Seepage through Dam Core

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Analytical solutions for seepage near material boundaries in dam cores: The

DavisonKalinin problems revisited

Anvar Kacimov a,, Yurii Obnosov b

a Department of Soils, Water and Agricultural Engineering, Sultan Qaboos University, Omanb Institute of Mathematics and Mechanics, Kazan University, Russia

a r t i c l e i n f o

Article history:

Received 3 September 2010

Received in revised form 14 July 2011

Accepted 27 July 2011

Available online 7 September 2011

Keywords:

Analytic functions

Free boundary problems

Lapalces equation

Seepage

Refraction

Hydraulic gradient

a b s t r a c t

Steady Darcian seepage through a dam core and adjacent shells is analytically studied. By

conformal mappings of the pentagon in the hodograph plane and triangle in the physical

plane flow through a low-permeable dam core is analyzed. Mass-balance conjugation of

flow in the core and downstream highly-permeable shell of the embankment is carried

out by matching the seepage flow rates in the two zones assuming that all water is inter-

cepted by a toe-drain. Seepage refraction is studied for a wedge-shaped domain where

pressure and normal components of the Darcian velocities coincide on the interface

between the core and shell. Mathematically, the problem of R-linear conjugation (the

RiemannHilbert problem) is solved in an explicit form. As an illustration, flow to a

semi-circular drain (filter) centered at the triple point (contact between the core, shell

and impermeable base) is studied. A piece-wise constant hydraulic gradient in two adja-

cent angles making a two-layered wedge (the dambase at infinity) is examined. Essentially

2-D seepage in a domain bounded by an inlet constant head segment, an outlet seepage-face curve, a horizontal base and with a straight tilted interface between two zones (core

and shell) is investigated. The flow net, isobars, and isotachs in the core and shell are recon-

structed by computer algebra routines as functions of hydraulic conductivities of two

media, the angle of tilt and the hydraulic head value at a specified point.

2011 Elsevier Inc. All rights reserved.

1. Introduction

Renewed interest to hydropower stations, dam reservoirs and large water supply schemes drives both civil/geotechncial

engineers and applied mathematicians, dealing with movement of water through porous materials, to revisit the legacy of

the founders of the specialism of subsurface mechanics: Bear, Casagrande, Cedergren, Dachler, Davison, Charny, Gersevanov,

Hamel, Muscat, Numerov, Pavlovsky, Riesenkampf, Polubarinova-Kochina[1]. What unifies these towering figures? All ofthem 50100 years ago worked on mathematical problems of seepage through earth dams and their contribution is now

in both manuals of geotechnical engineers and applied math books, where the dam problem became a shibboleth of a com-

munity of mathematicians working with free boundary problems (e.g.,[2]). The objective of this paper is to amend solutions

to the Laplace equation for the dam problem by an analytical study of seepage through dam heterogeneities.

Barrages, dikes, levees, weirs and embankments demarcating reservoirs, ponds, detention pools, canals and other hydrau-

lic and agro-engineering structures involve often an earth-, rock-filled element, which maintains a difference of water levels

on two sides of the structure. Dams made of local porous materials are cheap but permeable to water that seeps through the

0307-904X/$ - see front matter 2011 Elsevier Inc. All rights reserved.doi:10.1016/j.apm.2011.07.088

Corresponding author. Fax: +968 24413 418.

E-mail addresses: [email protected](A. Kacimov), [email protected](Y. Obnosov).

Applied Mathematical Modelling 36 (2012) 12861301

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dam soil from the upper pool (reservoir) with a higher water level to the tail water with a lower level. Permeable dams

according to the International Commission on Large Dams inventory make more than 80% of newly constructed largestructures in the world and the vast majority of smaller impoundments. In Oman, after the June-2007 Gonu cyclone and dev-

astating floods1 the government immediately invested into building new and upgrading the existing (31 as of 2009) protection,

groundwater recharge and storage dams, in particular, the Wadi Adei cascade of eight dams, Al-Khod dam and the first large

(70 m high, 100 mln m3 of reservoir capacity) Wadi Dayqah dam.

The tailing-dams have the length of up to several kilometers (e.g. the Al-Khod dam is 5.1 km long) and commonly consist

of the so-called side shells (shoulders), which offer structural resistance against failure, but have a relatively high

hydraulic conductivity k1 (i.e. little hydraulic resistance against seepage) and a relatively expensive core composed of a

material of low conductivity k2, which serves as a seepage-checking element (barrier, see, e.g. [3]). A typical vertical

cross-section is shown inFig. 1where the upper pool water level H1 is translated almost without any loss through the up-

stream shell to the left faceABof the core. From the right face DCof the core again almost no loss of the head occurs through

a coarse filling of the downstream shell to the tailwater, where the water level is H2.

Both the downstream and upstream shoulders may consist of two or more zones (e.g. an inner shell and outer shell) of

contrasting conductivity and then seepage occurs through a composite medium of conductivities k1, k2, k3,. . .

Seepage indownstream shell is often intercepted by toe drains such that a phreatic surface DC1 does not emerge on the slope of the

tailwater (Fig. 1). Water seeped through an element of a dam and collected by a drain is usually diverted to gutters installed

in a dam gallery. From there the leachate is discharged either gravitationally or by pumping to the tailwater.

The right face of the core (ADCin Fig. 1) is commonly subject to seepage erosion. A chimney drain (Fig. 1) or diaphragm is

then installed [4,5], forestalling a direct contact between the clay core and coarse shell and dislodging/migration of fine par-

ticulate into the downstream shell. The lack or fouling/degradation of filters may trigger sand boiling (as, e.g., with the New

Orleans levees) that may lead to a collapse with multi-billion damage and require potentially expensive repercussions.

Dam filters are often graded according to the Vicksburg Lab method[6]or an earlier published Soviet instructions for de-

sign of filters (see [7]for review), which stipulate thatdf60=d

f10 6 6, and d

f10=d

s60 6 6 where d

f and ds are the particle diameters

retrieved from the particle-size distribution curves of the filter and suffosion-prone dam material, correspondingly. The Ter-

zaghi-layering and gradation of the filter material according to US Department of Agriculture/American Society for Testing

Notations and nomenclature

b, bd the widths of the downstream shellGz,S1,S2,Gw,GV,Gf domains and subdomains in the physical, complex potential, hodograph and auxiliary planes (corre-

spondingly)h hydraulic headH head drop between the upper pool and tailwater

H1,H2 water levels in the upper pool and tailwaterk1,k2, kf hydraulic conductivities of conjugated dam zoneskr, Kr conductivity ratiosL, L1 the core and shell widthsp pressure headQ seepage flow rate through the dam (per unit width)vx, vy horizontal and vertical components of Darcian velocity~v vx;vy Darcian velocity vectorv= vxi vy complex velocityvn, vs normal and tangential (to the interface) velocity componentsw=/+ iw complex potentialx, y Cartesian physical coordinatesz=x+ iy= qexp[ix] complex physical coordinateap, bp,cp, dp angles of the tilted interfaces/ velocity potentialw stream functionf=n+ i g auxiliary complex variableDike a soil-filled embankment constructed in Holland for preventing inundationsPiping soil erosion due to seepage, which begins from the exit point (tailwater slope) by dislodging, mobilising and

flushing away soil particles (starting from the finest fraction) and propagating upgradient to the upper poolof the dam

Seepage heaving upward motion of a large soil volume resulting from a vertical component of the hydraulic gradientSuffosion seepage caused migration of fine particles into the void spaces between larger particles

1 The photos attached illustrate the collapsed section of a small dam in Wadi Bani Kharus, Oman. The dam was swept away by the Gonu flood.

A. Kacimov, Y. Obnosov / Applied Mathematical Modelling 36 (2012) 12861301 1287


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and Materials/Corps of Engineers protocols (e.g. [8]) still leads to a sharp transition of conductivity from k1,2 to one of the

filter, kf, across the seepage path. Numerous concerns were raised that the mentioned uniformity coefficients and their mod-

ifications for adjacent layers, as purely textural indicators, do not address any seepage field characteristics, e.g. hydraulic gra-

dients, which can vary both in magnitude and direction along and across the interfaces between porous media of different

composition. Moreover, the seepage field can vary with time owing to a sudden exposure of the structure to extreme flood/

tide reservoir head H1 (e.g., the Dutch St. Elizabeth-1421 and North Sea-1953 floods, American Katrina-2005, or Omani

Gonu-2007), gradual clogging of the filter or other seepage-induced re-texturalisation incidents/processes, which may befallthe dam. The analysis of failure occasions may require answering the question: was failure caused by heaving of a large vol-

ume of soil on the downslope of the earthwork (poor initial design of the whole structure due to the lack of a drain) or by a

focused piping due to clogging of an initially effective filter (poor design of the drain or its improper maintenance)? This calls

for a detailed analysis of the seepage field in the whole soil volume of the dam (global scale) and close to the interfaces (local

scale).

Zonation of the dam body as inFig. 1reduces seepage and makes the slope of the tailwater reservoir less susceptible to

sliding, but creates sharp changes of the hydraulic gradient along internal interfaces between dam zones (e.g. CD inFig. 1)

with potential piping (suffosion, see [9] for the nomenclature and recent review), which should be taken into account in

Oman and other Gulf countries, where most dam shells are made of poorly compacted, cohesionless materials with almost

no natural clay content.

In the design of the dam, even solely seepage-related criteria are often poorly compatible. Normally the upstream head in

Fig. 1 is fixed and if the core is absolutely impervious (Q= 0) the downstream shell is in the safest seepage-wise conditions. A

saturated zone, however, builds up to BA and, consequently, increases the structure-enfeebling pore pressure in upstreamshoulder (prone to, for example, liquefaction of the dam part adjacent to the upper pool). Similarly, drains/diaphragms in

downstream shell lower the phreatic surface as compared with a homogeneous dam body (stability-wise, an unsaturated

soil in the dam body is preferable, i.e. the lower is the free surface the better). Additionally, draindiaphragm intercept

the fine particles travelling with the seepage flow (ideally, no seepage erosion should occur). On another hand, from com-

parison theorems (see[10]) the draindiaphragm always increase the Darcian velocities and Q, compared to no-drain/di-

phragm regimes and this jeopardizes the dam stability. On one hand, the core clay in Fig. 1 should have as low k2 as

possible in order to impede the saturated seepage across it. On another hand, fine dispersing clays are erosion-prone and

seepage can dislodge too fine clay particles from the core, entrain them into downstream shell and toe drain with a gradual

clogging of the latter. Fine particles mobilized (bad phenomenon) by seepage from an upstream zone of the dam (core of

upstream shoulder) can be transported by flow to a downstream porous zone where they can self-heal piping channels

(good phenomenon). Seepage in heterogenous dam bodies/foundations and associated mechanical phenomena are so com-

plicated that post-failure litigations involve numerical and analytical mathematical models with often conflicting outcomes

(see, e.g.,[11]).In the old models of Pavlovsky [12], Dachler[13], Davison and Rosenhead[14], Casagrande[15], Nelson-Skornyakov

[16], Aravin and Numerov[17,18]2 seepage through embankments was typically analytically studied by assuming the whole

cross-section inFig. 1to be a homogeneous entity. Heterogeneities were taken into account in a simplistic way, by assuming the

boundaries of zones of high and low conductivity to be equipotentials, streamlines, or isobars that is mathematically sound if

the conductivity ratio of two adjacent zones is very high or very small. In this way, Polubarinova-Kochina[19],(further abbre-

viated as PK77) solved the dam problem by the theory of linear differential equations, paving a road to numerous applications

in other branches of applied mathematics[20,21]. The flow problem in the coreshells ofFig. 1is more complicated than in a

homogeneous dam due to the presence of refraction boundaries (ABandCD). Moreover, even if seepage in the downstream shell

zone is considered without a full refraction conjugation with flow in the core (we shall do it below), then unlike PK77 neither

pointD nor C1 of the phreatic surface are known, i.e. we have a hanging free boundary.

2 See the scanned images of Russian sources as supplementary materials.

Filter and chimney

drain,kf

Downstream

coarse shell, k1

Fine

core,k2

Upper pool

B C

H2

H1 Tailwater

Impermeable base

A

H

M

y

x

Toe

drain, kf

C1

D

L b

D1Gz

Upstream

coarseshell, k1

g

Fig. 1. Vertical cross-section of a physical plane of a zoned dam with a triangular core.

1288 A. Kacimov, Y. Obnosov / Applied Mathematical Modelling 36 (2012) 12861301
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In practical engineering, notwithstanding the availability of analytical solutions in homogeneous flow domains (e.g. trap-

ezoidal dam cross-sections with sloping tailwater and upper pool boundaries, PK77), most design manuals (e.g., [18,4]) rely

on simplified and even simplistic formulae, which treat seepage as a 1-D DupuitForchheimer process (e.g., the Casagrande

adjusted parabolas). This ignorance of more adequate 2-D models stems from the mathematical complexity of the PK77

final solutions. Clearly, resorting to 1-D approximations completely circumvents the 2- (or even 3-) D problem of internal

erosion on zonal interfaces, already obviated by the PK77-type homogenisation of the dam body in 2-D analytical models.

The proliferation of numerical techniques in the 1960th (e.g., [22]), which mince any zoned seepage domain into

standard grids, helped, on one hand, in modeling heterogeneous dam sections, taking into consideration 2- and 3-D satu-

ratedunsaturated transient flow conditions in porous media undergoing intricate compaction, ageing, swelling, etc., but,

on another hand, had an insidious effect on the analytical approach, which was denigrated for its lack of versatility and abil-

ity to deliver real seepage patterns. The analytical techniques indeed stumbled on the necessity to tackle the realistic

features of seepage/medium, in particular, the refraction boundary conditions on the interfaces between different zones

inFig. 1. Gradually, it became clear that the FDM-FEM codes despite their proclaimed superpower are caliginous in

describing the fine features of the flow field just in the vicinity of interfaces, where the jumps in Darcian velocity are of par-

amount importance for, say, the analysis of drain clogging and its long-term ability to check the phreatic surface and pore

pressure. So, the numerical solutions of the dam problem still recur to analytical classics (see, e.g., [23]). We recall that in

heterogeneous aquifers the same Darcian seepage does not normally effectuate any structural collapses and, hence, FDE-FEM

meshing although blurring the fine scale refraction gives an allegedly acceptable large-scale picture of groundwater

motion. If contaminant transport rather than bulk flow is of concern, then, as Bear [24]emphasized, homogenisation is

not suitable because local heterogeneities cause plume fingering similar to concentrated erosion patterns in dikes.

The recent advances in the theory of boundary value problems of R-linear conjugation [2528]made possible solving a

broad spectrum of seepage refraction problems (e.g. [2932]) for harmonic fields, scabrous for analytical treatment on the

days of the evoked dam classics. In this paper, we revisit the old schemes studied by Davison, Kalinin and Mikhajlov and

tackle those elements/peculiarities of flow in Fig. 1, which were overlooked/mathematically untractable on the days of dam

giants.

2. Seepage through arbitrary triangular core: hodograph and conformal mappings

We assume steady-state Darcian seepage in a homogeneous isotropic triangleABC(Gz) inFig. 1(to be shown in more de-

tails asFig. 2a). The unsaturated moisture movement in the shells outside ABCis neglected. Although H1 in Fig. 1does not

always rise to point A, the most common approach in analytical studies of seepage is to consider this extreme (most dan-

gerous) regime, which Nelson-Skornyakov[16]called the maximum one. IfH1is less than what is shown in Fig. 1, a phre-

atic surface in the core appears, seepage gradients, pore pressure and Qdrop (as compared with the maximum case, see

[10]for rigorous statements of the corresponding variational theorems), but mathematically the problem becomes prohib-

itively complicated (see PK77, pp. 5863). For the maximum regime with no free surface, classical books give a compen-

dium of analytical solutions: Nelson-Skornyakov [16]studied the case ofH2= 0, Davison[33, pp.261266], investigated a

special case of b = c = 1/4 and H2 0, and Mikhajlov (see PK77, pp. 288289) presented an approximate solution forH2 0 and arbitrary core slopes. PK77 mentioned that the general case in Fig. 1can be obtained by conformal mappings

but reported no results.

We fill in the lacunae in the Davison and Mikhajlov results and present a full solution for the case of arbitraryH2 0 and

arbitrary acute angles pbandpc,b + c > 1/2 (the latter inequality guarantees that a phreatic surface is not formed near pointAin Fig. 1). Most engineered cores are tall and narrow enough such that this inequality is satisfied. If the core apexAis above

H1 in Fig. 1, then a phreatic surface appears in the core and the problem becomes exceedingly complex. The focus of our

study is flow near points B andCin Fig. 1and we surmise that the phreatic surface near the tip A can be neglected.

Cu

v

GV(1/2)

V=u+i v

-ik2

-ik2/2

B M

A

D

(1/2)

A B M C DD

=+ i

-1 0 m c

A

B C

D

iQ

H

w=+i

D

Gw

G

A

B CM

k2H

D

Core of conductivity k2

Gz

(a) (b)

(c) (d)

Fig. 2. Flow through a triangular core. Physical domain (a), velocity hodograph pentagon (b), auxiliary half-lane (c), and complex potential domain (d).



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The objective of this section is to determine the seepage field as w zin a triangular core. For this purpose we use the

hodograph method. We introduce the complex physical planez=x+ iywith the origin at point Band show the core in Fig. 2a

which represents the corresponding dam element inFig. 1but without any filter/chimney or toe drain. The hydraulic head

h(x,y) obeys the Laplace equation inABC. The head drop fromABto CDisH= H1 H2, i.e. the reference level is H1 (tailwater).

In this sectionCD is a constant head boundary. The segment AD of the core is a seepage face along which the pressure head

(defined byp = h y) is zero, i.e. h =y. The foundation BMC is horizontal and impermeable. If the water level in the upper

pool (left of the core in Fig. 2a) drops, then a phreatic surface appears and then one will be on the safe side. Indeed, from

the comparison theorems (see, e.g.[47]) follows that the head drop along AB results in redcution of the size of the seepage

face, smaller pore pressure and Darcian velocities inside Gzin Fig. 2a.

The Darcian velocity vector ~v k2rhand complex potentialw(z) = /(x,y) + iw(x,y),/ = k2hare introduced as in PK77.The hodograph domain GVis a pentagon shown in Fig. 2b where the cut tip, point M, corresponds to the maximum of Darcian

velocity along BC. In GVAB is normal to the boundaryABin the physical plane, and both CDandADare normal toACin Gz(see

[33], and PK77). The complex potential domainGw is shown inFig. 2c where the thin lineAD is a curve, whose shape is not

known in advance.

Solution is obtained by conformal mapping ofGz andGV onto an auxiliary half-plane g > 0 (Gf) of a variable f = n + ig(Fig. 2d), wherem andcare the affixes found from solving a set of nonlinear equations (the details are given in Appendix A).

Example 1. Forb = 0.3,c = 0.4,L = 10 m, H2= 2 m andH= .75 m the parameters c= 7.76,m = 0.317,Q/(k2L) = 0.462.

Our solution is expressed as z(f) and dw/dz(f) via an auxiliary variable f involved in the hypergeometric function 2F1. On

the days of Davison and PolubarinvoaKochina tackling these functions, including their integration, was technically cumber-

some. Now days, computer algebra packages have special functions as built-in routines and analytical solutions, based on the

theory of holomorphic functions (e.g. mapping of a z-triangle onto a w-pentagon inFig. 2), are invigorated.

2.1. Primitive conjugation of seepage in core and downstream shell

In the above presented solution H2 was specified by the level in the tailwater and the only seepage-resisting element of

the dam was its core. In this subsection we consider the core conjugated with a downstream shell as shown inFig. 3andH2obtained from solution. We consider a toe drain, which forestalls exfiltration through the right slope of the dam and no chim-

ney drain. The core inFig. 3is relatively fine in texture and rectangular in shape. The distance between the right face of the

core and the drain is b.

The branchAD of a phreatic surface in the core is disconnected from the branch D1C1 in the downstream shell with a seep-

age face emerging on the dowstream side of the core. We assume that all water, which passed through the core and entered

downstream shell (both through the seepage face and through the constant head segment D1Cin Fig. 3), is collected in a sat-

urated zone below D1C1 in Fig. 3. All this water is intercepted by the drain, albeit moisture exuded from the core through DD1can temporarily loop through the unsaturated zone of downstream shell prior to joining the saturated zone through D1C1.

Head loss in the core and downstream coarse shell are Hand H2, respectively.

According to the Dupuit formula the flow rate through a rectangular dam with water levelsH1 and H2 is

Qk2H21 H

22

2L ; 1

that is an exact value in both the DupuitForchheimer approximation and full 2-D theory, as Charny showed (see PK77).

The downstream shoulder inFig. 3is usually relatively broad (b H2). Consequently, we can use the DupuitForchhei-

mer approximation for flow in this coarse zone, in congruity with which

Seepage in the finerectangular core

Upper pool

B C

H2

H1

Impermeable base

A

H

x

Toe drainC1

D

L b

D1

Seepage in the

coarse downstream

shell

GzGd

bd

g

k2 k1

Phreatic surfacein the core

Phreatic surface

in the downstream

shell

Seepage face of

the core

Fig. 3. Physical plane of a dam with a rectangular core and diaphragm.

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~v1 is perpendicular to the equipotential AO and the orientation of~v2 is determined by the refraction condition alongBO:

Krtanpb= tanpb1; 5

whereb1 is the angle that~v2 makes with the normal to BMO. Kalinin imposed the following no-phreatic-surface condition

b1 6 1=2 a: 6

An additional inequality:

V1 < cospa c= sinpb; 7

should be satisfied. Kalinin missed(7), which stipulates that flow on the lee-side of a low-permeable barrier should remain

saturated i.e. pore pressure to remain positive (seeAppendix B). Physically, in a relatively thick upstream shoulder or chan-

nel siltclay liner[35]the pressure head of flow passed through a low-permeable zone of sufficient thickness can become

negative. Then what is shown as refraction ofEM inFig. 4a is not true. Consequently, the unsaturated flow in k2 zone of

Fig. 4a becomes an infiltration shower (term used in PK77). In this case (possible fork2> k1only, see[36]for more details)

(7)does not hold. It is noteworthy that a cascade of intermittent saturatedunsaturated zones can emerge in intermittently

heterogeneous porous media[37].

Fig. 5a depictsV1 andV2 as functions ofb forKr= 0.3,a = 0.3,c = 0.1. This case corresponds to coredownstream shellconjugation (Kr< 1). The upper and lower curves in Fig. 5a correspond to the value of hydraulic gradient in the core (left

wedge) and downstream shell (right wedge) in Fig. 4, correspondingly. The vertical line cuts the solution (4)according to

(7), without which Kalinins [34] solution would extend to b 0.065, if limited solely by (6). In Fig. 5b the gradients are plot-

ted as functions ofa forKr= 3,b = 0.2, c = 0.1. Recurring to Fig. 4 this case corresponds to the conjugation between upstreamshoulder and core. The upper and lower curves inFig. 5b showV1 andV2 in upstream shell (now the right wedge inFig. 4)

and the core (the left wedge), correspondingly. For Kr< 1 the most dangerous (suffosion-wise) segment is the ray OB, i.e. V1in

Fig. 5a is critical. For Kr> 1 (upstream shoulder-core/chimney drain conjugation), the segment OC inFig. 4, (i.e. V2 from

Fig. 5b) is critical for assessments of piping.

Similarly to the Davison seepage scheme and in the sense of variational theorems discussed in the Introduction, the full-

saturation flow shown inFig. 4is most dangerous, i.e. in case of an emerging unsaturated zone on the right ofOB the pore

pressure and flow rates will drop and structural stability of the composite wedge ofFig. 4improves.

4. General solution for refraction on a wedge

In this section we consider those elements of the dam in Fig. 1, which are characterized by so-called triple points where

two adjacent porous zones are underlaid by an impermeable base. Near these points (B,Cand C1

in Fig. 1) the Darcian veloc-

ity vector has singularities if the angles b andc are unfavourable (we will specify later the deleterious slopes). Then

Fig. 5. Hydraulic gradients in the two components of the Kalinin wedge as functions ofb forKr= 0.3,a = 0.3,c = 0.1 (a) and as functions ofa for Kr= 3,

b = 0.2,c = 0.1 (b).

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suffosion takes place in the zones where the hydraulic gradient spikes above a certain limit. PK, 1977 recommends 1 as this

critical gradient, although other (higher and lower) empiric limits are used in geotechnical engineering.

This local increase of the gradient magnitude is caused by the impermeability or low permeability of the foundation (we

recall that below the line BCC1in Fig. 1 the soil is impervious) as has been recently described by Fox and Wilson [38]. Indeed,

an impermeable base prevents heaving but facilitates localization of high-gradient outseeps. For the sake of definiteness, we

consider the conjugation of upstream shoulder and core shown in Fig. 6 corresponding to b > 1/2 in Fig. 1 (ifb < 1/2 or equiv-

alentlyd > 1/2 inFig. 6then the gradient at point B is zero and the vicinity of this point is erosion-wise safe).

Here we study the vicinity of point B (Fig. 6), which represents a zoomed area of the corresponding point inFig. 1. Math-

ematically identical seepage refraction arises at triple points like B in Figs. 1 and 6 of conjugated coredownstream shell,

diaphragmdownstream shell, filtercore, filterdownstream shell, and downstream shellriprap (see, for example, points

CandC1 in Figs. 1 and 3). Our flow domain inFig. 6is bounded by curved linesMNandNC. It would be better to consider

two adjacent triangles:MNB andBNC. This geometry is, however, mathematically intractable and reconstruction ofMNand

NCas parts of solution is and inverse trick (see [16]).

In the two previous sections we conjugated seepage in the core and shell through either a simple mass-balance condition

(outflow from one homogeneous element of the flow domain equals inflow into an adjacent domain of contrasting conduc-

tivity) or by matching two 1-D flows in two wedges (Kalinins scheme with a rigorous conjugation of velocities and heads

along a ray). Now we shall study a fully-coupled, 2-D Darcian flows in the angles ABAu andEBEu separated by an interface

AuNBNEu.

We assume that the seepage domain in Fig. 6is bounded from below by an impermeable base, the horizon AMBCE. We

emphasize that if thetwosupplementary anglesABAu (domain S1) and EBEu (domain S2) were made of the sameporousmedium

and are considered separately, then their infinities would be mapped on one point for each Riemann sphere (PK77). This

means that in a homogeneous medium pointsA,Au,Euand Ecoincide. In Fig. 6 the infinities of the two angles are different.

The interface BNAu(Eu) is a ray making an angle dp with BAof the dam base. The origin of the (xy) Cartesian coordinates isnow at point B. In S2 we have a fully saturated Darcian flow with the complex potentials w2 and velocity~v2 v2x; v2y and in

S1 the corresponding functions are w1 and~v1. w1 and w2are holomorphic in S1 and S2. Along the interface we have a standard

refraction conditions (PK77):

v1n v2n; v1s=k1 v2s=k2; 8

where v1n, v2nare the components of~vnormal toA(E)Band v1s, v2sare tangential components. Generalizations of(8)to the

case of anisotropic media and two-phase flows are given by Mualem[39], Raats[40].

The objective of this section is to find w1andw2as explicit functions ofzfor an arbitrary sloped of a straight interface in

Fig. 6and to interpret these two conjugated holomorphic functions in a geotechnically meaningful manners. For these pur-

pose we select the classes ofw1andw2 that are either finite or infinite at point B and at infinity. We note that these re-

mote pointsA in S1andEin S2are different in the two zones (by contrast, in a homogeneous soil they would be imaged by

the same point on the Riemann sphere).

The character of singularity at pointB can be mathematically different. Grinberg [41]solved a mathematically equivalent

problem in electrostatics and in the class of two holomorphic functions with an essential (non-integrable) singularity at

pointB (see his Chapter XIV, Section 38). We restrict ourselves by a narrower class of integrable singularities atB. For exam-

ple, a logarithmic singularity forw1,w2atB, is usually interpreted as a drain (well) according to the well-known (PK77) rep-

resentation of a line 2-D sink or source. Clearly, for geotechnical applications we cut the corresponding vicinities ofB.

Obnosov[42]found two holomorphic functions (complexified velocities) vj(z) = vjx i vjy= dwj/dz,j = 1,2 in the two do-

mains S1 andS2 ofFig. 6as described below. At d 2 (0,1/2) the general solution is:

v1z XNj1

cjKhjzhj1; z2 S1;

v2z XN

j1

cjzhj 1; z2 S2;

9

k2k1

A B

Impermeable base

Au

y

xE

N

M C

Upper

pool

H1 Seepage face

Constant-head line

Interface

S1

S2

N1z=x+iy

Eu

g

L L1

M1M2

Fig. 6. Vertical cross-section of an upstream shell conjugated with core through a tilted interface.



9/16

wherecj,j = 1, 2 . . ., n, are arbitrary real parameters, and hj,j = 0,1,2,. . ., N, are anyNroots of the equation

sinph D sinp1 2dh 0; 10

where

D k1 k2=k1 k2:

In(9)the single-valued branches of the functionszhj and zh

j are fixed in the domains S2andS1by the condition that these

functions are real at z=x> 0 andz=x< 0, correspondingly. The function K(h) is defined as:

Kh sinp1 dh

sinpdh

k1 cosp1 dhk2 cospdh

; 11

Elementary inspection shows that Eq. (10) has an infinite number of roots (both positive and negative) which condense at

infinity. Each root corresponds to one term in(9). In(9)we retained non-negative roots only.

In what follows we consider two special cases of truncation of(9),viz. the so-called one-term solutions. Without any loss

of generality we assume k1> k2 and 0 < d < 1/2. Obviously, for d > 1/2 in the above-written solution one has to permute

v1? v2.

5. Drain/filter centered at the triple point

As we discussed above, near points B, Cand C1the porous continua of three different conductivities (k1, k2and 0) are most

susceptible to suffosion. Mitigation of internal erosion (translocation of soil particles from one zone to another and an ensu-ing decrease of structural stability) is achieved by constructing a drain/filter diagrammed inFig. 7. Here we consider a semi-

circular filter of a radius R centered at point B. Water seeps into the filter from both wedges ( S1 andS2) such that the filter

contour MNC(an interface between the gravel pack inside and the interior wedge media) is a constant-head line. Solution to

this problem follows immediately from(9)if we retain a single term corresponding to the root h0= 0, K = k1/k2. Then

v1z c0k1=k2z; z2 S1; v2z c0=z; z2 S2: 12

It is well-known (PK77) that a point sink in 2-D flow is determined by v= c0/zwhere c0 is an arbitrary real constant and v(z) is

a Darcian velocity field in a homogeneous porous plane. Our solution (12)is of the same sink type but for two adjacent

wedges. Indeed, we selectw2= 0 alongBCEand /2= 0 along the circleNC. Then the two complex potentials in(12)corre-

spond to a drain placed at the same pointB such that the head is counted from the drain contour (NC). Along this semicircle,

we have z= R exp[ix], 0 < x < p(1 c), where x is the angular coordinate. We integrate the second equation in(12)thatwith the selected references for the potential and stream function gives in polar coordinates:

w2z /2 iw2 c0 logqR ic0x; z2 S2; 13

whereq is the radial coordinate in the complex plane.Integration of the first equation in(12)yields:

w1z /1 iw1 c0k1=k2 logqR

U0 iW0; z2 S1: 14

The real part, U0, of the constant of integration is determined from the condition that /1= 0 alongMN(the segment of our

semicircle in the left wedge S1) because inside the gravel pack water is in hydraulically identical conditions, independent of

whether it touches the first or second medium in the exterior. The imaginary part,W0, follows from the conditionw1= w2at

point N. Actuallyw1 andw2 coincide along the whole interface rayNA (NE). Consequently,

w1z c0k1

k2

lnq

R

ic0p k1

k2

1 x 1 dk2 k1

k2 ; z2 S1: 15

A Interface (stream line)E

Constant

y

Constant-headsemiy semi-circle

SS2

k2 z=x+iy2N

S

z=x+iyR

k1S1 1

BM CB Impermeable baseM C

Filter filling with kf >>k1, k2 Line sinkf 1, 2

R

Fig. 7. Vertical cross-section of a circular drain centered at a triple point.



10/16

From(15)at point Mwe have

w1MQc0p1 dk1 k2

k2: 16

HereQis the total seepage inflow into our semicircular filter. From(16)we express the constant c0throughQ. Then the in-

flow throughCNis

w2N

Q

p

k2

k1 k2 :

(we recall that in(16)we assumedk1> k2 and 0 < d < 1/2).

This almost trivial solution, characterized by a purely radial inflow into the drain with no fluid motion across the interface

between S1and S2, will be used in the next section as a mode of inverse shaping of the domain boundaries. It is noteworthy

that the Kalinin [34]scheme in Fig. 4a can be obtained (by a similar inverse procedure) from Obnosov [42]who studied an

arbitrary orientation of a two-layered wedge with respect to an arbitrary flow. Indeed, inFig. 4b we are showing a standard

(e.g. [43]) problem of refraction on a straight interface. The equipotential lines (dashed lines in Fig. 4b) on the left and right of

the interface are straight. Any of these lines in the first medium can be selected as AO inFig. 4a. The isobars (solid lines in

Fig. 4b) are also straight. Therefore, by selecting one passing through point Oin Fig. 4b and setting pressure to be zero on this

ray we obtain the physical interpretation of the coreshell conjugation in the Kalinin scheme ofFig. 4a.

6. Shellcore conjugation

Now we come back toFig. 6and consider another special case of(9). We formally retain in the series one term, which

corresponds to the root of Eq. (10)on the interval 0 < h < 2, and will see what is the physical scheme of seepage correspond-

ing to this particular case. For positiveD Eq. (10) has one and only one root in the selected interval ofh. This root satisfies the

inequality 0 < h1< 1. Obviously, the value of the root depends ond and the ratio k1/k2. The corresponding one-term trunca-

tion of(9)is:

v1z Kh1c1zh11; z2 S1;

v2z c1zh11; z2 S2;

17

where K(h) is defined in(11)andc1 is a real parameter to be determined later.

In particular for d = 1/4 from(9)(11)we get

v1z 1 Dc1zh11; z2 S1;

v2z c1zh11; z2 S2; 18

where h1= (2/p)arccos(D/2), 0 < h1< 1.Obviously, atd = 1/2 the wedges degenerate into two adjacent half-planes and the corresponding trivial solution follows

from(9)as:

v1z v2z c1;

i.e. we obtain the known[43]unidirectional flow perpendicular to an interface.

Now we utilize(17)for reconstructing a finite-size domain MNCBMconsisting of a fine dam core and coarse upstream

shell. We implement an inverse approach of Nelson-Skornyakov [16]resuscitated by Obnosov[44]. The idea of this ap-

proach is to conjugate in a rigorous manner two complex potentials (or complex Darcian velocities) along a given curve

(an interface between two media of contrasting conductivity, e.g. parabola as in Kacimov and Obnosov [44]). The two ana-

lytic functions will have certain singularities at given isolated points. Then certain streamlines, equipotentials or isobars of

these functions are interpreted as boundaries (stream tubes) of seepage domains. Here we reconstruct a constant-headboundary MNand an isobar (seepage face) NC(Fig. 6) (again,(17)is exactly satisfied!). We notice that Nelson-Skornyakov

[16]himself used this inverse procedure for flows in homogeneous media only.

Pressure head in the two domains is (see PK77)

p1 /1

k1y cp

; p2

/2k2

y cp

; 19

where the real constantcp will be determined later.

Integration of(17)results in two complex potentials

w1z /1 iw1 c1h11 Kh1z

h1 ; z2 S1;

w2z /2 iw2 c1h11 z

h1 ; z2 S2:20

The constants of integration are zero in (20). This follows from adopting two reference values. First, at point B we set

w1= w2= 0 because bothAMBand BCEare streamlines and unlike the previous section there is no mass loss (generation)



11/16

at pointB. Second, at this point we set /1= /2= 0 because the velocity potentials in both media are determined up to a real

constant. We emphasize the difference between(18), (20)and the solution from the previous section where the drain solu-

tion atz= 0 has a logarithmic singularity for both w1and w2. We isolated this singularity by excluding a semi-circular vicin-

ity of point B (Fig. 7), interpreted as the contour of filter filling. The complex potentials in (20)at z= 0 are finite (zeros),

although the velocity is not. The singularity of velocity in(18)is said to be integrable (PK77). Unfortunately, unlike the pre-

vious section, Eq.(20)do not give simple known curves as equipotentials in S1andS2. Let us find out what are these curves.

If we use polar coordinates with the origin at Bin Fig. 6, thenz= qexp[ixp] where xp is the angle counted from Bx. In S2andS1 we have 0 < x < 1 dand 1 d 0). This point is located distanceL fromB (xM= qM= L). Then from the first Eq. (21)we obtain

c1 /0h1Lh1=Kh1:

Consequently, (21) and (22)become:

/1 /0q=Lh1 cosph1x 1; q> 0; 1 d 0; 0 0; 1 d 0; 0


12/16

From the second Eqs. (24) and (26)follows

w1N Q/0 tanph1d:

At last, from the second Eqs.(19) and (25)follows that/2C= k2(H1+ /0/k1) at pointC(we recall that CNis a seepage face

with p2= 0). At the same point we have/2C /0=Kh1qC=Lh1 due to(23). Wherefrom, using the relation(26), we get

qCxCL1 L k2Kh1

/0

L sinpd

cospdh11=h1

/0k1

!" #1=h1:

Fig. 8 shows the physical domain in dimensionless coordinates (X,Y) = (x/L,y/L) for k2/k1= 0.2, d = 0.4 and /0/(k1L) = 1 .

Lines 15 are the interface,MN,NN1, the streamlinew1 2.0 coming to pointNfrom the first medium, andNC (correspond-ingly). Curves 2, 4, and 5 extend to the adjacent medium (i.e. go beyond the intersection point N), where, of course, the

equipotentials, streamlines and isobars obey different equations. Fig. 9 shows the refraction of the streamlines

w1,2/(k1L) = 2.0, 1.5, 1.0, 0.5 (curves 14) on the interface.Fig. 10shows the contours of isotachs V2= 4, 3.8, 3.6 (curves 13, correspondingly). These contours evince point-wise and global criteria of stability of soil slopes [46].Fig. 11ac shows

H1/L,L1/(k1L) andQ/(k1L) as functions ofkr= k2/k1 for d = 0.45, 0.35, 0.25 (curves 13, correspondingly).

To summarize this most difficult section of the paper we can extend the Nelson-Skornyakov type reconstruction tech-

nique to other than(20)pairs of refraction-conjugated complex potentials. The selected solution (20)is in reality very sim-

ple: to generate the core and shell shapes we need actually one point MinFig. 6(and of course conductivities and the slope

of the interface). For more terms retained in (9)the reconstruction procedure may be more sophisticated.

-2 -1 0 1 2

0

0.5

1

1.5

2

2.5

3

3.5

12

3

4

5 X

Y

Fig. 8. Reconstructed shell-core for k2/k1= 0.2,d = 0.4 and /0/(k1L) = 1.Lines15are the interface, MN, NN1 ofFig. 6, the streamlinew1 2.0 comingto point

Nfrom the first medium, andNC.

-2 -1 0 1 2

0

0.5

1

1.5

2

2.5

3

3.5

1

2

3

4

X

Y

Fig. 9. Streamlinesw1,2/(k1L) = 2.0, 1.5, 1.0, 0.5 (curves 14) for the dam inFig. 8.



13/16

What is the practical value of obtaining boundaries that are unlikely to be practically constructed of exactly the shape

shown inFigs. 10 and 11, or to appear like these curves in nature? This kind of solution can be used as upper and lower

-2 -1.5 -1 -0.5 0 0.5 1

0.5

1

1.5

2

2.5

3

3

1

2

seepage face

X

Y

Fig. 10. Isotachs V2= 4, 3.8, 3.6 (curves 13) for the dam inFig. 8.

0.2 0.4 0.6 0.8 1 kr

1

2

3

4

5

6

7H1L

2

3

1

0.2 0.4 0.6 0.8 1 kr

2

4

6

L1 kL

2

3

1

0.2 0.4 0.6 0.8 1 kr

1

2

3

4

5

6

7

Q kL

2

3

1

Fig. 11. H1/L (a),L1/(k1L) (b) andQ/(k1L) (c) as functions ofkr= k2/k1 for d = 0.45, 0.35, 0.25 (curves 13).



14/16


15/16

GfofFig. 2b is carried out taking into account an additional vertex Mof the cutCMB and with the correspondence of points

M? m. Again from the SchwarzChrsitoffel formula we write:

dw

dz a2

Z f0

sb1=2c sc1=2s 1bc1m sds; 32

where the single-valued branch of the integrand function fb1/2(c f)c1/2(f + 1)b +c1(m f) is fixed in the upper half

plane by the condition of its positiveness at f = n, 0 < n < m.

The mapping parameter a2 in (32), is found from the given Darcian velocity at point A i.e. VA= k2 eip(1/2

b)

sinpc/sinp(b + c) that at f = 1 results in

a2 k2 sinpc

I3 sinpb c; I3

Z 10

tb1=2c tc1=2

1 tbc1

t mdt: 33

After simple derivations

I3 c1=2cB1=2 b;b c m1 1=cb1=2

1 2b

1 2cF3=2 b;1=2 c;3=2 c; 1=c

:

In order to find the last (and most difficult) mapping parameter m (seeGf,Fig. 2b) we integrate(32)as

wf Z f

0

dw

dz

dz

df

df: 34

In(34)dw/dz(f) is retrieved from(32)(taking into account (33)) and dz/dffollows from(29)as

dz

df

L

I1fb1c f

c11 f

bc: 35

From (34)atf = candw= k2H(seeFig. 2c) we obtain a non-linear equation with respect tom. We solve this equation by the

FindRootroutine using numerical integration by the NIntegrate routine ofMathematica.

Eventually, the flow rate is Q= w(1)/i. For finding Qwe again implement the NIntegrate.

Appendix B

In this appendix we solve the refraction problem shown in Fig. 4a.

All streamlines in both S1 and S2 are straight. Correspondingly, recalling that we assumed hAEO = 0, from the Darcy lawapplied to the segment EMand from the right-angled triangle OEM:

j~v1j k10 hM

jEMj k1

hMjOMj sinpb

; 36

wherehMand OMdepend, of course, on the position of point M(but their ratio does not).

The refraction condition (5) is actually a geometrical combination of two general conditions (8). The second condition (8),

and (3) and (5)applied to EMNgive:

j~v1j cospb j~v2j cospb1: 37

Next, we apply the Darcy law to the segment MN:

j~v2j k2hM hN

jMNj : 38

In (38) hN= yN= jONjcospc. From the triangle OMN we have jONj = jOMjcospb1/cosp(a + b1) and jMNj = jOMjsinpa/cosp(a + b1) that upon substituting into(38), with provision(4), gives:

V2cospb1 cospc

sinpa

hMcospa b1jOMj sinpa

: 39

Now, we excludehM/jOMj from(39)using(36)and get:

V2cospb1 cospc

sinpa V1

cospa b1 sinpbsinpa

: 40

Involvement of(37)leads to(4).The inequality(7) (overlooked by Kalinin [34]) follows from the condition that pressure

alongBMO must be positive. Otherwise, the infiltration shower regime (see PK77 and[35]) occurs in the downstream shell

and, hence, the analysis above is not valid.



16/16

Appendix C. Supplementary data

Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.apm.2011.07.088.

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http://dx.doi.org/10.1016/j.apm.2011.07.088http://dx.doi.org/10.1007/s10665-009-9289-8http://dx.doi.org/10.1007/s10064-007-0095-0http://dx.doi.org/10.1007/s10064-007-0095-0http://timecapsule.ecodev.ch/video.htmlhttp://dx.doi.org/10.1016/j.cageo.2010.01.014http://dx.doi.org/10.1016/j.cageo.2010.01.014http://timecapsule.ecodev.ch/video.htmlhttp://dx.doi.org/10.1007/s10064-007-0095-0http://dx.doi.org/10.1007/s10064-007-0095-0http://dx.doi.org/10.1007/s10665-009-9289-8http://dx.doi.org/10.1016/j.apm.2011.07.088

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