However, the preferential flow through fingering due to instability of the wetting front causes non-uniform and unstable flow, making field scale predictions based ...
Hydrological Sciences-Joumal-des Sciences Hydrologiques, 42(1) February 1997
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Simulation of stable and unstable flows in unsaturated homogeneous coarse sand M. S. BABEL Panya Consultants Co Ltd, Bangkok, Thailand
A. DAS GUPTA & R. LOOF Asian Institute of Technology, GPO Box 2754, Bangkok 10501, Thailand Abstract A numerical model developed to simulate stable and unstable flows in unsaturated porous media is described. Results of numerical studies carried out to simulate laboratory experiments with the assumption of stable flow demonstrate the occurrence of unstable flow for the initial conditions of both air dry and field capacity for unsaturated infiltration in sands. This indicates that the Richards flow equation based on moisture content and potential variables averaged over total crosssectional area may not be applicable for flow under instability-prone boundary conditions. The unstable flow due to wetting front instability is modelled using the steady-state theory proposed by Hillel & Baker (1988). Simulation results for fingered flux calculated with the theory represent the experimental data reasonably well. The pore water velocity remains constant irrespective of the incident flux as long as the flux is smaller than the hydraulic conductivity value at the water entry suction of the porous media. The effect of antecedent wetness on the unstable flow behaviour is to reduce the pore water velocity drastically from air dry to field capacity conditions.
Simulation d'écoulements stables et instables dans un sable grossier homogène et non saturé Résumé Nous avons développé un modèle numérique afin de simuler des écoulements stables ou non dans des milieux poreux non-saturés. Les résultats des études numériques entreprises pour simuler les expériences de laboratoire dans l'hypothèse d'un écoulement stable montrent que, pour l'infiltration dans des sables non saturés, un écoulement instable apparaît que le milieu soit initialement sec ou à la capacité au champ. Cela indique que l'équation de Richards qui concerne des variables de charge et d'humidité moyermées selon des sections n'est peut être pas applicable à des écoulements dont les conditions initiales favorisent l'instabilité. L'écoulement instable causé par l'instabilité du front d'humectation a été modélisé en utilisant la théorie en régime permanent proposée par Hillel & Baker en 1988. Les résultats des simulations de flux digités s'appuyant sur cette théorie reproduisent les résultats expérimentaux de façon satisfaisante. La vitesse de pore de l'eau demeure constante quel que soit le flux incident tant que ce flux reste inférieur à la conductivité hydraulique correspondant à la succion d'entrée du milieu poreux. L'effet de l'humidification antérieure sur le comportement instable de l'écoulement est une très importante réduction de la vitesse de pore entre l'état sec et la capacité au champ.
INTRODUCTION Several analytical and numerical models have been developed to predict water flow and solute transport between the land surface and the groundwater table. Most of these deterministic models are based on the Richards equation for unsaturated flow and the Fickian-based convection-dispersion equation for solute transport. In the recent past, the usefulness of these classical models has increasingly been questioned
Open for discussion until 1 June 1997
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due to their failure in predicting actual field-scale water and solute transport processes (Glass et al., 1989b; Gee etal., 1991; van Genuchten, 1991). Preferential flow phenomenon, i.e. the accelerated movement of water through isolated regions in the vadose zone, has been found responsible for the faster transport of the solute in the vadose zone (Beven & Germann, 1982; Starr et al., 1978, 1986; Richard & Steenhuis, 1988; Hallberg, 1989; Rice et al, 1991; Ghodrati & Jury, 1992; Babel et al., 1995). Preferential flow may take place either through macropores, such as decayed root channels, cracks, worm holes and other structural inhomogeneities, or by instability of the wetting front in porous media (Bouwer, 1991). Preferential flow caused by wetting front instability leading to fingered flow is termed here as "unstable flow". There are many situations occurring in nature which have been found to favour the initiation and development of an unstable wetting front in unsaturated porous media (Hill & Parlange, 1972; White et al., 1976, 1977; Diment & Watson, 1985; Samani et al., 1989; Glass et al, 1988, 1989a, 1989b; Baker & Hillel, 1990; Starr et al, 1978, 1986; Glass et al, 1988; van Ommen et al, 1989a,b; Tamai et al, 1987; Babel et al, 1995). The classical Richards flow equation, upon which most numerical models of water flow in unsaturated porous materials are based, was derived by averaging over a representative elementary volume flow properties which are continuously defined in the entire macroscopic flow region. These model developments have been based on the assumption that the one-dimensional flow process in the unsaturated zone is uniform and stable. However, the preferential flow through fingering due to instability of the wetting front causes non-uniform and unstable flow, making field scale predictions based on these equations invalid and inaccurate. There is, therefore, a need to develop numerical models which can simulate accurately the movement of water in soils susceptible to unstable flow so that a realistic prediction can be made. The present study attempts to model unstable flow occurring with a boundary condition of non-ponding infiltration in unsaturated sandy material and to compare the simulation results of water flow under stable and unstable flow conditions with those obtained from laboratory experiments for two initial conditions, namely air dry sand and sand at field capacity. The effects of incident flux and antecedent wetness on stable and unstable flow behaviour are also analysed and discussed.
THEORETICAL APPROACHES Modelling stable flow À two-dimensional water flow model has been developed based on the following pressure head form of the Richards equation to simulate stable flow (complete matrix flow) in unsaturated-saturated porous media (Bear, 1972): dT
/
a/i]
d"
. ,(dh
VI
dh
where C (= dQ/dh + Q/èSs) is the generalized specific water capacity [L"1]; 6 is the
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volumetric moisture content (L3 L3); h is the pressure head [L]; dO/dh is the slope of the soil water content-moisture tension curve [L"']; § is the porosity of the medium; 9/(|) is the index for saturation; Ss is the specific storage [L1]; t is the time [T]; W is the sink or source term [T1] (positive for sink and negative for source); x and z are the Cartesian coordinates [L] (z is taken positive downwards); and Kx(h) and Kz(h) are the hydraulic conductivities as a function of pressure head in the x and z directions respectively [LT1]. The specific storage, Ss, can be neglected in unsaturated flow (Neuman, 1973). Therefore, 9/(j> = 1 and dQ/dh = 0 in the saturated zone (h > 0). On the other hand the term containing Ss is zero in the unsaturated zone (h < 0). The initial and boundary conditions in their basic mathematical form can be written as: h(x,z,0)
= hQ(x,z)
h(x,z,t)
= hi\x,z,t)
-Kx(h)^nx-Kz(h{^
(x,z)eQ
(2a)
(x,z) e T,
(2b)
- l)n z j = q„{x,z,t)
{x,z) e T2
(2c)
where r, + F2 = T is the exterior boundary of the modelled domain Q; h0 is the initial head distribution; hx is the pressure head specified on the segment T,; and nx, nz are the normal vectors on the boundary segment T2 in the direction of x and z respectively. The details of the model development, the numerical scheme adopted and the validation of the developed model under stable (complete matrix) flow conditions are given in Babel (1993).
Soil hydraulic properties A closed-form retention model developed by van Genuchten (1980) and used in this study to describe the hydraulic properties of the sand is: 5e(A)
=
^ L ^
Se{h) = 1
=U+
(ahyTm
h
< o
(3a)
h > 0
(3b) 3
3
where 9 is the volumetric water content at pressure head h [L L/ ]; 9r is the residual water content [L3 L"3]; 9^ is the saturated water content [L3 L~3]; h is the pressure head [L]; Se is the effective saturation [dimensionless] ; « is a shape parameter which is inversely related to the air or water entry value [dimensionless]; a is a shape parameter which is inversely related to the width of the pore size distribution [L/1]; and m = (1 - lin). The parameters that appear in the van Genuchten (1980) retention model are used to derive analytical expression for the unsaturated hydraulic conductivity using
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Mualem's predictive conductivity model (Mualem, 1976). The relative conductivity function is given by (van Genuchten & Nielsen, 1985):
i-(i-s;T]
K, - S„
(4)
Modelling unstable flow Unstable flow in homogeneous coarse textured porous media under the boundary condition of unsaturated infiltration is a consequence of wetting front instability (Babel et al., 1995). The wetting front moves in the form of columns, generally called fingers, rather than as a plane. This indicates that only part of the soil matrix actively participates in the flow and transport processes. It is hypothesized that when flow becomes unstable, the flow takes place through isolated fingers and that the cross-sectional area of the individual fingers will add up to the fraction of total horizontal area necessary to transmit the flow. Further, it is assumed that the Richards equation is applicable as the governing partial differential equation to represent the flow of water in the fingered space. For steady state flows, the total fingered area and its influence on the flow velocity can be estimated by the following simple mass balance equation: f
qf = q
A
\
Af j
(5)
where qf is the flux through the fingered area (Af); q is the Darcy flux; and A is the total cross-sectional area. According to Hillel & Baker (1988), for a homogeneous coarse grained porous medium under steady state and unit hydraulic head gradient conditions, partial volume flow or unstable flow will occur if the conductivity of the medium at its water entry pressure head exceeds the flux given by equation (5). This may be expressed mathematically by the statement that unstable flow is not initiated unless: Kwe > q
(6)
where Kwe is the conductivity at water entry suction and q is the flux entering from the soil surface. The wetted fraction of the medium, F, required to conduct the incident flux can be predicted by the quotient of the two terms of equation (6): F = -f-
(7)
Combining equations (5) and (7) leads to: qf = Kwe
(8)
The average pore water velocity (VJ) through the fingered area during infiltration can be estimated by the following expression:
Simulation of stable and unstable flows
Vf = f*
53
(9)
where Qwe is the moisture content at the water entry pressure head of the medium. Once the fingered area is estimated, the steady state flux through the fingered area can be determined readily. This fingered flux can then be utilized to analyse the water flow and solute transport under unstable flow conditions. Measurement and/or prediction of water entry suction The characteristic water entry suction can be determined experimentally by measuring the limit of capillary rise from a free water surface into the porous medium in question. Baker & Hillel (1990) found that the median particle size of the porous medium is highly correlated with the water entry pressure head (r2 = 0.9855). For air dry sands, this empirical relationship is given as: hwe = 437
