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Hidrobiológica

versión impresa ISSN 0188-8897

Hidrobiológica vol.30 no.3 Ciudad de México sep./dic. 2020  Epub 06-Dic-2021

https://doi.org/10.24275/uam/izt/dcbs/hidro/2020v30n3/herrera 

Research articles

Determination of hydraulic transmissivity in coastal aquifer by optimal estimation of the Qe-T relationship using Kalman filter

Determinación de la transmisibilidad hidráulica en un acuífero costero mediante estimación óptima de la relación Qe-T usando el filtro de Kalman

Jaime Herrera-Barrientos1 
http://orcid.org/0000-0001-8077-9402

Héctor José Peinado-Guevara2  * 

José de Jesús Campos-Gaxiola3 

Adriana Cruz-Enriquez3 

Víctor Manuel Peinado-Guevara2 

María de los Ángeles Ladrón de Guevara-Torres4 

Salvador Isidro Belmonte-Jiménez4 

Leonardo Herrera1 

1Centro de Investigación Científica y de Educación Superior de Ensenada, B.C. Carretera Ensenada-Tijuana No. 3918, Zona Playitas, Ensenada, B. C., 22860. México

2Escuela de Ciencias Económicas y Administrativas, Universidad Autónoma de Sinaloa. Blvd. Juan de Dios Bátiz s/n, San Joachín, Guasave, Sinaloa, 81049. México

3Facultad de Ingeniería Mochis, Universidad Autónoma de Sinaloa. Fuente de Poseidón y Ángel Flores s/n, Jiquilpan, Los Mochis, Sinaloa, 81210. México

4Instituto Politécnico Nacional, Unidad CIIDIR-Oaxaca. Hornos No. 1003, Col. Noche Buena, Municipio de Santa Cruz Xoxocotlán. Oaxaca, Oaxaca, 71230. México


ABSTRACT

Background:

The knowledge on the management of water as a vital resource to develop agriculture allows having greater effectiveness in its use.

Goals:

The agricultural activity in the lower part of the Sinaloa River depends on the fresh water of the regional dams and the aquifer.

Methods:

The use of groundwater represents approximately 15% of the total water used. In the presence of prolonged periods of drought, new wells are drilled without the use of an appropriate guide for farmers on the location of aquifer areas with a greater hydraulic transmissivity with the purpose of exploiting them more rationally. The National Water Commission has registered more than 680 wells on both banks of the Sinaloa River.

Results:

The information of 205 of these pumping wells for agricultural or domestic use and the specific capacity information was analyzed. Then, 79 out of 205 wells have pumping tests. It is then determined that the objective of this research work was to find the relationship between the specific capacity (Qe) and hydraulic transmissivity (T) data of the study area using the Thiem formula, considering a fixed value of the radius of influence. This hypothetical consideration and the heterogeneities of the aquifer environment add to the T-Qe relationship an additional component, it is determined that it has a normal behavior. Using the Kalman filter it is possible to eliminate or reduce such a component, thus improving the determination of the T-Qe relation of an R-value of 0.95 (without filter) to 0.97 (with filter), for a linear and exponential relationship.

Conclusions:

The application of a T-Qe estimate allows characterizing the aquifer area, with this procedure a map was obtained on the distribution of T, which will serve as a guide for future exploitations of groundwater in the study area.

Keywords: groundwater; hydraulic parameters; Kalman filter; Thiem formula

RESUMEN

Antecedentes:

El conocimiento sobre el manejo del agua como recurso indispensable para desarrollar la agricultura permite tener mayor efectividad en su uso.

Objetivos:

La actividad agrícola en la parte baja del río Sinaloa depende del agua dulce de las presas regionales y del acuífero.

Métodos:

El uso del agua subterránea, representa aproximadamente el 15% del agua total usada. Ante la aparición de periodos de sequía prolongados, se perforan nuevos pozos sin el uso de una guía apropiada que oriente a los agricultores sobre la ubicación de las zonas acuíferas que tienen una mayor transmisividad hidráulica con el propósito de explotarlos de manera más racional.

Resultados:

La Comisión Nacional del agua tiene registrados más de 680 pozos en ambas márgenes del Río Sinaloa. Se analizó la información de 205 de estos pozos de bombeo para uso agrícola o doméstico y la información de capacidad específica. De este número, 79 pozos tienen pruebas de bombeo. Se determina entonces que el objetivo de este trabajo de investigación fue encontrar la relación entre los datos de capacidad especifica (Qe) y transmisividad hidráulica (T) de la zona de estudio usando la fórmula de Thiem, considerando un valor fijo del radio de influencia.

Conclusiones:

Esta consideración hipotética y las heterogeneidades del medio acuífero suman a la relación T-Qe una componente adicional, se determina que tiene un comportamiento normal. Mediante el filtro de Kalman es posible eliminar o reducir tal componente, mejorando así la determinación de la relación T-Qe de un valor r de 0.95 (sin filtro) a 0.97 (con filtro), para una relación lineal y exponencial. La aplicación de una estimación T-Qe permite caracterizar la zona acuífera, con este procedimiento se obtuvo un mapa sobre la distribución de T, que servirá de guía para futuras explotaciones del agua subterránea en la zona de estudio.

Palabras claves: agua subterránea; filtro de Kalman; fómula de Thiem; parámetros hidráulicos

INTRODUCTION

Hydraulic conductivity and transmissivity of an aquifer constitutes essential data for groundwater exploitation management and planning (Kazakis et al., 2016). Measurements of aquifer hydraulic parameters can be measured by laboratory experiments or in situ tests; however, the former are inaccurate, and the latter is expensive and difficult (Bateni et al., 2015).

Hydraulic transmissivity (T) determines the flow of groundwater that is transmitted through a vertical strip of aquifer unit width under a hydraulic gradient unit (Palafox-Avila, 2006). This parameter is required in numerical flow modeling processes (Painter et al., 2007; Asfahani, 2016); horizontal recharge of fresh water (Cruz-Falcón et al., 2013); in the determination of the radius of influence of the descent cone of the well (Vargas, 2016) in order to determine the perimeters for the protection from contamination of the well water, and water management, among others. It is useful to estimate the groundwater resource and its integral management (Tizro et al., 2012). Its determination can be from the specific well capacities (Qe), which is obtained from the pumping flow Q, static and dynamic level in a pump well once it is stabilized.

Through the Thiem’s formula, assuming a fixed influence radius and that there are no load losses in the wells, using a relation between transmissivity values obtained from pumping tests and their corresponding specific capacity, dependency relations are obtained between both parameters, which can be linear or exponential (Al Farrah et al., 2013).

Determinations of T in the manner indicated above have been made successfully in different geological environments (Chandra et al., 2008; Perdomo et al., 2014; Malík et al., 2015; Sanz et al., 2005; Sánchez et al., 2013). WRI Report 87-4034 (2008) states that estimates of transmissivity from specific capabilities provide values that are used to characterize transmissivity in certain local areas and may reveal trends or patterns. However, there are cases where this relation is not met due to the heterogeneity present in the aquifer, and erroneous transmissivities that do not correspond to the aquifer are obtained.

The groundwater of the coastal aquifer of the lower right and left bank of the Sinaloa River constitutes an important element of support for the development of agricultural activity in the region, since the water from local dams is insufficient to irrigate the Guasave valley, that is why the extraction of groundwater is required through wells and bored wells. Of the total water used in agriculture, groundwater accounts for 15% and surface water 85% (Peinado-Guevara et al., 2017).

The bed of the Sinaloa River is regulated by the Gustavo Díaz Ordaz Dam. In 2005, with the water from the dam, the left and right banks of the Sinaloa River were irrigated, 54,134 ha and 45,105 ha, respectively, corresponding to the Irrigation District No.63. Guillermo Blake Aguilar is another dam in the region, with this, 21,820 ha are cultivated. CONAGUA (2000), using the piezometric fluctuations method, determined an overexploitation of 97.3 million m³ for October 1997 and October 1998 periods.

The National Water Commission has registered more than 680 pumping wells distributed on both banks of the Sinaloa River, of the which, 79 pumping tests are analyzed with their respective information of specific capacity Qe and hydraulic transitivity T. The Kalman filter is applied to the Qe-T relation which is widely used to estimate the state of dynamic systems, as an optimization method, as an optimizer that eliminates or reduces the normal random component that is the use that will be given in this work to remove the Gaussian noise component produced naturally by assuming that the radius of influence of the wells is constant as well as by the influence of the heterogeneities of the aquifer, which deviate from the theoretical considerations of the Thiem formula which assumes that the aquifer is confined, homogeneous, isotropic, horizontal, among other considerations.

MATERIALS AND METHODS

Description of the study area. The study area lies between the coordinates 25°25´8.36” and 25°48’30.04” north latitude and 108°13’32.64” to 108 °35’38.65” west longitude (Fig. 1). The climate is very hot and warm dry with rain in summer. The average annual precipitation for the period 1986-2013 fluctuated from 300 to 400 mm (INEGI, 2014). The average annual temperature is of 22 to 24° for the serie1986-2013 (INEGI, 2014). The soils are of alluvial origin, Cenozoic era, Quaternary period, Vertisol soils predominate (62.55% of the surface of the municipality) (INEGI, 2009).

Figure 1 Location of the study area. With black circles are indicated wells with information of pumping tests and with black squares the wells with specific capacity information. 

Wells information. 205 pumping wells for agricultural or domestic use that have specific capacity information were analyzed. Of these, 79 wells have pumping tests carried out by the National Water Commission using different techniques.

Of the wells with pumping test it was obtained that 11.4% of the T values are between medium to high (100 < T < 500 m2/day), 8.9% in high (500 <T <1000 m2/day) and 79.7% in very high (T > 1000 m2/day) according to the classification of Villanueva & Iglesias (1984). Those values indicate that this is a coastal aquifer with high capacity to transmit water that, in the face of a scenario of overexploitation due to its high potential to be contaminated by saline intrusion or contamination by dissolution due to the presence of evaporite bodies in the study zone.

There is information of 30 lithological columns with depths ranging between 100 and 150 meters, with four lithological sections being constructed that show the heterogeneous distribution (horizontal and vertical variations) of the aquifer materials. The wells were geolocated with a portable GPS Magallanes brand.

Qe-T relation. Al Farrah et al. 2013 uses the Thiem equation which, by setting a value of R and substituting that of rw according to the radius of the well in question, Thiem’s formula can be written as

T=12πlnRrwQ/s

T=CQ/s

T=CQe

Where: T is hydraulic transmissibility (m2/day), R is the radius of influence of the pumping well (m) and rw (m) is the radius of the well, s (m) is the depression in the well. For a given R value and the corresponding radius of the well rw, the term 12πlnRrw is a constant, C.

Hamm et al. (2005) and Galvão et al. (2016) have established relations of the form:

T=CQ/sn

The range of the exponential coefficient is in a range of 0.6 to 1.4 and is related to the lithology and the aquifer (Al Farrah et al., 2013).

With 79 pairs of Qe values, T obtained from the National Water Commission, the Qe -T ratio was found without using the Kalman filter and using the filter. From the filtered T-Qe ratio, a map of equal hydraulic transmissibility was obtained, which is a guide for future drilling.

Uncertainties of Qe and T. The values of Qe and T present uncertainties due to the following: un aquifer tends to present heterogeneities and anisotropy, the well does not always cross the entire aquifer formation, part of the water pumped from the well reaches reincorporated into the aquifer, the pumping flow Q tends to present variations due to fluctuations in the electric current of the pumping system, the diameter of the well is finite and there is usually a head drop in the well due to the screen pipe or well face, the radius of influence is often unknown, among others factors. All these variables add uncertainties to the relationship between Qe and T expressed by Thiem equation, which establishes that it is that of a straight line that passes through the origin and that, due to the aforementioned uncertainties, it undergoes variations, so that the Kalman filter eliminates additive contributions whose behavior has a normal distribution (white noise uncertainty), thus achieving a better relationship between Qe and T given by the correlation coefficient.

Kalman Filter. The Kalman Filter (KF) is an optimal estimator (Kim, 2011). Faragher, 2012, indicates that it has extensive use as a noisy data smoother. Following Grewal-Mohinder & Andrews, 1995, the discrete-time model is established for a linear stochastical system which takes the form:

xk=Ak-l+uk-l (1)

yk=Ckxk+k (2)

The zero mean uncorrelated Gaussian random processes {uk-1 } and {vk } have matrices of variances Qk-1 and R k , respectively, at time tk ; x k describe the unknown signal; yk is the measurement with white gaussian noise and the matrices Ak-1 and C k are constants. The KF as a data softener has the form:

x^k-=Ak-lx^k-l+

Pk-=Ak-lPk-l+Ak-lT+Qk-l

Kk=Pk-CkTCkPk-CkT+Rk-1

x^k+=x^k-+Kkyk-Ckx^k-

Pk+=Pk--KkCkPk-

Where x^k- is the estimate signal of xk before processing the measurement y k in the instant t k ; Pk- is the variance of the estimation error x^k-; Kk is the optimization factor, usually named Kalman gain; x^k+ describes the estimate signal of xk after processing the measurement vk yk in the instant tk ; and Pk+ is the variance of the estimation error x^k+[/p] . For the particular case of this work, that correspond to the filtering of signals, the system (1) and (2) as well as the KF are specified with the matrices:

Ak-1=I2*2

Ck=I2*2,Qk-l=0.1 I2*2Rk=0.01 I2*2 and initial conditions

x^0+=173,7.5T and P0=I2*2

RESULTS

Geometry of the aquifer. With the information available from 30 lithological columns, the aquifer’s geometry was determined, as well as the distribution of the aquifer materials, which, as can be seen, has significant lateral and vertical heterogeneities. Figure 2 shows four profiles with the sequence of materials showing the abundance of gravel with clay-silt matrix, highlighting the presence of a gravel body, which are indicators of the ability of materials to yield water. The wells of 100 to 150 m partially penetrate the aquifer, since it cannot touch the geological or hydrological basement.

Figure 2 Perpendicular section to the Sinaloa River. 

Relation between specific capacity and hydraulic transmissivity. The empirical approach is based on determining the empirical relations between T and Qe, which are regarded as aquifer and area-specific (El-Naqa, 1994; Al Farrah et al., 2013). Table 1 shows the values of hydraulic transmissivity, and specific capacity Qe. Figure 5 shows the relation between transmissivity and specific flow rate with an adjustment of 0.989, so it is possible to estimate the transmissivity in function of its specific capacity. T is directly proportional to the specific flow, such as those obtained by Bosch, 2014, Chandra et al., 2008, Ebong et al., 2014 and Perdomo et al., 2014.

Table 1 Wells data from pumping tests and specific well capacities. 

Coordinates Hydraulic transmissivity Specific well capacities (Qe) Coordinates Hydraulic transmissivity Specific well capacities (Qe)
X Y m/day lps/m X Y m/day lps/m
749489 2844495 173 7.5 767015 2846992 3629 30.6
750195 2851435 207 3.2 767267 2848352 3646 29.4
758551 2840450 242 3.5 766837 2846157 4441 47
749330 2845569 251 2.9 752315 2849936 4454 37
744860 2845948 276 3.9 758790 2846828 4687 34.5
759225 2838831 302 3.8 756001 2839538 4700 34.6
747965 2851393 354 4.8 766321 2846824 5435 37.2
766554 2849107 492 4.2 768835 2848045 6834 75.9
763521 2847384 570 5 755058 2845062 8208 57.5
761184 2845644 829 6.9 751056 2824860 3410 27.506
751656 2847922 864 10.2 753500 2824020 2925 26.051
752414 2844704 924 20.8 754605 2825918 1588 13.136
769138 2849591 924 8 753768 2828632 1909 16.313
764518 2846326 1020 8.7 753040 2831360 2888 27.334
761854 2839960 1054 11.6 752519 2831428 3158 27.425
759801 2837919 1210 14.5 751240 2828174 3116 25.799
755019 2847063 1253 11.2 749198 2824766 4078 35.123
744330 2844398 1305 11.1 746824 2826422 4569 40.408
765560 2844468 1339 12.9 748387 2829222 4295 37.584
762087 2842274 1382 12 749063 2830475 2354 17.067
758964 2837902 1469 15.7 750466 2830934 2039 16.423
760495 2848063 1529 14.2 751524 2832152 1695 14.293
746357 2847915 1555 20.3 747628 2834115 1369 11.574
755981 2840616 1555 17.2 746611 2832315 1050 10.214
751052 2842985 1564 16.3 745434 2831474 2401 21.481
754746 2839514 1620 14.2 756419 2826252 1721 14.786
753243 2852417 1728 16.8 743042 2829906 2117 19.524
756610 2844169 1771 18 743186 2832156 452 3.908
753091 2831170 1901 21.5 743064 2834081 1676 14.466
753236 2830864 1901 21.6 744423 2836516 987 10.424
757544 2846342 2125 19 743590 2835017 715 5.937
774069 2856590 2195 21 755282 2826961 1725 16.063
772631 2856067 2316 22.4 747126 2833203 2319 19.399
769315 2847747 2411 25.9 749683 2832454 2325 21.208
765637 2846194 2635 28.1 751488 2831230 1254 11.335
747203 2839926 2730 23.2 752286 2830051 2685 25.924
766768 2848188 2981 29.4 751737 2829058 3139 29.707
753652 2845651 3041 27.3 752929 2827102 2063 17.914
752065 2848392 3283 37.4 754113 2827671 2207 18.492
771567 2853551 3473 37.3        

Relation is given by:

T=109.533Qe

Bosch (2014) states that the factor that relates to T with Q e oscillates from a range between 100 to 500, so the relation obtained is consistent with that obtained in other aquifers by other authors.

Figure 3 shows that it is possible to estimate the hydraulic transmissivity in function of specific capacity. The application of the Kalman filter to the T-Q e relation improved the correlation coefficient going from 0.95 to 0.97. Figure 3c shows how applying the Kalman filter reduces the dispersion of the data and therefore increases r, as is the case.

Figures 3a-d Regression line fitting specific discharge (Qe) and transmissivity (T) data from pumping tests in 79 wells: a) linear regression; b) exponential relation; c) linear regression with Kalman filter; d) exponential relation with Kalman filter. 

Practical application. In 126 pumping wells Q e was calculated by knowing the expense, static and dynamic level of the water in each well, estimating T from a relation T =109.533Q e . The map of equal values of hydraulic transmissivity shows that it is lower on the left bank of the Sinaloa River, which is consistent with that established by Norzagaray-Campos (2003), who built four profiles parallel to the river, indicating that the lithological changes are explained by the migration of the Sinaloa River from East to West.

Figure 4 Transmissivity Isocontorms in m2/day. 

Figures 5a-b Hydraulic transmissivity behavior (m2/day) in the study area: a) from the linear relation; b) from the Kalman filter application. 

DISCUSSION

Specific capacity is directly proportional to T, such as those obtained by [Ebong et al., 2014; Perdomo et al., 2014, Bosch, 2014, Zeferino et al., 2016, Hamm et al., 2005 y Galvão et al., 2016]. The relation T/Qe = 109.533 is in the range proposed by Bosch (2014) who established that the factor that relates T with Qe oscillates in a range between 100 and 500, reason why the relation obtained is consistent with that obtained in other aquifers, as reported by Perdomo et al. (2014) T = 135.36Qe - 50 and Zeferino et al. (2016) of T = 100.23 Qe - 7.126 in different geological environments. In other studies, this relation has been of the exponential form as reported by Hamm et al. (2005) T=0.99 Qe 0.89 where T and Qe are in m2/day. Galvão et al. (2016) also proposes an empirical relationship in karst systems in Sete Lagoas, MG, Brazil. T=330 Qe 0.21 where T and Qe are in m2/day, the coefficient of determination R2 was 0.55.

It has been found that the Kalman filter is a useful tool in the determination of the relation T-Qe, since it improved the relation between both parameters by increasing the correlation coefficient. The Thiem formula has a practical application that, although it is a relation for a homogeneous and isotropic medium, works in areas that present heterogeneities as is the present case.

Heterogeneities that are considered to have an effect with normal behavior in the T-Qe relation, which can be reduced by the Kalman filter, as indicated, the correlation increased from 0.95 to 0.97.

The T-Qe relation has practical application since it allows to characterize the aquifer environment concerning T.

ACKNOWLEDGMENT

Our gratitude to the General Direction of Research and Post graduate of the Autonomous University of Sinaloa for supporting the project and generate the suitable conditions to fulfill the present work.

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Received: February 14, 2019; Accepted: October 10, 2020

*Corresponding author: José Peinado-Guevara: e-mail: Hpeinado75@hotmail.com

To quote as: Herrera-Barrientos, J., H. J. Peinado-Guevara, J. de J. Campos-Gaxiola, A. Cruz-Enriquez, V. M. Peinado-Guevara, M. de los Á. Ladrón de Guevara-Torres, S. I. Belmonte-Jiménez & L. E. Herrera. 2020. Determination of hydraulic transmissivity in coastal aquifer by optimal estimation of the Qe-T relationship using Kalman filter. Hidrobiológica 30 (3): 211-219.

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