Journal of Hydrology 598 (2021) 126413

Available online 4 May 20210022-1694/© 2021 Elsevier B.V. All rights reserved.

Research papers

Analytical traveling-wave solutions and HYDRUS modeling of wet wedges propagating into dry soils: Barenblatt’s regime for Boussinesq’s equation generalized

A.R. Kacimov a,*, J. Simůnek b

a Department of Soils, Water and Agricultural Engineering, Sultan Qaboos University, Oman b Department of Environmental Sciences, University of California Riverside, CA, USA

A R T I C L E I N F O

This manuscript was handled by Corrado Cor-radini, Editor-in-Chief, with the assistance of Stephen Worthington, Associate Editor

Keywords: Dry dike subject to seepage from a reservoir with a water level rising at a constant rate Similarity solution Transient complex potential Translating phreatic surface Isobars-streamlines-isotachs in HYDRUS

A B S T R A C T

The classical Barenblatt solution of an initial-boundary value problem (IBVP) to the parabolic Boussinesq equation, which gives a rectangular triangle of full saturation, propagating from a reservoir into an adjacent porous bank with a vertical slope, is shown to coincide with a solution of IBVP to the elliptic Laplace equation with a phreatic surface along which both isobaricity and kinematic conditions are exactly met. For an arbitrary bank slope, a saturated wedge, which propagates (translates) into dry soil, is also explicitly found. The analytical solutions favorably compare with the results of HYDRUS-2D modeling, i.e., with the FEM solutions of the same IBVPs to the Richards equation. Applications to geotechnical engineering of dykes subject to the impact of flash floods are discussed by comparisons of phreatic lines, loci of the fronts, isobars, equipotential contours, vector fields of Darcian velocity, isotachs, and streamlines in the three models. For example, it is shown that a rapid drawup of the reservoir level induces hydraulic gradients, which may cause seepage-induced erosion of the porous medium, in particular, lessivage.

1. Introduction

The water level rise in dam reservoirs, ponds, rivers, lakes, oceans, and other surface water bodies with porous banks has hydrological, geotechnical, and geopolitical implications. Transient infiltration into these banks mitigates flash flood waves and smoothens peaks in hydrographs of watercourses (e.g., wadis in Oman and other arid countries or rivers in humid countries; Denkers, 2021), albeit one usu-ally does not differentiate “transmission losses” into vertical infiltration through the bed and banks of unregulated ephemeral streams (see, e.g., Noorduijn et al., 2014, Shanafield et al., 2021). The river stage is a boundary condition for bank filtration or infiltration from temporarily filled ponds in managed aquifer recharge (MAR) systems (Abdeldayem et al., 2020, Alam et al., 2021, Maliva, 2020). Dikes, levees, and earth dams, as well as natural soil slopes, when in contact with open water bodies with rapidly rising water levels, are prone to large-scale collapses and miniature (local) losses of structural stability of adjacent imbibed soil massifs (see, e.g., Cheng et al., 2021; Denkers, 2021; Pasetto, 2014;

Yan et al., 2019). During the run-up tide stages, the rising seawater level has a cyclostationary hydroecological impact on coastal biosystems (McLachlan and Defeo, 2017).

In this paper, we use analytical solutions and HYDRUS-2D modeling (Simůnek et al., 2016) to investigate transient seepage from an open water body with a rising water level into an adjacent slope of a porous earth body (hereafter called a reservoir and a dike, respectively). Steady and transient seepage from reservoirs through porous dykes (levees, earth filled embankments) have been studied both analytically and numerically as a part of design, construction, operation, maintenance, and forensic investigation in case of collapses of these earth works (see, e.g., Aravin and Numerov, 1953; Butera et al., 2020; Casagrande, 1937; Cedergren, 1989; Chahar, 2004; Chapuis and Aubertin, 2002; Desai, 1972; Design, 1987; Fell et al., 2005; Freeze, 1971; Hansen and Rosh-anfekr, 2012; Havenith et al., 2015; Jafari et al., 2019; Jaswal et al., 2020; Jia et al., 2009; Jiao and Post, 2019; Kacimov and Brown, 2015; Kacimov and Obnosov, 2012; Kacimov and Yakimov, 2001; Kacimov et al., 2021; Kacimov et al., 2020; Kreuter, 1921; Liggett and Liu, 1979;

* Corresponding author at: Department of Soils, Water and Agricultural Engineering, College of Agricultural and Marine Sciences, Sultan Qaboos University, Al- Khod 123, PO Box 34, Oman.

E-mail addresses: anvar@squ.edu.om (A.R. Kacimov), jsimunek@ucr.edu (J. Simůnek). URL: https://www.squ.edu.om/agriculture/Academic-Department/Soils-Water-and-Agricultural-Engineering (A.R. Kacimov).

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Journal of Hydrology

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https://doi.org/10.1016/j.jhydrol.2021.126413 Received 5 February 2021; Received in revised form 17 April 2021; Accepted 29 April 2021

Journal of Hydrology 598 (2021) 126413

2

Masciopinto et al., 1994; Loaiciga, 2020; Mishra and Singh, 2005; Navas et al., 2016; Neuman and Witherspoon, 1970; Nelson-Skornyakov, 1949; Nichiporovich, 1973; Nouridjanyan et al., 1988; Peter, 1982; Rehbinder, 1997; Soil Mechanics, 1979; Stark et al., 2017; Sun et al., 2017; Tayfur et al., 2005; The International Levee Handbook, 2013; Yang et al., 2019; Zhou et al., 2020). Mathematical models of Darcian flows through dikes should answer the following questions related to the seepage process:

• What is the shape of the moving phreatic surface, and where is the front’s locus (the tip of this surface)?

• What are the hydraulic gradients in the flow domain, especially in their “critical zones”?

• Do these gradients exceed a limit of safety against seepage erosion?

To address these questions, we use three different models: the Dupuit-Forchheimer model for purely saturated, vertically-averaged flow with a front (the Boussinesq PDE), the nonlinear potential model for purely saturated phreatic 2D seepage in a vertical plane (the Laplace PDE with a moving free boundary), and the saturated–unsaturated flow model in the same plane (Richards’ PDE without any fronts or free boundaries). We used these three different models to capture all essen-tial processes (transiency of reservoir’s level as a driver of seepage, gravity, Darcian resistance of dike’s soil and its capillarity) for the considered surface water-pore water hydrological system.

In all three models, we assume that seepage is one-phase (water- moisture motion), Darcian, and transient, taking place in a homoge-neous and isotropic soil. The soil has the saturated hydraulic conduc-tivity k and fillable porosity m. The latter is defined in Polubarinova- Kochina (1962, hereafter abbreviated PK-62) and is related to the HYDRUS soil characteristics in the van Genuchten (hereafter VG) model, in which m = θs − θr, and where θ s and θ r are the saturated and residual volumetric water contents, respectively. For transient phreatic (uncon-fined) seepage in aquifers and earthworks, the main factor, which con-trols the change of pore water storage is the motion of the phreatic surface (see, e.g., Barenblatt et al., 1984, and PK-62), which is caused by variations of the boundary conditions. We also assume that both water and porous skeleton are incompressible.

Although most studies of transient seepage through dykes focus on the regimes with a drawdown of the reservoir level (most dangerous in terms of a rapid collapse of the reservoir-facing slope of the dyke), below the drawup scenarios (for which the threat of seepage-induced gradients is subtly pernicious) are modeled. In all problems below, the horizontal free water level JF (Fig. 1a) in the reservoir rises at a constant rate r, 0 <r = const [m/day]. The engineering novelty of our work is based on the

modeling of a dyke, having an arbitrary reservoir slope, with an increasing reservoir water level.

2. Barenblatt’s solution revisited for a dike with a vertical slope

In the 1950ies, Barenblatt (see Aravin and Numerov, 1953; Bare-nblatt et al., 1984, 1990; Polubarinova-Kochina, 1962, 1977, 1996; Kochina and Kochina, 1994, 19961) obtained a number of similarity solutions to the Boussinesq equation, which mathematicians ascribed to the class of “porous medium equations” (see, e.g., Samarskii et al., 1995; Vazquez, 2007), which stems from the Polubarinova-Kochina legacy (see, e.g., Ockendon and Howison, 2002). The initial-boundary value problem (IBVP) to this nonlinear diffusion PDE for seepage studied below reads:

m∂hs

∂t= k

∂∂x0

(

hs∂hs

∂x0

)

, 0⩽x0⩽xIf (t),

hs(x0, 0) = 0, 0⩽x0 < ∞,

hs(0, t) = r t

(2.1)

where hs(x0,t) is the water table (phreatic surface) height above a hor-izontal bedrock M1L2 (Fig. 1a). A static Cartesian coordinate system x0y0 originates at the triple point O of the contact of initially (t < 0) dry dike soil, empty reservoir, and bedrock. The abscissa, xIf (t), of the tip, If, of the wetting front (at which both hs and seepage flux are zero) indicates the maximum extent of imbibition into dike’s soil, owing to the rise of the water level in the reservoir. In the parlance of mathematicians, the expanding segment OIf is a “finite support” of a “wetting signal” (a rectangular triangle Gz in Fig. 1a), which propagates inland with a “finite speed”2.

The Boussinesq equation eliminates (by the Dupuit-Forchheimer approximation) the vertical coordinate y0, i.e., posits a constant

Fig. 1. a) Barenblatt’s rectangular triangle as a right-ward propagating saturated zone; b) Barenblatt’s complex potential triangle propagating left-ward. x0 and y0 are hori-zontal and vertical coordinates, respectively, z(t) = x + i*y is a complex traveling variable, g is gravitational acceleration, r is the rate of the water level rise, k is the saturated hy-draulic conductivity, m is fillable porosity, and w(t) = φ(t)+iψ(t) is a transient complex potential, where φ = − k h is the velocity potential and ψ is the stream function.

1 In her latest, 1994 and 1996 books in Russian, Polubarinova-Kochina P. Ya. changed her name to “Kochina P. Ya.”.

2 The first analytical solution to a nonlinear heat conduction problem (the Boussinesq equation belongs to this class) was obtained by Zel’dovich and Kompaneets (1950). Finiteness of this speed was historically surprising to those who worked with nonlinear diffusion and heat conduction. For soil physicists working with wetting of absolutely dry soils, the “finite speed” was trivial. The excitement with a mathematically discovered (in the 1950ies) ”finite speeds” was strange, albeit the “localization” and “stopped fronts” in the so-called “blow-up” regimes (Samarskii et al., 1995) of reservoirs’ filling are indeed counterintuitive.

A.R. Kacimov and J. Simůnek

Journal of Hydrology 598 (2021) 126413

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piezometric head along vertical lines in Gz and purely horizontal seepage along the bedrock (Fig. 1a). Barenblatt’s exact solution to IBVP (2.1) (see, e.g., Barenblatt et al., 1990, PK-62) reads

hs(t, x0) = r t −r m/k

√xo (2.2)

i.e., the phreatic surface FCIf in Fig. 1a is a segment of a straight tilted line that caps Gz. This triangle expands into the dike as a traveling wave. Barenblatt’s flow domain Gz(t) is bounded from the right by a right-ward moving phreatic surface FCIf.

We show now that Barenblatt’s solution (2.2) is also exact in the full 2D transient potential model, in which the Darcian velocity vector V→(t, x, y) is not apriori assumed to be horizontal but, generally speaking, has both horizontal, u(t, x, y), and vertical, v(t, x, y), components. To illustrate this, we select a system of Cartesian traveling-wave co-ordinates xFy, which move vertically upward at the same rate r as the reservoir level, i.e., the traveling abscissa axis always coincides with this ascending ray JF (Fig. 1a).

We recall that Darcy’s law states V→ = − k∇h, where the piezometric head h(t, x, y) for incompressible media obeys Laplace’s equation:

∇2h(t, x, y) = 0 (2.3)

We follow PK-62 and Strack (1989) and introduce a complex trav-eling variable z(t) = x + i*y and a transient complex potential w(t,z) =

φ(t, x, y) + iψ(t, x, y), where φ = − k h is the velocity potential. This potential is harmonic-conjugated with the stream function ψ , which also obeys Laplace’s equation. Our complex potential w is a holomorphic function. The complexified Darcian velocity, V(t, z) = u + i * v, is an antiholomorphic function of z, while V = dw/dz (an overbar indicates complex conjugation) is holomorphic. The pressure head, p(t, x, y) = h-y, is another harmonic function, p = P/( ρ g), where P is the gauge pore pressure, ρ is water density, and g is gravity acceleration.

Along any moving water table, two boundary conditions must be satisfied (PK-62; Wang et al., 2011). The first, isobaricity (or zero gauge pore pressure), condition is:

ϕ+ k y = 0 (2.4)

The kinematic condition (see, e.g., Dicker and Babu, 1974; Eq. (2.10), Chapter XV in PK-62; Eq. (3) of Strack in Wang et al., 2011) is:

m∂ϕ∂t

+

(∂ϕ∂x

)2

+

(∂ϕ∂y

)2

+ k∂ϕ∂y

= 0 (2.5)

An intrinsic nonlinearity stems from the term

|V|2 =

(∂ϕ∂x

)2

+

(∂ϕ∂y

)2

= u2 +v2 on the LHS of Eq. (2.5). Both Eqs. (2.4)

and (2.5) must hold on an apriori unknown moving interface (FCIr in our case, Fig. 1a) between fully saturated (positive pressure) and dry (zero pressure) soil that is another source of nonlinearity. PK-62 discussed both the full nonlinear model with Eq. (2.5) and its various lineariza-tions. Nonlinearity makes the IBVP difficult (see Crank, 1984 for other IBVPs to elliptic PDEs with moving boundaries).

For our complex potential, we select the tip of the in-dyke creeping saturated wedge, If, as a fiducial point, i.e., a horizontally translating point in which w = 0 at any time. Using the Cauchy-Riemann conditions, we get for the Barenblatt wedge in Fig. 1a the following potential (2D) solution:

wB = − r k t + zr k m

√,

φB(t, x, y) = − r k t + xr k m

√, ψB(t, x, y) = y

r k m

√,

VB(z) = ComplexConjugate(dw/dz) =r k m

√

vB = 0, uB =r k m

√= const, xIf =

r k/ m

√t, tanβ =

k/(r m)

√

(2.6)

Solution (2.6) obeys both the governing PDE (Laplace’s equation for h, ϕ, ψ, and p) inside the “swelling” saturated triangle in Fig. 1a and all boundary conditions on the sides of this triangle, including conditions (2.4) and (2.5).

The transient complex potential domain for solution (2.6), Gw, is another rectangular triangle depicted in Fig. 1b. Gw is obtained from Gz

by a scaling factor -r k m

√and a translation –r k t to the left. Clearly, the

image of Gz in the hodograph plane degenerates into a single point because the Darcian velocity in solution (2.6) does not depend on z and t.

The phreatic surface and Darcian velocity in solutions (2.6) and (2.2) are identical. Kacimov et al. (2020) showed that Barenblatt’s solution

Fig. 2. a) Saturated tongue propagating into a dry dike with an arbitrary bank slope, b) upper part of the tongue, a traveling-wave wedge, c) Seepage from a reservoir over a wedge-shaped impermeable bedrock.

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(2.2) is close to what HYDRUS-2D models, provided dike’s soil has a relatively low VG capillarity, and the size of the positive pressure tri-angle OFIf in Fig. 1a is large enough (that is a common situation in geotechnical engineering applications if zoned earth-filled embank-ments have reservoir-facing shoulders with a rip-rap or other course porous cover and the main threat to dike’s stability occurs at the rela-tively late time of reservoir filling for a relatively high rates r).

We notice that the Boussinesq PDE (and all its linearizations, see Basha, 2013, PK-62) can model the dynamics of only “normal” phreatic lines (surfaces), for which hs(x) at any t is a function. In many applications to seepage through dikes and for unconfined aquifers in their near-shore zone, which are subject to transient drivers of the reservoir-tailwater levels and seawater tides-ebbs, this is not the case: neither hs(x) nor the inversion of hs(x) are - at a certain t - functions in a rigorous mathematical sense (see illustrative examples in Denkers, 2021; Pasetto, 2014). In other words, a part of the phreatic surface is located not above but below an “upper” positive pressure zone, which “overhangs” a dry soil zone, which in its own term “overhangs” another segment of a “normal” phreatic surface, which caps a “lower” positive pressure zone. Consequently, a wellbore (piezometer) screened across the whole depth of such a homo-geneous (!) dyke aquifer would detect a transient inflow through several horizons of a porous medium that is usually attributed to stratification (heterogeneity) in hydraulic properties. Such fancy flow topology, with dry soil zones sandwiched between “zigzagged” or “fingered” saturated zones, may seem weird to groundwater hydrologists indoctrinated by “standard models” and software packages like MODFLOW.

3. A constant-rate rise of a water level for a dike with an arbitrary slope

In this Section, we generalize Barenblatt’s solution for a vertical bank to an arbitrary bank slope.

3.1. Analytical solution for saturated seepage with a straight tilted water table propagating in soils without capillarity

In this Subsection, we study a large size dike (Fig. 2a). Its slope I1BFU, adjacent to the reservoir, is slanted at an angle ω,0 ≤ ω ≤ π/2, to the horizontal dike bed M2M1, which can be impermeable or highly permeable (these two cases will be examined later by HYDRUS). We study a transient seepage zone in the vicinity of the “climbing” tip F of the wetting front (phreatic surface) FI2M2. In other words, we ignore what happens near the segment M2M1 where perturbation of the flow by the dike bed makes seepage essentially two-dimensional (2D) and sim-ilarity conditions break. We also assume that the dike is broad enough, i. e., its right boundary, schematically depicted by a dashed line L1L2 in Fig. 2a, does not affect seepage near the slope. In Section 3.2, we will model by HYDRUS-2D dikes of a finite horizontal width.

In this Subsection, we ignore capillarity, which, we repeat, is acceptable in relatively coarse earth works such as blitz-constructed (e. g., by emergency bulldozing) embankments made of local topsoil (often sand or local alluvium) for exigency protection against flash floods. In gravitational earth dams, the dam shoulders (shells), buttressing the clay core, are also commonly coarse (Kacimov and Obnosov, 2012).

Thus, we obtain a similarity solution for an analytic element zoomed in Fig. 2b. Here, at the dike slope I1BF, we select a water-submerged point B and a system of traveling Cartesian coordinates xFy. Point B is the tip of the propagating surface and pore water at the time instance t =-r T, where T is an arbitrary time instance. At the snapshot time t = 0, depicted in Fig. 2b, the tip F is the point where air, reservoir water, and dry and saturated soils make four conterminous wedges.

The ponded water level above B is H(T) = r T. Along the phreatic surface FCM2, the pressure is atmospheric (zero). In the “similarity zone” of Fig. 2b, FCI2, this surface is a straight line at an angle β, 0 < β <π/2, with the slope M1I1BF. The β value is to be found from the solution below. The wedge I1BFI2 is a fully saturated flow domain.

Similarly to Section 2 above, we introduce the piezometric (total) head h(t, x, y) and stream function ψ(t, x, y). Both satisfy the Laplace equation inside I1BFI2. The streamlines (three are shown as bold arrowed segments in Fig. 2b) of Darcian flow from the reservoir into the soil are straight and perpendicular to the slope along the whole segment FBI1 (except, of course, two singular endpoints, F and I1). In the “simi-larity zone” (Fig. 2b), the constant total head lines (two are dashed in Fig. 2b) are parallel to the slope.

Although the whole problem is essentially transient due to the rise of the reservoir water level, the traveling wave solution is surprisingly simple. In this solution, the piezometric head (as well as the stream function and the pressure head) exactly satisfies the governing (Lap-lace’s) PDE inside the expanding ”seepage wedge” I1BFI2. The solution also meets the boundary conditions along its both rays, which bound the wedge, viz. along I1BF and I2CF (see Appendix I) :

β = arctansin2ω +

sin22ω − 8m r/kcos2ω + 8m r/k

√

4m r/k, V =

r mtanβsinω

(3.1)

where V is the magnitude of the Darcian velocity vector (constant in the whole saturated wedge). In the limit of a vertical slope, ω = π/2, Eq. (3.1) is reduced to the Barenblatt similarity solution. Fig. 3 shows functions β(ω) calculated using Eq. (3.1) for the VG loam from the HYDRUS Soil Catalogue having m = 0.352 (0.43–0.078) and different ratios of r/k.

PK-62 considered the inequality in the hydraulic gradient, Ig = V/k >1, as a criterion of seepage-induced instability of the porous skeleton, which is a key design criterion for earthen dams (De Mello, 1977). Eq. (3.1) allows one to immediately check this criterion in the similarity zone of Fig. 2b and, hence, along the whole slope M1F in Fig. 2a, because Darcian velocity along M1I1 is higher than along I1F.

Capillarity can be added to the above used analytical solutions in the same manner as in Chen and Young (2006) by imposing the condition p = -pc (a constant pressure head along I1F, a capillary fringe boundary in the Green-Ampt model).

3.2. HYDRUS FEM modeling

In this subsection, we use HYDRUS-2D and compare this FEM code’s results with the analytical solution above. There is no free surface in the HYDRUS domain, and the Richards equation is solved in the entire flow domain (Simůnek et al., 2016). This domain is finite, i.e., infinite wedges assumed in the analytical solutions have to be bounded, i.e., the domain has to be large enough so that the external boundaries do not affect the solution. We recall that semi-infinite domains, i.e., a quadrant and a half-plane, were used in the analytical solutions for seepage flows in

Fig. 3. Functions β(ω) for the VG (HYDRUS-default) loam having m = 0.352 and r/k = 0.04, 4.0, 32, and 128 (curves 1–4, respectively) calculated from Eq. (3.1).

A.R. Kacimov and J. Simůnek

Journal of Hydrology 598 (2021) 126413

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Fig. 1a and 2b, respectively. We select a HYDRUS trapezoid shown in Fig. 4. The VG soil hydraulic functions and parameters are taken from the HYDRUS Soil Catalogue. We recall that the total head h = p + Z. The notations h and p are different from the HYDRUS manual, where p stands for the pore pressure head (modulo matrix potential).

We use X and Z3 for the horizontal and vertical coordinates (fixed) in our HYDRUS simulations and corresponding figures, with the origin at point M1.

We use most of the default HYDRUS options, such as iteration criteria and time step controls. The initial (i.e., at t = 0, t is the HYDRUS simulation time, 0 < t < 10 days) pressure head p = -100 cm throughout the modeled 2D domain (i.e., we assume an initially relatively dry dike soil).

The HYDRUS domain size in Fig. 4 is indicated by the X and Z co-ordinates of points U and L1. The reservoir bank slope is 10:1. The dyke is made of the VG loam with k = 0.25 m/day, θs = 0.43, θr = 0.078, and the VG constants α = 3.6 1/m and n = 1.56. The spatial discretization of the transport domain with an element size of 0.5 m involves 23,171 nodes and 581 1D and 45,759 2D finite elements. Water starts rising in the dike reservoir from the trapezoid’s impermeable bottom in Fig. 4 with the rate of r = 8 m/day. We assume that the horizontal dike crest UL1 and the vertical tailwater slope L1L2 are impermeable, i.e., no evaporation occurs from these segments. This condition can be changed to the atmospheric one in HYDRUS. However, even in hyperarid Oman, evaporation is much smaller than the effect of the rapid drawup of the reservoir level, and the zero flux condition is thus a good approximation. Along M1FU, a time-variable boundary condition with a gradually increasing water level is imposed for the pressure head. The segment FU is assumed to be impermeable, using one of the HYDRUS special boundary options imposing no flux on the section of a boundary with a negative pressure head.

First, we assume that M1M2L2 is impermeable (as in Section 2). In Fig. 5a, isobaric contour lines are plotted for t = 10 days. Four contours are highlighted. The contour line p = 0 (θ = 0.43) is sandwiched be-tween the lines p = 4 m and p = -1 m and indicates the locus of the propagating phreatic surface. All isobars, located sufficiently far from M2, are almost parallel to the phreatic surface. The abscissa of M2 is 33.2 m. For comparisons, the Barenblatt solution (2.6) gives 23.84 m and from Eq. (3.1) we get xM2 = yF sin β/[sinω sin(β + ω)] = 23.6 m. Further propagation of the phreatic surface in the HYDRUS simulation compared to the analytical solutions is due to the capillary forces considered by the numerical model and neglected by the analytical solutions. This discrepancy will be larger for fine-textured soils than for coarse-textured soils and reflects the relative importance of gravitational and capillary forces characterized by the soil capillary length.

Fig. 5b illustrates simulated vectors of Darcian velocity at the same

instance. Far above M2 in the saturated tongue, these vectors have an almost constant magnitude (0.62 m/day) and the same direction, showing a slight downward dip. For comparisons, Eqs. (2.6) and (3.1) give 0.84 m/day and 0.85 m/day, respectively. The hydraulic gradients in all three models are much higher than 1. Moreover, HYDRUS (Fig. 5b) gives even higher values of the hydraulic gradient along the slope close to point F as one moves from below. In the vicinity of this point, HYD-RUS velocity has a blip (maximum at F) due to increased inflow in both horizontal and upward direction into the initially dry dyke and is zero along FU.

We emphasize that the similarity solution (3.1) is not valid suffi-ciently close to the bed of the dyke in Fig. 2a. However, (3.1) gives exact V for a rather extravagant geometry shown in Fig. 2c. Here, an imper-meable boundary is made of two rays: BM0 and BC. The ray BU bounds from above a “tilted quadrant”, UBC, of initially dry soil. The reservoir, the wedge M0BU, is initially empty. The water level there starts rising with a constant rate r at t = 0. As a result, a positive-pressure rectangular triangle FBIf propagates into the “tilted quadrant” in exactly the same manner as the Barenblatt triangle in Fig. 1a. Seepage is 1D along the tilted bedrock BC in Fig. 2c (see Youngs, 1974, his Fig. 3 for an analo-gous steady-state flow). Now we apply a variational theorem (Ilyinsky et al., 1998) to the Laplace-equation governed seepage with a moving phreatic surface (no capillarity). Specifically, let us “press-in” (rotate counter-clockwise) the tilted ray BC of Fig. 2c into the horizontal ray M1L2 in Fig. 2a. According to the variational theorem, this deformation reduces the Darcian velocity along M1L2 (Fig. 2a) compared to its con-stant value along BC (Fig. 2c). This reduction is significant near point M1 (B) and minor near point F. Consequently, the instantaneous discharges obey the inequality:

QM1F(t) =∫ F

M1

VM1F(t, s)ds < QBF(t) =∫ F

BVBF(t)ds = VBF(t)|BF| (3.2)

The HYDRUS simulations (Fig. 5b) qualitatively concur with the variational theorem. The inequality (3.2) is useful for coarse soils (minor capillarity) because it gives an upper bound for QM1F, which requires a numerical solution to the Laplace equation with a moving phreatic surface via an elementary calculated QBF. It is noteworthy that, for the example in Fig. 5b, the whole slope M1F is prone to seepage erosion (according to the PK-62 criterion) induced by the drawup. Hence, fine fractions of the loam can be agitated, entrained by seepage, and trans-located into the dyke interior (the phenomenon of lessivage).

When the dyke is made of the VG sand, i.e., from a much more permeable (k = 7.13 m/day) porous medium with smaller capillarity (α = 14.5 1/m and n = 2.68), then the width of the HYDRUS trapezoid has to be made larger than for loam (see Fig. 5c). This forestalls the impact of the tailwater (rightmost) boundary on the encroaching saturated tongue. At t = 10 days, HYDRUS gives XM2 = 123 m, and the velocity magnitude ranges 4.7 m/day < V < 5.1 m/day along the ponded reservoir slope M1F in the range 50 m < Z < 80 m. Assuming sand’s fillable porosity m = 0.43–0.045, the Barenblatt solution (2.6) gives XM2 = 121.72 m, V = u = 4.68 m/day, while Eq. (3.1) yields XM2 = 113.4 m, V = 5.05 m/day. As discussed above, since the extent of capillarity is significantly lower in the sand than in the loam, the numerical and analytical results’ correspondence is much higher in the sand. It is noteworthy that the Barenblatt solution for a vertical slope is a good approximation for a sufficiently steep non-vertical reservoir slope, pro-vided the rate of drawup in the reservoir is large enough.

Next, we considered the effect of the boundary condition along M1M2, i.e., along the bottom of the “saturated tongue” in Fig. 2a. Instead of a no-flow condition, we applied the HYDRUS free drainage (a unit vertical hydraulic gradient) condition along M1M2 for the same loamy trapezoid as shown in Fig. 4 and the same r. The results are shown in Fig. 5d, e. At the end of the simulation, the left flux QM1F(10) is about 48 m2/day for the case with the impermeable bottom and 53 m2/day for the permeable bottom, while the free drainage flux in this latter case is

Fig. 4. HYDRUS-2D trapezoidal flow domain.

3 Do not confuse the complex variable z(t, x, y) in Section 2 with the vertical coordinate Z in HYDRUS.

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Journal of Hydrology 598 (2021) 126413

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about 7 m2/d. Thus, only about 46 m2/d contributes to the change in storage (very similar to 48 m2/d for the impermeable bottom), which explains a relatively small difference in the front’s advancement. For the case of an impermeable bed of the dyke, the water storage added by infiltration of reservoir water is defined as:

S(t) =∫ t

0QM1F(u)du (3.3)

For the Barenblatt solution (2.6), Eq. (3.3) gives:

S(t) = r3/2k m

√t2/2 (3.4)

One can see from Fig. 5d, e that the boundary condition at M1M2 has a relatively small impact on the “traveling wave” zone near point F. For example, at t = 10 day, the abscissa of the HYDRUS tip M2 is XM2 = 33.2 m and 28.2 m for the impermeable and permeable beds of the dyke, respectively. Along the ponded slope FM1 in the range 50 m < Z < 80 m, the HYDRUS velocity magnitude ranges 0.63 m/day < V < 0.633 m/day for either boundary condition along M1L2. However, the boundary’s

Fig. 5. Pressure heads (a, c, d), the vector field of Darcian velocities (b), and streamlines (e) at t = 10 days for r = 8 m/day in a trapezoidal dyke made of the VG loam (a, b, d, e) or sand (c). The bed of the dyke is either impermeable (a–c) or permeable (d, e) with a free drainage boundary condition.

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impact is significantly larger on the velocity magnitude in the lower part of the inflow boundary.

Next, we selected a triangular dike with another, milder bank slope 1:1 (ω = π/4), r = 4 m/day, a permeable (free drainage) bed M1L2, and compared VG’s loam and sand as porous media. Fig. 6a shows the pressure head contours for a dike made of the VG sand at t = 10 days and r = 4 m/day. The contour lines between 5 m and − 1 m are plotted to capture the water table sandwiched in between. HYDRUS gives the front position XM2 = 52 m. Along the ponded slope FM1 in the range 30 m < Z < 40 m, the HYDRUS velocity magnitude ranges 5.34 m/day < V < 7.36 m/day (decreasing to the dyke crest F). Eq. (3.1) yields a pretty close value of XM2 = 51.43 m and the velocity V = 6.68 m/day.

Fig. 6b presents the same contour lines for a dike made of the VG loam at the same time instance. As one can see, the less permeable soil results in an “inverted water table” (Sophocleous, 2002). Obviously, this inverted water table indicates a positive pressure wedge spreading above dry soil (an “unstable” situation prone to vertical “fingering”). In the analytical model (Subsection 3.1), the condition of such a seepage regime is ω + β < π/2. For r/k = 32 and the VG loam, from Eq. (3.1) we obtain that if ω < 1.281, the water table will overhang a dry soil beneath the saturated wedge.

Let us return to a loamy trapezoid on an impermeable base (Fig. 5). We placed four observation nodes: the first one at the bottom of the

reservoir (X = 0, Z = 0), the second at the bottom of the dyke (22.5 m, 0), and the third (22.5, 30.2) m and fourth (22.5, 75.2) just above the second and third. Fig. 7 shows p(t) for the first three observation nodes. This figure shows that the water table and the ensued positive pressure head zone arrive at observation nodes 2 and 3 at t = 6.7 days and 9.8 days, respectively. The front does not arrive at all (in 10 days of simu-lations) at observation node 4.

Pressure tracking at specified HYDRUS points can be used for the design of screens’ length and depth of vertical (horizontal, directional) wells to be drilled (dug) in porous banks of ephemeral streams (wadis, in Oman). Indeed, local Omani farmers tap ephemeral perched aquifers adjacent to wadis and reservoirs of recharge dams. The main questions are: Where and how deep to drill (dig) a well? Shallower wells are cheaper but may not intercept the encroached (essentially “ephem-eral”4) saturated tongue. For example, a well screened to a depth of observation node 4 will be redundant (for the flash flood simulated in Fig. 5b, c).

Fig. 6. Pressure heads, p, at t = 10 days for r = 4 m/day and an impermeable bed of a triangular dyke made of the VG sand (a) and loam (b).

4 Obviously, a constant-rate water level rise in the reservoir will cease at a certain time. After that, drawdown of the reservoir water level will commence. The saturated triangle will slump and a “dry” observational point may remain dry forever.

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We simulated other reservoir slopes ω, VG soils, drawup rates r, and other boundary conditions along M1M2.

4. Concluding remarks

Three models (the parabolic 1D Boussinesq equation, the elliptic 2D Laplace equation with a moving phreatic surface, and the parabolic 2D Richards equation), solved analytically and numerically (by HYDRUS- 2D), concur in their solutions of IBVPs if the water level in a reservoir rises with a sufficiently high but constant rate and the imbibed soil is initially dry. Specifically, the shape of the saturated zone advancing into the dyke near the rising free water-soil point (tip F in Fig. 1a and 2) is a traveling wedge, where Darcian velocities, and therefore the hydraulic gradients and threats of lessivage, are pretty close in all three models.

The intricacy of Darcian flows with fronts and phreatic surfaces “overhanging” dry soil zones should precaution modelers not to be obfuscated by “standard hydrological concepts”.

There are Abbreviations

1) PK-62 = reference to Polubarinova-Kochina, P.Ya., 1962. Theory of Ground Water Movement. Princeton University Press, Princeton +second edition of the book in Russian (published in 1977)

2) VG = Van Genuchten

CRediT authorship contribution statement

A.R. Kacimov: Conceptualization, Methodology, Software, Formal analysis, Resources, Writing - review & editing, Visualization, Project administration, Funding acquisition. J. Simůnek: Conceptualization, Methodology, Software, Formal analysis, Resources, Writing - review & editing, Visualization.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported by the Sultan Qaboos University via the grant DR/RG/17. Helpful comments of three anonymous referees are appreciated.

Appendix I

In this Appendix, we derive Eq. (3.1). We consider an arbitrary rectilinear streamline BC in Fig. 2b, which is far above the dike foundation M1M2 (Fig. 2a). Close to the foundation,

seepage is 2D, and I2M2 is not a straight line. Obviously, from the rectangular triangle ABF, the hypotenuse |BF|=|AB|/sin ωπ. From the condition of a constant rate r, |AB|=r T. We now select point F in Fig. 2b as fiducial, i.e., the total head h(t, x, y) is counted from F, viz. h(0,0,0) = 0. The pressure head p = h-y. At point B, pB = r T, yB = -r T, and hB = 0. From the right triangle AFB, we have:

|BC| = |BF|tanβ =rT

sinωtanβ (A1)

From the right triangle BCN:

yC = − r T − |BC|cosω = − r T −r Ttanβ

tanω (A2)

Fig. 7. Pressure heads p(t) at three observation nodes [(0,0), (0,22.5), (22.5, 30.2)] in a loamy trapezoidal dyke (Fig. 5a, b).

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Journal of Hydrology 598 (2021) 126413

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At point C on the phreatic line, pC = 0, and hence:

hC = yC = − r T −r Ttanβ

tanω (A3)

Eqs. (A1), (A3), and Dacry’s law (we recall V→ = − k∇h = ∇φ) for the straight (but time-stretching) streamline BC yield:

|VBC| = khB − hC

|BC|= k

(sinωtanβ

+ cosω)

(A4)

From the conservation of mass (we recall that water and soil are posited to be incompressible), we have:

VC = md|BC|

dT= mr

tanβsinω (A5)

Eq. (A5) is a kinematic boundary condition for this 1D “titled” infiltration along BC. In the same form, this condition is used in the Green-Ampt model of 1D vertical infiltration (see, e.g., PK-62).

We equate the right-hand sides of Eqs. (A4) and (A5) and get an equation:

m r/ktan2β − tanβ sinωcosω − sin2ω = 0 (A6)

Eq. (A6) is a quadratic equation with respect to tan β. We retain only a positive root of this equation that, together with Eq. (A5), gives Eq. (3.1).

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