saturated and unsaturated water flow in inclined porous media

Environmental Modeling and Assessment 9: 91–102, 2004. 2004 Kluwer Academic Publishers. Printed in the Netherlands.

Saturated and unsaturated water flow in inclined porous media

Scott W. Weeks a, Graham C. Sander b,∗, Roger D. Braddock c and Chris J. Matthews c

a Department of Natural Resources Mines and Energy, Brisbane 4068, Australiab Department of Civil and Building Engineering, Loughborough University, Leicestershire LE11 3TU, England

c Faculty of Environmental Sciences, Griffith University, Nathan 4111, Australia

This paper considers the two-dimensional saturated and unsaturated flow of water through inclined porous media, namely a waste dumpor hill slope. Since the partial differential equation governing this water flow transforms from being parabolic to elliptic as the water flowvaries from unsaturated to saturated, an iterative, finite differencing scheme is used to develop a numerical solution. The model can be usedto investigate the effects that hill slope angle, depth of soil cover and hilltop width have on water accumulation in the dump and the timerequired for saturation to occur at different areas in the dump domain. The accuracy and reliability of the computer based solution is testedfor two different boundary conditions – (1) no flow on all boundaries (i.e., the internal redistribution of soil moisture to steady state) and(2) a constant rainfall flux on the dump surface. Numerical studies then show the effects of changing the hill slope angle, depth of layer,and dump geometry on the flow characteristics in the dump.

Keywords: porous media, saturated, unsaturated, sloping layer, capillary barrier

1. Introduction

Waste management is becoming increasingly impor-tant in today’s industrialized world. The heavy relianceof today’s society on natural resources has repercussionsthroughout the environment and for its sustainability. Onemajor concern in preserving the environment is the preven-tion of groundwater contamination. Acid mine drainage(sometimes known as acid rock drainage) occurs when con-taminants within waste dumps produced as a result of miningoperations, react with oxygen in the atmosphere, and witheach other, to form, among other contaminants, sulphuricacid. Rain water can penetrate the waste dump and transportthe sulphuric acid and other contaminants into the environ-ment resulting in severe detrimental effects.

Acid mine drainage is a problem throughout the miningworld and is “potentially the single largest cause of detri-mental environmental impact resulting from the mining ofsulphidic ores” [1]. Acid mine sites are located in Aus-tralia, Papua New Guinea, Sweden, Norway, eastern andnorth western United States and western Canada, partic-ularly British Columbia [2,3]. In the Republic of SouthAfrica, 34% of lime manufactured there is used to neutraliseacid mine drainage leachates [4].

Several options exist for the control of acid mine drainageand the three main groups are: (i) Primary control measures,(ii) Secondary control measures, which aim to prevent con-taminant migration by diverting surface water, interceptingor isolating groundwater or by preventing the infiltration ofrainfall into the waste and (iii) Tertiary control measures [1].The primary control measures are often impractical, un-economical or have questionable long-term performance,while the tertiary control measures require that collection

∗ Corresponding author.

and treatment of acid mine drainage be maintained in thelong-term.

The secondary control measures show the most promisein preventing acid mine drainage by effectively sealing thesurface of the waste material and thus preventing water fromentering. Sealing the dump can be achieved through the useof artificial and/or soil covers. Artificial covers have theproblem of questionable long-term performance (cracking,tearing, leaking, etc.) and cost. Covers using a combina-tion of synthetic membranes and soil layers can provide ex-tremely effective infiltration controls [5] but, again, the long-term effectiveness of such covers is unproven [1].

Capillary barriers offer a viable option in preventing acidmine drainage. These barriers consist of fine-over-coarsesoil layers. The contrast in hydraulic conductivities betweenthe fine and coarse soils in the layers, allow capillary barriersto function in preventing downward flow [6]. Water is heldin the fine-grained soil layer by capillary forces until it isremoved by evapotranspiration, or by lateral drainage alongthe interface between the fine and coarse soils. Through thismechanism of lateral migration (also known as the “wickingeffect”), capillary barriers can be used to divert rainwateraway from the waste material. Hence contaminants remainwithin the dump and are not transported into the environ-ment. Once the dump is sealed, topsoil can be placed overthe dump allowing the establishment of vegetation.

An effective capillary barrier as one in which “the com-bined effects of evaporation, transpiration and lateral di-version exceeds the infiltration from precipitation, therebykeeping the soil sufficiently dry so that appreciable break-through into the coarse soil does not occur” [6]. In order todesign the waste dump effectively and at minimal cost, it isnecessary to understand how water moves and/or accumu-lates throughout the layer of fine-grained soil covering thewaste material.

92 S.W. Weeks et al. / Flow in inclined porous media

The aim of the following model is to determine whichphysical characteristics of the dump are most effective inpromoting water flow from precipitation away from thewaste material. A desirable soil layer is one that maximizesthe time of arrival of rainwater to the boundary between thesoil and the waste material as well as maximizing the de-gree of lateral spread [7]. The following model considersthe two-dimensional saturated and unsaturated flow of waterthrough the homogeneous fine grained soil layer covering adump of coarse grained mining waste material. This is donefor two reasons. Firstly, the homogeneous case for this slop-ing dump has not been thoroughly studied with respect todiffering boundary conditions. Secondly, homogeneous soillayers are likely to be much easier to construct over wastedumps.

2. Model development and method of solution

2.1. Physical system and flow equation

Waste material from mining is usually placed in longrows with slightly flattened tops as depicted in figure 1 [8].These rows are long enough that they can be modelled assemi-infinite in the y direction (along the dump), and there-fore the majority of water flow taking place will be two-dimensional, with x being the horizontal co-ordinate (pos-itive right), z being the vertical co-ordinate (positive down)(figure 2). Only one such dump is considered in this study,as the dumps are assumed to be independent, with no flowconditions from dump to dump (on the vertical sides in fig-ure 2). The dimensions of the dump include height, length,slope angle and depth of the covering layer (see figure 2).These properties of the dump can be adjusted, as can hill-

Figure 1. Waste material over which soil is placed.

Figure 2. Waste dump domain (the shaded area represents the waste mater-ial whilst the area enclosed by ABCDEFGH is the covering soil).

top width AB and soil depth AH, in order to investigate howthe system responds to various parameter regimes. The hillslope domain itself is taken as symmetric about the line AO,with CD being the line of symmetry between one dump andthe next. A piece-wise linear domain is considered in theinterests of simplicity.

The soil moisture pressure based Richards’ equation forwater transport through soil is

Cw(h)∂h

∂t= ∂

∂x

(K

∂h

∂x

)+ ∂

∂z

(K

(∂h

∂z− 1

)), (1)

where h is the water pressure head, K is the hydraulic con-ductivity, Cw is the specific water capacity, t is time, x isthe horizontal spatial coordinate and z is the vertical spa-tial coordinate taken as positive downwards [9]. The watercontent is �(h), with �sat as the saturated water content.Equation (1) represents flow in both the unsaturated domain(where Cw = Cw(h),K = K(h),� = �(h) for h < hae,hae being the air entry value) as well as in the saturated do-main (where Cw = 0, K = K(hae) = Ks , � = �sat forh � hae). Note that Richards’ equation is a Partial Differ-ential Equation (PDE), which is parabolic in the unsaturatedzone, and elliptic in the saturated zone.

The first quasi-analytic solutions of Richards’ equationfor unsaturated flow within a semi-infinite hill slope werepresented by Philip [10]. This solution was achieved by us-ing a rotated coordinate system and then ignoring the effectof water content variations at the top of the hill slope. Byignoring the slope crest, the problem becomes identical inform to the classical one-dimensional infiltration problemexcept K (the hydraulic conductivity of water) is replacedwith K cos γ (where γ is the hill slope angle). The total hor-izontal flow per unit cross-slope length, U , is independentof time, meaning that throughout the infiltration process,the rate of total horizontal discharge into the slope is con-stant [10]. Secondly, for a homogeneous, isotropic soil, thedown slope flow Ud (flow parallel to the soil surface) is pro-portional to t1/2 for small time, and is proportional to t forlarge time. Usually these two flow components are discussedin terms of anisotropy or soil layering [11] but for a homo-geneous, isotropic soil, horizontal and down slope flows aresimple physical consequences of capillary and gravitationalforces [10,12].

The domains used in most studies are rectangular, whilethe domain in figure 2 is quite complex. An excellent com-parison of six finite differencing methods used to model one-dimensional vertical flow of water through soils is givenin [13]. The schemes which have the widest range of ap-plicability for predicting saturated and unsaturated watermovement in soils, were the implicit schemes with implicitor explicit evaluation of the hydraulic conductivity and wa-ter capacity functions [13]. Some studies have used Alter-nate Direction Implicit methods (ADI) methods for circulartype domains [14–16]. In addition, [17] and [12] conductednumerical and experimental studies for two-dimensional soilmoisture flow in a sloping, but still rectangular, region underthe condition of rainfall infiltration.

S.W. Weeks et al. / Flow in inclined porous media 93

The Alternating Direction Implicit (ADI) iterative schemeis chosen since (1) is parabolic for unsaturated flow but el-liptic for saturated flow, where it reduces to

∂2h

∂x2+ ∂2h

∂z2= 0. (2)

The ADIPIT scheme can handle both elliptic and parabolicforms [9]. The ADIPIT scheme is an iterative adaption ofthe ADI, which discretises the flow equation into two simul-taneous systems of difference equations which are solved it-eratively. The details of the numerical scheme are presentedin appendix.

Each full time step is achieved by iterative correction oftwo forward discretization passes. The first forward passis based on a horizontal, or x, discretization to determineapproximate values of h, at the new time line. This passuses the best currently available values of h, and leads to alinear difference equation for an iterative value of h. Thesecond forward corrective pass is based on a vertical, or z,discretization for the full time step, using approximate val-ues of h leading to a linear difference equation for an newapproximation of h at the next time line. The scheme thusinvolves performing a pair of passes (one horizontal pass fol-lowed by one vertical pass) to complete a full iteration. Thisiteration procedure is repeated until the difference betweenthe results of a horizontal pass and its subsequent verticalpass is negligible. The ADIPIT then moves on to the nextforward time step.

2.2. Boundary conditions

Physically, rain is falling onto the surface of the dump(ABC) with AH and CD acting as axes of symmetry withinthe dump and between the dumps, respectively. Rain fallsat the rate R directly onto the top boundary AB while waterfalling onto BC flows normally across the top sloping sur-face [10]. The surface CD has no flow, as this boundarycondition isolates the dumps shown in figure 1. The capil-lary barrier is represented by the lower surface of the soilcover (EFGH) and is impermeable to water flow until satu-ration (or the water entry potential of the underlying waste)occurs at some point along it. A natural drainage conditionis applied along the bottom boundary DE.

Referring to figure 2, the boundary conditions are,

AB: − K

(∂h

∂z− 1

)= R, (3a)

BC, FG: sin γ

(K

∂h

∂x

)− cos γ

(K

∂h

∂z− K

)

= R cos γ (on FG, R = 0), (3b)

CD, EF, AH: ∂h

∂x= 0, (3c)

GH: − K

(∂h

∂z− 1

)= 0, (3d)

ED: ∂h

∂x= 0. (3e)

All dumps had a uniform initial distribution of h(t = 0) =h0 = −38, a value which corresponds to approximately halfof the saturated moisture content θsat.

3. Model validation

3.1. Hydraulic functions

For the purpose of testing the numerical scheme, the hy-draulic functions

θ(h) = (θsat − θr)

(1 + (−m1h)m2)m3+ θr , (4a)

K(θ) = m4θm5 (4b)

are used where m1 = 0.044, m2 = 2.22, m3 = 0.55,m4 = 18 130 cm/hr, m5 = 6.07, θsat = 0.312 cm3/cm3

(the saturated water content), and θr = 0.0265 cm3/cm3

(the residual water content) [18].

3.2. No flow boundary conditions

Several tests can be made to assess the performance of thenumerical solution scheme. No flow boundary conditionscan be applied by setting R to zero and taking ∂h/∂z = 1 forthe boundary DE. The no flow boundary condition is chosenfirst as it will give some indications on how well the ADIPITscheme has been implemented. Firstly, if there is no waterentering or leaving the hill slope domain, the volume of wa-ter W0 in the domain should remain constant. After severalsimulations were completed, it was found that the final nu-merically calculated W0 had less than 0.01% error with theinitial numerically calculated volume of water. Agreementbetween the numerical and analytical volumes was obtainedto at least four significant figures.

Secondly, the internal redistribution of the water shouldeventually run from the initial profile h(t = 0) = h0 = −38,to steady state, where ∂h/∂t = 0, in (1). At steady state, thewater will settle in such a way that there is no variation inthe horizontal direction thus ∂h/∂x = 0 and from (1)

h(z) = c1 + z, (6)

where c1 is determined by matching the volume of water inthe domain. Since there is no flux across the boundaries, thesoil water potential h is due only to gravity which is repre-sented by z in (6). Equation (6) shows that the distributionof h throughout the hill slope domain is linear in z [19]. Sev-eral simulations for this set of boundary conditions verifiedthis result, and again, at least four figure accuracy was ob-tained.

While implicit finite differencing schemes are generallyconsidered to be unconditionally stable compared to explicitfinite differencing schemes, it is found that good accuracyis not always maintained as �x,�z and �t are decreasedunless the following condition is satisfied [20]:

σ = �t

�x�z< 0.5. (5)


After varying �x,�z and �t in the computer simulations,it was found that the scheme is stable for a very largerange of σ but the results are not necessarily accurate.For σ > 1/2, values obtained could be unrealistic. Thisindicates that the numerical scheme does indeed requireσ < 1/2, for accurate results to be obtained.

3.3. Constant flux boundary conditions

Another check of accuracy arises from testing the nu-merical results of the scheme with available analytical so-lutions. Analytical solutions for infiltration into long planarhill slopes of homogeneous isotropic soil have been obtainedfor a slope angle γ (measured from the horizontal axis), andboundary conditions θ = θsat = constant [10]. Beyond acertain distance downwards from the slope crest, the infiltra-tion into the slope along rotated co-ordinates (x∗, z∗), wherex∗ = x cos γ −z sin γ and z∗ = −x sin γ +z cos γ , acts likethe one-dimensional vertical infiltration case. Physically thismeans that the water flow into the soil cover is initially nor-mal to the sloping surface and that the wetting front advancesparallel to and away from the sloping surface. Knowing thatthe water flow into the dump along the top sloping surfaceis independent of x∗ (except for a small upper region of thetop slope), (1) may be written as

∂θ

∂t= ∂

∂z∗

(Dw

∂θ

∂z∗

)− dK

dθ

∂θ

∂z∗ cos γ, (7)

where Dw = K dh/dθ is diffusivity. In rotated coordi-nates [10].

t = 0, z∗ � 0, θ = 0, (8a)

t > 0, z∗ = 0, −Dw∂θ/∂z∗ + K cos γ = R cos γ.

(8b)

Equations (7) and (8) are identical to equation (1a) of [21]with z replaced by z∗ and K replaced by K cos γ . For Dw =D0(1 − ν1θ)−2, K = (K1 + K2θ + K3θ

2)/(1 − ν1θ) (withD0, ν1,K1,K2,K3 being constants). An exact solution tothis system is given by equations (10) and (11) of [21] butwith dimensionless variables x∗ and τ redefined as x∗ =R cos γ z∗/D0, τ = R2 cos2 γ t/D0 .

The constant flux case is applied by assigning non-zerovalues to R and selecting the natural drainage boundary con-dition (3e) along DE. Direct comparisons between the nu-merical scheme applied to this problem, and the exact so-lution [21] can now be made. Comparing the exact so-lution with those obtained numerically gave agreement tothe fourth significant figure. Here the simulations involvingvariations of �x, �z and �t revealed that σ = 0.01, yieldedstable solutions and four figure accuracy.

4. Results

The model can be used to investigate the effects that hillslope angle γ , depth of soil cover AH and hill top width

AB have on water flow in the dump domain. The followingtwo sections investigate both water accumulation and waterdrainage within the dump. Both these features are examinedsince time spent in residence and the ability of the dump todischarge uncontaminated water once rainfall has ceased, isof great importance and interest.

Waste dumps are mechanically inherently stable with an-gles of repose ranging from 1◦ to 38◦ (with most being in therange 11◦ to 30◦) so γ has been restricted to this range [22].The cases where R > 0, can involve quite steep wetting gra-dients, requiring �x,�z and therefore �t to be smaller toensure σ < 1/2.

4.1. Water accumulation

Saturation is most likely to first occur at point H, in thedump, for constant rainfall. Investigation of water accumu-lation and drainage at this point should be beneficial in deter-mining which particular characteristics of the dump are mostuseful in promoting water flow away from the waste. Let tsrepresent the time required for water to collect to saturationat point H. Table 1 contains details of the geometry of sev-eral test dumps, along with the corresponding times to satu-ration ts , at H. All dumps are 5 meters high (i.e., Lz = 5 m),resulting in a half width of 12.4 meters for 22◦ dumps (i.e.,Lx = 12.5 m), 8.32 meters for the 31◦ dump and 7.14 metersfor the 35◦ dump. The rainfall is expressed in dimensionlessform where Ks = K(θsat). All dumps had an uniform initialdistribution of h(t = 0) = h0 = −38.

As time progressed, the numerical solution showed thatthere was an accumulation of water at the bottom of the hillslope and also along the top of the no flow sloping bound-ary (FG). Examining the behaviour of ts alone can be mis-leading. This will be seen in the next section where wa-ter drainage is considered. Taking Dump 1 as the standarddump design and comparing the resulting ts after modify-ing a certain physical characteristic of Dump 1 leads to thefollowing:

(i) Lengthening hilltop width AB (Dump 2) results in aslightly higher value of ts . As AB continues to lengthen,ts approaches the one-dimensional vertical infiltration value.This occurs since the greater the length from H to G, (i.e.,the larger the length of AB), then the flow around G has lessinfluence on the water accumulating at H. Water flow fromthe sloping boundary BC is initially normal to that bound-ary before flowing vertically in the layer. When H is rel-atively close to G (i.e., small hilltop width), there is a rel-

Table 1Characteristics of the test dumps.

γ R/Ks AB (m) AH (m) ts (hr)

Dump 1 22◦ 0.5 0.5 0.6 0.9485Dump 2 22◦ 0.5 1.5 0.6 0.9526Dump 3 22◦ 0.5 0.5 1.2 1.8741Dump 4 31◦ 0.5 0.5 0.6 0.9451Dump 5 22◦ 0.9 0.5 0.6 0.6099Dump 6 35◦ 0.9 0.5 1.2 1.22


atively greater volume of water converging towards H andwater collects at H more quickly than for the longer hilltopwidth.

(ii) Increasing soil depth AH (Dump 3) obviously leadsto an increase in ts since there is more soil for the infiltrat-ing water to travel through. There is also a greater soil voidvolume per unit length of AB, to store and hold water. In-creasing AH by a factor of two has basically doubled ts .

(iii) Increasing the hill slope angle γ (Dump 4) hasslightly reduced ts , i.e., water is collecting faster at H. Sincewater flow from the sloping boundary is initially normal tothat boundary, increasing the angle of the sloping boundaryincreases the horizontal flow components into the dump, es-pecially on the sloping face near B.

(iv) A larger flux (Dump 5) reduces ts due to the quickerinfiltration of water. This case has been included for com-parison in section 2.

4.2. Water drainage

The longer water resides in the top section of the wastedump (ABGH), the greater the likelihood for penetration ofthe capillary barrier. Examining the behaviour of h throughtime at various points in the domain can provide insight intohow water drains from the dump once rainfall has ceased.

Figures 3, 4, 5 and 6 are graphs of h(t) taken from thesimulations of the dumps 1, 2 and 5 in table 1. The corre-sponding graphs for Dumps 3 and 4 are very similar and arenot presented here. In each case, rainfall is set to zero oncewater collects to saturation at H, i.e., R = 0 for t > ts . Ofcourse, once saturation has occurred, water penetrates thecapillary barrier but in these simulations, the barrier is left

as impermeable to investigate drainage and transport proper-ties of the capillary barrier itself. Since artificial barriers andgeomembrane liners are sometimes used [5], these dumpscan be regarded as always having a no flow condition on thebottom boundary.

4.2.1. Dump 1Figures 3 and 4 show h(t) at all corner points of the do-

main of Dump 1 (A through to H) as well as the points Tm

(mid-point of the top sloping boundary) and Bm (midpoint ofthe bottom sloping boundary). figure 3 is an enlarged viewof the first section of figure 4 and shows water accumulationat the above points during rainfall.

(a) For t < ts (where R �= 0), the graphs for points Aand B follow a typical time-to-ponding curve obtained fromexperiments for one-dimensional infiltration [23]. Note thatfor R/Ks = 0.5, the h values at both points A and B leveloff to h = −12.36, a value of h that corresponds to 0.5Ks .Points Tm and C, being on the top surface of the dump, alsofollow this type of curve but lag slightly behind A and B.Point Tm lags behind A and B at first since, at early times, itis subject to a rainfall rate of R cos γ whereas A and B aresubject to a rainfall of R. At later times however, more waterhas infiltrated the top sloping surface causing ∂h/∂x → 0and ∂h/∂z → 0. This leads to Tm being subject to the sameboundary condition as A and B and hence, h(t) at Tm ap-proaches h(t) at A and B. Like A and B, point C receivesrain at the rate of R. However, it then starts to feel the influ-ence of drainage taking place at D which causes it to beginto lag behind A and B. At later times, ∂h/∂x → 0, and

Figure 3. Graph of h(t) for points in Dump 1, showing details for short time scale.


Figure 4. Graph of h(t) for points in Dump 1, for longer time scale.

Figure 5. Graphs for h(t) for points in Dump 2, for longer time scale.

∂h/∂z → 0, as in the case of Tm, resulting in the value ofh(t) at C , approaching the values of h(t), at A and B.

At point D, h(t) remains essentially constant initially asany moisture present at this point can drain freely. The mois-ture content at D does not increase until the foot of the wet-ting front arrives. This behaviour also occurs at points Eand F, both of which lag behind D since the wetting fronthas further to travel to reach these two points. Point E stays

drier than F since water can drain freely from point E whileat F, water arrives from both the wetting front from the topsloping surface and from water movement along the bottomsloping boundary.

There are two parts to the wetting curves at points Bm, Gand H for t < ts . These three points sit on the impermeablebarrier and, hence, the initial part of the h(t) curve (wheret < 0.7 hours) is due to the natural redistribution of water


Figure 6. Graphs of h(t) for points in Dump 5, for longer time scale.

in the soil. For t > 0.7, the wetting front has arrived atthese points and water begins to accumulate. Close inspec-tion shows that H is wetter then G at first since water hassome ability to drain down the bottom slope away from Gbut the wetting front hits G just slightly before reaching H.This is due to water flow from the top sloping surface ini-tially being normal to that surface and thus water at point Garrives from both the AB boundary and the BC boundary.However, water drainage from G then occurs which cannothappen at H and, thus, saturation first occurs at H. Redistrib-utive flow also occurs initially at Bm but lags behind that ofG and H since Bm lies on a sloping boundary.

(b) For t > ts , where R = 0, the h(t) curves decreaserapidly for the points which lie on the top surface (A, B, Cand Tm) (see figure 3). Water drains away from B slightlyfaster than from A. At A, water can only drain vertically andinto an area that has a lower impermeable barrier while waterat B has a higher degree of access to any horizontal flowcomponents that have developed around point G and thuscan drain away both vertically and parallel to the slopingsurface.

Water at C and Tm also starts to drain immediately. How-ever, Tm only drains to a certain level. This level is reachedwhen water flow parallel to the sloping surface developsand constantly replenishes the water at Tm that drains away.Once the water supply above Tm has been exhausted, thevalue of h at Tm starts to decrease again; i.e., near t = 15to 20 hours (see figure 4). This double step drainage pro-file does not happen at C (and therefore D) since, due tothe drainage condition along the bottom boundary, the waterflow becomes vertical before reaching C.

Drainage from D lags behind that of C since water drain-ing from D is resupplied by that water in the soil above D.Once this water has passed through D and out of the dump,

then the value of h(t) at D drops away. Water continues tocollect at the points on the bottom surface (E, F, Bm, G andH) after rainfall has ceased until drainage has a chance totake effect and h(t) starts to decrease. The largest amount ofwater accumulates at H. Once water starts to drain from G,it also starts to drain from H almost immediately, though Gremains drier than H at all times. As water from these twopoints drains down slope, it replaces the water that drainsfrom Bm, hence the value of h(t) at Bm levels off until h atH and G returns to negative values. This phenomenon alsocauses water at G and H to drain more quickly than at Eand F.

4.2.2. Dump 2The results for Dump 2 (figure 5) are similar to the re-

sults for Dump 1 (figure 4), both during rainfall and afterrainfall has ceased. However, there are differences arisingfrom the effect of lengthening the hill top width AB, andthese are seen by comparing h(t) for the two dumps at par-ticular points. By lengthening AB, drainage effects aroundG are further away from H and hence, not as easily felt at H.This reduces the size of any horizontal flow components thatmight develop at or near H. Thus a larger amount of wateraccumulates at H after rainfall has ceased (i.e., h(t) peaksat a higher value) and secondly, water drainage from H isconsiderably slower (figure 5). As a result, the difference inmoisture between H and G and between A and B as drainageoccurs, is greater for the longer hilltop width.

In figure 5, the rate at which h decreases at A slows sud-denly at approximately 1.5 hours due to water along AHbeing further away from G. Thus water drains more slowlyfrom A as a result of water draining more slowly along H toG, i.e., the vertical gradients in h are smaller in magnitudeas a result of increasing the distance from H to G.


4.2.3. Dump 3In this case, the depth of soil cover AH has been doubled.

For t < ts , the values of h(t) at all points are basically thesame as for 1 (figure 3) and are not presented here. However,once saturation has occurred at H and rainfall ceases, thereexists a much larger volume of water in the soil layer andtherefore, h reaches higher values at all points on the lowerboundary (points H, G, Bm, F and E). The most noticeablefeature is that h(t) peaks higher at Bm than at H. Again, thisis due to the larger amount of water in the soil cover.

4.2.4. Dump 4The graphs for this case are similar to the others and will

not be presented here. A steeper hill slope angle γ allowsfor quicker drainage of water from the top region ABGH.This reduces the time of residence of the water along theboundary HG. Water drainage at Tm and Bm occurs earlierthan for Dump 1. Points D and C after ts for γ = 31◦ arealways wetter at any given time than for D and C after ts forγ = 22◦. This occurs because increasing γ will increasedown slope flow, causing a faster resupply of water downslope to C and D.

4.2.5. Dump 5A higher flux of R/Ks = 0.9 causes h at A and B to level

off at a higher value prior to ts . This value, h = −5.60,corresponds to 0.9Ks . Even though saturation at H occursearlier (due to the higher rainfall rate) and therefore rain-fall ceases earlier, a greater amount of water has actuallyinfiltrated into the soil cover. An interesting feature of fig-ure 6 is that h(t) at H still drains as quickly as for Dump 1even though there is a greater volume of water present in thesoil above HG at the time of cessation of rainfall. This is inagreement with [11] in that when rainfall ceases, a changein the boundary conditions occurs and horizontal moistureand pressure gradients develop earlier, inducing down slope

drainage and resulting in the diversion of a greater amountof water.

4.3. Water movement

The contour plots of h give a clearer picture of how infil-trating water enters, accumulates and drains from the dumpdomain. Figures 7, 8, and 9 show contour plots for Dump 6where the parameters are given in table 1. Dump 6 has a rel-atively steep slope, a thick soil layer and high rain fall ratecompared to the other dumps listed in table 1. The uniforminitial distribution is h0 = −38. The figures show the con-tour plots of h at times ts (figure 7), at ts +4 hours (figure 8),and at ts + 8 hours (figure 9).

In figure 7, a wetting front has carried water down to thelower boundary and there is a saturated zone near the pointG. As the wetting front moves further from the upper sur-face, the contour lines gradually curve around to becomevertical near D due to the drainage condition along DE. Con-tour lines also develop along the lower boundary FGH, ad-vancing away from FGH as water redistributes and collectsalong the capillary barrier. In figure 7, the dump is driestdown near the edge EF due to the boundary condition onDE allowing water to drain freely. As more water collectsalong the capillary barrier, it advances down EF as waterflows along the lower sloping surface FG.

Unlike Dumps 1 to 5, saturation first occurs at G inDump 6. This is due to a combination of factors, the deepersoil depth, higher flux and steeper hill slope angle, all ofwhich contribute to water infiltrating from the top slopingsurface converging on G. Figure 7 shows that, at ts there is asaturated zone forming at G.

Figures 8 and 9 show how the short hilltop width (0.5 m)and steeper hill slope angle (35◦) interact to increase thedrainage of water from the top region of the dump. As thewater drains, the saturated zone splits into two zones, onealong HG, the other along FG. Since HG is horizontal, the

Figure 7. Contours of h(t) for Dump 6, at t = ts hours.


Figure 8. Contours of h(t) for Dump 6, at t = ts + 4 hours.

Figure 9. Contours of h(t) for Dump 6, at t = ts + 8 hours.

water collected there needs to be as close as possible to Gin order to drain quickly and effectively. The shorter hill-top width encourages greater horizontal flow components todevelop along HG, and water can drain more readily fromH. The steeper hill slope angle increases down slope flowwhich in turn, magnifies the flow components around G.Only 8 hours after ts , there is no saturated zone remaining inthe top region of the dump, i.e., no saturated zone anywhereon HG. Here the bulk of the water is located down the lowersloping interface and down to the free drainage boundary.

5. Conclusions

The model presented here predicts water transportthrough a homogeneous soil representing the capillary bar-

rier for the simplified waste dump problem. The model hasthe capability to vary the dimensions and angle of the wastedump as well as the depth of the soil cover. A local accuracycriteria was applied for the waste dump problem, namelythat σ < 1/2. Stability is still maintained for σ > 1/2 butaccuracy is lost.

The model revealed that different combinations of hillslope angle and hilltop radius can result in different behav-iour of water flow through the waste dump. Examinationof ts alone suggested that a steeper hill slope angle andshorter hilltop width can have a detrimental effect on wa-ter flow around the waste material. Water accumulates at Hmore quickly due to water infiltrating from the top corner ofthe sloping surface. However, an examination of the waterdrainage from the top region of the dump (ABGH) shows


that both a steeper hill slope angle and shorter hilltop widthare actually beneficial in preventing acid drainage since boththese factors allow larger and more effective water flow com-ponents to develop around G. A shorter hilltop width placesany water near H in closer proximity to the drainage fluxesthat develop around point G.

A steeper hill slope angle allows quicker down slopedrainage of water and also encourages stronger flow com-ponents to develop around G. A deeper soil layer results inhigher peak values of h(t) at H and G and, at first glance,this appears to be contrary to the objective at hand. Note,however, that once saturation has occurred anywhere alongthe capillary barrier (EFGH), water will penetrate into thecoarse material.

For the case where there is a membrane present alongEFGH, then these results can be taken further. Concerns overthe integrity and long term performance of artificial mem-branes have been reported [1]. Where the integrity of theartificial membrane has been compromised through cracksand ruptures, preferred flow paths develop into the waste.The objective of reducing acid mine drainage then centreson removing as much water from the top region as possiblebefore it has a chance of entering the waste material throughthese openings. While higher peaks in h(t) will cause higherpressures on the membrane and may increase the probabil-ity of cracks and ruptures, these same peaks will also causestronger down slope driving forces. A shorter hilltop widthand steeper hill slope angle both help down slope drainage.This drainage will occur more quickly if the soil cover is setrelatively deep and thereby takes advantage of the strongerdown slope driving forces that develop. This conclusion is inagreement with [7] which stated that “actual liner thicknessat the site should be as large as possible and only limited byengineering and economic constraints”. The opposing op-tion of shallow soil covers may reduce the peaks in h(t) afterrainfall has ceased but this happens at the cost of horizontaland vertical flow components that are smaller in magnitude.This, in turn, would result in saturation zones existing in thetop region of the dump for longer periods of time and thusincreasing the likelihood of water entering the waste mater-ial.

Note that this model does not require extensive modifica-tions to handle multiple layers of anisotropic soils. Finally, ifwater does break through into the waste material, then accu-rate and reliable prediction of water flow through the wastealong with the transport of subsequent contaminants, willdepend on an accurate model of the flow field in the wastematerial itself.

Appendix

The hill slope domain is overlaid with a rectangular mesh,where i is the vertical counter (positive downwards), m

is the horizontal counter (positive right) so that any point(zi, xm) = (i�z,m�x) is represented by (i,m) on the grid.Time is also discretized with tj = j�t , and j is the time

counter. Then hji,m represents the value of h at the node

(i,m) at time j . Central differencing of all space derivativesis used to discretize the right hand side of (1), while forwarddifferences are used to represent the time derivative [9].

Applying the ADIPIT scheme to (1) in the horizontal di-rection means ∂h/∂x is discretized on the 2k+1 iterate while∂h/∂z is discretized on the previous iterate 2k. This gives

Cj+1/2,(2k)wi,m

hj+1,(2k+1)

i,m − hj

i,m

�t

+ IkKj (

hj+1,(2k+1)i,m − h

j+1,(2k)i,m

)

= 1

�x2

[K

j

i,m+1/2

(h

j+1,(2k+1)

i,m+1 − hj+1,(2k+1)

i,m

)− K

j

i,m−1/2

(h

j+1,(2k+1)i,m − h

j+1,(2k+1)

i,m−1

)]

+ 1

�z2

[K

ji+1/2,m

(h

j+1,(2k)i+1,m − h

j+1,(2k)i,m

)

− Kji−1/2,m

(h

j+1,(2k)i,m − h

j+1,(2k)i−1,m

)+ �z

(K

ji−1/2,m − K

ji+1/2,m

)]. (A.1)

The second term in (A.1) is a correction term represent-ing the difference in the estimates of h

ji,m from consecutive

sweeps. For the vertical direction, ∂h/∂z is discretized onthe 2k + 2 iterate while ∂h/∂x is taken from the 2k + 1 iter-ate, giving

Cj+1/2,(2k)wi,m

hj+1,(2k+2)i,m − h

ji,m

�t

+ IkKj (

hj+1,(2k+2)i,m − h

j+1,(2k+1)i,m

)

= 1

�x2

[K

j

i,m+1/2

(h

j+1,(2k+1)

i,m+1 − hj+1,(2k+1)i,m

)− K

j

i,m−1/2

(h

j+1,(2k+1)

i,m − hj+1,(2k+1)

i,m−1

)]

+ 1

�z2

[K

j

i+1/2,m

(h

j+1,(2k+2)

i+1,m − hj+1,(2k+2)i,m

)

− Kj

i−1/2,m

(h

j+1,(2k+2)i,m − h

j+1,(2k+2)

i−1,m

)+ �z

(K

j

i−1/2,m − Kj

i+1/2,m

)]. (A.2)

The conductivities between nodal points are evaluated as thearithmetic mean of the conductivities at the neighbouringnodes, e.g.,

Kj

i+1/2,m = 1

2

[K

(h

j

i+1,m

) + K(h

ji,m

)]. (A.3)

In (A.1) and (A.2),

Kj = K

j

i−1/2,m +Kj

i+1/2,m +Kj

i,m+1/2 +Kj

i,m−1/2, (A.4)

where, for k � 1,

Cj+1/2,(2k)wi,m = dθ

dh= θ

j+1,(2k)i,m − θ

ji,m

hj+1,(2k)

i,m − hj

i,m

, (A.5)

otherwise

Cj+1/2,(2k)wi,m = dθ(h

j+1/2,(2k)i,m )

dh. (A.6)


Here, we take

hj+1/2,(2k)i,m = 1

2

(h

j+1,(2k)i,m − h

ji,m

). (A.7)

In the above scheme, Kji,m denotes the conductivity K at

node (i,m), and time j . Terms including the iteration pa-rameter Ik = Nk are added to speed convergence of the it-erative procedure. Following [9], numerical testing showedquick convergence with N = 0.55.

Thus, ADIPIT generates a linear tridiagonal matrix sys-tem

Hj,(2k)hj+1,(2k+1) = b − V j,(2k)hj+1,(2k) (A.8)

for the horizontal sweep and

V j,(2k)hj+1,(2k+2) = b − Hj,(2k)hj+1,(2k+1) (A.9)

for the vertical sweep. The vector b contains known valuesfrom evaluating the boundary conditions (using central dif-ferencing methods). However, the two sloping boundariesBC (top slope) and FG (bottom slope), require careful con-sideration. Where a grid point (i,m) lies on a boundary, then(A.1) and (A.2) involve artificial grid points which lie out-side of the domain. These boundaries need to be evaluateddifferently for the two directional sweeps in order to preservelinearity. Thus, for the horizontal sweep, the x-direction(or x-derivative) will be evaluated at the 2k + 1 iterationwhile the z-direction (or z-derivative) will be evaluated at theknown 2k iteration. For the vertical sweep, the x-directionis known at the 2k + 1 iteration and the z-direction will beevaluated at the 2k + 2 iteration.

Performing central differencing on (3b) and rearrangingfor the artificial points, the horizontal sweep case gives

hj+1,(2k+1)

i,m+1 = hj+1,(2k+1)

i,m−1

+ 2�x

tan γ

[R

Kji,m

− 1 + hj+1,(2k)

i+1,m − hj+1,(2k)

i−1,m

2�z

], (A.10)

while the vertical sweep gives

hj+1,(2k+2)

i−1,m = hj+1,(2k+2)

i+1,m

+ 2�z

[R

Kj

i,m

− 1 − tan γh

j+1,(2k+1)i,m+1 − h

j+1,(2k+1)i,m−1

2�x

].

(A.11)

Equation (A10) shows that artificial points arising from thex-derivative (i,m + 1), iteration (2k + 1) depend upon val-ues from the z-derivative of the previous iteration (i −1,m),iteration 2(k). Similar behaviour is evidenced by (A.11).This means that there is a “doubling-up” of the artificialpoints outside the sloping boundaries in that it plays differ-ent roles for the vertical or horizontal sweep. Each artifi-cial point requires two values for each full iteration; a valuefrom the x-derivative and a value from the z-derivative. Ifthe doubling-up of artificial points does not take place, thescheme converges only partially before oscillations occur at

the boundary points. The bottom sloping boundary FG canbe handled in exactly the same manner.

As stated earlier, the ADIPIT scheme involves perform-ing pairs of sweeps (one horizontal sweep followed by onevertical sweep) until the difference between the results of ahorizontal sweep and its subsequent vertical sweep is negli-gible, i.e.,

max∣∣hj+1,(2k+2)

i,m − hj+1,(2k+1)

i,m

∣∣ < ε ∀i,∀m (A.12)

with ε = 10−4 [9] for this study. Once this is achieved,the resulting values of h throughout the domain are takenas the solution for that particular time and the time step isadvanced.

References

[1] A.MacG. Robertson and J.P. Barton-Bridges, Management options forthe short-term and long-term control of acid mine drainage, in: Envi-ronmental Workshop Proceedings, Vol. 2 (Australian Mining IndustryCouncil, 1988).

[2] J.R. Harries, Acid drainage from waste rock dumps at mine sites (Aus-tralia and Scandinavia), Australian Nuclear Science and TechnologyOrganization ANSTO/E692 (1990).

[3] I.P.G. Hutchinson and R.D. Ellison, Mine Waste Management (Cali-fornia Mining Association, Lewis, Michigan, 1992).

[4] O.O. Hart, Water for the mines and mine water – a perspective in treat-ment and re-use of water in the minerals industry, The South AfricanInstitute of Mining and Metallurgy (1989).

[5] A.MacG. Robertson, Long term prevention of acid mine drainage, In-ternational Conference on Control of Environmental Problems fromMetal Mines, Federation of Norwegian Industries and State PollutionControl Authority, Norway (June, 1988).

[6] J.C. Stormont, C.E. Morris and R.E. Finley, Capillary barriers for cov-ering mine wastes, in: Proceedings of the Third International Confer-ence on Tailings and Mine Waste, January (Fort Collins, Colorado,1996).

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[9] G. Vachaud, M. Vauclin and R. Haverkamp, Towards a comprehen-sive simulation of transient water table flow problems, IFIP workingconference on modelling and simulation of water resources systems,Univ. of Ghent, Belgium (1974).

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Sci. 29 (1978) 22–31.[16] Y.Z. Boutros, H. Mansour El-Saadany and I. El-Awadi, Computer

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187.[19] G. Pantelis, J. Appl. Math. and Phys. (ZAMP) 36 (1985) 648–657.[20] G.D. Smith, Numerical Solutions of Partial Differential Equations:

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[21] G.C. Sander, J.-Y. Parlange, V. Kuhnel, W.L. Hogarth, D. Lockingtonand J.P.J. O’Kane, J. Hydrol. 97 (1988) 341–346.

[22] A.R. Milnes, W.W. Emerson, B.G. Richards, R.W. Fitzpatrick andA.B. Armstrong, The long term stability of waste-rock dumps in theRanger Project Area, Northern Territory, Australia, in: Environmental

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saturated and unsaturated water flow in inclined porous media

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