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## Hydrodynamics in coastal aquifers

**Dynamic balance of fresh water and salt water in a coastal aquifer**

The intrusion of seawater in coastal aquifers was first conceptualized independently by Badon-Ghijben (1889) and Herzberg (1901) assuming hydrostatic equilibrium, immiscible fluids and the existence of a sharp interface between fresh and saltwater in a homogeneous unconfined aquifer. They found that the depth of the freshwater–saltwater interface below sea level (zs) is shown in Figure 2 and given by:

Where ρf is the density of freshwater, ρs is the density of saltwater, and hf is the elevation of the water table above sea level. When the equation is applied correctly, the estimated depth closely approximates the real one (Cheng and Ouazar, 2004); it is still widely used to simulate saltwater intrusion (Essaid, 1992, Cheng and Chen, 2001) and, especially for educational purposes, to gain clear insight into the behavior of fresh and saline groundwater in coastal aquifer systems (Essink, 2001, Duque et al., 2008). Due to molecular diffusion and hydrodynamic dispersion, fresh and salt water are actually miscible liquids: the contact between the two fluids is therefore a transition zone rather than a sharp interface (Gambolati et al., 1999). The situation is further complicated by the fact that the saltwater intrusion itself changes the fluid density, so that this parameter varies in space and time as a function of changes in concentration, temperature and pressure in the fluid. Furthermore, the porous medium itself is usually stochastically heterogeneous. In order to properly reproduce the mechanism of saltwater encroachment, a variable density flow and transport modeling approach is therefore currently adopted (Voss and Souza, 1987, Holzbecher, 1998a, Koch and Zhang, 1998, Bear, 1999, Diersch and Kolditz, 2002). The medium- and long-term effects of water management are difficult to foresee due to interaction between numerous elements and variables having different natures (Bear, 1999, Cheng and Ouazar, 2004, Custodio, 2005). Groundwater management thus requires the use of numerical models to test current and alternative exploitation scenarios taking into account technical aspects as well as economic, legal, social and political ones (Bear and Verruijt, 1987, Anderson and Woessner, 2002, Bear, 2004).

**The pattern of fresh water flow in a coastal aquifer**

In homogeneous media, the interface between salt and fresh water follow the Ghyben-Herzberg relation which assumes, under hydrostatic conditions, the weight of a unit column of freshwater extending from the water table to the salt-water interface is balanced by a unit column of salt water extending from sea level to that same point on the interface. Also, for every unit of groundwater above sea level there are 40 units of fresh water between sea level and the saltwater interface at depth.

Figure 2. Interface between fresh water and salt water in a coastal aquifer in which the salt water intrusion is static (according to the Ghyben-Herzberg relation).

Figure 3. Circulation of salt water from the sea to the zone of diffusion and back to the sea in homogeneous coastal aquifer (Cooper, 1959).

This dynamic is more complex and very difficult to characterize and simulate when groundwater pumping through the freshwater lens induces a salt water upconing.

Although the presence of salinity in coastal aquifers has been widely studied, the source of this salinity remains in many cases unclear (Lloyd, 1992) Monitoring of saline domains is essential to determine and predict groundwater and soils deterioration, and assess the groundwater resources management in coastal aquifers. Monitoring involves designing a field survey strategy and methodology to obtain a reliable dataset (Melloul and Goldenberg, 1997).

Monitoring surveys can be implemented with direct (boreholes and sampling) or indirect (geophysical, geochemical or isotope survey) methods or a combination of both. Direct methods are generally expensive, especially for large study areas. It is in that context that inexpensive electrical techniques, such as the geophysical prospecting methods, are the most suitable (Land et al., 2004). With the development of computing science, we can characterize and predict saltwater intrusion with numerical modeling software such as SEAWAT, FEFLOW, SUTRA and FEMWATER.

In this research, the groundwater flow modeling will be validated using isotope measurements and geo-electrical investigations.

### Effects of density in coastal groundwater

Many field and laboratory studies have shown that fluid density gradients caused by variation in concentration and/or temperature can play an important role in the transport of solutes in groundwater systems (Bear, 1988, Simmons et al., 2001). Simulation of density driven flow problems is very complicated due to the non-linear coupling between flow and transport equations and requires long computational time and/or powerful computer (Post et al., 2007, Ackerer and Younes, 2008). When corrections for density variations are ignored or not properly taken into account, misinterpretation of both groundwater flow direction and magnitude may result (Lusczynski, 1961, Post et al., 2007).

The hydraulic head is one of the most important metric in hydrogeology as it underlies the interpretation of groundwater flow patterns, the quantification of aquifer properties and the calibration of flow models (Post and von Asmuth, 2013). Hydraulic head observations in aquifers and surface water/sea water are very important to determine the interactions between groundwater and surface water bodies and research on salt intrusion. This parameter determined based on water level measurements in wells and piezometers.

**Groundwater head of variable density**

**Head and Pressure equation**

Hydraulic head is a measure of mechanical energy per unit weight of water at everywhere within a groundwater aquifer. At any point ‘i’ in the groundwater body, hydraulic head can be quantified by the following equation: hi is the hydraulic head (m), zi is the elevation head (m), represents the level of the well screen, and hp,i the pressure head is the length of the water column in the well relative to zi. For stagnant water condition in the well, i is the density of water at the measurement point (kg/m3), g is the gravitational acceleration (m/s2), and the pressure Pi of the groundwater at the measurement point (Pa) is equal to the weight of the overlying water per unit cross-sectional area:

Pgh p ,i hp,i related to the pressure of the ground water at the well screen Pi or the height of the water column.

Figure 4. Schematic representation of a piezometer in a groundwater system with a constant density (edited from Musczynski,1961 and Post, 2013).

The absolute pressure Pabs is the sum of the atmospheric pressure and the groundwater pressure:

Pasb = Patm + Pi (6)

In groundwater of constant-density, water table is defined as the surface where the absolute pressure equals the atmospheric pressure, Pasb = Patm and Pi = 0. With the equation(5), hi=zi, therefore, water table measured in situ equals the hydraulic head at the depth of water table. In this case, if all heads are compare to the same reference datum and groundwater have the same density , hydraulic head could be compare directly to define the flow direction, groundwater will flow along the hydraulic head gradient from high to low hydraulic head.

If groundwater system with varies over an area or with depth, value of hp,i do not correctly represent partially variations of pressure P or the same P can correspond to different values of hp,i depending on groundwater density. The concept of point-water head have been used (Lusczynski, 1961) as water level in a well filled with water coming from a point in an aquifer and which is sufficient to balance the pressure in the aquifer at that point. Therefore, density corrections must be made to the head that is measured in a well (Fetter, 2001) so the density of water in each piezometer must be estimated.

**The fresh-water head**

The fresh water head at any point in the ground is defined (Lusczynski, 1961, Acworth, 2007) as the head that would be developed in a variable density system if the pressure at the piezometer intake was generated by an equivalent column of fresh water. In other words, this point is defined as the water level in a well filled with fresh water from that point to a level high enough to balance the existing pressure at that point. Therefore, for the waters that have higher salinity contents than fresh water, the fresh water head will be higher than the point-water head and if we are in a fresh water aquifer, all point water heads are also fresh water heads. Figure 5 shows the head relationship in water of variable density for point-water head and fresh-water head in an unconfined aquifer with saline water overlain by fresh water.

Figure 5. Schematic representation of head definitions in variable-density groundwater systems (modified from Musczynski 1961). Lightest shading corresponds to fresh water and darker shading represents increasing salinity.

**Variable-density flow**

Fluid flow is classified as variable-density flow if the flow pattern is affected by density differences in the fluid system, mainly influenced by temperature and salinity (Holzbecher, 1998b). Some other scientists use the term as “density-driven flow” or “density-dependent flow”.

**Darcy’s Law**

Actually, flow in aquifer cannot be directly measured. Darcy’s law is used to define the relation between specific flow and hydraulic head which can be measured.

Where: q denotes specific discharge (volume of fluid per unit cross-sectional area of porous medium per unit time, m/s) and also call as Darcy velocity K is hydraulic conductivity or permeability (m/s), and considered as isotropic, i.e. constant in all directions x,y,z. h is the hydraulic gradient, which is the driving force of groundwater flow per unit weight of groundwater (dimensionless).

The equation (9) is a simplified form of general physical law for fluid flow in a porous medium, which also applies to variable-density fluids (Bear, 1988, Post et al., 2007): Where kx, ky, kz, are the principal directional permeabilities in the x, y, z directions (m2), a property of the porous medium; is dynamic viscosity (kg/m/s) of the groundwater; P is fluid pressure (kg/m/s2), is fluid density (kg/m3) and g is the gravitational acceleration (m/s2).

From equation (11) shows the two basic driving forces for groundwater flow should be know: P and g . This is the main reason why quantification of groundwater flow from field data, which normally occurs in the form of head measurements requires a special treatment (Post et al., 2007). In this case, it is necessary that the hydraulic head refers to the same density, normally to the density of freshwater. Combining equation (7) and equation (12) gives the following: principal directions x, y, z. It is assumed here that salinity variations have a negligible effect on so that f/ 1, which is a very good approximation for most practical applications. The difference between Kf and field-measured values of hydraulic conductivity, which are for ambient values of and , is much smaller than the uncertainty associated with this parameter. Therefore, in the equation for two components x and y, no special correction of existing hydraulic conductivity is required. Whereas in vertical flow component, the term involving local groundwater variable-density is needed (Post et al., 2007). This equation is used in most of well-known variable-density flow and transport code such as FEFLOW (Holzbecher, 1998a, Kolditz et al., 1998, Diersch and Kolditz, 2002) or SEAWAT (Guo et al., 2002).

Horizontal flow component:

When using equations (10) and (13) to calculate horizontal flow, it is very important that the freshwater head gradient (or pressure gradient) is evaluated using freshwater heads at the same depth (Post et al., 2007). In variable-density groundwater, freshwater head may vary with depth, even for hydrostatic conditions (Fetter, 2001). Therefore, all the measurements are taken from piezometers with screens at different depths, fresh water heads need to be calculated at a suitable reference depth and with the same density. At a point i, the pressure at reference depth Pr has a relation to pressure measured at screen Pi

**Table of contents :**

**Chapter 1 INTRODUCTION **

1.1 General context

1.2 Research objectives

1.3 Outline of thesis

**Chapter 2 SALTWATER INTRUSION INTO THE COASTAL AQUIFER AND THE STUDY AREA **

2.1 Hydrodynamics in coastal aquifers

2.1.1 Dynamic balance of fresh water and salt water in a coastal aquifer

2.1.2 The pattern of fresh water flow in a coastal aquifer

2.2 Effects of density in coastal groundwater

2.2.1 Groundwater head of variable density

2.2.2 Variable-density flow

2.3 The Crau aquifer and study area

2.3.1 Introduction

2.3.2 Geological and hydrogeological setting

2.3.3 Topography

2.3.4 Climate

2.3.5 Artificial recharge and discharge

2.3.6 Origin of salinity of Crau aquifer

**Chapter 3 METHODOLOGIES **

3.1 Methods

3.2 Hydrodynamic measurements

3.3 Geophysical investigation

3.3.1 Electromagnetic (EM) mapping

3.3.2 The Electrical Resistivity Topography (ERT)

3.3.3 Applying geophysics to validate flow and transport modeling

3.3.4 Application to the study area

3.4 Short-lived natural radioactive isotopes as tracers

3.4.1 General introduction

3.4.2 Estimate of groundwater velocities using the decay of 222Rn in a single well

3.4.3 Estimate of groundwater discharge in to surface water using 222Rn continuous monitor

3.5 Numerical modeling of saltwater intrusion

3.5.1 Introduction

3.5.2 Heterogeneity and variable-density flow

3.5.3 Governing equations

3.5.4 Numerical modeling of heterogeneity and variable-density flow

3.5.5 FEFLOW Code

**Chapter 4 RESULTS **

4.1 Geophysics survey

4.1.1 Electromagnetic mapping (EM34)

4.1.2 Electrical Resistivity Tomography (ERT)

4.1.3 Partial conclusions

4.2 Isotopic surveys

4.3 Groundwater numerical modeling

4.3.1 Conceptual model

4.3.2 3D density-dependent flow model

4.3.3 Information on saltwater intrusion

4.4 Coupling geophysical and isotopic approach to calibrate and validate saltwater intrusion modeling

4.5 Effect of recharge and pumping on saltwater intrusion

**GENERAL CONCLUSION **

**Bibliography**