Advances in Water Resources 145 (2020) 103732

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Advances in Water Resources

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Seepage to ditches and topographic depressions in saturated and

unsaturated soils

A.R. Kacimov a , ∗ , Yu.V. Obnosov b , 1 , J. Š im ů nek c

a Department of Soils, Water and Agricultural Engineering, Sultan Qaboos University, Muscat, Oman b Institute of Mathematics and Mechanics, Kazan Federal University, Kazan, Russia c Department of Environmental Sciences, University of California Riverside, CA, United States

a r t i c l e i n f o

Keywords: Analytic and HYDRUS solutions for Darcian 2-D and axisymmetric flows in saturated and unsaturated soils towards drainage ditches and topographic depressions Evaporation and seepage exfiltration from

shallow groundwater Complex potential and conformal mappings Method of images with sinks and sources for the Laplace equation and ADE Boundary value problems involving seepage faces on Earth and Mars Isobars, isotachs, constant piezometric head, and Kirchhoff potential lines

a b s t r a c t

An isobar generated by a line or point sink draining a confined semi-infinite aquifer is an analytic curve, to which a steady 2-D plane or axisymmetric Darcian flow converges. This sink may represent an excavation, ditch, or wadi on Earth, or a channel on Mars. The strength of the sink controls the form of the ditch depression: for 2-D flow, the shape of the isobar varies from a zero-depth channel to a semicircle; for axisymmetric flow, depressions as flat as a disk or as deep as a hemisphere are reconstructed. In the model of axisymmetric flow, a fictitious J.R. Philip’s point sink is mirrored by an infinite array of sinks and sources placed along a vertical line perpendicular to a horizontal water table. A topographic depression is kept at constant capillary pressure (water content, Kirchhoffpotential). None of these singularities belongs to the real flow domain, evaporating unsaturated Gardnerian soil. Saturated flow towards a triangular, empty or partially-filled ditch is tackled by conformal mappings and the solution of Riemann’s problem in a reference plane. The obtained seepage flow rate is used as a right-hand side in an ODE of a Cauchy problem, the solution of which gives the draw-up curves, i.e., the rise of the water level in an initially empty trench. HYDRUS-2D computations for flows in saturated and unsaturated soils match well the analytical solutions. The modeling results are applied to assessments of real hydrological fluxes on Earth and paleo-reconstructions of Martian hydrology-geomorphology.

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. Introduction

“I had to live in the desert before I could understand the full value of grassn a green ditch. ”Ella Maillart Analytical models of 2-D seepage towards drainage ditches and

renches, constructed by civil, geotechnical, and agricultural engineers,sed the machinery of complex variables ( Anderson, 2013 , Aravin andumerov, 1953 , Bear, 1972 , Kirkham and Powers, 1972 , Polubarinova-ochina, 1962 , 1977, hereafter abbreviated as PK-62,77, Skaggs et al.,999 , Strack, 1989 , Vedernikov, 1939 ), in particular, by tackling freeoundaries of Darcian flows, the so-called phreatic surfaces. We recall (see, e.g., Radcliffe and Š im ů nek, 2010 ) that transient, 3-

, saturated-unsaturated flows in porous media (when both water andoil are incompressible) obey the Richards equation:

𝜕𝜃 = ∇ ( 𝐾( 𝑝 )∇ ℎ ) (0)

𝜕𝑡

Abbreviations: ADE = , advective dispersion equation; BVP = , boundary value probl , Ordinary Differential Equation; OSDP = , Optimal Shape Design Problem; PK-62,7rinceton University Press, Princeton. The second edition of the book (in Russian) wa∗ Corresponding author at: Department of Soils, Water and Agricultural Engineerin

E-mail addresses: anvar@squ.edu.om (A.R. Kacimov), yobnosov@kpfu.ru (Yu.V. O1 ORCID ID:orcid.org/0000-0001-9220-7989

ttps://doi.org/10.1016/j.advwatres.2020.103732 eceived 5 May 2020; Received in revised form 21 July 2020; Accepted 11 August 2vailable online 12 August 2020 309-1708/© 2020 Elsevier Ltd. All rights reserved.

here 𝜃( t, x, y, Z ) is the volumetric moisture content, ∇ is the nablaperator (in Cartesian or cylindrical coordinates), K ( p, x, y, Z ) is theydraulic conductivity function, h ( t, x, y, Z ) = p + Z is the total head, is the pressure head, Z is a vertical coordinate, and p ( 𝜃) is a capillaryressure (water retention) function, fixed for each soil. Eq. (0) involves

he Darsy law

→𝑉 = − 𝐾( 𝑝 )∇ ℎ , where

→𝑉 is the Darcian flux vector, and the

rinciple of mass conservation. Boundary value problems (hereafter abbreviated as BVPs) are solved

or Eq. (0) by specifying initial conditions, e.g., 𝜃(0, x, y, Z ), as well asmposing physically meaningful boundary conditions (e.g., Dirichlet’s,eumann’s). Only numerical codes like HYDRUS-3D ( Š im ů nek et al.,016 ) tackle such problems for arbitrary 3D transient flows. Eq. (0) is highly nonlinear parabolic PDE. For steady flows, its LHS vanishes,nd the equation becomes elliptic. If the flow is purely saturated andhe porous medium is homogeneous, then K ( p ) = K s , where K s is the sat-

em; DF = , Dupuit-Forchheimer; LHS, RHS = , left hand side, right hand side; ODE 7 = , Polubarinova-Kochina, P.Ya., 1962. Theory of Ground Water Movement. s published in 1977, Nauka, Moscow.; VG = , van Genuchten. g, Sultan Qaboos University, Al-Khod 123, PO Box 34, Sultanate of Oman. bnosov), jsimunek@ucr.edu (J. Š im ů nek).

020

A.R. Kacimov, Yu.V. Obnosov and J. Š im ů nek Advances in Water Resources 145 (2020) 103732

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rated hydraulic conductivity, and Eq. (0) is reduced to a linear Laplacequation. For a special class of soils with an exponential K ( p ) function Gardner, 1958 ), the Kirchhoff transform makes possible a linearizationuch that Eq. (0) is reduced to a linear advective dispersion equationADE) with respect to a Kirchhoff potential. The ADE can be analyticallyolved for solitary or systematically placed singularities ( Kacimov, 2007 ,hilip, 1968 , 1971 , Pullan, 1990 , Raats, 1971 , 1972 ). More details onVPs for the Laplace equation and the ADE are given in Sections 2 - 5 ofhe MS. The total head for saturated seepage in undeformable porous media

beys the Laplace equation and, therefore, the mathematical common-lity between flows of ideal fluids ( Zhukovskii, 1948 ) and pore waterotion has been widely explored (PK-62,77, Strack,1989 ). The methodf images and the theory of BVPs ( Henrici, 1993 ) have been engaged. Inhis paper, we combine these traditional analytical and new numericalHYDRUS-2D, Š im ů nek et al., 2016 ) methods to problems of ground-ater and soil water movement from shallow confined and unconfinedquifers towards ditches and topographic depressions, common in desertandforms of arid-hyperarid climates of Oman and Mars. A practical motivation of our work stems from a daunting prob-

em of groundwater inundation caused by a rapid rise of the wa-er table in perched unconfined and shallow confined aquifers, de-ected in many urban areas of the world (see, e.g., Attard et al., 2016 ,arron et al., 2013 , Chaudhary, 2012 , Coda et al., 2019 , Howard andsrafilov, 2012 , Jha et al., 2012 , Kazemi, 2011 , Lerner, 2003 , Lerner andarris, 2008 , Medovar et al., 2018 , Naik et al., 2008 , Porse et al.,016 , Preene and Fisher, 2015 , Quan et al., 2010 , Schirmer et al.,013 , Vázquez-Suñé et al., 2005 , Vogwill, 2016 ). Surprisingly, the aridountries of Arabia – despite the hydrological mantra on water deficitnd groundwater depletion – encounter the same problem of water-ogging of urban structures by shallow groundwater as the cities inumid zones (e.g., Abu-Rizaiza et al., 1989 , Abu-Rizaiza, 1999 , Al-awas and Qamaruddin, 1998 , Al-Sefry and Ş en, 2006 , Al-Senafy, 2011 ,l-Senafy et al., 2015 , Alsharhan et al., 2020 , Bob et al., 2016 ,reibich and Thieken, 2008 ). It is noteworthy that urbanization in theouthwest of the USA caused similar negative hydrological impacts (am-lification of flash flood intensities due to reduced evapotranspiration,ee, e.g., Goudie, 2013 ). In the metropolitan area of Muscat, at the campus of Sultan Qaboos

niversity (SQU), the groundwater inundation has become evident evenithout any piezometry. In recent years, subsurface water seeped out inatural topographic depressions, man-made excavations, ditches, andrainage channels called wadis. Large wet spots or ponded areas at SQUre bio-marked by adjacent lush, wild vegetation. In the desert climatef Oman, this vegetation in some places pops up virtually as Maillart’sgrass in green ditches ” (see the epigraph). The studied SQU field isocated in a hyperarid zone of the Batinah Coast, Oman, Arabia (e.g.,lsharhan et al., 2001 , 2020 ). The investigated wadi section was about km long, with several excavated pits-trenches and natural depressions.or instance, one of the excavations has the following coordinates: Nor-hing: 619721, Easting: 2608729, 47.5 m above the level in the Sea ofman. Urbanization of the field started in the mid-1980s, before whichhe site was a desert with scattered shrubs and trees. Currently, a sig-ificant part of the campus area is paved, which reduced natural evapo-ranspiration. Waterlogging and associated lush vegetation are geotech-ically deleterious, putting foundations of the SQU buildings, roads,ubsurface cables, etc., in jeopardy, even when the salinity of the ris-ng groundwater is small. Secondary salinization of the topsoil is alsoidespread in Oman when the rise of the water table is from an aquiferf poor groundwater quality. Fieldwork was carried out in November-une 2020 at the SQU campus: auger holes, ditches, and excavationsere constructed, and groundwater monitored, to assess soil moistureozing fluxes above shallow perched and confined aquifers detected inhe study area. The motivation of this work also stems from our intention to attract

he attention of Planet Mars hydrologists, who model seepage of Mar-

ian groundwater into remotely scanned relief features of the Red Planet,nd the legacy of experts in irrigation and drainage engineering, whoduring the 20th century - assembled a good arsenal of analytical andumerical methods for investigations of the motion of pore water to-ards drains ( Bear, 1972 , Kirkham and Powers, 1972 , Skaggs et al.,999 , 2012 , Strack, 1989 , Vedernikov, 1939 , Zhukovskii, 1948 ). Mars is a hyperarid planet where the remotely analyzed landforms

craters, dunes, endorheic lakes and playas, pits, alcoves, gullies,ecurring slope lineae, fluvio-lacustrine basins, fans, shallow dustorizons, streaks, amphitheater ‐headed valleys, sapping valleys,tc.) and hydrological elements (springs, outseeps, deep fractureonduits, seepage spots), which convey liquid water, brine, andartian gases, are studied. The alleged paleo and current dynamicsf Martian pore fluids are elaborated, for example, in Abotalib andeggy (2019) , Bhardwaj et al. (2019) , Boatwright and Head (2019) ,dwards and Piqueux (2016) , Goldspiel and Squyres (2011) ,rimm et al. (2014) , Hobbs et al. (2014) , Kereszturi et al.,010 , Kochel and Piper (1986) , Luo et al. (2011) , Malin andarr (1999) Malin and Edgett (2000) , Marra et al. (2014 , 2015 ),ichalski et al. (2013) , Mukherjee et al. (2020) , and Salese et al. (2019) .For hypothetical Martian aquifers (both unconfined and artesian),

s is assessed from proxy data (see, e.g., the posited “nominal ” value inoldspiel and Squyres (2011) ; Luo et al. (2011) evaluated the hydrauliconductivity from the solution of inverse problems of the hydrologicalycle on Mars; Boatwright and Head (2019) assumed that the conductiv-ty exponentially decreases with the Martian depth). We note that evenhe Darcy law is sometimes not correctly formulated by Martian hydrol-gists (see, e.g., an erroneous Eq. (1) in Abotalib and Heggy, 2019 ).lthough purely unsaturated flows are also hypothesized ( Edwards andiqueux, 2016 , Grimm et al., 2014 ) to occur on Mars, to the best of ournowledge, Martian hydrologists (e.g., Luo et al., 2011 ) have so far usednly flow models for saturated soils. Models of Darcian flows towards drainage ditches commonly con-

eptualize the ditches as rectangles in vertical cross-sections hav-ng vertical slopes ( Afruzi et al., 2014 , Barua and Alam, 2013 ,hahar and Vadodaria, 2008, 2012 , Gureghian and Youngs, 1975 ,armah and Tiwari, 2018 , Youngs, 1975 , 1990 , 1994 ). However, nat-ral ephemeral river channels, both on Earth and Mars, as well asonstructed drainage ditches and trenches, often have non-rectangularhapes ( Grotzinger et al., 2014 , Kocurek et al., 2020 ). Excavationspits) and natural depressions are often axisymmetric and mild-slopedsee, e.g., Goldspiel and Squyres, 2011 ). Only in few studies havehreatic seepage and flows in the unsaturated zone to non-rectangulartrapezoidal and curvilinear) draining entities been analytically ex-mined (PK-77 reported solutions by Bazanov, p. 150 of her book,nd Vedernikov, pp.181-182, see also Ilyinsky and Kacimov, 1992a ,acimov, 2005 , 2006a , Kacimov and Obnosov, 2002 ). We note that al-hough Luo et al. (2011) sketched trapezoidal draining channels in theirodels of groundwater discharge into these channels, they posited theupuit-Forchheimer (hereafter abbreviated as DF) model, which actu-lly ignores the shape of channels. Marra et al. (2014 , 2015 ) conceptual-zed, and studied experimentally in sandboxes, seepage into real trape-oidal channels, allegedly occurring on Mars, but unfortunately did notefer to a plethora of theoretical studies of this type of seepage to drain-ng channels on our own planet (see, e.g., Aravin and Numerov, 1953 ,K-62,77, Vedernikov, 1939 ). This work is organized as follows. In Section 2 , we develop a sim-

le analytical solution for a 2-D (not DF!) confined flow to a curvilinearitch draining a saturated, confined aquifer. Some natural wadi channelsn our study area are arcuate, like ones on Mars ( Malin and Carr, 1999 ),ut as the first approximation, we ignore the 3-D effect. Our ditch con-our is an isobar generated by a line sink. In Section 3 , we obtain a sim-lar solution for a point sink in an axisymmetric flow towards a crater-haped isobar, the piezometric head being still a harmonic function. Inection 4 , we assemble an infinite array of J.R. Philip’s sinks-sourcesf the advective dispersion equation (ADE) for the Kirchhoff potential,

A.R. Kacimov, Yu.V. Obnosov and J. Š im ů nek Advances in Water Resources 145 (2020) 103732

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hich models an axisymmetric, steady water movement from a horizon-al water table towards a topographic depression in a partially saturatedo-called Gardner soil. In Section 5 and Appendix I, 2-D seepage towards triangular empty or partially-filled ditch is studied by the machineryf holomorphic functions. All analytical solutions of Sections 2 - 5 areested against HYDRUS-2D numerical simulations. In Section 6 , we out-ine applications to arid zone hydrology, and in particular, to Martianydrology.

. A line-sink-generated seepage face ditch draining a confined

quifer

Unlike Hobbs et al. (2014) , who could not identify confining layers,n the form of bedrocks and caprocks, of hydrostratigraphic units in theartian regolith massif, we - when making excavations on our SQU site

indeed crashed a low-permeable caliche layer at a depth of 1.5-2 m.ypical values of K s for low-permeable (cemented) layers (also calledalcrete, petrocalcic horizons, see, e.g., Duniway et al., 2010 ), formed byhe precipitation of carbonates or gypsum due to intensive evaporationn semi-arid, arid and hyper-arid regions, are given in the literature:0 − 4 to 10 − 6 cm/s ( Al-Yaqout and Townsend, 2004 ), 0.0002 to 0.008/day ( Heilweil and Watt, 2011 ), 2.8 ∗ 10 − 7 m/s ( Mujica and Bea, 2020 ),mong others. In most reported cases, these strata hydrologically cap theadose zone rather than a perched or confined aquifer in our situation.We could not excavate deeper to the next low-permeable horizon,

nd therefore, in Fig. 1 a, the aquifer is not confined from below. Weote that genesis of the caliche in the SQU soil profile is similar tohat Michalski et al. (2013) argued about the impact of the upwellingartian groundwater on the sequence of cemented sediments of theed Planet. Specifically, a periodic “upwelling ” (rise) of groundwatern SQU shallow perched unconfined aquifers, and intensive evaporation

ig. 1. A vertical cross-section of an empty ditch B 0 MC 0 a); the right half of the empf the flow in the domain G z , generated by a fictive line sink at the origin O( x,y ) b);

rom the water table has – allegedly – formed the caliche and “self-onfined ” this aquifer by a caprock ( Fig. 1 a). When we “punctured ” thealiche by the JCB scoop, groundwater gushed into the excavation andlled it to the level of ℎ ∗

𝐴 ( Fig. 1 a) within 2 hours. We collected soil-

ediment samples, measured K s using standard techniques (see, e.g., Al-hukaily et al., 2020b ), and ascribed the obtained value to the texturallass of HYDRUS-2D sands. Fig. 1 a shows a vertical cross-section of an empty ditch B 0 MC 0 ,

hich drains a dry soil layer above a horizontal aquifuge and a confined,omogeneous and isotropic aquifer (having at saturation a hydrauliconductivity K s ) beneath an originally impermeable layer (caprock)BCA . Point A in Fig. 1 a is at infinity on the Riemann sphere. If thexcavation depth H

∗ , counted from ABCA (point M is the deepest pointf the ditch), is below the aquifuge, groundwater discharges into theitch BMC (symmetric and having the width 2 L ) with the total flowate 𝑄

∗ 𝑑 (m

2 /s). We assume that flow takes place only below the aquifuge.

oldspiel and Squyres (2011) call this layer “aquiclude ”, that is – inhe vernacular of groundwater hydrology ( Strack, 1989 ) – a “Martianisnomer, ” because Goldspiel and Squyres (2011) model impermeableather than leaky confining layers. At a point A h ( Fig. 1 a), located a cer-ain distance 𝐿 ∗

𝐴 ( 𝐿 ∗ 𝐴 >> 𝐿 ) | from O, a remote piezometer shows an el-

vation ℎ ∗ 𝐴 . For simplicity, we ignore groundwater exfiltration upward

hrough the caliche in Fig. 1 a, i.e., leaky layer (aquitard) flow scenarios,s considered by Kacimov and Obnosov (2008 , 2019 ). We introduce Cartesian coordinates O x ∗ y ∗ and the complex physical

oordinate z ∗ = x ∗ + i y ∗ ( Fig. 1 b). In this section, we study the case of anmpty ditch, i.e., the pressure head, p ∗ , along BMC is zero. The empti-ess of the ditch can be ensured by either a high Manning slope in theirection perpendicular to the plane of Fig. 1 a, so that all seeped waterapidly flows away as surface water (see, e.g., Al-Shukaili et al., 2020a ),

ty ditch constructed such that the bottom of the ditch, BMC, is a seepage face the complex potential domain G w for the whole ditch c).

A.R. Kacimov, Yu.V. Obnosov and J. Š im ů nek Advances in Water Resources 145 (2020) 103732

Fig. 2. a) Half-contours of sink-generated ditches for Q d = 0.2, 1, and 5.86 (curves 1, 2, and 3, respectively). b) The depth H of a sink-generated ditch and its area A (curves 1 and 2, respectively) as functions of the seepage flow rate Q d .

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r by intensive evaporation, provided H

∗ is relatively small and climates arid enough. We note that surface water flow perpendicular to thelane of Fig. 1 , i.e., along the ditch, can be easily tracked on Earth. In oureldwork, we detected what Malin and Edgett (2000) called a Martianseepage-fed surface runoff, ” slow Hortonian motion downslope a ditch,ith velocities of few mm/s. On Mars, the juxtaposition of groundwaternd surface water flows, reconstructed from the contemporary Martianandforms, is purely hypothetical ( Malin and Carr, 1999 ). As we have already pointed out in Section 1 , a saturated 2-D flow

rom a homogeneous aquifer into a ditch ( Fig. 1 a) obeys Laplace’s equa-ion:

ℎ ∗ ( 𝑥 ∗ , 𝑦 ∗ ) = 0 , (1)

here ⃖⃖⃗𝑉 ∗ ( 𝑢 ∗ , 𝑣 ∗ ) = − 𝐾 𝑠 ∇ ℎ ∗ is the Darcian velocity vector, u ∗ and v ∗ arets horizontal and vertical components, h ∗ is the total piezometric head,

∗ = p ∗ - y ∗ , where the pore water pressure is P ∗ = 𝜌d gp ∗ , g is grav-ty acceleration (9.8 m/s 2 on Earth and3.7 m/s 2 on Mars), and 𝜌d isore fluid’s (groundwater, brine, soil moisture) density. We emphasizehat Eq. (1) and all the results below do not assume a quasi-horizontalore water motion, as it is postulated, for instance, in the DF model ofoatwright and Head (2019) and Luo et al. (2011) . We introduce the complex potential w

∗ = 𝜑 ∗ + i 𝜓 ∗ , 𝜑 ∗ = − K s h ∗ . The

tream function 𝜓 ∗ and a velocity potential 𝜑 ∗ are conjugate harmonicunctions. Due to symmetry, we consider only the right half of the phys-cal domain G z ( Fig. 1 b). The following dimensionless quantities arentroduced:

𝐻, 𝐿 𝐴 , 𝑥, 𝑦, 𝑧, 𝑟, ℎ, ℎ 𝐴 , 𝑝 ) = ( 𝐻

∗ , 𝐿 ∗ 𝐴 , 𝑥 ∗ , 𝑦 ∗ , 𝑧 ∗ , 𝑟 ∗ , ℎ ∗ , ℎ ∗

𝐴 , 𝑝 ∗ )∕ 𝐿, ( 𝑤, 𝑄 𝑑 )

( 𝑤

∗ , 𝑄

∗ 𝑑 )∕( 𝐾 𝑠 𝐿 ) , ( 𝑉 , 𝑢, 𝑣 ) = ( 𝑉 ∗ , 𝑢 ∗ , 𝑣 ∗ )∕ 𝐾 𝑠

here r (0 ≤ r < ∞) is the modulus of point z = x + i y in the aquifer Fig. 1 a). The complex potential domain G w is depicted in Fig. 1 c. The piezo-

etric head is assumed to be zero at points B and C . The image of theMC isobar in G is a specific curve whose shape is unknown. Far awayrom the ditch, the piezometric head is constant ( h A ) along a circle of radius L A in G z ( Fig.1 a). The image of this circle is a segment A h A v in w ( Fig. 1 c). Now we use the trick implemented by Zhukovskii (1948) in fluid

echanics of ideal fluids (see recent applications in Belotserkovsky andifanov, 1992 ), also employed in problems of subsurface mechan-cs of steady and transient Darcian flows ( Fujii and Kacimov, 1998 ,acimov, 1992 , 2000a , 2007 , 2009 , Kacimov et al., 2009 , Kacimov andbnosov, 2002 , PK-62,77, Strack, 1989 , 2020 , Vedernikov, 1939 ). Welace a line sink of intensity Q = 2 Q at the origin of coordinates. The

d

omplex potential of this sink is:

𝑤 = −i 𝑄 𝑑 2 −

𝑄 𝑑 𝜋ln 𝑧, 𝑧 = 𝑟 exp [ i 𝜈] , 𝑟 =

√𝑥 2 + 𝑦 2 ,

ℎ =

𝑄 𝑑 𝜋ln 𝑧, 𝜓 = −

𝑄 𝑑 2 −

𝑄 𝑑 𝜋

v , |𝑉 | =

𝑄 𝑑 𝜋𝑟 ,

(2)

here 𝜈 is the argument of z . It is evident from Eq. (2) that the stream-ines of this flow are rays converging to point O in Fig. 1 b. The isotachsnd equipotential lines are semicircles centered at the same point. Now we restrict r in Eq. (2) to only a positive-pressure part of the

alf-plane in Fig.1 a, i.e., we use Eq. (2) for the reconstruction of BMCy plotting the curve, which is determined by the following explicit for-ula:

𝑑 ln √𝑥 2 + 𝑦 2 = 𝜋𝑦, 0 ≤ 𝑥 ≤ 1 , − 𝐻 ≤ 𝑦 ≤ 0 . (3)

The depth H of the ditch is found from the solution of the equation

𝑑 Log |𝑦 | = 𝜋𝑦, (4)

nd is equal to

= − 𝑄 𝑑 𝑊 (− 𝜋∕ 𝑄 𝑑 )∕ 𝜋, (5)

here W stands for the Lambert W-function (implemented as Product-

og in Wolfram’s ( 1991 ) Mathematica ) . Fig. 2 a shows the MC contours for sink-generated ditches ( Fig. 1 b)

lotted for Q d = 0.2, 1, and 5.86 (curves 1, 2, and 3, respectively). Curve in Fig. 2 b shows H ( Q d ) . From Eq. (3) it is obvious that for small Q d ,he ditch degenerates into a horizontal segment, while in another limit,lim

𝑑 →∞𝐻 = 1 , BMC becomes a semicircle. The analytical solution (2)-(5)

s similar to those obtained by Bazanov (see PK-62, 77) and Kacimov andbnosov (2002) for empty ditches reconstructed by the specification ofoundaries corresponding to BMC in the hodograph or other auxiliarylanes (e.g., the plane of the Zhukovskii holomorphic function, whichs defined as w–i z, see PK-62,77). We also recall the analytical andumerical solutions of Dagan (1964) and Gjerde and Tyvand (1992) ,ho studied potential transient 2-D (not DF) flows towards horizontalrains (modeled by linear sinks and empty circles, correspondingly). Curve 2 in Fig. 2 b shows A ( Q d ) . The cross-sectional area, A , of BMC

in Fig. 1 a) is evaluated by the NIntegrate routine of Mathematica as:

= −2 ∫1

0 𝑦 ( 𝑥 ) d 𝑥 , (6)

Fig. 3 shows the results of modeling with HYDRUS-2D Š im ů nek et al.,016 ). We selected Q d = 0.2/ 𝜋 in the analytical solution (3). With theelp of Eqs. (3) and ( (5) we generated 10 points on MC , imported themrom Mathematica into HYDRUS, and made a spline curve in dimensionaluantities used by HYDRUS. Also, HYDRUS uses z for the vertical coor-inate. Fig. 3 a shows the HYDRUS modeling domain bounded by an

A.R. Kacimov, Yu.V. Obnosov and J. Š im ů nek Advances in Water Resources 145 (2020) 103732

Fig. 3. The HYDRUS flow domain for a half-ditch generated by a line sink a), isotachs b), and streamlines c).

Fig. 4. An axial cross-section of an axisymmetric crater drain- ing a confined aquifer.

i

t

M

s

c

S

w

v

i

t

s

i

w

3

c

i

O

2v

n

ℎ

w

𝜌

e

e

F(

ℎ

s

i

h

h

h(

c

s

𝜌

p

c

w

sobaric curve MC , for which we selected L = 100 cm. Soil is loam fromhe HYDRUS soil catalog. The horizontal and vertical segments CA h andA v in Fig. 3 are no-flow lines. We selected 𝐿 ∗

𝐴 = 500 cm, i.e., we as-

umed a quarter-circle A h A v with a radius of 500 cm to be a line with aonstant piezometric head (we recall that HYDRUS, unlike PK-62,77 andtrack, 1989 , uses h as a notation for the pressure head), along whiche set ℎ ∗

𝐴 = 32 . 2 cm. This value was found at r ∗ = 500 cm for a selected

alue of 𝑄

∗ 𝑑 from the logarithmic variation of h ∗ ( x ∗ ) along CA , count-

ng from the fiducial point C where ℎ ∗ 𝐶 = 0 , see Eq. (2) . Fig. 3 b shows

he isotachs (also constant h ∗ and 𝜑 ∗ contours), and Fig. 3 c shows thetreamlines. The origin of the spatial coordinates is retained in Fig. 3 tondicate the position of the generating sink. HYDRUS results match veryell the analytical solution.

. A point-sink-generated dimple draining a confined aquifer

In this section, we use the same procedure as in Section 2 for thease of axisymmetric flow towards a depression BMC shown in Fig. 4n an axial cross-section. We select a system of cylindrical coordinates z ∗ 𝜌∗ 𝜅, where z ∗ is a vertical coordinate (the same now as in HYDRUS-D), 𝜌∗ =

√𝑥 ∗2 + 𝑦 ∗2 , 𝑟 ∗ =

√𝜌∗2 + 𝑧 ∗2 and the angular coordinate 𝜅

anishes from the analysis due to axial symmetry. At the origin O , weow place a point sink of intensity 𝑄

∗ 𝑑 (m

3 /s). The piezometric head caused by a point sink is (PK-62,77):

∗ =

𝑄

∗ 𝑑

2 𝜋𝐾

( 1 𝑅

−

1 𝑟 ∗

), (7)

𝑠

here R is the radius of the depression that coincides with the r ∗ (and∗ ) coordinate of points B and C in Fig. 4 . Eq. (7) sets up the total headqual to zero in these reference points. The depression of a depth H

∗ ismpty and, therefore, the pressure head 𝑝 ∗

𝐵𝑀𝐶 = 0 along this curve in

ig. 4 (a surface in 3-D). At a remote point A h , located at a distance 𝑅

∗ 𝐴

𝑅

∗ 𝐴 >> 𝑅 ) from point O ( Fig. 4 ), we have:

∗ 𝐴 =

𝑄

∗ 𝑑

2 𝜋𝐾 𝑠 𝑅

. (8)

Point A h is a surrogate infinity of the infinite point A on the Riemannphere. The head of Eq. (8) is illustrated in a remote piezometer sketchedn Fig. 4 . We recall (PK-62,77) that for flow to a 3-D sink, the piezometricead at infinity is finite, unlike for a 2-D sink ( Section 2 ), for which theead at infinity (point A) in Fig. 1 b logarithmically blows up to infinity.We introduce dimensionless quantities: ( H, z , 𝜌, r, h, p,

A ,R A ) =( 𝐻

∗ , 𝑧 ∗ , 𝜌∗ , 𝑟 ∗ , ℎ ∗ , 𝑝 ∗ , ℎ ∗ 𝐴 , 𝑅

∗ 𝐴 ) , 𝑄 𝑑 = 𝑄

∗ 𝑑 ∕(2 𝜋𝐾 𝑠 𝑅

2 ) , ( 𝑉 , 𝑢, 𝑣 ) =

𝑉 ∗ , 𝑢 ∗ 𝜌, 𝑣 ∗ 𝑍 )∕ 𝐾 𝑠 , where ( u,v ) are now the radial and vertical velocity

omponents. From Eq. (7) , a zero-pressure isobar (seepage face) BMC is recon-

tructed, similarly to Eq. (3) , by an equation:

=

√

1 (1 − 𝑧 ∕ 𝑄 𝑑 ) 2

− 𝑧 2 , 0 ≤ 𝜌 ≤ 1 , − 𝐻 ≤ 𝑧 ≤ 0 . (9)

We note that some Martian hydrologists use the term “aquifer sap-ing face ” and “crater wall ” ( Goldspiel and Squyres, 2011 ) for what isalled a “seepage face ” in terrestrial civil engineering ( Strack, 1989 ). Eq. (9) defines a quartic line. The depth H is found from Eq. (9) ,

hich at 𝜌 = 0 is reduced to a quadratic equation having the following

A.R. Kacimov, Yu.V. Obnosov and J. Š im ů nek Advances in Water Resources 145 (2020) 103732

Fig. 5. a) Half-contours of a sink-generated axisymmetric topographic depression draining a confined aquifer for Q d = 0.2, 1, and 2 (curves 1, 2, and 3, respectively). b) Depth and volume (curves 1 and 2, respectively) of an axisymmetric sink-generated depression in a confined aquifer.

Fig. 6. HYDRUS isobars corresponding to Q d = 0.2 a), isotachs b), and streamlines c).

p

𝐻

B

𝑉

s

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c

A

r

E

H

h

ℎ

D

s

t

p

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t

4

a

u

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𝑘

w(

1

a

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a

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d

g

hysically meaningful solution:

=

√

𝑄

2 𝑑 + 4 𝑄 𝑑 − 𝑄 𝑑

2 . (10)

We used Eqs. (9 - 10 ) and calculated the volume of the depressionMC in Fig. 4 as a body of revolution:

= 𝜋 ∫0

− 𝐻 𝜌2 ( 𝑧 ) d 𝑧 =

𝜋𝑄 𝑑

6

(𝑄 𝑑

2 + 6 𝑄 𝑑 + 6 − (4 + 𝑄 𝑑 ) √𝑄 𝑑 (4 + 𝑄 𝑑 )

),

(11)

Fig. 5 a shows the isobars MC for Q d = 0.2, 1, 2 (curves 1-3, corre-pondingly). In Fig. 5 b, the curves (10) and (11) are plotted. Obviously,or lim

𝑄 𝑑 →0 𝐻 = 1 and lim

𝑄 𝐷 →∞𝑉 = 2 𝜋∕3 , BMC becomes a hemisphere in this

imit, while for small Q d , we get a disk. The results of HYDRUS-2D simulations (in dimensional quantities)

orresponding to the analytical case of Q d = 0.2 are presented in Fig. 6 .s in Section 2 Fig. 3 ), we considered a loamy soil. We selected theadius R of the HYDRUS axisymmetric depression to be 100 cm. Usingqs. (9) and ( (10) , we generated ten discrete points on MC and made aYDRUS spline (seepage face). The radius 𝑅

∗ 𝐴 of a constant piezometric

ead quarter-circle A h A v was 500 cm. According to Eq. (8) the total head

∗ 𝐴 = 𝑝 ∗

𝐴ℎ = 20 cm at point A h and, therefore, along A h A v , i.e., the HY-

RUS pressure head is 𝑝 ∗ 𝐴𝑉

= 520 cm at point A v . Fig. 6 a presents steady-tate pressure head isolines. Fig. 6 b illustrated isotachs and Fig. 6 c plotshe streamlines. Fig. 6 bc perfectly fit the analytical solution. In the next section, we focus on an opposite case of steady flow in a

urely unsaturated Gardner soil, albeit the same trick will be used, viz.e place a fictitious sink “in the air ” and use real isobars to constructopographic craters (dimples).

. An isobaric depression reconstructed from Philip’s point-sink

rray and HYDRUS-2D simulations

Analytical solutions of Sections 2 and 3 ignored flow in the unsat-rated zone and the capillary fringe, which are important in the casef small-scale drainage entities (e.g., Abit et al., 2008 , Silliman et al.,002 ). In this section, we incorporate into an analytical quasilinearodel all three physical factors, which control seepage: the gravitationalorce (on Earth or Mars), Darcian resistance of the soil, and capillarity. We use a constitutive relation for the phase conductivity:

𝑢𝑛 = 𝐾 𝑠 exp [𝛼𝑝 𝑝

∗ ], (12)

here k un ( p ∗ ) for p ∗ ≤ 0 is an unsaturated hydraulic conductivity, 𝛼p

const > 0) is the sorptive number (1/m) ( Gardner, 1958 , see also Raats,973). J.R. Philip (1968) pioneered utilizing Eq. (12) for modeling 2Dnd 3D unsaturated flows (see, e.g., Pullan, 1990 , Raats et al., 2002 ).he soil constant 𝛼p is related to the VG-HYDRUS parameters 𝛼 and via well-known empiric relations ( Acharya et al., 2012 ); we usedq. (10) from Ghezzehei et al. (2007) . The evaporation arises from a steady upward flow from a horizontal

shallow) water table W 1 W 0 W 2 to a hot and dry soil surface BMC ( Fig. 7 )here the pressure head is constant ( 𝑝 ∗ = − 𝑝 ∗ 𝑠 ,𝑝

∗ 𝑠 > 0 ). This surface has

topographic depression with the deepest point M at a depth H

∗ > 0ounted from the soil surface, which is horizontal and flat, far awayrom the topographic trough in Fig. 7 , i.e., the water table is at a givenepth b ∗ above a flat soil surface (near remote points B and C ). BMCn Fig. 7 can also be related to the “sapping valleys ” on Mars (see, e.g.,alese et al., 2019 ), albeit there is a difference in scales: the Martianepth of the water table ( 𝑍

∗ 𝑠 − 𝑍

∗ 1 ) is 4-5 km, while at SQU, the shallow

roundwater table is only several tens of cm deep.

A.R. Kacimov, Yu.V. Obnosov and J. Š im ů nek Advances in Water Resources 145 (2020) 103732

Fig. 7. An axial cross-section of evaporative flow in an unsaturated soil from a horizontal water table W 1 W 0 W 2 to a topographically de- pressed isobar BMC .

o

d

s

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o

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t

(

i

w

i

a

s

t

l

s

t

t

i

e

𝑝

t

i

o

fi

t

𝜃

f

o

s

t

B

m

c

Ω

i

o

s

n

t

(

𝑄

r

Δ

w

a

O

o

w

t

i

Ω

The depression in Fig. 7 is axisymmetric, like in Section 3 . The axisf symmetry coincides with the OZ ∗ axis of a cylindrical system of coor-inates OZ ∗ 𝜌∗ (introduced similarly as in Section 3 above). To be con-istent with Philip, we orient OZ ∗ downward. Thus, the unsaturated soiln 3-D is sandwiched between the W 1 W 0 W 2 plane and the soil surfacebtained by the revolution of BMC with respect to the axis OZ ∗ in Fig. 7 .here are no physical singularities representing drains, subsurface emit-ers, or plant roots in this soil layer. The contour BMC in Fig. 7 is obtained in the following manner

Philip, 1989 , Obnosov and Kacimov, 2018 ). A fictitious point sink ofntensity 𝑄

∗ 𝑝 (m

3 /s) is placed at the origin of coordinates, such that theater table is beneath this sink at a depth of 𝑍

∗ 1 > 0 . A fictitious hor-

zontal isobaric plane F 1 F 0 F 2 is placed at the level 𝑍

∗ = − 𝑍

∗ 2 ,𝑍

∗ 2 > 0 ,

s illustrated in Fig. 7 . At this plane, the pressure head 𝑝 ∗ = − 𝑝 ∗ 0 ispecified. Points B and C in Fig. 7 of an “intermediate isobar ” be-ween two sandwiching isobaric planes F 1 F 0 F 2 and W 1 W 0 W 2 are at theevel of 𝑍

∗ 𝑠 , which can be either positive or negative. Fictitious point

ources and sinks of intensity 𝑄

∗ 𝑝 mirror each other with respect to

he planes F 1 F 0 F 2 and W 1 W 0 W 2 . Only the first fictitious source underhe water table, placed on the OZ ∗ axis at a depth of 2 𝑍

∗ 1 , is shown

n Fig. 7 .The array of an infinite number of such sinks-sources gen-rates a family of isobars, some of which are real (two dashed lines

∗ = − 𝑝 ∗ 2 , 𝑝 ∗ = − 𝑝 ∗ 1 ,𝑝

∗ 2 < 𝑝

∗ 1 < 𝑝

∗ 𝑠 are sketched in Fig. 7 ), and others are fic-

itious (two dotted lines 𝑝 ∗ = − 𝑃 ∗ 2 , 𝑝 ∗ = − 𝑃 ∗ 1 ,𝑝

∗ 𝑠 < 𝑃

∗ 2 < 𝑃

∗ 1 are sketched

n Fig. 7 ). Mathematically, instead of BMC in Fig. 7 , we can select an-ther soil surface, for example, any of the two dashed isobars. Practically, we proceed in the following manner. One goes to the

eld with a theta-probe and collects the moisture content, 𝜃s , from theopsoil of a real depression, having certain geometrical sizes b ∗ , H

∗ . If

s (and hence 𝑝 ∗ 𝑠 ) is almost constant in the depression and on a flat sur-

ace away from the depression, then the position ( 𝑍

∗ 1 , 𝑍

∗ 2 ) of the origin

f coordinates (sink’s locus) and the strength 𝑄

∗ 𝑝 of Zhukovskii’s sinks-

ources array should be mathematically adjusted in such a manner thathe physical (field) depression becomes close to a mathematical isobarMC in Fig. 7 . In what follows, we illustrate this algorithm.

e

Routinely (see Philip, 1968 , Raats, 1970 , Obnosov and Kaci-ov, 2018 ), we introduce the Kirchhoff potential, Ω∗ , which is oftenalled the matric flux potential:

∗ ( 𝜌∗ , 𝑍

∗ ) = ∫𝑝 ∗

−∞𝑘 𝑢𝑛 ( 𝑢 ) d 𝑢 = 𝑘 𝑢𝑛

[𝑝 ∗ ( 𝜌∗ , 𝑍

∗ ) ]∕ 𝛼𝑃 , Ω∗

∞ = 𝐾 𝑠 ∕ 𝛼𝑃 , (13)

.e., Ω∗ ∞ is the Kirchhoff potential along W 1 W 0 W 2 . Obviously, the lines

f a constant Kirchhoff potential are also isobars (the values of corre-ponding pressure heads are found directly from Eq. (13) ) and isowet-ess curves (the corresponding water content is found from Eq. (13) andhe soil-water retention function). We introduce dimensionless quantities: ( 𝑧, 𝑏, 𝐻, 𝑧 𝑠 , 𝑧 1 , 𝑧 2 , 𝜌, 𝑟, ℎ, 𝑝 )=

𝑍

∗ , 𝑏 ∗ , 𝐻

∗ , 𝑍

∗ 𝑠 , 𝑍

∗ 1 , 𝑍

∗ 2 , 𝜌

∗ , 𝑟 ∗ , ℎ ∗ , 𝑝 ∗ ) × 𝛼∕2 , Ω = Ω∗ / Ω∞, 𝑄 𝑝 =

∗ 𝑝 𝛼∕ Ω∞, ( 𝑉 , 𝑢, 𝑣 ) = ( 𝑉 ∗ , 𝑢 ∗ 𝑝 , 𝑣

∗ 𝑍 )∕ 𝐾 𝑠 , for which the Richards Eq. (0) is

educed to a linear ADE:

Ω∗ = 𝛼𝑃 𝜕 Ω∗

𝜕 𝑍

∗ , 𝑢 ∗ 𝜌 = −

𝜕 Ω∗

𝜕 𝜌∗ , 𝑣 ∗

𝑍 = 𝛼𝑃 Ω∗ −

𝜕 Ω∗

𝜕 𝑍

∗ , (14)

here Δ is the Laplacian operator in ( 𝜌,Z ), and 𝑢 ∗ 𝜌 and 𝑣 ∗ 𝑍 are the radial

nd vertical velocity components. Then, we adapt Philip’s ( 1989 ) andbnosov and Kacimov (2018) solutions to the case of flow in Fig. 7 andbtain a series:

Ω=

Ω0 [exp

[2( 𝑧 1 + 𝑧 2 )

]− exp

[2( 𝑧 + 𝑧 2 )

]]+ exp

[2( 𝑧 + 𝑧 2 )

]−1

exp [2( 𝑧 1 + 𝑧 2 )

]−1

−

𝑄 𝑃 exp 𝑧 8 𝜋

×[ exp (− 𝑟 ) 𝑟

−

exp (− 𝑟 1 ) 𝑟 1

−

exp (− 𝑟 2 ) 𝑟 2

+

exp (− 𝑟 3 ) 𝑟 3

+

exp (− 𝑟 4 ) 𝑟 4

] − … ,

𝑟 =

√𝜌2 + 𝑧 2 , 𝑟 1 =

√

𝜌2 + ( 𝑧 − 2 𝑧 1 ) 2 , 𝑟 2 =

√

𝜌2 + ( 𝑧 + 2 𝑧 2 ) 2 ,

𝑟 3 =

√

𝜌2 + ( 𝑧 + 2 𝑧 1 + 2 𝑧 2 ) 2 , 𝑟 4 =

√

𝜌2 + ( 𝑧 − 2 𝑧 1 − 2 𝑧 2 ) 2 , … (15)

here the constant Ω0 = exp [ − 𝛼p z 2 ],0 ≤ Ω0 ≤ 1 is the Kirchhoff po-ential along F 1 F 0 F 2 . At points B and C , which are located on a physicalsobar, whose remote «wings» are above the sink in Fig. 7 , we have= Ωs = exp [ − 𝛼p z 2 ] for Ω0 ≤ Ωs ≤ 1. The first term on the RHS of Eq. (15) determines 1-D unsaturated

vaporative flow from W 1 W 0 W 2 to F 1 F 0 F 2 in the layer z 1 < z < − z 2 .

A.R. Kacimov, Yu.V. Obnosov and J. Š im ů nek Advances in Water Resources 145 (2020) 103732

Fig. 8. Analytically computed isolines of Kirchhoff’s potential for ( z 1 , z 2 , Ω0 , Q P ) = (1, 1, 0.2, 1).

T

a

t

O

i

fi

m

i

a

c

t

m

n

t

t

t

e

F

f

u

z

F

w

1

m

a

𝑝

d

a

D

H

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a

a

T

o

c

s

a

i

w

o

s

R

F

m

F

t

A

fi

s

p

s

t

(

b

i

t

m

a

6

L

i

e

u

b

i

fl

i

i

i

g

i

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t

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a

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D

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i

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s

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h

r

he terms in the square brackets of Eq. (15) are responsible for sinksnd sources, and consequently, for mirrored (imaged) with respect tohe plains z = z 1 and z = − z 2 . We emphasize that Philip (1989) andbnosov and Kacimov (2018) studied flow for one real source represent-ng a subsurface irrigation emitter, while all in Fig. 7 and Eq. (15) arectitious. A trick similar to ours was used by Zhukovskii (1948) in his fa-ous formula for aerodynamics of ideal fluids. He combined mathemat-cal singularities of a characteristic holomorphic function (two dipolesnd vortex), the latter placed inside an airfoil (e.g., an impermeableylinder). Zhukovskii also ignored a fictitious flow inside an airfoil (inhe vicinity of the vortex) and only considered an exterior of a mathe-atical separatrix-streamline (cylinder). We confine real Darcian flowot by streamlines (for the sake of brevity, we have not even introducedhe Stokes stream function, see Obnosov and Kacimov, 2018 ), but bywo isobars ( BMC and W 1 W 0 W 2 in Fig. 7 ). The results of computations based on Eq. (15) , where the series was

runcated to 5 terms, are plotted by the ContourPlot routine of Math-matica in Fig. 8 for ( z 1 , z 2 , Ω0 , Q P ) = (1, 1, 0.2, 1). As an example, inig. 8 , we selected an isobar BM as the real depressed isobaric soil sur-ace maintained at the Kirchhoff potential Ωs = 0.26. For this isobar, wesed the FindRoot routine of Mathematica and evaluated z M

= 0.389, B = z s = − 0.177, i.e., the dimensionless depth of the depression inig. 8 is H = 0.566. HYDRUS-2D simulations are presented in Fig. 9 . We used silt loam

ith the HYDRUS triad of the VG soil hydraulic parameters ( 𝛼, n, K s ) = (2/m, 1.41, 1.08 m/day). We followed Ghezzehei et al. (2007) andade Gardners’s soil having 𝛼P = 1.3 𝛼 n , i.e., 𝛼P = 3.67 1/m. In Fig. 9 a, HYDRUS axial section is shown for the case of the pressure head

∗ 𝑀𝐶

= −0 . 82 m . The flow domain, which corresponds to the analyticalepression, has the HYDRUS coordinates z M

= − 0.13 m, z C = 0.23 m,nd z W 2 = − 0.55 m. The analytical isobar MC is converted to the HY-RUS spline curve similarly as in Sections 2 and 3 . Fig. 9 b and c show theYDRUS isobars and streamlines. As is evident from Fig. 9 c, the ditchunnels up the flux of moisture from the water table that also evaporatest the almost flat part of the soil surface (near point C ). In other words, simple flow tube is realized in Fig. 9 c. In Fig. 9 d, we assumed a wetter soil surface with 𝑝 ∗

𝑀𝐶 = −0 . 63 m .

he origin of coordinates O is again, like in Figs. 8 and 9 a, at the locusf the analytic imaginary sink. The geometry of the flow domain has

hanged as compared to Fig. 9 a. The corresponding analytical depres-ion MC has now the HYDRUS coordinates z M

= − 0.1 m, z C = 0.176 m,nd z W 2 = − 0.55 m. The isobars are shown in Fig. 9 d. They seem un-nteresting, like ones in Fig. 9 b. Fig. 9 e presents HYDRUS streamlines,hich are more interesting. Indeed, the flow topology is different fromne in Fig. 9 c. Specifically, like in Kacimov and Youngs (2005) , whotudied a seemingly trivial flow domain having a nontrivial two-sheetiemann surface in the hodograph plane (see also Anderson, 2013 ). Inig. 9 e, we see a separatrix S s S w S d , which divides the unsaturated do-ain into three subdomains. In the first one (counted from the left inig. 9 e), moisture ascends from segment W 0 S w of the water table tohe closest (deepest) section MS d of the ditch surface (like in Fig. 9 c).t the rest of the water table, segment S w W 2 , moisture descends (in-ltrates) from S s C on the soil surface (third subdomain). Point S w is atagnation point. Stagnation points and separatrix streamlines also ap-eared in Raats (1977) for steady flows to an array of the parallel lineinks in unsaturated Gardner soils. In HYDRUS, the numerical value ofhe velocity vector at this point ( x = 0.35 m) attains a sharp minimum1.3 × 10 − 5 m/day). The second, and most interesting, subdomain in Fig. 9 e is bounded

y the separatrix (streamline) and a curve segment S d H p S s . Along thissobar (we recall that 𝑝 ∗

𝑀𝐶 = −0 . 63 m ), moisture infiltrates into the soil

hrough H p S s and exfiltrates back into the atmosphere through S d H p ,aking a U-turn. Therefore, the whole flow domain in Fig. 9 e represents complex flow tube similar to one in the potential flow shown in Fig.2 of PK-77. It is noteworthy that in potential flows governed by theaplace equation, this type of separatrix (complex flow tubes) emergesn situations when the boundary of the flow domain has three or morequipotential components. Fig. 9 c and e illustrate that in ADE-governednsaturated flow, a complex topology can appear in a domain boundedy two lines (surfaces) of a constant Kirchhoff potential. A hinge point H p in Fig. 9 e is similar to ones in the classical Toth-

an (see, e.g., Kirkham, 1947 , Toth, 2009 ) topology of purely saturatedows controlled by gravity, Darcian resistance, and topography of thesobaric ground surfaces (postulated by Toth), along which groundwaternfiltrates-exfiltrates (but without taking into account unsaturated flown the vadose zone). At the point H p ,the Darcian velocity vector is tan-ential to the soil slope. We rephrase: if the velocity vector is decouplednto the components normal and tangential to the local soil surface, thent H p , the normal component is zero. Along S d H p , this normal compo-ent is oriented from the soil to the air, whereas along H p S s , the normalomponent points into the soil. To summarize: in the analysis of flow in an unsaturated soil repre-

ented in Fig. 9 d and e, we amended the Tothian (2009) physical fac-ors by capillarity of the soil. We then got a topological pattern shownupside-down) at the forefront of the Freeze and Cherry (1979) and in normal way in Figs. 2.25 – 2.26 of Radcliffe and Š im ů nek (2010) .

. A triangular ditch in a confined aquifer

In this section, we return to steady, and capillarity-free flow in satu-ated soil. We consider 2-D potential flow in a vertical plane ( Fig. 10 a)nd use the notations of Section 2 , i.e., the complex physical coordinates again z ∗ = x ∗ + i y ∗ . A caprock of a confined aquifer is now breached by a triangular

sosceles ditch B 0 MC 0 , having a bank slope 𝜋𝜔 , 0 ≤ 𝜔 ≤ 1/2. The flowomain, G z , is now the half-plane y ∗ < 0 with a triangular indent BMC .itch’s depth is H

∗ = L ∗ tan 𝜋𝜔 . Unlike in Section 2 , we allow water to accumulate in the ditch up the

orizon B 1 C 1 at the level 𝐻

∗ 𝑤 . The piezometric head at infinity (point A

n Fig. 10 ab) is infinite but, as in Section 2 , we replace this infinity by asurrogate infinity, ” viz. an equipotential line A v A h (shown as a dashedemicircle and a segment in Fig. 10 a and b, respectively). Practically, in remote piezometer located distance 𝐿 ∗

𝐴 from the ditch axis, the total

ead, ℎ ∗ 𝐴 , counted from points B and C , is measured. The overall flow

ate into the ditch is 𝑄

∗ . The domain G w of the complex potential w is

𝑡

A.R. Kacimov, Yu.V. Obnosov and J. Š im ů nek Advances in Water Resources 145 (2020) 103732

Fig. 9. HYDRUS simulations of purely unsaturated flow between a water table and an analytically found isobaric depression: a) finite element discretization of the flow domain and the boundary conditions, b) and d) pressure head contours, c) and e) streamlines. b) and c) flow to a relatively dry soil surface, d) and e) evaporation-infiltration at a relatively wet soil surface.

s

⎧⎪⎪⎨⎪⎪⎩(

p

A

o

t

B

t

w

h

K

fl

t

s

t

𝑄

d

g

K

o

r

a

t

u

5

t

a

c

H

v

i

p

a

s

d

a

d d

ketched in Fig. 10 b. The boundary conditions for w are

𝜙∗ = − 𝐾 𝑦 ∗ along 𝐵 𝐵 1 and 𝐶 𝐶 1 , 𝜙∗ = 𝐾𝐻

∗ 𝑤 along 𝐵 1 𝑀 and 𝑀 𝐶 1 ,

𝜓 ∗ = 𝑄

∗ 𝑡 ∕2 , along 𝐴𝐵,

𝜓 ∗ = − 𝑄

∗ 𝑡 ∕2 , along 𝐴𝐶.

(16)

To the dimensionless quantities introduced in Section 2 , we add

𝐻 𝑤 ′𝐿 𝐴 , ℎ 𝐴 ) = ( 𝐻

∗ 𝑤 , 𝐿

∗ 𝐴 , ℎ ∗ 𝐴 )∕ 𝐿, 𝑄 𝑡 = 𝑄

∗ 𝑡 ∕( 𝐾 𝑠 𝐿 ) .

We map the tetragon G z onto a ha(a.lf-plane 𝜂 ≥ 0 of a referencelane 𝜁 = 𝜉 + i 𝜂 ( Fig. 10 c) with the correspondence of points M → 0, →∞, B → 1, C → − 1, B 1 → 𝛽, C 1 → − 𝛽 . Appendix I presents the detailsf our solution. An alternative method for the analytical solution (seehe taxonomy of these methods in Aravin and Numerov, 1953 , see alsoear, 1972 ) involves a conformal mapping of G z onto a mirror-image ofhe pentagon in the hodograph plane. Fig. 10 d sketches such a half-planeith two semi-infinite cuts for the case of an empty ditch. Fig. 11 shows the function Q t ( H ) (curve 1) computed by Eq. (A9) for

A = 3 and L A = 5. Similarly to Al-Shukaili et al. ( 2020b ) andacimov (1985) , we introduce the Morel-Seytoux dimensional seepageow rate 𝜇 = 𝑄

∗ 𝑡 ∕( 𝐾 𝑠

√𝐴

∗ ) , 𝐴

∗ = 𝐻

∗ 𝐿 ∗ . Curve 2 in Fig. 11 plots the function 𝜇( H ). The minimum of the func-

ion is 𝜇 ≈ 6.085 and H ≈ 1.177. This minimum can be interpreted as aolution to the following optimal shape design problem, OSDP (similaro ones solved in Kacimov, 2005 , 2006b ):

OSDP 1. Determine an empty ditch, into which a minimal quantity

∗ 𝑡 of water seeps, provided the ditch area A

∗ and remote piezometricata ( ℎ ∗

𝐴 and 𝐿 ∗

𝐴 ) are fixed.

Curve 2 in Fig. 11 provides evidence that there is a unique andlobal solution to OSDP1. The minimum is “mild ” (similar to ones inacimov, 1985 ), i.e., triangular ditches having slopes close to the bestne will not deviate much from the best (minimal) seepage exfiltrationates. Problems similar to OSDP1 are common in civil engineering when pit (e.g., for a building foundation) is excavated, and minimum exfil-ration is desired, or pit’s drainage measures are planned. For comparisons, in Fig. 12 , we present the results of HYDRUS sim-

lations for the tetrad of parameters ( H

∗ , L ∗ , 𝐿 ∗ 𝐴 ,ℎ ∗ 𝐴 ) = (50 cm, 100 cm,

00 cm, 300 cm). Fig. 12 a shows the isobars. Fig. 12 b illustrates the iso-achs. In the analytical solution, the Darcian velocity approaches infinityt points M and C . At the same time, it attains a minimal value, whichorresponds to the tip of the cut in the hodograph domain ( Fig. 10 d).YDRUS shows the same. In Fig. 12 c, we plot the distribution of theelocities along the ditch side MC , where a minimum | V | = 43.1 cm/days attained at s d = 66.7 cm, where s d is the arc coordinate counted fromoint M in Fig. 10 a. Curves like the one in Fig. 12 c are useful for thessessment of erosional stability of slopes of ditches, since heaving anduffusion are determined by exit values and directions of hydraulic gra-ient vectors ⃗𝑖 . For example, PK-62,77 considers the condition |⃗𝑖 | > 1 as criterion of instability, according to which Fig. 12 c illustrates that theitch side MC in Fig. 12 is absolutely unstable at any s .

A.R. Kacimov, Yu.V. Obnosov and J. Š im ů nek Advances in Water Resources 145 (2020) 103732

Fig. 10. A vertical cross-section of seepage flow towards a triangular ditch in a confined aquifer a), a complex potential domain b), a reference plane c), and a mirror-image in the hodograph plane d).

Fig. 11. Dimensionless seepage flow rates Q t (curve 1) and 𝜇(curve 2) as func- tions of H for an empty ditch, h A = 3, and L A = 5.

t

c

v

l(

t

s

F

Q

a

S

c

(

f

a

t

s

t

fi

F

i

(

s

e

c

w

a

w

i

f

E

2

1

t

i

F

l

f

s

d

i

a

t

The HYDRUS-computed Morel-Seytoux factor 𝜇 is equal to 6.97 forhe trench in Fig. 12 , while the analytical solution gives 𝜇 = 7.08. Weomputed other triangular ditches in HYDRUS-2D, by using a discreteariation of the slope angle. We evaluated 𝑄

∗ 𝑡 from the steady-state

imit of the cumulative flux into the ditch from a “feeding semicircle ” Fig. 10 a, dashed curve). Then we converted this dimensional 𝑄

∗ 𝑡 into

he dimensionless 𝜇, which fits well the analytical curve 2 in Fig. 11 . Collation of solutions for seepage into a triangular ditch and

ink-generated ditch in Section 2 is done in the following manner.or the case presented in Fig. 11 , we use Eq. (2) and evaluate d = 𝜋h A /Ln L A = 3 𝜋/Ln5 = 5.86. The corresponding isobar is showns curve 3 in Fig. 2 . Using Eq. (6) , we evaluated the area and the Morel-eytoux factor 𝜇 = 5.94. We surmise that this is a global optimum in thelass of arbitrary ditch shapes. Next, we considered a general case of a partially-filled ditch

Fig. 10 a). In Fig. 13 , curves 1-3 show the graphs of the functions Q t ( H w )or three ditch slopes: 𝜋/6, 𝜋/4, and 𝜋/3, and the same h = 3 and L = 5

A A

s in Fig. 11 . In the regimes of partial filling, we can solve OSDPs similaro OSDP1. Regional groundwater flow can be easily taken into account,imilarly to Ilyinsky and Kacimov ( 1992b ). The analytical solution tohe steady problem can be readily extended to a transient regime of ex-ltration into a gradually-filling ditch. We assume that the value ℎ ∗

𝐴 in

ig. 10 a is high enough such that when we make a ditch, seepage intot does not reduce much ℎ ∗

𝐴 , and the soil in Fig. 10 a remains saturated

an aquifer remains confined). Let us assume, similarly as in HYDRUS, that during the transient

eepage phase, the porous skeleton and water are incompressible. Ifvaporation from the ditch is ignored, then from the principle of massonservation, the following ODE follows:

d 𝐴

∗ 𝑠

d 𝑡 ∗ = 𝑄

∗ 𝑡

[𝐻

∗ ( 𝑡 ) ], (17)

here t ∗ is dimensional time and 𝐴

∗ 𝑠 ( 𝑡 ) is the area of the swelling tri-

ngle B 1 C 1 M in Fig. 10 a (for comparisons see Al-Shukaili et al., 2020b ,here emptying of triangular trenches has been studied). Obviously, thenitial condition in Eq. (17) is 𝐴

∗ 𝑠 (0) = 0 , i.e., seepage-filling commences

rom an empty ditch stage. We put 𝐴

∗ 𝑠 ( 𝑡 ) = ( 𝐻

∗ − 𝐻

∗ 𝑤 ( 𝑡 ) )

2 cotan 𝜋𝜔 intoq. (17) and get the following Cauchy problem:

cotan 𝜋𝜔 (𝐻

∗ 𝑤 ( 𝑡 ) − 𝐻

∗ ) d 𝐻

∗ 𝑤 ( 𝑡 )

d 𝑡 ∗ = 𝑄

∗ 𝑡

[𝐻

∗ 𝑤 ( 𝑡 )

], 𝐻

∗ 𝑤 (0) = 0 , 0 < 𝑡 < ∞.

(18)

We used the NDSolve routine of Mathematica to solve the nonlinear-st order ODE (18) numerically, with the RHS taken from Eq. (A9) . Forhis purpose, we interpolated 51 point values of 𝑄

∗ 𝑡 [ 𝐻

∗ 𝑤 ] (see curve 1)

n the steady-state problem. The results of computations are shown inig. 13 in dimensionless quantities, to which we added a dimension-ess time t = t ∗ K s / L. In Fig. 14 , we plotted the functions H w ( t ) obtainedrom the solution of Eq. (18) for h A = 3 and L A = 5, and the same ditchlopes 𝜔 = 1/6, 1/4, and 1/3, (curves 1-3, correspondingly). As is evi-ent from these draw-up curves, the ditches are filled out in finite timentervals, t f = (0.115, 0.175, 0.24), respectively. At t > t f , seepage into ditch continues due to an artesian pressure in an aquifer, and exfil-rating water has to be removed (e.g., pumped out) if we want to keep

A.R. Kacimov, Yu.V. Obnosov and J. Š im ů nek Advances in Water Resources 145 (2020) 103732

Fig. 12. HYDRUS simulations for a triangular empty ditch having H

∗ = 50 cm, L ∗ = 100 cm, 𝐿 ∗ 𝐴 = 500 cm, and ℎ ∗

𝐴 = 300 cm: isobars a), isotachs b), and Darcian velocities

along the ditch side MC c).

Fig. 13. The dimensionless seepage rate Q t into a triangular partially filled ditch as a function of H w for three slopes ( 𝜔 = 1/6, 1/4, and 1/3), h A = 3, and L A = 5.

p

T

(

T

Fig. 14. Dimensionless draw-up curves H w (1, 2, and 3) as functions of dimen- sionless time t for different values of 𝜔 (1/6, 1/4, and 1.3,respectively).

(

u

fi

s

oints C and B in Fig. 10 a at zero pressure heads (i.e., not ponded).hat is equivalent to the condition that the horizon BC remains staticsee in Kacimov, 1991 , a similar flow regime in a horizontal plane).he type of curves in Fig. 14 can be used for the determination of K

s

similarly to Al-Shukaili et al., 2020b and Kacimov, 2000b ) that is alsoseful for deep structure constructors when pumping tests in deep con-ned aquifers are not feasible. If an aquifer is unconfined, then for auddenly emptied ditch (or an auger hole, Kacimov 2000,b), t f = ∞

A.R. Kacimov, Yu.V. Obnosov and J. Š im ů nek Advances in Water Resources 145 (2020) 103732

Fig. 15. A draw-up curve computed by HYDRUS for 𝜔 = 1/6, H

∗ = 50 cm, and L ∗ = 100 cm.

o

(

s

B

o

o

d

t

t

o

a

r

T

s

L

T

1

W

a

F

w

6

s

L

I1

fi

Q

a

a

t

h

B

S

l

s

(

o

a

(

t

fi

m

7

A

e

(

t

u

o

o

(

l

t

h

c

e

fl

l

a

l

m

v

f

fi

A

i

c

T

b

s

i

D

i

t

A

g

D

o

e

t

S

t

A

h

𝑧

A transient analytical solution can be compared with a numericalne, which involves a new reservoir boundary condition of HYDRUS Š im ů nek et al., 2018 ). This new HYDRUS option allows for the con-ideration of variable storage of a furrow channel, or wellbore (e.g.,ristow et al., 2020 , Sasidharan et al. 2018 , 2019 , 2020 ), the contoursf which are subject to the condition of a constant piezometric headn the submerged part and a seepage face on the empty part of theraining entity. HYDRUS then considers flow into or out of the reservoirhrough its interface with the subsurface transport domain to depend onhe prevailing conditions in the transport domain, such as the positionf the groundwater table or piezometric heads, and external fluxes, suchs pumping, or injection, or evaporation, or precipitation. The furroweservoir ( Š im ů nek et al., 2018 ) can be adapted for our triangular ditch.he problem considered in Fig. 12 was rerun under transient conditions,tarting with an empty ditch with dimensional sizes of H

∗ = 50 cm and

∗ = 100 cm, corresponding to the case illustrated by curve 1 in Fig. 14 .he HYDRUS draw-up curve 𝐻

∗ 𝑤 ( 𝑡

∗ ) is shown in Fig. 15 . There is a perfect match between the results in Fig. 15 and curve

in Fig. 14 from the analytical solution in dimensionless variables.e also compared the transient fluxes into the half-ditch, i.e., thenalytical Q t /2 from Fig. 11 (curve 1) and one computed by HYDRUS.or example, at t ∗ = 0.1day, the HYDRUS flux is about 6075 cm

2 /day,hereas the analytical flux 𝑄

∗ 𝑡 ∕2 is 6063 cm

2 /day.

. Comparison of terrestrial and martian evaporating spots

As an example, let us consider depression 1 ( Q d = 0.2) in Fig. 5 . As-ume R

∗ = 100 cm, H

∗ = 35 cm, and a loamy soil having K s = 25 cm/day.et the water evaporate from the disk 𝜋R

∗ 2 with a constant intensity e .n order to keep the crater empty, this value has to be e = 0.2 × 25/ 𝜋 ≈.6 cm/day, which is - according to the Penman-Monteith equation andeld measurements of evaporation - a too high value even in the Emptyuarters of Oman. If one wants to reconstruct the desertification of Mars or intercept

scending moisture fluxes from the deep water table, such as, for ex-mple, in the process of Mars colonization, then the topography likehe one in Fig. 9 should be taken into account similarly as what weave on Earth. Hummocky terrains or even earth dams ( Kacimov andrown, 2015 ) generate complex 2D and 3D Toth-type Darcian fluxes.pecifically, flat landforms with a homogeneous vadose zone and a shal-ow water table are easy to examine for water storage and fluxes: ituffices to measure the depth of the water table and the pressure headmoisture content) of the topsoil to infer what is the value and directionf the flux. If the vadose zone is heterogeneous, the flux magnitudes canttain nontrivial extrema, even of a simplest type of a two-layered soil

Kacimov et al., 2019 ). Fig. 9 e illustrates that routine measurements ofhe boundary condition on soil surfaces with depressions are not suf-cient for the determination of the directions and magnitudes of theotion of water governed by the Richards equation.

. Concluding remarks

The mainstream groundwater hydrology focuses on MAR (Managedquifer Recharge), a method of replenishing aquifers, especially in aridnvironments. We are putting forward the opposite concept of MADManaged Aquifer Discharge) of shallow aquifers, which have to be in-elligently drained to mitigate the damage inflicted by groundwater in-ndation. The analysis in this paper will be useful in planned drainagef water-logged built-up areas and further studies in dryland ecohydrol-gy ( Camporeale et al., 2019 ). Maillart’s desert ditches of Central Asiasimilarly as ones in humid Holland), can be viewed as a technique ofowering a shallow water table or piezometric surface. The shape ofhese ditches can be mathematically approximated by isobars made byydrodynamic sinks, comparable with Zhukovskii’s method of a smartomposition of singularities in aerodynamics. The Darcian flows gov-rned by the Laplace equation in saturated media or ADE for steadyows in unsaturated media can be viewed as created by intelligentlyocated singularities. The resulting isobars model the outlets in seep-ge to empty curvilinear ditches and depressions. Seepage to triangu-ar ditches, both empty and partially filled, is modeled by conformalappings and the BVP method. The transient stage of filling, i.e., theariable storage in ditches, is examined by solving a Cauchy problemor a nonlinear ODE with type curves showing how rapidly the ditch islled up. HYDRUS-2D simulations well agree with analytical modeling.nalytical solutions and HYDRUS reconstruct jointly isobars, isohumes,sotachs, and streamlines - the theoretical arsenal of soil physicists andivil engineers working with Earth and Mars soil and water systems.he proper design of ditches, pits, and other artificial excavations toe operated in shallow water table environments, as well as an under-tanding of subsurface hydrology of natural wadis and depressions, canncorporate the findings of this paper.

eclaration of Competing Interest

The authors declare that they have no known competing financialnterests or personal relationships that could have appeared to influencehe work reported in this paper.

cknowledgments

This work was supported bySultan Qaboos University via therant “Rise of Water-table and Its Mitigation at SQU Campus ”, grantR\RG\17 and it was carried out as part of the development programf the Scientific and Educational Mathematical Center of the Volga Fed-ral District, agreement No. 075-02-2020-1478. Helpful comments bywo anonymous referees are highly appreciated.

upplementary materials

Supplementary material associated with this article can be found, inhe online version, at doi: 10.1016/j.advwatres.2020.103732 .

ppendix

The Schwarz-Christoffel integral maps conformally the referencealf-plane on G z :

( 𝜁 ) + i 𝐻 = − 𝑐 1 (1 − i 𝐻) ∫𝜁

0 (1 − 𝜏2 ) − 𝜔 𝜏2 𝜔 d 𝜏

= −(1 − i 𝐻) B 𝜁2 (1∕2 + 𝜔, 1 − 𝜔 ) B(1∕2 + 𝜔, 1 − 𝜔 )

, (A1)

A.R. Kacimov, Yu.V. Obnosov and J. Š im ů nek Advances in Water Resources 145 (2020) 103732

w

t

s

z

a

𝑦

w

y

𝑊

f⎧⎪⎨⎪⎩

𝑊⎧⎪⎨⎪⎩

6

c

i

𝜉

p

(

𝑊

wi

b

𝑊

P∈

(

𝜑

𝑥

p

𝐿

W

−

𝑄

R

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

B

B

B

B

B

B

B

B

C

C

C

C

here B and B 𝜁2 are the complete and incomplete beta-functions, respec-ively ( Abramowitz and Stegun, 1969 , formula 6.6.1). The positive con-tant 𝑐 1 = 2∕ B

(1∕2 + 𝜔, 1 − 𝜔 ) in (A1) is determined from the condition (1) = − 1. Note that z ( 𝜁) obeys the symmetry condition 𝑧 (− ̄𝜁) = − 𝑧 ( 𝜁 )s it should be. From Eq. (A1) , along two seepage faces ( BB 1 and CC 1 ), we have:

( 𝜉) = 𝐻

( B 𝜉2 (1∕2 + 𝜔, 1 − 𝜔 ) B(1∕2 + 𝜔, 1 − 𝜔 )

− 1 )

, 𝛽 ≤ |𝜉| ≤ 1 , (A2)

here − 𝛽 is the affix of point C 1 . We determine 𝛽 from the relation ( − 𝛽) = − H w , that gives:

B 𝛽2 (1∕2 + 𝜔, 1 − 𝜔 ) B(1∕2 + 𝜔, 1 − 𝜔 )

=

𝐻 − 𝐻 𝑤

𝐻

. (A3)

Next, we introduce a holomorphic function:

( 𝜁 ) = 𝑤 ( 𝜁 ) −

i 𝑄 𝑡

𝜋arcsin 𝜁. (A4)

The function (A4), in accordance with (16) and properties of theunction arcsin, satisfies the conditions:

Re 𝑊 ( 𝜉) = − 𝑦 ( 𝜉) , 𝜉 ∈ (−1 , − 𝛽) ∪ ( 𝛽, 1) , Re 𝑊 ( 𝜉) = 𝐻 𝑤 , 𝜉 ∈ (− 𝛽, 𝛽) , Im 𝑊 ( 𝜉) = 0 , 𝜉 ∈ (−∞, −1) ∪ (1 , ∞) .

Here y ( 𝜉) is determined in Eq. (A2) . Then the function 𝑊 0 ( 𝜁 ) = ( 𝜁 )∕

√1 − 𝜁2 obeys the following conditions along the 𝜉-axis:

Re 𝑊 0 ( 𝜉) = − 𝑦 ( 𝜉)∕ √1 − 𝜉2 , 𝜉 ∈ (−1 , − 𝛽) ∪ ( 𝛽, 1) ,

Re 𝑊 0 ( 𝜉) = 𝐻 𝑤 ∕ √1 − 𝜉2 , 𝜉 ∈ (− 𝛽, 𝛽) ,

Re 𝑊 0 ( 𝜉) = 0 , 𝜉 ∈ (−∞, −1) ∪ (1 , ∞) . (A5)

Eqs. (A5) comprise a Riemann (Schwartz) BVP ( Henrici, 1993 , PK-2,77). The function W ( 𝜁) is finite at two transition points 𝜉 = ± 1,ontinuous at points ± 𝛽, and has a logarithmic singularity at infin-ty. Hence, the function W 0 ( 𝜁) has integrable singularities at points= ± 1 and vanishes at infinity. Therefore, the unique solution of theroblem (A5) gives the Schwartz operator for the upper half-plane Henrici, 1993 , PK-62,77). Thus, after simple algebra, we get

( 𝜁 ) =

2 𝜁√1 − 𝜁2

𝜋i

(

𝐻 𝑤 ∫𝛽

0

d 𝜏√1 − 𝜏2 ( 𝜏2 − 𝜁2 )

− ∫1

𝛽

𝑦 ( 𝜏) d 𝜏√1 − 𝜏2 ( 𝜏2 − 𝜁2 )

)

,

(A6)

here, for the sake of definiteness, the branch of the radical √1 − 𝜁2

s fixed positive for 𝜁 = 𝜉 ∈ ( − 1, 1) in the upper half-plane. Thisranch and, therefore, the function (A6) satisfy the symmetry condition (− 𝜁) = 𝑊 ( 𝜁 ) . The integrals in (A6) are evaluated by the Sokhotski-lemely formula: the first for 𝜁 → 𝜉 ∈ (0, 𝛽), and the second for 𝜁 → 𝜉

( 𝛽, 1) ( Henrici, 1993 ). We illustrate the computations for the case of an empty ditch

H w = H ). We use the piezometric data at point A h ( Fig.10 a), i.e., A = − h A First, from Eq. (A1) along the ray CA we get:

( 𝜉) = 1 + 2 √1 + 𝐻

2 ∫− 𝜉

1 ( 𝜏2 − 1) − 𝜔 𝜏2 𝜔 d 𝜏∕B(1∕2 + 𝜔, 1 − 𝜔 ) ,

− ∞ ≤ 𝜉 < −1 . (A7)

From Eq. (A5) we determine the affix − 𝛾 of point A h in the referencelane ( Fig. 10 c):

𝐴 = 𝑥 (− 𝛾) . (A8)

We use the FindRoot routine of Mathematica to solve Eq. (A8) for 𝛾.e put the found root of Eq. (A8) into Eq. (A6) , and at 𝜙𝐴 = 𝜙(− 𝛾) = 𝑄 𝑡 log ( 𝛾 +

√𝛾2 − 1 )∕ 𝜋 + Re 𝑊 (− 𝛾) arrive at the equality:

𝑡 =

2 𝛾√𝛾2 − 1

log ( 𝛾 +

√𝛾2 − 1 )

(

𝐻 𝑤 ∫𝛽

0

d 𝜏√1 − 𝜏2 ( 𝛾2 − 𝜏2 )

− ∫1

𝛽

𝑦 ( 𝜏) d 𝜏√1 − 𝜏2 ( 𝛾2 − 𝜏2 )

)

−

𝜋𝜙𝐴 √ . (A9)

log ( 𝛾 + 𝛾2 − 1 )

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