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 m3 sec - 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 mul-tiaquifer 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 one-dimensional 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 to simulate multiaquifer systems and modified to 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 rep-resentativity 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.

THE 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 ID (vertical) in the aquitards.

For these conditions, the governing equations for any given aquifer is :

&{T$£) + ii(Tl%)=S% + Q(x,y,t) + qL (1)

where h is the hydraulic head, T is the isotropic transmissivity, S is the storativity, t is t ime, 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.

47 Nonlinear modeling of groundwater flow and total subsidence

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) :

A. Rivera et al. 48

K' = K' n ( l -

o 1 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) = J^Pd^ZiaiAh'iit) (8)

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.

> -o

5

4

3

2

1

(Ie

Mill

-

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ons

—m— —«—-¥—"

- —

Water consumption .

Millions

-"/--

L / Urban growth

20

18

16

14

12 £ c

1 0 +J

a £

6 —

4

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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 m3 s'1 with more than 600 wells. Figure 2 presents, as an histogram, the pumping data in Mexico city for the period of 1934-1986.

49 Nonlinear modeling of groundwater flow and 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.

-3 ( m )

-4

- 6

^ ^ N ^

^

(A) Catedral

(B) Alameda

(C) Carlos IV

" ' I ' " 1 ' " 1 ' " 1 ' "

Observed subsidence

V ^ \ T ~ ^ \ . A

c | " 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

A. Rivera et al. 50

•:;:£[ 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 4pkm T i r • i

FIG. 4 The Mexico city basin and location of study area.

A '

1 T O O ­LS TBAZA ——1821 "

i—was—i M W Ansa • «**•—*

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:

A. Rivera et al. 52

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.

A Q U I T A R D

^ 5 S 4 4 4 4

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 dis-cretized 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

FAS

CD

FAI

LAYER

1-6

7

8

9-22

K' (m s""1)

2xl0~ 8

l x l O - 8

910-5

lxlO"-8

A. Rivera et al. 54

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 sim­

ulations 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 Cc, 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

UCF

CD

LCF

LAYER

1-7

8

9-22

n

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

55 N

onlinear modeling of groundw

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A. Rivera et al. 56

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:

Recharge

Aquifer

storage

Pumping

Leakage of

aquitards

Sources

INS

(m3 s'1)

8.1

12.5

2.1

OUTS

(m3 s'1)

20.6

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 ad-ministracion cientifica del agua subterrânea de la cuenca de Mexico. Symposium "El Sistema Acuifero de la Cuenca de Mexico", Mexico city; Geoffsica Interna-cional 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 saturated-unsaturated flow in deformable porous media, 1. Theory, Water Resour. Res. 13(3) : 657-664.

NEUMAN, S.P., C. PRELLER, & T.N. NARASIMHAN, 1982. Adaptive explicit-implicit 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 271-294.

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.