Simple procedure to simulate karstic aquifers

HYDROLOGICAL PROCESSESHydrol. Process. 22, 1876–1884 (2008)Published online 28 August 2007 in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/hyp.6772

Simple procedure to simulate karstic aquifers

Alberto Padilla1 and Antonio Pulido-Bosch2*1 ALJIBE, S.L.L., Granada, Spain

2 Department of Hydrogeology, University of Almeria, Almeria, Spain

Abstract:

A procedure to simulate karstic aquifers is presented. It is based on a simulation of spring discharge using precipitation and,where necessary, temperature as input data. The karstic aquifer system is considered to be divided into three zones: the surfacezone, the unsaturated zone (UZ) and the saturated zone (SZ). Each of these is described by a transfer function that determinesthe water supplied from the overlying zone. Water loss through evapotranspiration is calculated empirically and subtractedfrom the total precipitation in order to obtain the effective infiltration into the UZ. The transfer function characterizing the UZcan be expressed as a convolution function. The UZ acts as a buffer, delaying effective infiltration into the SZ. Water dischargefrom the SZ is described by the recession function of the spring, and this becomes the transfer function that characterizes theemergence of water from the SZ. The model permits the simulation of the influence of pumped abstractions from the systemby a simple modification of the transfer functions involved. Copyright 2007 John Wiley & Sons, Ltd.

KEY WORDS karstic aquifer; conceptual model; convolution; recession function; pumping

Received 25 September 2006; Accepted 20 March 2007

INTRODUCTION

Karstic aquifers have often been studied by analysingthe discharge rate of the main spring, either through itsrecession characteristics or by field measurements overa complete discharge cycle (spectral and autocorrelationanalyses; Mangin, 1984; Padilla et al., 1994; Padilla andPulido-Bosch, 1995; Manga, 1999), with precipitationacting as an input to the system. Black-box (convolution)models (De Marsily, 1978) and reservoir models havealso been devised to simulate the functioning of karsticaquifers, and these simplify the processes somewhat byidentifying only the transfer functions between a series ofreservoirs. Noteworthy examples include MERO, CREC,and BEMER (Cormary and Guilbot, 1971; Mero, 1978;Avias and Joseph, 1984). This type of approach, thoughgreatly simplifying the physical reality, represents acertain advance on earlier methodology.

Hydrograph analysis is another method, like the black-box model or the recognition of changes, where the inputparameter (precipitation or effective rainfall) controls thedischarge from a spring or other conduit (Eisenlohr et al.,1997a,b; Grasso, 1998; Jeannin and Sauter, 1998; Tamet al., 2005). The structure of the time-series itself canalso be studied as a process identifier. Brackish coastalsprings can thus be simulated (Lambrakis et al., 2000;Arfib et al., 2002), including their mixing mechanism.The specific cases analysed are Almiros spring in Herak-lion in Crete, Greece, and Makaria spring in Attica (Mara-mathas et al., 2005). However, there has also been a lot

* Correspondence to: Antonio Pulido-Bosch, Department of Hydrogeol-ogy, University of Almeria, La Canada de San Urbano, E04120 Almeria,Spain. E-mail: [email protected]

of progress in the simulation of karstification processes,from the formation of the initial fissures to the devel-opment of the massif (Dreybrodt, 1996; Palmer et al.,2006).

A karstic aquifer can also be simulated by estimat-ing the spatial distribution of the parameters using aprocedure that takes the variable hydraulic conductiv-ity of the karstic networks into account (Pulido Boschand Padilla, 1988). The simulation can use either finitedifferences (Huyakorn et al., 1983) or finite elements(Kiraly, 1988, 1998). Another factor that must be takeninto account is that the flow of water through a karsticnetwork changes its characteristics, developing certainconduits to the detriment of others, so establishing a hier-archy in the system. In complex Mediterranean areas, aconceptual model of a homogeneously karstified aquiferhas been devised based on observations that overexploita-tion in one part of these aquifers causes water levels todrop throughout the aquifer (Pulido Bosch et al., 1999).

This paper describes a procedure that allows karsticaquifers to be simulated using the rainfall, temperatureand discharge data of the system. It describes how thekarstic aquifer system of Torcal de Antequera (Figure 1)was classified into three domains: the superficial zone, theunsaturated zone (UZ) and the saturated zone (SZ), andhow the transfer functions between them were defined.

METHODS

Principle

The rainfall–discharge model comprises three zones,each characterized by a separate transfer function

Copyright 2007 John Wiley & Sons, Ltd.

KARSTIC AQUIFERS 1877

Figure 1. A view of El Torcal massif

(Figure 2). In the surface zone, run-off and evapotran-spiration are subtracted from rainfall to give effectiveinfiltration. In the second, the UZ, the infiltration rateslows, and in the third, the SZ, the discharge functionis denoted by the recession function of the hydrograph.Below, we describe the various functions involved ineach of these zones.

Let us apply a recession function QiC1 D f�Q1� to thehydrograph in Figure 3. The volume of water supplied tothe SZ under natural conditions (that is, with no pumping)during interval i as a result of effective infiltration is

VAi D F�Qi� � F�QŁi � for QŁ

i D f�Qi�1�

where

F�Qi� D∫ 1

if�Qi� dt �1�

represents the dischargeable volume stored in the SZ,Qi represents the mean discharge at interval i, Qi

Ł isthe discharge at interval i when there is no input to thesystem, and Qi�1 is the discharge during the preceding

Figure 3. Scheme showing the procedure to calculate the volume of watersupplied to the SZ. ES: runoff; EV: evapotranspiration; VI infiltratedvolume; �: convolution kernel parameters; VA: volume transferred to the

SZ; V: dischargeable water volume stored at the SZ; Q: discharge

interval. Thus, from a series of N discharge data, a newseries of N � 1 data points is obtained that is a functionof the mean discharge at interval i and the precedinginterval i � 1.

Let us suppose that the UZ exercises a control overthe effective infiltration into the SZ. The input intothe SZ can be discretized over a series of time stepsfollowing the initial input. Let us also assume that thisseries is linear over time, and that the infiltration and thevolume entering the SZ are proportional. In this way therelationship between the two zones can be expressed asa convolution integral:

VIi D∫ 1

0VAi�t�t dt

Figure 2. Scheme of the relationship between the three zones used in the model. P: rainfall; VI: effective infiltration; ES: runoff; EV:evapotranspiration; VA: amount of water supplied to the SZ; V: dischargeable water volume stored at the SZ; �: convolution kernel parameters

Copyright 2007 John Wiley & Sons, Ltd. Hydrol. Process. 22, 1876–1884 (2008)DOI: 10.1002/hyp

1878 A. PADILLA AND A. PULIDO-BOSCH

Using Hankel’s transform (Watson, 1966):

VAi D∫ 0

�1VIi�t�t dt

where VIi�t is the volume infiltrated at time i � t, VAi isthe volume supplied to the SZ during interval i, and �t

are the parameters of the convolution kernel.In a discrete form, and presuming that the convolution

parameters vanish after interval k, the integral may beexpressed as

VAi Dk∑

jD0

VIi�j�j �2�

The convolution kernel coefficient can be calculatedeither by applying a Fourier transform or by solvingthe series of equations that results from an iterativeminimization of the mean quadratic error between thecalculated (Equation (1)) and theoretical (Equation (2))volumes, or by any other suitable method. If we combineEquations (1) and (2), then

Qi D F�1

F�QŁ

i � Ck∑

jD0

VIi�j�j

�3�

This allows us to obtain the discharge for interval i asa function of the discharge at the immediately precedinginterval and the effective infiltration during the precedingk intervals. Pumped abstractions from the aquifer can beaccommodated in this equation in the following manner:

Qi D F�1

[F�QŁ

i � � VBi] Ck∑

jD0

VIi�j�j

�4�

where VBi is the volume pumped at interval i.If effective infiltration and rainfall have a linear

relationship, then one could substitute one for the other inthe equation above. However, since this is not generallythe case, infiltration must be estimated empirically, e.g.by using the methods of Thornthwaite (1948), Kessler(1965) or Penman–Monteith (Monteith, 1965). Thesemethods describe the transfer function of the surfacezone.

Theoretical development of the functions

One of the most important steps in fitting the model isthe choice of recession function, since this characterizesthe SZ and allows us to calculate the flow supplied fromthe UZ. A prior condition is that the defined integralof this function should be relatively simple in orderto be able to apply Equations (1)–(4). In most karsticaquifers, recession satisfactorily fits Maillet’s formulaQt D Q0 e�˛t where Qt is the discharge at time t andQ0 is the discharge at the beginning of recession t D 0.

This equation is quite satisfactory for obtaining thefunctions in the model; but, given that certain aquifers,particularly karstic ones, show a very rapid initial reces-sion after a downpour prior to the control effect taking

over, we prefer the formula devised by Coutagne (1968).The initial postulation of this formula is

Q D CVn

where Q is the discharge of the spring, C is a constant, Vwould be the volume of water stored in the SZ if recessionwere complete and uninterrupted, and n is an exponentthat can vary between 0 and 2. Furthermore, Q D dV/dt.The solution for this differential equation for n 6D 1 is

Qt D Q0[1 C �n � 1�˛0t]n/�1�n� �5�

where ˛0 is the recession constant at time t D 0 for Q0.For two consecutive intervals, i and i C 1, on a

hydrograph showing a recession, we may assume that

Qi D Q0[1 C �n � 1�˛0t1]n/�1�n� �6�

andQiC1 D Q0[1 C �n � 1�˛0tiC1]n/�1�n� �7�

and that QiC1 is a function of Q1:

QiC1 D Q0[1 C �n � 1�˛it]n/�1�n� �8�

The recession constant ˛i depends on the volume ofdischargeable water stored in the SZ during interval i;thus, it depends on Qi. Equation (8) can then be writtenas (

QiC1

Qi

)�1�n�/n

D ˛i�n � 1�t C 1 �9�

When Equation (7) is divided by Equation (6), we obtainan equation where ˛i becomes a function of Qi:

˛i D ˛0

�Qi/Q0��1�n�/n �10�

If we then include ˛i in Equation (8) we get

QiC1 D Qi

[1 C �n � 1�Q�1�n�/n

0 ˛0t

Q�1�n�/ni

]n/�1�n�

If our hydrograph is one of discrete flows, where tis constant and ˛0 is established for a constant Q0 in theaquifer system, then we can then obtain the discharge atinterval i C 1 (when there is no input to the system) usingthe expression

QiC1 D Qi

(1 C A

Q�1�n�/ni

)n/�1�n�

�11�

where A is a constant equal to �n � 1�Q�1�n�/n0 ˛0t.

Calculation of the supply to the saturated zone

If the hydrograph in Figure 3 takes the volume ofwater supplied to the SZ during interval i into account,then, according to Equation (1), this will be the volumestored at interval i minus the volume stored if there were



no input to the system. This can be determined by therecession function, i.e.:

VAi D F�Qi� � F�QŁi � D Vi � VŁ

i

If we take t D i D 0 as the time of origin, we have,according to Equation (8):

Vi D Qi[1 C �n � 1�˛it]n/�1�n� dt

Solving this equation gives

Vi D � Qi

˛i[1 C �1 � n�˛it]n/�1�n� �12�

For n < 1 the water discharge is nil:

Qt D Qi[1 C �n � 1�˛it]n�1�n� D 0

when t D 1

�1 � n�˛i

In this case:

Vi D � Qi

˛i[1 C �n � 1�˛it]1/�1�n�˛i

which, if we solve for n < 1, is

Vi D Qi/˛i

For n > 1, [1 C �n � 1�˛it]1/�n�1� D 0, if t ! 1; Thus,Equation (12) can be rewritten for n > 1 as

Vi D Qi/˛i

Thus, for any case where n 6D 1, the stored volume abovethe discharge height can be represented as

Vi D Qi/˛i

If we substitute ˛i from Equation (10), then

Vi D Q1/ni

�˛0Q0��1�n�/n

Likewise, for VŁi we obtain

VŁi D QŁ1/n

i

˛0Q�1�n�/n0

Thus, the supplied volume will be

VAi D Q1/ni � QŁ1/n

i

˛0Q�1�n�/n0

�13�

where QiŁ, according to Equation (11), is then

QŁi D QiC1

[1 C A

Q�1�n�/niC1

]n/�1�n�

�14�

When n D 1, Coutagne’s formula becomes the same asthat of Maillet:

Qt D Q0 e�˛t

where ˛ is constant (˛ D ˛0 D ˛i). This function can berewritten as

QŁi D BQi�1

where B D constant D e�˛t, and

VAi D Qi � QŁi

˛�15�

Simulation

As described above, once we have established the˛0 parameters for Q0 and n, which are constant in thesystem, Equations (13) and (15) can be used to calculatethe volume of water supplied to the SZ for any interval ias a function of the spring discharges at intervals Qi andQi�1.

The next step is to calculate the � parameters ofthe convolution kernel in Equation (2) to obtain thevolume supplied to the SZ as a function of the effectiveinfiltration. Finally, the estimated discharge for any timeinterval Q0

i will be expressed by solving Equation (3);thus, according to Equation (13), for n 6D 1

Q0i D

˛0Q�1�n�/n

0

k∑jD0

VIi�j�j C QŁ1/ni

n

�16�

and, according to Equation (14), for n D 1

Q0i D ˛0

k∑jD0

VIi�j�j C QŁi �17�

Considering pumped extractions (VB), Equations (16)and (17) become

Q0i D

˛0Q�1�n�/n

0

k∑

jD0

VIi�j�i � VBi

C QŁ1/n

i

n

�18�and

Q0i D ˛

k∑

jD0

VIi�j�i � VBi

C QŁ

i �19�

respectively.

APPLICATION AND RESULTS

The Torcal de Antequera aquifer, in the northeast ofthe province of Malaga, is an isolated aquifer with acatchment area of some 28 km2; it is drained principallyby La Villa spring (Figures 4 and 5). Since October 1974,the Instituto Geologico y Minero de Espana (IGME) hasbeen measuring the flow from this spring (Figure 6).There are also daily records of temperature and rainfallfor the catchment. As input data to the model, werelied on the daily effective infiltration, calculated by theThornthwaite method and integrated on a weekly basis.

A cross-correlogram between rainfall and effectiveinfiltration (Figure 6) reveals certain interesting charac-teristics of the system. First, its discharge shows consid-erable periodicity throughout the year. Moreover, despite



Figure 4. The Torcal de Antequera karstic aquifer. (1) Impervious Triassic Keuper facies; (2) a: karstified Jurassic limestones; b: imperviousCretaceous marls; (3) Sierra de las Cabras Unit; aquifer; (4) very low permeability Aguila Complex; (5) very low permeability Flysch Units;(6) pervious post-orogenic Miocene rocks; (7) normal fault; (8) inverse or thrust fault. A–A0 and B–B0 represent cross-sections included in Figure 5

Figure 5. Two representative geological cross-sections. Karstic Jurassic terrains comprising: (1) oolitic limestones with dolostones at the bottom,(2) nodular limestones and (3) white micritic limestones; (4) impervious Cretaceous marls; (5) impervious Flysch; (6) pervious Miocene terrains;

(7) talus slopes; (8) impervious Triassic Keuper facies

Figure 6. La Villa spring hydrograph. Weekly average data



being a karstic aquifer, the response time of the systemis quite high, between 14 and 19 weeks, a feature moreakin to a porous medium than a fractured one. The cross-correlogram shows that the maximum response of thisspring occurs some 4 weeks after input.

Bearing these considerations in mind and observingthe recessions, the function that best fits the recessionprediction is Maillet’s formula, derived from Coutagne’sexpression (developed above), for n D 1 and for arecession constant ˛i that is independent of the volumeof water stored in the aquifer.

The first step in the experiment was to calculate thevolume of water supplied to the SZ using Equation (14),in which ˛ was given a value of 0Ð0125 days�1. It isuseful to compare the cross-correlogram between rainfalland effective infiltration shown in Figure 7 with that inFigure 8. In the latter, the response of the UZ is from 3to 7 weeks, reaching its maximum about 1 week after theinput of water into the system, compared with the 14- to19-week response in Figure 7.

The next step was to obtain a series of volumessupplied to the SZ and then calculate the coefficientsfor the convolution kernel in Equation (2). To do this,we took rainfall to be the same as effective infiltration,since runoff is practically negligible in this area. Inthe evaluation we used a parameter-sensitive, iterativemethod to minimize the mean quadratic error betweenthe observed and the calculated values. For a lag time ofzero the value of parameter is 1.2 ð 10�2 (Figure 9).

The theoretical hydrograph, obtained usingEquation (17), is shown in Figure 10, together with the

Figure 7. Rainfall–discharge cross-correlogram for La Villa spring

Figure 8. Cross-correlogram for rainfall–volume supplied to the SZ

Figure 9. Convolution kernel parameters in the transfer function betweenthe UZ and SZ

experimental hydrograph. The first four years correspondto the period of fit of the parameters in the model, andthe last two years represent the simulation period. Oncethe model was calibrated, a series of pumping simulationswas made in the SZ, for which we applied Equation (19).The simulation served the dual purpose of observing thecondition in the flow rate of La Villa spring and also theprogression of the volume stored in the SZ, including itsoccasional deficit when the water table was lower thanthe elevation of the spring.

The results obtained should be understood to representan average for the system; under no circumstances arethey intended to show a real situation. It is well knownthat a well located nearby would immediately affect theflow rate of a spring. In contrast, a well located far awaywould affect the discharge rate only after a certain periodof time, which would depend on the system’s hydraulicconductivity. On the other hand, it is believed that there isalways enough saturated thickness to ensure the pumpingyield would not be cut. Table I summarizes the resultsfrom four different simulations.

The first case involved continuous pumping of0Ð2 m3 s�1 over the whole year, which is approximately50% of the average discharge; this amount coincides

Table I. Summary of the results obtained with the simulationscarried out at the Torcal: (A) continuous pumping of 0Ð2 m3 s�1;(B) the same as A plus 0Ð3 m3 s�1 from June to September;(C) the same as A, but only when the demand is not met bythe spring; (D) the same as B but only when the demand is not

met

A B C D

Mean spring discharge(m3 s�1)

0Ð215 0Ð13 0Ð312 0Ð17

Mean yield frompumping (m3 s�1)

0Ð20 0Ð3 0Ð104 0Ð256

Maximum difference(Mm3)

�3Ð0 �7Ð5 �3Ð0 �6Ð5

Withdrawal (%) 48 72 25 62



Figure 10. Real and calculated hydrographs, obtained on the basis of weekly recordings

Figure 11. The first case simulated at El Torcal: (a) hydrograph of the spring; (b) the evolution of the water volume in the aquifer

with the water demand for the town of Antequera. Asshown in Figure 11a, La Villa spring would run dryin some autumn months and for longer periods in verydry years, such as 1974–1975 and 1980–1981, but flowwould return to normal during the wet months. The aver-age discharge of the spring under these conditions was0Ð215 m3 s�1. (The sum of the average discharge of the

spring added to the volume pumped is not equal to themean of the discharge rate under natural conditions since;as shown in Figure 11b, there is a deficit during thelast year.) Figure 11b shows the dynamic volume on thepositive side. The greatest deficit of 3 Mm3 occurred in1975–1976. If we calculate an average aquifer porosityof 2% and 28 km2 of constant piezometric surface area,



this would equate to a maximum drawdown of 5 m inwater levels over the period simulated. Given the geo-logical characteristics, this could be borne perfectly wellby the aquifer.

We could conclude that with an abstraction of 50% ofthe discharge from the spring, equivalent to the aquifer’sresources (this being an isolated system), the averagedrawdown caused is very small.

In the second case simulated, pumping is increased to0Ð5 m3 s�1 during the low water months, from June toSeptember, coinciding with the time of greatest waterdemand for agriculture. Under these conditions, thespring would yield water only during the months of high-est rainfall; during low rainfall years, such as 1980–1981,the spring would be completely dry for the entire year.The average discharge rate is 0Ð13 m3 s�1 and abstrac-tion from the aquifer proves to be very high, at around72% of the discharge. The water shortage in the aquiferis acute during the dry summer months (Figure 12). Itreaches a maximum during years when reserves are notrecovered. For example, during the last year simulated,the deficit amounted to 7Ð5 Mm3, which would equate,with the same readings as the previous case, to an averagedrop in water level of 13Ð5 m.

The two cases described above would not be consistentwith the supposed real conditions of the zone, sincethe spring emerges at a relatively high elevation. Thus,

pumping would occur only when the discharge rate didnot meet the demand. Therefore, another two cases wereconsidered; these were similar to the previous ones,except that abstraction was varied subject to the springdischarge.

For the third case, we supposed that pumping is carriedout only when the spring supplies less than 0Ð2 m3 s�1.The spring would dry up only for a few months in yearswith average rainfall, between October and December;only in dry years would it stop supplying water for longerperiods. The average discharge rate is 0Ð312 m3 s�1.Abstractions would be reduced to approximately halfof the time, since it would be necessary to pump atan average rate of only 0Ð104 m3 s�1, which equates toa withdrawal of 25%. The shortages caused would besubstantially lower in average years than in the first case,although dry years would give the same readings.

The fourth hypothetical case considers a demand equalto that in the second case, but with the extractionsbeing based on the spring discharge rate. Under theseconditions, it would be necessary to desist from pumpingwater from the aquifer for only a few months a year;the deficits caused by the heavy summer pumping arenot overcome until well into the rainy period. The meandischarge of the spring is 0Ð17 m3 s�1 and the amountneeded to be pumped is 0Ð256 m3 s�1, which is 44 l s�1

less than in the second simulated case. The shortages

Figure 12. The second case simulated at El Torcal: (a) hydrograph of the spring; (b) the progression of the water volume in the aquifer



produced in the aquifer, though somewhat lower, are ofthe same magnitude as in the case of continuous pumpedabstractions.

FINAL CONSIDERATIONS

The methods proposed here constitute an approach to thestudy of karstic aquifers involving the prior acceptance ofa series of hypotheses that may, in some cases, be opento debate. Nevertheless, an application of the procedureto a real karstic aquifer supports the reliability of theapproach.

A positive element that is worth emphasizing is theversatility of this approach to deal with many differentsituations, due to the establishment of transfer functionsbetween each of the three zones considered.

It is relevant that, by suitably modifying the physicalsignificance of the reservoir zones involved, this proce-dure can equally well be applied to the study of drainagesystems at surface level.

ACKNOWLEDGEMENTS

We are grateful to the three anonymous reviewers whocontributed to improve the original manuscript. ChristineLaurin and Sarah Steines corrected the draft Englishversions. This work was carried out in the context of theResearch Group RNM-189 with the economic support ofthe Junta de Andalucıa (Andalusian Government).

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Simple procedure to simulate karstic aquifers