#### 2.1. Barometric Pressure Analysis

Water table variations show an inverse and proportional correlation to barometric pressure fluctuations. Water table level variations (

WL) are related to barometric pressure changes (

BP) through the barometric efficiency (

$BE)$, which can be obtained as follows [

24]:

Atmospheric pressure variations generate a uniformly distributed stress on the ground surface. Part of the stress is absorbed by deformation of the materials that constitute the aquifer rock and the rest is transmitted to the fluid in the porous medium [

3]. If the materials exhibit a high transmissivity or specific yield, a drained condition is favored (i.e., mass transfer through flow), so that a response to atmospheric pressure fluctuations may not be observed [

25]. However, it is common practice to consider the lateral flow as negligible because of the vast extension of rock formation and the almost uniform effect of atmospheric loading on the ground surface. Water level variations in the borehole can be conceptualized as changes in the aquifer pore pressure, except with a well open to the atmosphere when the barometric pressure also exerts a uniform pressure on the water surface [

26]. In the case of a well penetrating an unconfined aquifer, barometric pressure variations will affect the water level within the well instantaneously. Meanwhile, if this effect is observed in the aquifer, it will show a delayed time because of the presence of air and other gases contained in the unsaturated zone. This lag time creates pressure differences between the well and the aquifer which generate outflow and inflows to the borehole, and therefore water-level fluctuations [

12].

Many methods have been developed to estimate barometric efficiency. Some of them consider it independent of frequency and calculated it by using linear regression techniques [

24,

27,

28,

29], while other methods consider barometric efficiency as frequency dependence, through transfer functions computed in just some segments or along the entire spectrum of frequencies. Additionally, barometric pressure can be evaluated alone or coupled with solid-Earth tide [

8,

9,

10,

30].

#### 2.2. Solid-Earth Tide Analysis

Solid-Earth tides are small, periodic variations in the earth’s shape due to forces of expansion and compression caused by the gravitational attraction of the celestial bodies, mainly the moon and the sun. These gravitational forces are balanced by changes in pore pressure in aquifer materials. The induced stress modifies the groundwater level in the wells located in both confined and unconfined aquifers [

3,

25]. Pore pressure (

p) is related to applied stress (

${\sigma}_{33}$) using the tidal efficiency (

${\gamma}_{e}$), which is obtained as follows [

1]:

The fluid contained in the porous medium responds to a three-dimensional stress field,

${\sigma}_{v}$. However, for the imposed deformation by solid-Earth tide or tectonic events, the three-dimensional stress field is not well-known as a priori knowledge. The vertical deformation observed on the terrain surface is approximately equal to one component of the horizontal tidal strain but with an opposite sign. The volumetric deformation of the surface rocks expressed as the sum of normal components is approximately equal to one of the horizontal tidal strain components, with an approximate amplitude of

$1\times {10}^{-8}$ [

26]. Thereby, it is better to examine the water level response to the areal strain (

${\epsilon}_{a}={\epsilon}_{11}+{\epsilon}_{22}$). Additionally, because

${\sigma}_{33}$ presents an opposite sign,

${\epsilon}_{v}\ll {\epsilon}_{a}$, which implies that the response of water level to solid-Earth tide will be substantially lower.

Strain field due to solid-Earth tide can be estimated from the theoretical gravitational potential [

31]. This theoretical strain field may differ from the real field one, mainly because of the local effects of geological faults and topography [

32,

33]. Geological and topographic influence is difficult to correct as a priori knowledge, and in the absence of measured strain, it is recommended that the theoretical value of strain field be used [

30].

#### 2.3. Aquifer Properties Estimation

In 1950, Jacob [

34] derived expressions that relate barometric and tide efficiencies to the elasticity of the medium and can be written as:

where

${\beta}_{k}$ is the rock-matrix compressibility (

$\mathrm{L}\xb7{\mathrm{T}}^{2}/\mathrm{M})$ (reciprocal of the rock bulk modulus

${K}_{k}$ (

$\mathrm{M}/\mathrm{L}\xb7{\mathrm{T}}^{2}$);

$\phi $ is the porosity (

$\%/100$); and

${\beta}_{f}$ is the water compressibility (

$\mathrm{L}\xb7{\mathrm{T}}^{2}/\mathrm{M}$), where units are expressed in fundamental units. If both expressions are added, the result is unity, therefore, in calculating any of these parameters, it is possible to define the other (

$BE=1-\gamma $).

Poroelastic properties of the rock formation can be estimated because the response of the water level to atmospheric loading and solid-Earth tide stress is a result of rock aquifer deformation in undrained conditions, assuming a static-confined response of the rock formation [

3,

4,

6,

35].

Rojstaczer and Agnew [

6] defined the areal strain sensitivity

${A}_{s}$ as the response of the water level

$WL$ to the areal strain

${\epsilon}_{a}$ as:

where

${\beta}_{u}$ is the rock-matrix compressibility in undrained conditions (

$\mathrm{L}\xb7{\mathrm{T}}^{2}/\mathrm{M}$); and

${v}_{u}$ is the Poisson’s ratio of the formation in undrained conditions and can be expressed as follows [

36]:

where

$B$ is the Skempton’s coefficient, dimensionless;

$\rho $ is the fluid density (

$\mathrm{M}/{\mathrm{L}}^{3}$);

g is the gravity acceleration (

$\mathrm{L}/{\mathrm{T}}^{2}$);

$v$ is the drained Poisson’s ratio; and

$\alpha $ is the A Skempton’s coefficient, dimensionless, also known as the Biot-Willis coefficient. The Skempton’s coefficients can be calculated using the following expressions:

Once values for barometric efficiency and areal strain sensitive have been calculated, the compressibility, Skempton’s coefficient, porosity, and specific storage (

${S}_{s}$ (

$1/\mathrm{L}$)) are computed as follows [

6,

37]:

The estimation of ${\beta}_{k},B,\phi ,\mathrm{and}{S}_{s}$ is through an iterative process and requires a priori values of ${\beta}_{u},v$.

On the other hand, if the compressibility of the rock materials that constitute the aquifer is not considered (

${\beta}_{u}=1/{K}_{u}=0$), it is possible to determine the porosity and specific storage based on the water level fluctuations due to solid-Earth tide and barometric pressure. Water level variation produced by aquifer dilatation as a result of solid-Earth tide is a function of the specific storage of the aquifer and can be calculated by measuring water level fluctuations (

$dh$), and assuming the drained Poisson’s ratio [

3], the specific storage can be obtained from the following equation [

4]:

where,

$\overline{h}=0.6032$ and

$\stackrel{=}{l}=0.0839$ are the Love numbers [

38]; and

$a$ is the Earth’s radius (L).

The ratio of the theoretical gravitational potential

$d{W}_{2}$ to the water level variation

$dh$, is proportional to the ratio of the amplitude of the harmonic component of the theoretical gravitational potential

${A}_{2}\left(\tau ,\theta \right)$ to the amplitude of water level variation

${A}_{w}\left(\tau \right)$ at the same period (

$\tau $). This is considering that the derivative can be approximated by a finite differential, small finite change

$\Delta {W}_{2}$ and

$\Delta h$ in a small period of time

$\Delta t$, the specific storage can be expressed as follows [

7]:

where

${A}_{2}\left(\tau ,\theta \right)$ is given by:

The general lunar coefficient

${K}_{m}$, the amplitude factor

b for each harmonic component with a period

$\tau $, and the latitude function

$f\left(\theta \right)$ values were obtained by [

7]. The classical method of representing the theoretical gravitational potential is to represent it as composed of a finite set of harmonic functions [

39]. Each

k-tidal harmonic component has a distinct amplitude

${A}_{k}$, frequency

${f}_{k}$, and phase

${\mathsf{\Phi}}_{k}$. Melchior [

40] concluded that only five of them are associated with fluctuations in groundwater level, and these harmonic components are responsible for 95% of the variation of gravitational potential [

41]. They are the

${M}_{2}$ and

${N}_{2}$ semidiurnal lunar tides, the

${S}_{2}$ semidiurnal solar tide, the

${O}_{1}$ diurnal lunar tide, and the

${K}_{1}$ diurnal lunar-solar tide.

Estimates of amplitude

${A}_{wk}$ and phase

${\phi}_{wk}$ for water level variations at frequencies that correspond with the components of the theoretical tidal potential are calculated from the regression coefficients

${a}_{wk}\mathrm{and}{b}_{wk}$ obtained from implementing regression techniques that minimize the mean square error, where the amplitude and phase are obtained as follows:

Therefore, the areal strain sensitivity for a particular component

${A}_{sk}$ is given by:

For incompressible aquifer rock materials, the change in the volume of the aquifer in response to a variation of strain could be approximated as a change in porosity [

1]. This assumption is suitable for many of the types of aquifers studied in hydrogeology, except those aquifers present in low porosity rocks [

3]. Thus, the porosity can be obtained as:

Analyses of water level fluctuations have generally focused on the amplitude response. Cooper et al. [

2] showed that the harmonic amplitude response depends on: the transmissivity (

$T$), storage coefficient (

${S}_{C}$), radius of the well casing (

${r}_{c}$), radius of the screened or open portion of the well (

${r}_{w}$), periodicity of the pressure head disturbance (

$\tau $), and the inertial effects of water in the well. From these relations, a set of dimensionless parameters was derived by Hsieh et al. [

42], as follows:

However, the amplitude of the response is generally different from that of the disturbance, and there is also a shift in phase. Hsieh et al. [

42] derived expressions to estimate

$T$ from the time lag between the earth tide dilatation of an aquifer and water level response in a well. The lag in time, referred to as the phase shift (

${\eta}_{k}$) of the

$k\mathrm{th}$ tidal constitute, is determined by [

42]:

The least square fitting procedure to estimate the regression coefficients, Equations (16) and (17), was applied to determine

${\mathsf{\Phi}}_{tk}$ for the

${O}_{1}$ and

${M}_{2}$ lunar components in the calculated areal strain tide. Graphs of the amplitude ratio and phase shift as a function of Equation (20) for selected values of the parameter in Equation (21) were obtained by Hsieh et al. [

42]. Values of dimensionless transmissivity can be estimated and converted into standard units (

${\mathrm{L}}^{2}/\mathrm{T})$ if the phase shift and an order of magnitude estimate of

${S}_{C}$ are calculated.

#### 2.4. Study Area

The Mexicali Valley is located northeast of the peninsula of Baja California. Its main economic activities are agriculture and geothermal electric power generation. The primary sources of water for irrigation are surface water from the Colorado River and groundwater from the Mexicali aquifer. The geothermal steam was obtained from the geothermal reservoir (confined and deeper aquifer) at the Cerro Prieto Geothermal Field located in the central west of the Mexicali Valley.

Tectonically, both the shallow aquifer and reservoir are located in the Salton Trough, which is part of the San Andreas–Gulf of California fault system that corresponds to the boundary between the Pacific and North American tectonic plates [

43]. On a local scale, the Salada-Cucapá, Cerro Prieto, and Imperial faults created a basin up to 5000 m deep filled with sediments supplied by the Colorado River, as well as transported eroded debris from the Colorado Plateau basin margins [

44,

45]. The fault systems are currently active and have a significant amount of associated seismic activity [

46,

47]. The transmissive sediments in the basin have been categorized into two main units—consolidated and unconsolidated sediments–separated by strata of a very low permeability because of hydrothermal alteration [

48,

49]. The consolidated deepest unit consists of sandstones that comprise the geothermal reservoir. The shallow unconsolidated unit consists of fine to coarse sands with intercalations of gravel, clays, and silts. The shallow unit has a variable thickness of 400–2500 m and contains the aquifer of the Mexicali Valley [

50]. Groundwater pumping, irrigation seepage, return flows, and recharge from episodic Colorado River flows are the primary hydrologic processes affecting the potentiometric surface of the Mexicali Valley aquifer [

51]. The Mexicali Valley aquifer is an unconfined and non-homogeneous aquifer of variable thickness, formed by a sequence of unconsolidated granular sediments mostly of deltaic origin. Discontinuous layers of materials with different permeabilities cause local confinement, but together behave as a single hydrogeological unit [

21]. The general map of the study area including tectonic, geology, and hydrology context is shown in

Figure 1.

#### 2.5. Data

A set of three monitoring wells (C-03, PZ-01, and PZ-03) located in the central-west area of the Mexicali Valley were equipped with pressure transducer data loggers for recording the water level every 5 min for 83 days, from 23 May–14 August 2007, and their location is shown in

Figure 1. The boreholes were drilled into Quaternary alluvial deposits, which mainly consisted of water-saturated sand, gravels, and clays. The main characteristics of the well design are presented in

Table 1. The water table level fluctuations were recorded using pressure transducers (Solinst Levelogger

^{®}, Solinst Canada Ltd, Georgetown, ON, Canada) with a measuring range of 5.00 m and accuracy of 0.05% of the full range. Barometric pressure fluctuations were also recorded every 5 min using a Solinst Barologger

^{®} (Solinst Canada Ltd, Georgetown, ON, Canada) barometer installed in PZ-03 with a measuring range of 1.50 m and an accuracy in the order of 0.001 m. Water level and atmospheric pressure time series are shown in

Figure 2a.

The sampling frequency election was based on the water level response to the seismic activity on Mexicali Valley, analyzed and reported in a previous study [

23]. Detailed inspection of the water level did not show a significant response to local seismic activity (local magnitude ≤ 4.0) recorded by IRIS [

52]. Therefore, in this work, we assumed that groundwater fluctuations are only caused by barometric pressure and solid-Earth tide effects.

The areal strain associated with the theoretical gravitational potential for the vicinity of each well, expressed in nanostrain (1 nstr = 1 ppb), was calculated using Some Programs for Ocean-Tide Loading (SPOTL, version 3.3.0.2) [

53,

54]. This computer code does not consider geological and topographic discontinuity effects for the calculation; also, the ocean tide was ignored. The areal strain is shown in

Figure 2b.