¼ K s sin q K s cos q; ð3þ which is the well-known Darcy s law. The introuction of equation (3) in equation (1) leas to ð1þ Copyright 2004 by the American Geophysical Union /04/2003WR ¼ K s sin q þ K h cos q; ð4þ 1of5

2 DALY AND PORPORATO: TECHNICAL NOTE that is usually referre to as the Boussinesq equation for the general case of an unconfine aquifer on a sloping be. A few exact similarity solutions of equation (4) have been erive for horizontal be, i.e., sin q = 0 [e.g., Polubarinova- Kochina, 1962; Aravin an Numerov, 1965; Bear, 1972; Gratton an Minotti, 1990]. [6] A common simplification of equation (2) is to assume that the hyraulic graient is equal to the be slope [e.g., Beven, 1981]; with this hypothesis, equation (4) is simplifie to a linear kinematic wave equation, whose solutions are waves moving ownhill at a constant spee with unaltere shape. In fact, the presence of linear convection suggests the use of a travelling wave coorinate also to transform equation (4). Thus introucing a new system of coorinates whose origin moves with the velocity c 0 = K s sin q/n, i.e., equation (4) becomes x ¼ x c 0 t; t ¼ t; ¼ h ; ð6þ where K is equal to K s cos q/n. Obviously, the initial an bounary conitions associate to equation (4) must be transforme too. It is interesting to note that c 0 represents the effective mean velocity in the soil pores when / =0, i.e., c 0 = u/n, so that the origin of the new coorinate system follows the movement of the water along the hillslope. 3. Grounwater Moun [7] The evolution in time an space of a given water volume per unit with, V 0, initially concentrate in the section x = 0 of a completely ry soil can be escribe by equation (4), with the conition Z þ1 hx; ð tþx ¼ V 0 =n: ð7þ Clearly, thanks to the transformation (5), such a problem is equivalent to the classical case of a grounwater moun spreaing on a horizontal be. Following Barenblatt [1996], the similarity solution can be elegantly foun by imensional analysis. The level of the phreatic surface h epens on the governing parameters, t, x, V 0 /n, an K. Using the characteristic length in the irection orthogonal to the be, H, the length in the flow irection, L, an a characteristic timescale, T, the imensions of h an of the governing parameters can be expresse as ½hŠ ¼ H; ½tŠ ¼ T; ½Š¼L; x ½V 0 =nš ¼ HL; ½KŠ ¼ L 2 T H : ð8þ Dimensional analysis leas to the relation c = (h), where h c ¼ ðv 0 =nþ 2=3 K =3 t =3 h ¼ x ðv 0 =nþ 1=3 K 1=3 t 1=3 : ð9þ The substitution of (9) in (6) leas to the orinary ifferential equation h h þ h ð10þ 3 which, integrate twice (the first integration constant is zero by symmetry), yiels c ¼ ðhþ ¼C 1 h2 6 ; ð11þ where C 1 is equal to (3/32) 1/3 for the conition (7). [8] Returning to the initial coorinate system, the equation escribing the evolution of the phreatic surface along the hillslope of the initial water volume V 0 thus reas hx; ð tþ ¼ 3 1=3 V 2= ðktþ ð x c 0tÞ 2 1=3 6Kt ; ð12þ which escribes a parabola that moves ownhill with velocity c 0 expaning its base an reucing its peak. As the be slope, q, increases so oes c 0, while the spreaing of the moun base, which is proportional to (cos q) 1/3, becomes slower; as a consequence, the water volume being constant, the peak of the moun also ecays more slowly, i.e., proportionally to (cos q) /3. It shoul be notice, moreover, that the previous solution is not limite to small be slopes, but only by the Dupuit approximation (i.e., small curvatures of the streamlines). [9] The corresponing expression of the ischarge is Qx; ð tþ ¼ " # 3 1=3 V 2= ðktþ ðx c 2 0tÞ 1=3 6Kt nx þ 2K s sin q t : ð13þ 3t As appears from Figure 1a, uring the initial perio the water front moves uphill, leaing to a negative flux in the sections upstream x = 0. From equation (12), the position of the upper front moves in time accoring to " x u ¼ 1 2n 2K s sin q t 6 2=3 n V # 0K s cos q t 1=3 n 2 : ð14þ The uppermost point, x u,m, is touche by the front at pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t u ¼ nv 0 tan q sin q pffiffi ; 6 Ks ð15þ as shown in the example of Figure 2. [10] The combine analytical expressions of h an Q (equations (12) an (13)) escribe the so-calle loop-rating curve [e.g., Henerson, 1966], which summarizes the temporal ynamics of the grounwater wave in a given section (Figure 3). [11] The extension to the case with cylinrical symmetry is also of interest. Assuming an isotropic an homogeneous soil, the two-imensional continuity equation an the two equations of motion along a hillslope are ð hu xþ hu ð16þ 2of5

3 DALY AND PORPORATO: TECHNICAL NOTE Using cylinrical coorinates, x = r cos f an z = r sin f, equation (22) may be written ¼ 1 2 h 2 þ 1 2 þ 2 h 2 r 2 ; ð23þ where the last term on the right han sie is zero by raial symmetry. [12] Being now [V 0 /n] =HL 2, imensional analysis leas to c = (r), where h c ¼ ðv 0 =nþ 1=2 K =2 t =2 r r ¼ ðv 0 =nþ 1=4 K 1=4 t ; 1=4 ð24þ which substitute in equation (22) gives the orinary ifferential equation whose solution is r 2 r r þ r2 2 ð25þ Figure 1. Flow of a grounwater moun along a hillslope: (a) phreatic surface an (b) water flux per unit with. Parameters: n = 0.4, K s =0.5m, q =15, V 0 =1m 2 (initial time 0.2 ; final time 3.8 ; time step 0.4 ). u x ¼ K s sin q K s cos q; ð17þ u y ¼ K s cos ð18þ where u x an u y are the seepage velocity along respectively x an y. Substituting the two expressions of the velocities in equation (16), the two-imensional Boussinesq equation is obtaine as ¼ K s sin q þ K s cos : ð19þ ¼ C 2 r2 8 ; ð26þ p where C 2 is equal to 1/(2 ffiffiffi p ), as follows from equation (20). [13] Therefore, in terms of the original coorinates system, the evolution of the phreatic surface, hx; ð y; tþ ¼ V " # 1=2 0=n 1 p Kt 2 ffiffiffi ðx c 0tÞ 2 þ y 2 ; ð27þ p 8ðV 0 =nþ 1=2 ðktþ 1=2 represents a paraboloi that subtens a constant volume equal to V 0 /n, an that moves ownhill expaning its circular base an reucing its peak. 4. Traveling Waves [14] Since equation (4) is translationally invariant, it amits travelling wave solutions of type [e.g., Gratton an Minotti, 1990] h ¼ hðþ; x x ¼ x ct; ð28þ Introucing a water volume V 0 concentrate in the origin of a ry soil, such that Z þ1 Z þ1 hx; ð y; tþxy ¼ V 0 =n; ð20þ the substitution yiels x ¼ x c 0 t; z ¼ y; t ¼ t; ¼ : ð22þ 3of5 Figure 2. Space-time evolution of the upper front of a grounwater moun flowing along a hillslope. The two terms of equation (14) are also shown. Parameters as in Figure 1.

4 DALY AND PORPORATO: TECHNICAL NOTE Figure 3. Loop-rating curve showing the temporal evolution of the grounwater wave at section x =4 m. Parameters as in Figure 1. where c is a constant etermine by the bounary conitions. Such solutions are closely connecte with selfsimilarity [Barenblatt, 1996], an represent phreatic surface profiles which move along the hillslope without changing their shape. [15] Using equation (28), equation (4) may be written as ðc c 0 Þh þ Kh h x x which after a first integration leas to ð29þ ðc c 0 Þh þ Kh h x ¼ C 3: ð30þ The solution corresponing to C 3 = 0 is simply h ¼ h 0 þ c 0 c x; ð31þ K which escribes a straight line that avances with spee c with a front locate at x =(h 0 K)/(c c 0 ), where h 0 is the water level at x = 0 (Figure 4a). The flow might be imagine as prouce by a piston, whose axis is parallel to the be, pushing at constant spee a given volume of water [e.g., Gratton an Minotti, 1990]. [16] A ifferent solution is obtaine when C 3 6¼ 0. Letting C 3 =(c c 0 )h 0,itis x þ C 4 ¼ K ½ c 0 c h þ h 0 lnðh h 0 ÞŠ; ð32þ Figure 4. Traveling wave solutions efine by equation (30). (a) C 3 = 0 an (b) C 3 6¼ 0. Parameters: n = 0.4, K s = 0.5 m, q =45, h 0 =1m,C 4 chosen such that h(x =0) =3m. 4of5

5 DALY AND PORPORATO: TECHNICAL NOTE where C 4 is etermine choosing a value of h for a certain x. This case might represent the asymptotics of a flow prouce by a piston [Gratton an Minotti, 1990] pushing a layer of flui of initial thickness h 0 (Figure 4b) or, equivalently, a bounary conition h = h 0 moving with constant spee c. All the profiles for c < c 0 ten to h 0 when x!, while they are asymptotic to Kx/(c 0 c) for positive x (i.e., horizonal for c = 0). The situation is reverse for c > c 0. References Aravin, V. I., an S. N. Numerov (1965), Theory of Flui Flow in Uneformable Porous Meia, Isr. Program for Sci. Transl., Jerusalem. Aronson, D. G. (1986), The porous meium equation, in Some Problems in Nonlinear Diffusion, eite by A. Fasano an M. Primicerio, Lect. Notes Math., 1224, Barenblatt, G. I. (1996), Scaling, Self-Similarity, an Intermeiate Asimptotics, Cambrige Univ. Press, New York. Bear, J. (1972), Dynamics of Fluis in Porous Meia, Elsevier Sci., New York. Beven, K. (1981), Kinematic Subsurface Stormflow, Water Resour. Res., 17(5), Brutsaert, W. (1994), The unit response of grounwater outflow from a hillslope, Water Resour. Res., 30(10), Dagan, G. (1967), Secon orer theory of shallow free surface flow in porous meia, Q. J. Mech. Appl. Math., 20(4), Diez, J. A., R. Gratton, an J. Gratton (1992), Self-similar solution of the secon kin for a convergent viscous gravity current, Phys. Fluis, 4(6), Fan, Y., an R. L. Bras (1998), Analytical solutions to hillslope subsurface storm flow an saturation overlan flow, Water Resour. Res., 34(4), Gratton, J., an F. Minotti (1990), Self-similar viscous gravity currents: Phase-plane formalism, J. Flui Mech., 210, Henerson, F. M. (1966), Open Channel Flow, Macmillan, New York. Li, L., D. A. Barry, F. Stagnitti, J.-Y. Parlange, an D.-S. Jeng (2000a), Beach water table fluctuations ue to spring-neap ties: Moving bounary effects, Av. Water Resour., 23, Li, L., D. A. Barry, C. Cunningham, F. Stagnitti, an J.-Y. Parlange (2000b), A two imensional analytical solution of grounwater responses to tial loaing in an estuary an ocean, Av. Water Resour., 23, Lockington, D. A., J.-Y. Parlange, M. B. Parlange, an J. Selker (2000), Similarity solution of the Boussinesq equation, Av. Water Resour., 23, Mizimura, K. (2002), Drought flow from hillslope, J. Hyrol. Eng., 7(2), Peletier, L. A. (1971), Asymptotic behavior of solutions of the porous meia equation, SIAM J. Appl. Math., 21(4), Polubarinova-Kochina, P. Y. (1962), Theory of Grounwater Movement, Princeton Univ. Press, Princeton, N. J. Sanfor, W. E., J.-Y. Parlange, an T. S. Steenhuis (1993), Hillslope rainage with suen ryown: Close form solution an laboratory experiments, Water Resour. Res., 29(7), Stoker, J. J. (1957), Water Waves, Wiley-Interscience, Hoboken, N. J. Szilagyi, J., an M. B. Parlange (1998), Baseflow separation base on analytical solutions of the Boussinesq equation, J. Hyrol., 204, Telyakovskiy, A. S., G. A. Braga, an F. Furtao (2002), Approximate similarity solutions of the Boussinesq equation, Av. Water Resour., 25, Troch, P. A., F. P. De Troch, an W. Brutsaert (1993), Effective water table epth to escribe initial conitions prior to storm rainfall in humi regions, Water Resour. Res., 29(2), Troch, P., E. van Loon, an A. Hilberts (2002), Analytical solutions to hillslope kinematic wave equation for subsurface flow, Av. Water Resour., 25, E. Daly, Dipartimento i Iraulica Trasporti e Infrastrutture Civili, Politecnico i Torino, Torino, Italy. A. Porporato, Department of Civil an Environmental Engineering, Duke University, 127 Huson Hall, Durham, NC 27708, USA. (amilcare@ uke.eu.) 5of5