hydrogeology

Hydrogeology (''hydro-'' meaning water, and ''-geology'' meaning the study of rocks) is the part of hydrology that deals with the distribution and movement of groundwater in the soil and rock (geology)rocks of the Earth's crust (geology)crust (commonly in aquifers). The term geohydrology is often used interchangeably. Some make the minor distinction between a hydrologist or engineer applying themselves to geology (geohydrology), and a geologist applying themselves to hydrology (hydrogeology).

Introduction - Hydrogeology (like most earth sciences) is an interdisciplinary subject; it can be difficult to account fully for the chemistrychemical, engineeringphysical, biologybiological and even legal interactions between soil, water, biospherenature and urban planningman. Although the basic principles of hydrogeology are very intuitive (e.g., water flows "downhill"), the study of their interaction can be quite complex. Taking into account the interplay of the different facets of a multi-component system often requires knowledge in several diverse fields at both the experimentexperimental and theorytheoretical levels. This being said, the following is a more traditional (Ockham's Razorreductionist viewpoint) introduction to the methods and nomenclature of saturated subsurface hydrology, or simply hydrogeology.

Hydrogeology in relation to other fields - Hydrogeology, as stated above, is a branch of the earth sciences dealing with the flow of groundwaterwater through aquifers and other shallow porous formations (typically less than 450 metrem or 1,500 feetft below the land surface.) The very shallow flow of water in the subsurface (the upper 3 m or 10 ft) is pertinent to the fields of soil science, agriculture and civil engineering, as well as to hydrogeology. The general flow of fluids (water, hydrocarbons, geothermal fluids, etc.) in deeper formations is also a concern of geologygeologists, geophysicsgeophysicists and petroleum geologypetroleum geologists. Groundwater is a slow-moving, viscosityviscous fluid (with a Reynolds number less than unity); many of the empirically derived laws of of groundwater flow can be alternately derived from the field of fluid mechanics. The mathematical relationships used to describe the flow of water through porous media are the diffusion equationdiffusion and Laplace equationLaplace equations, which have applications in many diverse fields. Steady groundwater flow (Laplace equation) has been simulated using electrical, elastic and heat conduction analogies. Transient groundwater flow is analogous to the diffusion of heat in a solid, therefore some solutions to hydrological problems have been adapted from heat transfer literature.Traditionally, the movement of groundwater has been studied separately from hydrologysurface water, climatology, and even the chemistrychemical and microbiologymicrobiological aspects of hydrogeology (the processes are uncoupled). As the field of hydrogeology matures, the strong interactions between groundwater, riversurface water, geochemistrywater chemistry, soil moisture and even climate are becoming more clear.

Definitions and material properties - ''Main article: Aquifer''In order to further characterize aquiferaquifers and aquitards some primary and derived physical properties are introduced below. Aquifers are broadly classified as being either confined or unconfined (water table aquifers), and either saturated or unsaturated; the type of aquifer affects what properties control the flow of water in that medium (e.g., the release of water from storage for confined aquifers is related to the storativity, while it is related to the specific yield for unconfined aquifers).

Hydraulic head - ''Main article: !Hydraulic_head_%28hydrology%29 Hydrualic head (hydrology)''Changes in hydraulic head (''h'') are the driving force which causes water to move from one place to another. It is composed of pressure head (''ψ'') and elevation head (''z''). The head gradient is the change in hydraulic head per length of flowpath, and appears in Darcy's law as being proportional to th discharge.Hydraulic head is a directly measurable property; ''ψ'' can be measured with a pressure transducer, and ''z'' can be measured relative to a surveyed datum (typically the top of the well casing). Commonly, in wells tapping unconfined aquifers the water level in a well is used as a proxy for hydraulic head, assuming there is no vertical gradient of pressure. Often only ''changes'' in hydraulic head through time are needed, so the constant elevation head term can be left out (''Δh = Δψ'').A record of hydraulic head through time at a well is a hydrograph or, the changes in hydraulic head recorded during the pumping of a well in a test are called drawdown.

Porosity - ''Main article: Porosity''.Porosity (''n'') is a directly measurable aquifer property; it is a fraction between 0 and 1 indicating the amount of pore space between unconsolidated soil particles or within a fractured rock. Typically, the majority of groundwater (and anything dissolved in it) moves through the porosity available to flow (sometimes called efective porosity).Porosity does not directly effect the distribution of hydraulic head in an aquifer, but it very strongly effects the migration of dissolved contaminants, since groundwater flow velocities are controlled by it.

Water content - ''Main page: water content''Water content (''θ'') is also a directly measurable property; it is the fraction of the total rock which is filled with liquid water. This is also a fraction between 0 and 1, but it must also be less than the total porosity.The water content is very important in vadose zone hydrology, where the hydraulic conductivity is a strongly nonlinear function of water content; this complicates the solution of the unsaturated groundwater flow equation.

Hydraulic conductivity - ''Main article: Hydraulic conductivity.''Hydraulic conductivity (''k'') and transmissivity (''T'') are indirect aquifer properties (they cannot be measured directly). ''T'' is the ''k'' integrated over the vertical thickness (''b'') of the aquifer (''T=kb''). These properties are measures of an aquifer's ability to transmit water. permeability_(geology)Intrinsic permeability (''κ'') is a secondary medium property which does not depend on the viscosity and density of the fluid (''k'' and ''T'' are specific to water); it is used more in the petroleum industry.

Specific storage and specific yield - ''Main article: Specific storage.''Specific storage !(''S_s'' )? and its depth-integrated equivalent, storativity !(''S=S_s b''),? are indirect aquifer properties (they cannot be mesured directly); they indicate of the amount of groundwater released from storage due to a unit depressurization of a confined aquifer. They are fractions between 0 and 1.Specific yield !(''S_y'' )? is also a ratio between 0 and 1 which indicates the amount of water released due to drainage, from lowering the water table in an unconfined aquifer. Typically ''S_y'' is orders of magnitude larger than ''S_s''. Often the porosity or effective porosity is used as an approximation to the specific yield.

Governing equations -

Darcy's Law - ''Main Article: Darcy's law''Darcy's law is a Constitutive_equation (derived by Henri Darcy, in 1856) which states the amount of groundwater discharging through a given portion of aquifer is proportional to the cross-sectional area to flow, the hydraulic head gradient and the hydraulic conductivity.

Groundwater flow equation - ''Main Article: Groundwater flow equation''The groundwater flow equation, in its most general form, describes the movement of groundwater in a porous medium (aquifers and aquitards). It is known in mathematics as the diffusion equation, and has many analogs in other fields. Many solutions for groundwater flow problems were borrowed or adapted from existing heat transfer solutions.It is often derived from a physical basis using Darcy's law and a conservation of mass for a small control volume. The equation is often used to predict flow to wells, which have radial symmetry, so the flow equation is commonly solved in polar coordinatespolar or cylindrical coordinates.The Aquifer testTheis equation is one of the most commonly used and fundamental solutions to the groundwater flow equation; it can be used to predict the head around a pumping well, due to the effects of pumping one or a number of wells.

Calculation of groundwater flow - To use the groundwater flow equation to estimate the distribution of hydraulic heads, or the direction and rate of groundwater flow, this partial differential equation (PDE) must be solved. The most common means of analytically solving the diffusion equation in the hydrogeology literature are:

Laplace transformLaplace and Fourier transformFourier transforms (to reduce the number of dimensions of the PDE),

similarity transform (also called the Boltzmann transform) is commonly how the Aquifer testTheis solution is derived,

separation of variables, which is more useful for non-Cartesian coordinates, and

Green's functions, 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 use to solve the groundwater flow equation, we need both initial conditions (heads at time (''t'') = 0) and boundary conditions (representing either the physical boundaries of the domain, or an approximation of the domain beyond that point). Often the initial conditions are supplied to a transient simulation, by a corresponding steady-state simulation (where the time derivative in the groundwater flow equation is set equal to 0).There are two broad categories of how the (PDE) would be solved; either mathematical analysisanalytical methods, numerical analysisnumerical methods, or something possibly in between. Typically, analytic methods solve the groundwater flow equation under a simplified set of conditions ''exactly'', while numerical methods solve it under more general conditions to an ''approximation''.

Analytic methods - Analytic methods typically use the structure of mathematics to arrive at a simple, elegant solution, but the required derivation for all but the simplest domain geometries can be quite complex (involving non-standard coordinates, conformal mapping, etc.). Analytic solutions typically are also simply an equation, which can give a quick answer based on a few basic parameters. The Aquifer test#Theis solutionTheis equation is a very simple (yet still very useful) analytic solution to the groundwater flow equation, typically used to analyze the results of an aquifer test or slug test.

Numerical Methods - The topic of numerical analysisnumerical methods is quite large, obviously being of use to most fields of engineering and science in general. Numerical methods have been around much longer than computers have (In the 1920s Lewis Fry RichardsonRichardson developed some of the finite difference schemes still in use today, but they were calculated by hand, using paper and pencil, by human "calculators"), but they have become very important through the availability of fast and cheap personal computers. A quick survey of the main numerical methods used in hydrogeology, and some of the most basic principles is below.There are two broad categories of numerical methods: gridded or discretized methods and non-gridded or mesh-free methods. In the common finite difference method and finite element method (FEM) the domain is completely gridded. The analytic element method (AEM) and the boundary integral equation method (BIEM — sometimes also called BEM, or Boundary Element Method) are only discretized at boundaries or along flow elements (line sinks, area sources, etc.), the majority of the domain is mesh-free.General Properties of 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 (squares, rectangles, triangles, blocks, tetrahedrontetrahedra, etc.) and solving the flow equation for each element (all material properties are assumed constant or possibly linearly variable within an element), then linking together all the elements using conservation of mass across the boundaries between the elements (similar to the divergence theorem). This results in a system which overall approximates the groundwater flow equation, but exactly matches the boundary conditions (the head or flux is specified in the elements which intersect the boundaries).Finite differences are a way of representing continuous differential operators using discrete intervals (''Δx'' and ''Δt''), and the finite difference methods are based on these (they are derived from a Taylor series). For example the first-order time derivative is often approximated using the following forward finite difference, where the subscripts indicate a discrete time location, : $\backslash frac\; =\; h\text{'}(t\_i)\; \backslash approx\; \backslash frac\; \}$.The forward finite difference approximation is unconditionally stable, but leads to an implicit set of equations (that must be solved using matrix methods, e.g. LU decompositionLU or Cholesky decompositionCholesky decomposition). The similar backwards difference is only conditionally stable, but it is explicit and can be used to "march" forward in the time direction, solving one grid node at a time (or possibly in parallel computingparallel, since one node depends only on its immediate neighbors). Rather than the finite difference method, sometimes the Galerkin finite element methodFEM approximation is used in space (this is different from the type of FEM often used in structural engineering) with finite differences still used in time.Application of Finite Difference Models
water.usgs.gov - MODFLOW is a well-known example of a general finite difference groundwater flow model. It was developed by the US Geological Survey (USGS) in 1988 as a modular and extensible simulation tool for modeling groundwater flow. It is free software developed, documented and distributed by the USGS. Many commercial products have grown up around it, providing graphical user interfaces to its text file based interface, and typically incorporating pre- and post- processing of user data. Many other models have been developed to work with MODFLOW input and output, making linked models which simulate several hydrologic processes possible (flow and transport models, surface water and groundwater models and chemical reaction models), because of the simple, well documented nature of MODFLOW.Application of Finite Element Models
Finite Element programs are more flexible in design (triangular elements vs. the block elements most finite difference models use) and there are some programs available (water.usgs.gov - SUTRA, a 2D or 3D density-dependent flow model by the USGS; hydrus2d.com - Hydrus, a commercial unsaturated flow model; and comsol.com - COMSOL Multiphysics (FEMLAB) a commercial general modeling environment), but they are not as popular in with practicing hydrogeologists as MODFLOW is. Finite element models are more popular in university and laboratory environments, where specialized models solve non-standard forms of the flow equation (vadose zoneunsaturated flow, density dependant flow, coupled heat transferheat and groundwater flow, etc.)Other Methods
These include mesh-free methods like the Analytic element methodAnalytic Element Method (AEM) and the Boundary Integral Equation Method (BIEM), which are closer to analytic solutions, but they do approximate the groundwater flow equation in some way. The BIEM and AEM exactly solve the groundwater flow equation (perfect mass balance), while approximating the boundary conditions. These methods are more exact and can be much more elegant solutions (like analytic methods are), but have not seen as widespread use outside academic and research groups.

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