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METHODOLOGY FOR DRENABILITY STUDIES OF STACKED MINING
COARSE TAILINGS
J. C. Machado Júnior
Departamento de Engenharia Civil, Universidade Federal de Ouro Preto
W. L. Oliveira Filho
Departamento de Engenharia Civil, Universidade Federal de Ouro Preto
Abstract: This paper presents a methodology for drenability studies of coarse mining tailings.
These studies have a particular impact in the wall stability and construction methods of stacked
mining coarse tailings structures (Gomes et al. 1999a). In that context, flow processes on
unsaturated soil are the major complicated factor. For that purpose, a short review of this flow
problem is presented, including theory and flow material functions needed in the analysis and their
determination. Some common scenarios of flow problems related to tailings piling up are discussed
and analyzed using a commercial code, SEEP/W Ò, as a numerical tool. This paper also aims to
make a contribution to the diffusion of unsaturated soil mechanics out of the academic circles.
1. INTRODUCTION
Nowadays the mining tailings disposal
management is often an activity of primary
importance for the success of a mining
operation. The economics and also the
environmental issues associated with that
activity have made the mining industries to
optimize the disposal areas and to develop new
processes to discard tailings. To pile up coarse
tailings is one of these new techniques for the
coarse fraction of the waste generated by the
iron ore processing plants (Gomes et al.
1999a). In this regard, one of the key issues of
the stability analysis of these structures is the
question of drenability of the tailings. The
project will be more economical and safe,
depending on how precise a model can be
made of the flow processes taking place in the
tailings.
However, a factor that complicates the flow
analysis is that the process of drainage needs to
deal with unsaturated porous media, and a
complete new field of study for geotechnical
engineers has to be faced. In this paper, an
effort to popularize unsaturated flow analysis
out of the academic circles is made by
addressing a methodology to study a real
problem on mining tailings disposal.
2. FUNDAMENTALS OF UNSATURATED
FLOW
 brief review of unsaturated flow theory is
worth to be presented here since there is a lack
of general understanding in that subject within
our community of geotechnical engineers.
2.1. THE GOVERNING EQUATION
The governing equation of the flow
processes in soils is well known from
textbooks of basic soil mechanics (Craig 1996,
Lambe and Whitman 1979, Ortigão 1996).
This equation can be expressed in a vectorial
form like,
 ( )
dt
dM
dzdydx ww
T =r vÑÑ (1),
where, rw is the water density, v is the darcian
velocity vector, ÑÑ is a differential operator so
that, ÑÑT = [ ¶/¶x ¶/¶y ¶/¶z ], MW is the water
mass, and t is the time.
234
Equation (1) relates the mass balance in a
three dimensional set, x, y and z, with the
storage term on the right hand side. The latter
can be expressed in terms of soil porosity, n,
and degree of saturation, S, so that,
 dxdydznSM ww r= (2)
and (1) becomes
 ÑÑ T W
WSn
t
( )
( )
r
¶ r
¶
v = (3).
Depending on the way that the properties
rw, S, and n are considered, different soil flow
processes are described by (3). The usual
simplification for most of soil engineers is to
consider rw, S and n as constants, and
therefore their derivatives are equal to zero.
The right hand side then is equal to zero and
this is the case of saturated, steady state flow
analysis.
 ÑÑ T ( )v = 0 (4).
In another case, when hypotheses of
isothermal conditions and saturated porous
media are introduced, the derivatives on rw
and S are zero, and (3) becomes
 ÑÑ T
n
t
( )v =
¶
¶
 (5),
which is the general equation for (large strain)
consolidation analysis.
The less known form of (3), at least for
geotechnical engineers, appears when the
derivatives of rw and n are zero, and (3)
becomes
 ÑÑ T n
S
t
( )v =
¶
¶
 (6).
Considering the Darcy’s law, (6) becomes
 ÑÑ ÑÑT h n
S
t
( )K =
¶
¶
 (7),
where K is the hydraulic conductivity tensor
and h is the total head.
The governing equation in this form
describes the transient analysis of unsaturated
flow of incompressible porous media and
isothermal conditions.
2.2 Richard's equation
Further developments of the governing
equation for unsaturated flow are possible. The
concept of volumetric water content, q, is very
often needed in such cases. The relation
between water volume in the voids, Vw , and
total soil volume, V, defines this quantity.
Thus,
 q = = ×
V
V
n Sw (8).
Also, the total hydraulic head, h, can be
divided in two other terms, the elevation head,
z, and pressure head, y. Thus, 
 h z= + y (9).
Equation (7) then becomes,
 ÑÑ ÑÑT
t
( )Ke K+ =y
¶q
¶
 (10),
where e is a vector with the component z
unitary, so that eT=[ 0 0 1 ].
The last development of the flow equation
is to take into account that K and q are
functions of pressure head in unsaturated flow.
It means that,
 K = K (y) (11)
and
q = q(y) (12),
which implies,
 
¶q
¶
y
¶y
¶t
c
t
= ( ) (13),
where c(y)=¶q/¶y is called specific capacity
retention.
By considering these relationships, the
primary dependent variable in the flow
equation becomes y and the governing
equation assumes the form
235
 ÑÑ ÑÑT C
t
( ( ) ( ) ) ( )K e Ky y y y
¶y
¶
+ = (14).
This y-based equation of flow for transient
flow through an unsaturated media is called
Richard's equation (Freeze and Cherry 1979).
It’s a nonlinear second order partial differential
equation. The nonlinearity on (14) is because
the derivative coefficients are not constants but
dependent on the primary variable, y. The
solution of (14) must satisfy the problem
boundary conditions. These can be a
prescribed pressure head boundary condition
(Dirichlet condition) so that,
 y=y )t,(x on G1 (15),
or a prescribed normal flux boundary condition
(Neuman condition) so that,
 [ ] vKeKn =yy+y ÑÑ)()(T on G2 (16),
where G=G1+G2, is the boundary of the volume,
V; y andv are prescribed values, nT is the
unit normal vector on G2 and x is a position
vector.
The solution of (14) must also satisfy the
initial condition,
y y( ,0)x = 0 (17),
where y 0 is a known value.
To solve (14), flow material functions (11)
and (12) are required. The determination of
these functions are discussed in section 3.
Closed form solutions of (14) exist only in
very few cases what makes the use a numerical
scheme a necessity.
2.3 Applications
To illustrate the use of the flow equation,
the analyses of two very common field
situations involving flow in unsaturated media
are presented.
2.3.1 Infiltration in a soil column
To study this situation, an unsaturated soil
layer is thought to receive on its top boundary
a constant rain precipitation, R, with
dimensions [L/T]. Because the problem can be
considered one-dimensional, the governing
equation, (14), can be written as
 
¶
¶
y
¶y
¶
y
¶y
¶z
k
z
c
tz
( )( ) ( )1+
é
ëê
ù
ûú
= (18).
From Darcy's law, v=ki, where i = ¶h/¶z
and ¶h/¶z = 1+¶y/¶z, hence at top boundary
 R
zk= +æèç
ö
ø÷1
¶y
¶
y( ) (19)
or
 
¶y
¶ yz
R
k
= -
( )
1 (20).
At the bottom boundary, a Dirichlet
prescription is usually assumed, like
 y = 0 (21).
(18), (20) and (21) constitute the boundary
value problem for infiltration in a soil column.
2.3.2 Drainage in a soil column
This situation can also be considered one-
dimensional and therefore the governing
equation looks the same as in the previous
case.
 
¶
¶
y
¶y
¶
y
¶y
¶z
k
z
c
tz
( )( ) ( )1+
é
ëê
ù
ûú
= (22).
The top boundary is now impervious which
implies R = 0 or
 ¶y/¶z = -1 (23).
At the bottom, it is usually assumed again,
 y = 0 (24).
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(22), (23) and (24) defines the boundary value
problem for drainage in a soil column.
3. DETERMINATION OF THE FLOW
MATERIAL FUNCTIONS
As it was mentioned, the determination of
k= k(y) and q = q(y) is required for the
solution of the governing equation. Those
functions are called characteristic curves. The
first is the hydraulic conductivity function and
the second is the soil-water characteristic curve
(SWCC). These curves often have sigmoid
shapes and present hysteresis for infiltration
and drainage cycles.
The determination of characteristic curves
can be achieved by specific laboratory or field
tests or even by correlation (especially the
hydraulic conductivity function). Benson and
Gribb (1997) present a comprehensive review
of the procedures available.
In the laboratory the water retention curve
can be obtained through the pressure plate test
according to the ASTM D2325-68. The setup
of the test is schematically shown in Fig. 1.
A chamber houses the soil specimen that is
placed on a porous disc of high air entry value.
The disc is in contact with a water reservoir at
atmospheric pressure. The difference between
the pore-air, ua, and pore-water, uw, pressures is
called matric suction and it’s created by
pressurizing the chamber with compressed air.
The amount of water that leaves the specimen
with the increasing pressure is monitored,
allowing the determination of a relationship
between the degree of saturation and suction.
Both drainage and infiltration cycles are run in
the test.
It is possible to measure the permeability
function directly but intrinsic test difficulties
and lack of accuracy very often prevent its
large use and lead the practitioner to use
indirect methods. The usual procedure is to
estimate the hydraulic conductivity function
from the SWCC. The greater reliability and
short period of time required for the SWCC
determination favor this approach. The
correlation also has its strength in the intimacy
of the two relationships. The SWCC describes
the amount of water present in a soil under
various matric suctions. This curve essentially
indicates the space available for the water to
flow through the soil at various matric suctions
since water can only flow through the water-
filled pores. Therefore the shape of the curve
can be used to estimate the permeability
function. The function decreases
monotonically from the saturated permeability,
ksat, at zero matric suction and assumes lower
values as the matric suction increases.
Figure 1 – Schematic diagram of a pressure
plate extractor (Rahardjo and Leong, 1997)
There are many empirical equations that
relate the hydraulic conductivity to the
volumetric water content or degree of
saturation of the soil (Gardner, 1958, Brooks
and Corey, 1964; van Genuchten, 1980). The
van Genuchten equation is given below:
 kr e e= - -
æ
è
ç
ö
ø
÷
é
ë
ê
ê
ê
ù
û
ú
ú
ú
-
-
q q
b
b
b
1 2 1
1
1 2
1 1/ (21)
where,
 kr – relative hydraulic conductivity,
 b - constant,
and hydraulic conductivity is given by
 k = ksat ×kr (22).
The b - coefficient of (21) is a fitting
parameter obtained from the experimental data
of SWCC and the theoretical model given
below also due to van Genuchten,
 
( )[ ]
q
a b b
e
h
=
+
-
1
1
1
1 (23)
237
and
 q
q q
q qe
r
s r
=
-
-
 (24)
where,
 q - volumetric water content,
 qe – equivalent volumetric water content,
 qr – residual volumetric water content,
 qs – saturated volumetric water content,
 h = ( ua - uw ) – matric suction,
 a, b - constants.
4. NUMERIC ANALYSIS
In the following subitems a complete
example of a numeric simulation is given. The
analyses were performed using a commercial
computer program called SEEP/WÒ. This is a
finite element software used to model seepage
problems, developed by GEO-SLOPE
International Ltd.
4.1 Flow material functions
The characteristic curves used in the
analyses were obtained from the piece of
literature (SEEP/WÒ user’s guide) and
correspond to a fine sand, ksat = 4.3 x 10
-6,
n=0.22, d10 = 0.093 mm. Figure 2 and 3
illustrate the flow material functions used in
the analyses.
4.2 Scenarios for the numeric analyses
Several field situations can be analyzed
within a tailings management strategy. In this
paper three scenarios were envisioned for the
study of tailings drenability. These case studies
are very often found in practice and the flow
processes happen to be gravitational and one-
dimensional (Collins and Znidarcic, 1997).
The boundary conditions for these situations
are presented in terms of total head, H, total
nodal flow, Q, and flow per unit length, q.
In the first scenario, a one-meter thick
tailings column, initially saturated, is underlain
by a drainage layer, and is left to drain freely.
This situation is shown in Figure 4(a).
Figure 2 – Soil water characteristic curve
(SEEP/WÒ user’s guide)
Figure 3 – Hydraulic condutivity function
(SEEP/WÒ user’s guide)
The second scenario is a follow-up to the
previous case with the initial conditions
corresponding to the end of that analysis. In
the sequence, a constant flow rate is imposed
in the upper boundary, simulating rain
precipitation over the drained layer. Figure 5
(a) is a schematic drawing of that case study.
The last scenario allows the evaluation of
the flow pattern considering multiple layers in
different drainage stages. In the actual case, the
behavior of a newly disposed layer (saturated)
and its iteration with the underlain layer at
certain drainage stage are analyzed. This
scenario is sketched in Figure 6 (a).
4.3 Analysis Results
The results of the numeric simulation of a
draining soil column are shown in Figure 4 (b)
in terms of pressure head x depth x time. The
instant t = 0 reflects the initial conditions of the
problem (a hydrostatic water column). It can
be seen that for t > 0 the imposed boundary
conditions are Q = 0 at top boundary and H = 0
238
at the bottom boundary. Hydrostatic conditions
firstly develop at top portion of the column and
progress downwards with time. At the end
steady state conditions are reached with a
hydrostatic suction profile throughout the
entire column.
The results of the infiltration case are shown
in Figure 5 (b). At t = 0, the initial conditions
corresponding to a hydrostatic suction profile.
For t > 0, the boundary conditions are q = 4.3 x
10-6 m/s at the top boundary, and H = 0 at
bottom boundary. At the beginning of the
simulation, at the top portion, it can be seen
that high gradients are necessary to overcome
the low permeability values associated with
high matric suctions and to allow seepage to
take place. As the degree of saturation
increases, the entrancegradients are smaller,
tending towards a gravitational value of 1.
Because q = ksat, the steady state solution gives
a profile with constant suction from top to
bottom. The corresponding degree of
saturation profile would be similar with a
constant S = 1.0 from top to bottom.
The last case results are shown in Figure 6
(b). The initial conditions reflect two different
situations. From 0 to 1m a suction profile
corresponding to a layer at certain stage in a
drainage process is shown. From 1 to 2 m, a
hydrostatic positive porewater pressure profile
is seen. This one corresponds to a newly
disposed saturated layer. In the initial lapses of
time, t > 0, a drainage process of the top layer
is taking place while the bottom layer is
experiencing a saturation phase. At a certain
point in those flow processes, a typical steady
state infiltration profile can be seen but only
instantaneously, because for later lapses of
time, a soil column drainage pattern governs.
In the last stage, the two layers behave like
one, in a very similar way to the results of the
first case study. The boundary conditions in
this case are Q = 0 at the top boundary and
H=0 at the bottom.
Figure 4 - (a) Finite element mesh and boundary conditions to scenario #1; (b) SEEP/WÒsimulation
results to scenario #1.
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Figure 5 - (a) Finite element mesh and boundary conditions to scenario #2; (b) SEEP/WÒ
simulation results to scenario #2.
Figure 6 - (a) Finite element mesh and boundary conditions to scenario #3; (b) SEEP/WÒsimulation
results to scenario #3.
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4.4 Discussion
Questions about flow rate pressure head and
time required for water to flow can now be
easily and rationally addressed. Some
variations of the case studies described before
can be envisioned. Even more sophisticated
situations can also be considered such as two-
dimensional flow and different materials,
without introducing any major change in the
methodology here described.
This kind of analysis can be used in the
design of stacked mining coarse tailings
structures in areas such as wall stability
studies, monitoring programs and construction
methods (Znidarcic, 1999). These aspects will
be object of future work.
5. CONCLUSION
From the simple cases presented in this
paper it is apparent that the methodology has
great capability to expand the horizons to study
flow processes in unsaturated soils. The
requirements for this development include: to
perform new but not sophisticated tests, to use
a competent and reliable computer code such
as SEEP/WÒ and others, and most of all to get
acquainted with unsaturated flow processes.
An effort to bring about this issues has been
made in this work.
6. REFERENCES
Benson, C. H. and Gribb, M. M.
(1997).Measuring Unsaturated Hydraulic
Conductivity in Laboratory and Field,
Unsaturated Soil Engineering Practice.
Geothecnical Special Publication (68)
ASCE, 113-165.
Brooks, R. H. and Corey, A. T. (1964).
Hydraulic Properties of Porous Medium.
Hydrology Paper No.3, Civil Engineering
dept., Colorado State University, Fort
Collings, Colorado.
Craig, R. F. (1997) Soil Mechanics, Chapman
& Hall, sixth edition.
Collins, B. D., and Znidarcic, D. (1997),
Triggering Mechanics of Rainfall Induced
Debris Flows, Proceedings of 2nd Pan-
American Symposium on Landslides and
2nd Brazilian Conference on Slope Stability,
II PSL/COBRE, Vol.1, 277-286.
Freeze, R.A. & Cherry,J.A. (1979).
Groundwater, Prentice-Hall, Inc.,
Englewood Cliffs, New Jersey.
Gardner, W. R. (1958). Some Steady State
Solutions of the Unsaturated moisture Flow
Equation with Aplications to Eva poration
from Water Table . Soil Science, 85, 228-
232.
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SEEP/W – for finite element seepage
analysis. Version 3
Gomes, R. C., Araújo,L. G., Oliveira, W. L.F.,
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em Pilhas de Rejeito de Minério de Ferro
em Cava Exaurida de Mineração. VI
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Van Genuchten, M.T. (1980). A Closed Form
Equation for Predicting the Hydraulic
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