Land Subsidence (Proceedings of the Fourth International Symposium on Land
Subsidence, May 1991). IAHS Publ. no. 200,1991.
Nonlinear Modeling of Groundwater Flow and Total Subsidence
of the Mexico City Aquifer-Aquitard System
A. RIVERA & E. LEDOUX
Paris School of Mines- CIG. 35, Rue
Saint-Honoré 77305 Fontainebleau, France
G. de MARSILY
University P. et M. CURIE 4, Place Jussieu, 75230,
Paris Cedex 05, France
ABSTRACT The water budget in the Mexico city closed basin was
in equilibrium until the begining of the thirties, natural recharge, of
approximately8 m 3 s e c - 1 for a 1500 km2 surface area, was the main
source of some springs to the west and groundwater exploitation only
represented less than 1 %. In 1980 the water pressure had diminished
under the effect of a heavy groundwater pumping which exceeded natural recharge by more than 260 %. As a consequence the aquitards in
the system have compacted to a total land subsidence of about 6.5 meters in downtown area for that period. In this study a numerical model
is presented to simulate groundwater flow and total subsidence of multiaquifer systems. The model accounts for the non-linear compaction
and total subsidence on multilayered systems by coupling a simultaneous numerical solution of the groundwater flow equation with the onedimensional consolidation equation of aquitards through the Terzaghi's
effective-stress concept. An important issue of this investigation is the
model's capabilities to simulate the phenomenon at a regional scale.
The application of this model to the Mexico city's basin has proved of
great interest,, its aim is to quantitatively analyse: (1) the water budget in the basin, (2) the land subsidence due to overexploitation, and
(3) the response of the system to artificial recharge. The analysis of
the water budget in the basin permits the calibration of,the model in
steady-state. Simulations in unsteady-state are perforated for the period of 1930-1986 using the steady-state results as theiaitial conditions.
The calibration process for this period is done through two variables:
observed groundwater heads and land subsidence. The simulations are
done with the linear and non-linear versions of the model. The observed subsidence is reproduced with a great detail with the non-linear
version of the model. The linear version fails to reproduce the observed
phenomenon.
INTRODUCTION
The compaction of clayey layers is a phenomenon generally studied through the optic
of soil mechanics. In hydrogeology "the aquitard drainage models" are succesfully
applied if previously known levels fluctuations are imposed as boundary conditions.
In this work, we propose a deterministic model combining the mechanics of the
compaction of semi-pervious layers with the three-dimensional groundwater flow.
In Mexico, as is the case in general, reliable data are scarce when dealing with
aquitard parameters. Due to the time dependent response of aquitards to pumping in
aquifers and to the lack of adequate technics to analyse pumping tests in leaky aquifers,
45
A. Rivera et al.
46
many modelers are "forced" to infer aquitard parameters or to use the soil mechanics
techniques to obtain them. In the best of the cases vertical permeability (Kf) of clay
samples can be obtained from consolidation tests through the consolidation coefficient
Cv (see for instance Scott, 1963), and compressibility (a) derived graphically from void
ratio-applied effective stress (e — loga) curves (Jorgensen, 1980). The latter may in
turn be used to estimate the storativity of the aquitard in which the sample was taken.
A problem associated with this is the non-linear response of the clay sample to the
applied stress wich obviously also happens in the whole aquitard itself. In the last few
years some modelers have circumvented this problem by using different constitutive
relationships between soil engineering equations and groundwater flow equations so
that both K' and a can be stress-dependent parameters (Helm, 1976; Narasimhan and
Witherspoon, 1977; Neuman et al, 1982). Unfortunately most of this type of models
have only been applied at a local scale.
Various efforts to simulate the aquifer-aquitard system of the Mexico basin have
already been made (Cruickshank, 1982; Herrera et al, 1989; and Rudolph et al, 1989).
Only the last of these deals with the non- linearities of K' and a but is restricted to
a local analysis. Compaction and land subsidence from groundwater pumping have
been documented in many areas (see Poland, 1984) but only few models of regional
groundwater flow include the effects of permanent compaction.
The present model is based in the Paris School of Mines code NEWS AM (Marsily
et al, 1978) originally developed t o simulate multiaquifer systems and modified t o
account for aquitard storativity changes and compaction, distribution of stresses in
aquitards through the discretization of these layers with nested square meshes, and
the non-linearities of vertical permeability and compressibility (hence storativity) of
the aquitards (Rivera, 1990).
The application of this methodology to the case of Mexico city has shown the representativity of the model allowing the reproduction of the observed land subsidence,
to a regional scale, during more than 50 years. A complete study of the Mexico basin
including the effects of pumping , in the last 50 years, on groundwater levels decline, on
land subsidence, and on the response of the system to artificial recharge was proposed
by Rivera (1990). In this paper we present a summary of the results.
T H E MODEL
The classical approach for quasi-3D models is used here. Let us consider a multiaquifer
system consisting of a number of aquifers separated by aquitards. All aquifers are
heterogeneous and isotropic in the horizontal plane, aquitards have variable thickness
in the horizontal direction, and their hydraulic parameters are functions of the vertical
coordinates, time and effective stress. Flow is then assumed to be 2D (horizontal) in
the aquifers, and I D (vertical) in the aquitards.
For these conditions, the governing equations for any given aquifer is :
&{T$£) + ii(Tl%)=S% + Q(x,y,t) + qL
where
h is the hydraulic head,
T is the isotropic transmissivity,
S is the storativity,
t is time,
Q is the strength of a source or sink function
(e.g. pumping from the aquifer and areal vertical
recharge of top aquifer),
ÇL is a term representing the drainage.
(1)
Nonlinear modeling of groundwater flow and total subsidence
47
If one wishes to account for storage changes in compressible aquitards, then qi represents the aquitard areal leakage flux into the aquifer (variable in time). The complexity
of the qL term is dependent on whether or not head changes in aquifers can be assumed
to occur throughout the aquitard layers within the time of integration.
Equation (1) is supplemented by appropriate initial and boundary conditions in
the (a;, y) plane.
In any given aquitard groundwater flow is supposed to take place in the vertical
direction. This flow is governed by the one-dimensional vertical diffusion equation :
U«»%)
= S',(n,W)%
(2)
where z is the coordinate taken vertically upward as positive,
h! is the hydraulic head in the aquitard,
K' is the vertical permeability,
S's is the specific elastic storage coefficient in the aquitard,
n is porosity,
<f is the effective stress.
The effective stress is related to the total vertical stress, a, and to the pore pressure p
by Terzaghi's relationship :
W = cr-p
(3)
Since the hydraulic parameters in (2) are function of n and W which depend at the same
time of the pressure, equation (2) is then non-linear. This equation which represents
the vertical flow including the compaction of aquitards, is solved simultaneously with
equation (1) by the finite-difference method with an implicite-explicite scheme.
The specific storage coefficient S's related with the compressibilities of the fluid
and the porous matrix, is defined as :
S't =
Pg(a
+ n/3)
(4)
where p is the mass per unit volume of water, g is the acceleration due to gravity,
and P is the water compressibility. In general the contribution of water elasticity to
specific storage is relatively small and is ignored. We will nontheless keep it in equation
(4) in order to be consistent with volume changes during compaction when porosity
diminishes.
The values of the coefficient a vary as a function of effective-stress (a) and void
ratio (e). This coefficient is estimated from consolidation tests from the slope of a
semi-log curve void ratio-applied effective stress (e — logâ), mathematically defined
as :
a(e,a) = 0,434p£j=
(5)
The coefficient C takes the value of Cc (compression index) when â is greater than the
"preconsolidation" stress; and the value of Cs (swelling index) when it is smaller.
Combining equations (4) and (5), using the relation n = j ^ , we get :
5 i ( n , â ) = />ff(0,434f(l-n) + »J8)
(6)
An empirical relationship was used to account for permeability changes as a function
of porosity [K1 = /(re)] (Rivera, 1990) :
48
A. Rivera et al.
K' = K'o 1
n(l-
(?)
n0(l-n
K'0 is theinitialpermeability, n0 is the initial porosity, m is an exponent, which for the
case of Mexico city is equal to 3.
Finally, the total land subsidence L(t) at a regional scale, is computed as the sum
of the compaction of each layer, in the vertical direction, as :
L(t) =
(8)
J^Pd^ZiaiAh'iit)
AZi is the thickness of layer i.
This methodology was tested and verified through the comparison with various
existing analytical solutions; the non-linear model was validated precisely reproducing
a real case: the observed subsidence at Pixley, California (Rivera et al, 1990).
WATER SUPPLY AND OBSERVED LAND SUBSIDENCE IN MEXICO CITY
Potable water supply to the city of Mexico was assured by sources and springs located
to the west and south of the city until the end of the last century. Between 1900 and
approximately 1930, when the city's population was less than one million inhabitants,
the water supply shifted progressively to the use of artesian wells. With time these
wells, and other new wells were drilled deeper and deeper and were equipped with
pumps, rapidly modifying the regional piezometry.
In order to catch up with the problem of a bigger water supply, needed for
economic growth (figure 1), the city authorities created a very ambitous program of
groundwater exploitation. From 1934, the construction of deep wells (-50 m) started
in downtown area and to the north and west, and later continued to the south, towards
the fifties.
5
Millions
Mill ons
20
18
4
-
-
-"/--
—
16
14
12 £
c
3
>
-o
Water consumption
10
.
2
1
+J
a
£
6
—
4
L /
Urban
growth
2
—«—-¥—"
(I e I — —m—
1880 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990
FIG. 1 Urban growth and water consumption in Mexico city.
In 1980, total pumping rate exceeded 21 m 3 s'1 with more than 600 wells. Figure
2 presents, as an histogram, the pumping data in Mexico city for the period of 19341986.
49
Nonlinear modeling of groundwaterflowand total subsidence
1934
1940
1950
1960
1970
1980
1986
FIG. 2 Groundwater pumping for the period of 1934-1986.
In the same period of time, a maximum of more than 6 meters of land subsidence
was observed at some locations (see figure 3), constituing one of the most remarquable
cases in the word due to its magnitudeand its extension.
It has been shown since the forties, that the principal cause of this phenomenon,
observed at a regional scale, is the groundwater exploitation.
The evolution of subsidence is marked by three periods (fig. 3). A first phase,
between 1935 and 1948, is marked by a weak slope in a plot of the observed subsidence:
the average speed is approximately 8 cm year-1.
Between 1947 and 1957-58 the
observed subsidence averaged around 29 cm year-1. The third phase, after 1959, is
characterized by a decrease of the subsidence: ia a long period of 27 years the observed
subsidence averaged 5 to 6 cm year-1.
Observed subsidence
^ ^ N ^
-3
( m )
^
-4
V^\T~^\. A
(A) Catedral
(B) Alameda
-6
c
(C) Carlos IV
"
' I ' " 1 ' " 1 ' "
1 ' | ""
1 | ! 1 1 ] 1 1 1 1 ! ! 1 1 1 1 1 1 1 1 1 1 1 M
1 M 1
1934 1938 1942 1946 1950 1954 1958 1962 1966 1970 1974 1978 1982 1986
FIG. 3 Observed subsidence at three selected sites in downtown area.
A likely explanation is that the consolidation phenomenon alters the physical
properties of the aquitards and causes significative variations of the hydraulic parameters K' ans S's; this would affect, in turn, the hydraulic response of the whole
aquifer-aquitard system. The effect of the diminishing of these parameters during
50
A. Rivera et al.
•:;:£[ Alluvial aquifer, confined
under clayey sediments.
Alluvial aquifer, unconfined.
Fractured basalts, S;
Andesitic rocks, E,W,N.
Popocateptl-*
(5430 m asl )
Mexico City
Q
io
20
so
T
i
r
•
4pkm
i
FIG. 4 The Mexico city basin and location of study area.
A '
1 TOOLS TBAZA
——1821
M W Ansa
i—was—i
•
«**•—*
FIG. 5 Detailed cross section, A-A', of clayey sediments.
"
51
Nonlinear modeling of groundwater flow and total subsidence
consolidation has, as a consequence, a decrease of the aquitard leakage, implying a
longer time to reach steady-state, and a subsidence smaller than the one that would
be predicted by a standard linear analysis.
We will try to reproduce, with our model, this phenomenon of non-linear subsidence observed in Mexico city.
CONCEPTION AND IMPLEMENTATION OF THE MEXICO CITY SIMULATION
MODEL
Figure 4 shows a map of the hydrogeological formations in the basin of Mexico city.
This basin is formed by a series of closed sub-basins, more or less independent;
it is considered that the main aquifer underlying the city of Mexico forms an entity
relatively isolated from the aquifers in the north of the basin. This aquifer is separated
by a line forming the volcanic cone of Sierra Guadalupe traversing the basin from west
to east in its meridional part.
In the same figure is shown the study zone of approximately 2900 km2 surface
area. The main alluvial aquifer is confined in most of the valley by the lacustrine
sediments of clays and silts, 50 m thick in downtown area. Figure 5 shows a detailed
cross section, west-east, of these sediments.
The main assumptions adopted in the simulations are:
- the aquifer units, alluvial material (confined and unconfined, from the center of
the basin to the piedmonts), fractured basalts (to the south), and andesitic rocks
(to the east, west and north), function as a single continous aquifer;
- groundwater flow in the aquifer is considered horizontal, in the x,y directions, and
is represented by a single layer in the model;
- the distinction between the three types of rocks is done as a function of their
hydraulic characteristics (T, S, n);
- the lacustrine sediments constituing the aquitards are divided into three formations: the upper clay formation (UCF), the "Capa Dura" (or hard stratum, CD),
and the lower clay formation (LCF). The flow in the aquitard is considered as
essentially vertical and is represented by 22 horizontal layers in the model.
Figure 6 shows, in a west-east diagrammatic profil, the adopted conceptual model.
FIG. 6 Conceptual model of the aquifer-aquitard system.
Discretization of Study Area
Horizontal discretization
In order to better represent the boundaries and heterogeneities of the study area
we adopted a grid of variable size using nested square meshes of three different sizes:
52
A. Rivera et al.
a big square mesh of 2000 m side, aa intermediate mesh of 1000 m, and a small mesh
of 500 m (figure 7).
The small meshes are used at the boundaries between the three zones ("zone of
lakes", "zone of transition", and "zone of piedmonts", fig. 6), at the contact with
the volcanic cones (impermeable), and at the zone of the "ancient city" where the
maximum subsidence is observed.
The total number of elements used to discretized the aquifer layer is 1181 square
meshes.
AQUITARD
^ 5 S4 4 44
J
' ^ ^
1
AQUIFER
FIG. 7 Schematic perspective of the aquifer-aquitard system.
Vertical discretization
Since one of our principal objectives in this study is to analyze in detail, in
the vertical direction, the distribution of pressures associated to the compaction of
53
Nonlinear modeling of groundwater flow and total subsidence
aquitards, we used a very much finer grid with 22 layers to discretize the three aquitard
formations UCF, CD, and LCF.
Figure 7 shows a schematic perspective of the aquifer-aquitard system as it is discretized in the model. The hydraulic connection between the aquifer and the aquitard
is done exclusively in the lacustrine zone ("zone of lakes").
The total number of layers for each formation varies depending on the zone. For
the ULF, the number of layers may be from 3 to 7; the CD is represented by a single
layer (the 8th one); the number of layers used to discretize the LCF may vary from 1
to 14.
The 22 layers representing the aquitards have 6523 meshes, making, for the whole
model, a total of 7704 square meshes.
The adopted boundary conditions, with a possible combination between them
are:
- constant flux (aquifer)
- prescribed drainage elevation combined with a limited vertical flux (aquitards),
The model also includes the eventual loss of water pressure in the confined aquifer
(a change from a confined to an unconfined condition).
SIMULATIONS AND RESULTS
In order to identify and better understand the different components of the flow and
their spatial distribution through the aquifer-aquitard system, we first analysed the
water budget in the basin.
A simulation in steady-state permitted a first calibration of the model (before
the aquifer exploitation) using the available data, and the review of some of the initial
assumptions.
We considered in our simulations that a steady-state persisted until 1930.
Values of aquifer transmissivities, obtained from pumping tests (Lesser, 1985),
were used in the model. Initial values of permeability for the aquitard were obtained
from consolidation tests performed on clay samples (several references cited in Rivera,
1990).
In order to calibrate T and K' in steady-state, a twofold criteria was adopted.
Firstly, a water budget in the basin (INS and OUTS) was calculated with the model
to match a value of a classical hydrologie budget using data of mesured precipitation,
évapotranspiration and runoff. Unfortunately there exist no reference piezometric levels that would allow the calibration in steady-state using the piezometry.
Secondly, it was search to reproduce the discharge zones of the aquifer, that is,
the springs as well as the artesian zones which implied an upward vertical drainage
through the semi-pervious layers.
The results of the simulations for this stage are totally coherent as a whole, they
reproduce the orders of magnitudeof the different components of the water budget.
The water budget calculated by the model is schematically summarized in figure 8.
The aquifer recharge rate by infiltration from precipitation calculated by the
model was 8.1 m3 s _ 1 and is equivalent to a mean annual infiltration rate o f l 5 / s _ 1 —
km2. Other authors (Ortega and Farvolden, 1989) propose an infiltration rate of 10
to 29 / s'1
-km2.
The adopted initial permeabilities of the aquitards as calculated with the model
are summarized in the following table:
FORMATION
LAYER
K' (m s""1)
FAS
1-6
7
8
9-22
2xl0~ 8
lxlO-8
910- 5
CD
FAI
lxlO"- 8
54
A. Rivera et al.
FIG. 8 Components of the water budget simulated in steady-state.
From the thirties, and after the begining of the heavy pumping in the aquifer,
the behaviour of the aquifer-aquitard system has been marked by a progression of four
processes:
- gradual drying up of the springs and sources,
- downward vertical drainage (leakage) of the aquitards,
- compaction of aquitard layers and land subsidence observed at the surface, and
- dewatering of the confined aquifer (a change from a confined to an unconfined
condition).
The numerical model was adapted to account for these processes during the simulations in unsteady-state, for the period of 1934-1986. The observed values of the two
unknowns: piezometry and subsidence, were used to calibrate the model at this stage.
The model parameters (besides T and K' already calibrated in steady-state) for
this stage are the compression index C c , the swelling index Ca, and porosity, n. From
Cc and/or Cs, the model estimates the specific storage coefficient S'3 (eq. 6), the latter
will vary as a function of piezometry and porosity.
The input values of Cc and Cs were obtained directly from consolidation tests
performed in the laboratory on clay samples coming from approximately 65 borings in
the valley of Mexico city.
Porosity was determined from values of void ratio (e) obtained from the same
tests. The mean value of void ratio for the UCF is 6.7, with some maximum values
of 9 mesured in the Texcoco zone. For the LCF, the mean value of e is 4.7, and for
the Capa Dura, this value is 0.66. From those values, we input into the model the
following values of porosity:
FORMATION
LAYER
n
UCF
CD
LCF
1-7
8
9-22
0.87
0.4
0.82
Permeability will vary as a function of porosity (eq. 7) which, itself varies with
compaction.
Figure 9 shows a comparison between observed and calculated piezometric levels
for two selected piezometers. The evolution of the observed piezometry for PC190 is
V
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1
O
CD
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i
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*
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Nonlinear modeling of groundwater flow and total subsidence
7
rve
55
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CD
A
(ise IU 0002 +) M
(w) aouapisqns
A*
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L-
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O
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Catedral
56
A. Rivera et al.
adequately reproduced by the model; this piezometer has the largest series of observed
data available. The comparison of the other piezometer is less good but it should be
emphasized that the series of data is shorter. A comparison between the observed
piezometry and the calculated piezometry at a regional scale and for different years is
presented in Rivera (1990).
Figure 10 shows the results of the calculated subsidence compared with the observed subsidence at two selected sites. The agreement is very good. We believe
that the fact that the model allows the correct reproduction of the three different
periods of the subsidence, clearly shows the importance of considering the non-linear
phenomenon.
The effects of the non-linearities on the aquitard layers, the variation of A", S's, n,
and of the preconsolidation stress are presented in Rivera (1990).
The water budget estimated with the model is presented schematically on figure
11 for the year 1986, and is summarized in the following table:
INS
(m 3 s'1)
Recharge
Aquifer
storage
Pumping
Leakage of
aquitards
Sources
OUTS
(m 3 s'1)
8.1
12.5
20.6
2.1
2.1
In the light of these results, it is clear that the aquifer is overexploited.
The evolution in time of the water budget for the whole period of the study,
1934-1986, is shown in figure 12. It is noted that the dewatering of aquitards does
not start until 1940, identified by a downward drainage, with 0.217 m3 s - 1 flow rate;
the maximum flow rate coming from the aquitards appears in 1965 with a total of
2.35 m3 s - 1 . The overexploitation of the aquifer starts in 1958 when total pumping
exceeded natural recharge.
Rechsrgs 8.10 m3/a
FIG. 11 Components of the water budget simulated in unsteady-state,
year 1986.
57
Nonlinear modeling of groundwater flow and total subsidence
1934 1938 1942 1946 1950 19S4 1958 1962 1966 1970 1974 1978 1982 1986
FIG. 12 Components of the water budget for the whole period of simulation.
CONCLUSIONS
The implemented model NEWSAM-compaction may simulate:
-*• quasi-tridimensional groundwater flow,
—> the compaction of interbeds and total subsidence in a heterogeneous system,
—y the variation of hydromechanic properties as a function of applied stress.
• The results clearly show that the mechanical properties obtained from laboratory tests allow the explanation of the regional system. The conceptual model is then
acceptable.
• The simulations for the Mexico city case permitted a better appreciationof
both the groundwater flow and the water budget of the aquifer-aquitard system, both
quantitatively and qualitatively.
® The subsidence phenomenon observed from 1934 to 1986 was nicely reproduced
thanks to the non-linear hydraulic-mechanic coupled model.
• Predictions for a rational management of the Mexico city aquifer system will
be possible with the implemented model.
REFERENCES
CRUICKSHANK, V.C., 1982. Modelos matematicos para acuiferos del valle de Mexico. in : El sistema hidrâulico del Distrito Federal, DDF-DGCOH. pp. 6.1-6.23.
HELM, D.C., 1976. One-dimensional simulation of aquifer system compaction near
Pixley, California. 2. Stress-dependent parameters. Water Resour. Res. 12(3) ;
375-391.
HERRERA I., R. MARTINEZ, & G. HERNANDEZ, 1989. Contribucion para la administracion cientifica del agua subterrânea de la cuenca de Mexico. Symposium
"El Sistema Acuifero de la Cuenca de Mexico", Mexico city; Geoffsica Internacional vol 28, n° 2, pp 297-334.
JORGENSEN, D.G., 1980. Relationships between Basic Soils-Engineering Equations
and Basic Ground-Water Flow Equations. U.S. Geol. Survey Water-Supply Pap.
2064, p.40.
LESSER & Asoc, 1985. Actividades Geohidrologicas en el Valle de Mexico, Piano 3-5
(29.V.85), Mexico, D.F.
MARSILY, G. de, E. LEDOUX, A. LEVASSOR, D. POITRINAL, & A. SALEM,
1978. Modelling of large multilayered aquifer systems : theory and applications.
J. of Hvdrol. 3_6, 1-33.
A. Rivera et al.
58
NARASIMHAN, T.N., & P. WITHERSPOON, 1977. Numerical model for saturatedunsaturated flow in deformable porous media, 1. Theory, Water Resour. Res.
13(3) : 657-664.
NEUMAN, S.P., C. PRELLER, & T.N. NARASIMHAN, 1982. Adaptive explicitimplicit quasi three-dimensional Finite Element model of flow and subsidence in
multiaquifer systems. Water Resour. Res. 18(5) ; 1551-1561.
ORTEGA, A. & R.N. FARVOLDEN, 1989. Computer analysis of regional groundwater
flow and boundary conditions in the basin of Mexico. J. of Hydrol. 110 , pp 271294.
POLAND, J.F., 1984. Guidebook to studies of land subsidence due to ground-water
withdrawal. UNESCO. PHI Working Group 8.4. p.305.
RIVERA, B. A., 1990. Modèle hydrogéologique quasi-tridimensionnel non-linéaire
pour simuler la subsidence dans les systèmes aquifères multicouches. Cas de Mexico. Ph.D. thesis. Ecole Nationale Supérieure des Mines de Paris, CIG. Paris,
France.
RIVERA, B.A., E. LEDOUX, & G. de MARSILY, 1990. Modèle hydrogéologique quasi
tridimensionnel non-linéaire pour simuler la subsidence. Cas de Pixley, Californie
et de la ville de Mexico. Revue d'Hydrogéologie. BRGM. n° 1. pp. 27-39.
RUDOLPH, D., I. HERRERA, k R. YATES, 1989. Groundwater Flow and Solute
Transport in the Industrial well fields of the Texcoco Saline Aquifer Symposium
"El Sistema Acuffero de la Cuenca de Mexico", Mexico; Geofisica Internacional
vol. 28 n° 2, pp. 363-408.
SCOTT, R.F., 1963. Principles of Soil Mechanics. Addison-Wesley Reading, Mass.