Hydrogeology

Introduction

Hydrogeology as the majority of the Sciences of the ground is an interdisciplinary branch. As long as it can be difficult to take account completely of all the interactions between the ground, the Eau, the Biosphère and the man, as well on a chemical level as Physique, biology or even legal. Although the basic principles of hydrogeology are intuitive (for example: water runs downwards), the study of their interactions can be very complex. In a general way the fact of taking account the interactions of various facets of a system to several components asks for a knowledge of several branches as much at the experimental level that theoretical. These precautions being taken, this article will be interested rather in the methods and the nomenclature of hydrogeology.

Hydrogeology compared to other branches

As one previously saw, hydrogeology is a branch of sciences of the ground which deals with the flow of the Subterranean water through the Aquifère not very deep S and other mediums Poreux (generally less 1000 meters under surface). The flow of water far from deep (higher than 3 meters under surface) is a relevant branch for the Pédologie, the Agriculture and the Civil engineering, as much as for hydrology. The flow of Fluide S which one finds in major formations (such as water but also the Carbohydrates and the fluids Géothermique S) is also important for the Géologie, the Géophysique and the Géologie of oil. The subterranean water is a fluid viscous (with a Reynolds number smaller than 1) which runs slowly (except loan for particular geological environments like the karstic conduits traversed by the underground rivers, the fractured alluvia very coarse and rocks). Many laws deduced empirically from subterranean water can also be deduced from the Mécanique of the fluids by the particular case of flow of Stokes where one considers terms of viscosity and pressure, but not of inertia.

The mathematical relations used to describe the flow of water through a porous environment are the equation of diffusion and the equation of Laplace. These two equations have applications in several different fields. The regular flow of water, described by the equation of Laplace, was simulated thanks to analogies of electricity, elasticity and thermal Conduction. The flow of transition from subterranean water is similar to the diffusion of heat in a solid, so that certain solutions with hydrological problems were adapted those of the thermal Conduction.

Traditionally the movement of subterranean water was studied independently in Hydrologie, in Climatologie, Chimie and Microbiologie. With the maturation of hydrogeology, the strong interactions between water souteraine, the water surface, the Geochemistry, the moisture of the ground and the Climat become increasingly clear.

Definitions and properties

principal Article: Aquiferous

An aquifer is in a strict sense a permeable geological formation which contains water, in opposition to a Aquitard or a Aquiclude which are both not very permeable (but distinction between the two terms ready with confusion). One generally speaks about aquitard to indicate a not very permeable geological formation (relative with the formation considered as permeable). One will keep here with the spirit that the permeable character of a geological formation is relative. A sandstone can be regarded as permeable compared to an argillaceous level, but becomes not very permeable if one compares it with coarse gravelly deposits for example.

The flow of subterranean water can be not-confined (free) or confined (captive). In the first case, the level of the free face of the underground flow can move vertically unbounded to the top (to topographic surface). In the second case, the presence of a not very permeable level does not allow the rise in the level of the free face. In the case of a free flow, the hydraulic potential on the free face of the flow is equal to the altitude of this point. In the case of a confined flow, the hydraulic potential is equal to or higher than the altitude of the Mur of the not very permeable formation located at the top of the flow.

In the case of a confined flow, the aquifer is entirely saturated with water (saturation=1 or 100%). One speaks about saturated flow. In the case of a free flow, one distinguishes part of the aquifer saturated with water (it is the saturated zone), and a part for which saturation is lower than 1, it is the not-saturated zone which is well heard located at the top of the saturated zone. The zone of transition between the saturated zone and the not-saturated zone is called the capillary Frange.

Hydraulic potential

The hydraulic Gradient of potential (concretely, the difference in water level in two wells occupying the same tablecloth) is at the origin of the displacement of the water masses - water moving the most potential towards low. The Loi of Darcy, valid only for the saturated mediums, postulates that the water flow through a given surface of an aquifer is proportional to the hydraulic gradient. The relationship between the flow and the hydraulic gradient is hydraulic conductivity (Perméabilité being a deprecated term).

The hydraulic potential is a directly measurable property. It can be measured using a Transducteur of pressure. This value can be negative in the case of suction, but it positive in the aquifers is saturated. A recording of the hydraulic potential on a well and during a certain time is called a Hydrographe.

Porosity

principal article: Porosity .

Porosity ( N ) one is propiété directly measurable of an aquifer. It is a fraction between 0 and 1 which indicates the empty quantity of space between free particles of ground or in a fractured rock. Generally the majority of subterranean water moves through porosity available to run there.

Porosity directly does not affect the distribution of the hydraulic potentials in an aquifer, but it has a very strong effect on the migration of contaminants, because it affects the speed of the flow of subterranean water by a relation proportional opposite.

Water contents

The water contents ( θ ) are a directly measurable property. It represents the fraction of the rock which is full of liquid water. It is a fraction between 0 and 1, and it must be lower or equal to total porosity.

The water contents are very important in hydrology of the Zone vadose where hydraulic conductivity is strongly a non-linear function water contents. That complicates the solution of the equation of the flow not-saturated with subterranean water.

Hydraulic conductivity

Hydraulic conductivity ( K ) and the transmissivity ( T ) are indirect properties of the aquifer. T is equal to K integrated on the vertical thickness ( B ) of the aquifer. These properties are measurements of the capacity of an aquifer to lead water. The Perméabilité ( κ ) is a secondary property of the medium. It does not depend on the viscosity nor of the Densité of the fluid. K and T is specific to water. The permeability is especially used in oil industry.

Specific storage and specific output

Specific storage ( S_s ) and its equivalent integrated on the depth, the stockativity, are properties indirect of the aquifer: they cannot be measured directly.

They indicate the quantity of water of the ground evacuated by storage because of a unit of depressurization of a confined aquifer, They are fractions between 0 and 1.

The specific output ( S_y ) is also a fraction between 0 and 1 ( S_y ≤ porosity) which indicates the quantity of water evacuated by the drainage of to a lowering of the water table in an aquifer not conditioned. Generally S_y is several orders of magnitude larger than S_s . Porosity or effective porosity is often used as limits higher than the specific output.

Fundamental equations of state

Law of Darcy

Equation of water flow of the ground

The equation of water flow of the ground, in its most general form, described the movement of the water of the ground in a porous environment (an aquifer or a aquitard). It is known in mathematics under the name of equation of diffusion, and it has much analog in other branches. Many solutions of the water flow of the ground was borrowed or adapted existing solutions the thermal Conduction.

It is often derived from a physical base by using the Loi of Darcy and the conservation of the mass for a small volume of control. The equation is often used to predict a flow towards well, which have a radial symmetry, so that the equation of flow is commonly solved with polar or cylindrical coordinates.

The test of aquifer is one of the most used solutions and most fundamental of the equation of water flow of the ground. It can be used to predict the evolution of the head due to the effect of a pumping or several wells of pumping.

The solution of Thiem solves the equation of flow of the water of ground to balance (equation of Laplace). Real balance is seldom reached actually has less of the presence of broad close sources of water (a lake or a river).

Calculation off groundwater flow To uses the groundwater flow equation to estimate the distribution off hydraulic heads, however the direction and spleen off groundwater flow, this Partial differential equation (PDE) must Be solved. The most common means off analytically solving the diffusion equation in the hydrogeology literature are:

Laplace and Fourier transforms (to reduce the number off Dimension S off the PDE),
Similarity transform (also called the Boltzmann transform) is commonly how the Theis solution is derived,
Separation off variable, which is more useful for non-Cartesian coordinates, and
Green' S function S, which is another common method for deriving the Theis solution - from the Fundamental solution to the diffusion equation in free space.

No matter which method we uses to solve the Groundwater flow equation, we need both initial conditions (heads At time ( T ) = 0) and Boundary conditions (representing either the physical boundaries off the domain, but year approximation off the domain beyond that not). Initial Often the conditions are supplied to has transient simulation, by has corresponding steady-state simulation (where the time derivative in the groundwater flow equation is set equal to 0).

There are two broad categories off how the (PDE) would Be solved; either analytical methods, numerical methods, but something possibly in between. Typically, analytic methods solve the groundwater flow equation under has off simplified set conditions exactly , while numerical methods solve it under more general conditions to year approximation .

Analytic methods

Analytic methods typically uses the structure off Mathematics to arrives At has simple, elegant solution, goal the required derivation for all goal the simplest domain geometries edge Be quite complex (involving not-standard Coordinate S, Conformal mapping, etc). Analytic solutions typically are also simply year equation, which edge give has quick answer based one has few BASIC parameters. The Theis equation has very simple (yet still very useful) analytic solution to the Groundwater flow equation, typically used to analyze the results off year Aquifer test gold Slug test.

Numerical methods

Broad The topic off numerical methods is quite, obviously being off uses to most fields off Engineering and Science in general. Numerical methods cuts been around much to skirt than Computer S cuts (In the 1920s Richardson developed nap off the Finite difference designs still in uses today, goal they were calculated by hand, using paper and pencil, by human " calculators"), goal they cuts become very off important through the availability fast and cheap Personal computer S. has quick survey off the hand numerical methods used in hydrogeology, and nap off the most BASIC principles is below.

There are two broad categories off numerical methods: gridded gold discretized methods and non-gridded gold mesh-free methods. In the common Finite difference method and Finite element method (FEM) the domain is completely gridded (" cut" into has grid gold mesh off small elements). The Analytic element method (AEM) and the boundary integral equation method (BIEM - sometimes also called BEM, gold Boundary Method Element) are only discretized At boundaries gold along flow elements (line sinks, area sources, etc), the majority off the domain is mesh-free.

General properties off gridded methods

Gridded Methods like Finite difference and Finite element methods solve the groundwater flow equation by breaking the problem area (domain) into many small elements (public gardens, rectangles, triangles, blocks, will tetrahedra, etc) and solving the flow equation for each element (all material properties are assumed constant variable gold possibly linearly within year element), then linking together all the elements using Conservation off farmhouse across the boundaries between the elements (similar to the Divergence theorem). This results in has system which overall approximates the groundwater flow equation, goal exactly matches the boundary conditions (the head gold flow is specified in the elements which intersect the boundaries).

Finite differences are has way off representing continuous Differential operators using discrete intervals ( Δx and Δt ), and the finite difference methods are based one thesis (they are derived from has Taylor series). Derivative For example the first-order time is often approximated using the following forward finite difference, where the subscripts indicate has discrete time hiring,

$\ frac {\ partial H} {\ partial T} = h' (t_i) \ approx \ frac {h_i - h_ {i-1}} {\ Delta T}.$

The forward finite stable difference approximation is unconditionally, goal leads to year implicit set off equations (that must Be solved using matrix methods, e.g LU gold Cholesky decomposition). Stable The similar backwards difference is only conditionally, goal it is explicit and edge Be used to " march" forward in the time direction, solving one grid node At has time (gold possibly in parallel, since one node depend only one its immediate neighbors). Rather than the finite difference method, sometimes the Galerkin FEM approximation is used in space (this is different from the standard off FEM often used in Structural engineering) with finite differences still used in time.

Application off finite difference models

MODFLOW has well-known example off has general finite difference groundwater flow model. It was developed by the US Geological Survey (USGS) in 1988 aces has modular and extensible simulation tool for modeling groundwater flow. It is Free software developed, documented and distributed by the USGS. Commercial Many products cuts grown up around it, providing Graphical to use interface S to its text file based interface, and typically incorporating pre and post- processing off to use dated. Many other models cuts off been developed to work with MODFLOW input and output, making linked models which simulate several hydrologic processes possible (flow and models transport, Surface toilets and Groundwater models and chemical reaction models), because the simple, well documented natural off MODFLOW.

Application off finite element models

Finite Element programs are more flexible in design (triangular elements vs. the block elements most finite difference models uses) and there are summons programs available (SUTRA, has 2D gold 3D density-depend flow model by the USGS; Hydrus, has commercial unsaturated flow model; FEFLOW, has commercial modeling environment for subsurface flow, aqueous solution and heat processes transport; and COMSOL Multiphysics (FEMLAB) has commercial general modeling environment), goal unless they are gaining in importance they are still not ace popular in with practicing hydrogeologists ace MODFLOW is. Finite element models are more popular in University and Laboratory environments, where specialized models solve not-standard forms off the flow equation (unsaturated flow, dependant Density flow, coupled heat and groundwater flow, etc)

Other methods

Thesis include mesh-free methods like the Analytic Element Method (AEM) and the Integral Boundary Method Equation (BIEM), which are closer to analytic solutions, goal they C approximate the groundwater flow equation in nap way. The BIEM and AEM exactly solve the groundwater flow equation (perfect farmhouse balances), while approximating the boundary conditions. Thesis methods exact are more and edge elegant Be much more solutions (like analytic methods are), goal cuts not seen ace widespread uses outside academic and research groups.

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