Groundwater models are

computer models Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of, or the outcome of, a real-world or physical system. The reliability of some mathematical models can be deter ...

groundwater Groundwater is the water present beneath Earth's surface in rock and soil pore spaces and in the fractures of rock formations. About 30 percent of all readily available freshwater in the world is groundwater. A unit of rock or an unconsolid ...

flow systems, and are used by

hydrologist Hydrology () is the scientific study of the movement, distribution, and management of water on Earth and other planets, including the water cycle, water resources, and environmental watershed sustainability. A practitioner of hydrology is call ...

s and hydrogeologists. Groundwater models are used to simulate and predict

aquifer An aquifer is an underground layer of water-bearing, permeable rock, rock fractures, or unconsolidated materials (gravel, sand, or silt). Groundwater from aquifers can be extracted using a water well. Aquifers vary greatly in their character ...

conditions.

Characteristics

An unambiguous definition of "groundwater model" is difficult to give, but there are many common characteristics. A

model may be a

scale model A scale model is a physical model which is geometrically similar to an object (known as the prototype). Scale models are generally smaller than large prototypes such as vehicles, buildings, or people; but may be larger than small prototypes ...

or an electric model of a groundwater situation or

. Groundwater models are used to represent the natural groundwater flow in the environment. Some groundwater models include (chemical) quality aspects of the groundwater. Such groundwater models try to predict the fate and movement of the chemical in natural, urban or hypothetical scenario. Groundwater models may be used to predict the effects of hydrological changes (like groundwater pumping or irrigation developments) on the behavior of the aquifer and are often named groundwater simulation models. Groundwater models are used in various water management plans for urban areas. As the computations in

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

groundwater models are based on groundwater flow equations, which are

differential equations In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...

that can often be solved only by approximate methods using a

numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods th ...

, these models are also called ''mathematical, numerical, or computational groundwater models''. The mathematical or the numerical models are usually based on the real physics the groundwater flow follows. These mathematical equations are solved using numerical codes such as

MODFLOW MODFLOW is the U.S. Geological Survey modular finite-difference flow model, which is a computer code that solves the groundwater flow equation. The program is used by hydrogeologists to simulate the flow of groundwater through aquifers. The s ...

, ParFlow,

HydroGeoSphere HydroGeoSphere (HGS) is a 3D control-volume finite element groundwater model, and is based on a rigorous conceptualization of the hydrologic system consisting of surface and subsurface flow regimes. The model is designed to take into account all ...

OpenGeoSys
etc. Various types of ''numerical solutions'' like the

finite difference method In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial domain and time interval (if applicable) are ...

and the

finite element method The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat t ...

are discussed in the article on "

Hydrogeology Hydrogeology (''hydro-'' meaning water, and ''-geology'' meaning the study of the Earth) is the area of geology that deals with the distribution and movement of groundwater in the soil and rocks of the Earth's crust (commonly in aq ...

Inputs

For the calculations one needs inputs like: *hydrological inputs, *operational inputs, *external conditions:

initial In a written or published work, an initial capital, also referred to as a drop capital or simply an initial cap, initial, initcapital, initcap or init or a drop cap or drop, is a letter at the beginning of a word, a chapter, or a paragraph that ...

and

boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment * ''Boundaries'' (2016 film), a 2016 Canadian film * ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film *Boundary (cricket), the edge of the pla ...

conditions, *(hydraulic)

parameters A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...

. The model may have chemical components like

water salinity Salinity () is the saltiness or amount of salt dissolved in a body of water, called saline water (see also soil salinity). It is usually measured in g/L or g/kg (grams of salt per liter/kilogram of water; the latter is dimensionless and equal to ...

soil salinity Soil salinity is the salt content in the soil; the process of increasing the salt content is known as salinization. Salts occur naturally within soils and water. Salination can be caused by natural processes such as mineral weathering or by th ...

and other quality indicators of water and soil, for which inputs may also be needed.

Hydrological inputs

The primary coupling between groundwater and hydrological inputs is the

unsaturated zone The vadose zone, also termed the unsaturated zone, is the part of Earth between the land surface and the top of the phreatic zone, the position at which the groundwater (the water in the soil's pores) is at atmospheric pressure ("vadose" is f ...

vadose zone The vadose zone, also termed the unsaturated zone, is the part of Earth between the land surface and the top of the phreatic zone, the position at which the groundwater (the water in the soil's pores) is at atmospheric pressure ("vadose" is f ...

. The soil acts to partition hydrological inputs such as rainfall or snowmelt into

surface runoff Surface runoff (also known as overland flow) is the flow of water occurring on the ground surface when excess rainwater, stormwater, meltwater, or other sources, can no longer sufficiently rapidly infiltrate in the soil. This can occur when th ...

soil moisture Soil moisture is the water content of the soil. It can be expressed in terms of volume or weight. Soil moisture measurement can be based on ''in situ'' probes (e.g., capacitance probes, neutron probes) or remote sensing methods. Water that ent ...

evapotranspiration Evapotranspiration (ET) is the combined processes by which water moves from the earth’s surface into the atmosphere. It covers both water evaporation (movement of water to the air directly from soil, canopies, and water bodies) and transpira ...

and

groundwater recharge Groundwater recharge or deep drainage or deep percolation is a hydrologic process, where water moves downward from surface water to groundwater. Recharge is the primary method through which water enters an aquifer. This process usually occurs ...

. Flows through the

that couple surface water to

and

can be upward or downward, depending upon the gradient of

hydraulic head Hydraulic head or piezometric head is a specific measurement of liquid pressure above a vertical datum., 410 pages. See pp. 43–44., 650 pages. See p. 22. It is usually measured as a liquid surface elevation, expressed in units of length, ...

in the soil, can be modeled using the numerical solution of

Richards' equation The Richards equation represents the movement of water in unsaturated soils, and is attributed to Lorenzo A. Richards who published the equation in 1931. It is a quasilinear partial differential equation; its analytical solution is often limited ...

partial differential equation, or the ordinary differential equation Finite Water-Content method as validated for modeling

and

interactions.

Operational inputs

The operational inputs concern human interferences with the ''water management'' like

irrigation Irrigation (also referred to as watering) is the practice of applying controlled amounts of water to land to help grow crops, landscape plants, and lawns. Irrigation has been a key aspect of agriculture for over 5,000 years and has been devel ...

drainage Drainage is the natural or artificial removal of a surface's water and sub-surface water from an area with excess of water. The internal drainage of most agricultural soils is good enough to prevent severe waterlogging (anaerobic conditio ...

, pumping from

wells Wells most commonly refers to: * Wells, Somerset, a cathedral city in Somerset, England * Well, an excavation or structure created in the ground * Wells (name) Wells may also refer to: Places Canada * Wells, British Columbia England * Wel ...

watertable control Watertable control is the practice of controlling the height of the water table by drainage. Its main applications are in agricultural land (to improve the crop yield using agricultural drainage systems) and in cities to manage the extensive underg ...

, and the operation of retention or infiltration basins, which are often of an hydrological nature.
These inputs may also vary in time and space. Many groundwater models are made for the purpose of assessing the effects

hydraulic engineering Hydraulic engineering as a sub-discipline of civil engineering is concerned with the flow and conveyance of fluids, principally water and sewage. One feature of these systems is the extensive use of gravity as the motive force to cause the ...

measures.

Boundary and initial conditions

''Boundary conditions'' can be related to levels of the

water table The water table is the upper surface of the zone of saturation. The zone of saturation is where the pores and fractures of the ground are saturated with water. It can also be simply explained as the depth below which the ground is saturated. T ...

, artesian pressures, and

along the boundaries of the model on the one hand (the ''head conditions''), or to groundwater inflows and outflows along the boundaries of the model on the other hand (the ''flow conditions''). This may also include quality aspects of the water like salinity. The ''initial conditions'' refer to initial values of elements that may increase or decrease in the course of the time ''inside'' the model domain and they cover largely the same phenomena as the boundary conditions do. The initial and boundary conditions may vary from place to place. The boundary conditions may be kept either constant or be made variable in time.

Parameters

The parameters usually concern the

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

of and

distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over") ...

s in the domain to be modelled and those physical properties of the aquifer that are more or less constant with time but that may be variable in space. Important parameters are the

topography Topography is the study of the forms and features of land surfaces. The topography of an area may refer to the land forms and features themselves, or a description or depiction in maps. Topography is a field of geoscience and planetary sc ...

, thicknesses of soil / rock layers and their horizontal/vertical

hydraulic conductivity Hydraulic conductivity, symbolically represented as (unit: m/s), is a property of porous materials, soils and rocks, that describes the ease with which a fluid (usually water) can move through the pore space, or fractures network. It depends on ...

(permeability for water), aquifer transmissivity and resistance, aquifer porosity and storage coefficient, as well as the

capillarity Capillary action (sometimes called capillarity, capillary motion, capillary rise, capillary effect, or wicking) is the process of a liquid flowing in a narrow space without the assistance of, or even in opposition to, any external forces l ...

of the unsaturated zone. For more details see the article on

hydrogeology Hydrogeology (''hydro-'' meaning water, and ''-geology'' meaning the study of the Earth) is the area of geology that deals with the distribution and movement of groundwater in the soil and rocks of the Earth's crust (commonly in aq ...

. Some parameters may be influenced by changes in the groundwater situation, like the thickness of a soil layer that may reduce when the water table drops and/the hydraulic pressure is reduced. This phenomenon is called

subsidence Subsidence is a general term for downward vertical movement of the Earth's surface, which can be caused by both natural processes and human activities. Subsidence involves little or no horizontal movement, which distinguishes it from slope move ...

. The thickness, in this case, is variable in time and not a parameter proper.

Applicability

The applicability of a groundwater model to a real situation depends on the accuracy of the input data and the

. Determination of these requires considerable study, like collection of hydrological data (

rainfall Rain is water droplets that have condensed from atmospheric water vapor and then fall under gravity. Rain is a major component of the water cycle and is responsible for depositing most of the fresh water on the Earth. It provides water ...

) and determination of the parameters mentioned before including pumping tests. As many parameters are quite variable in space, expert judgment is needed to arrive at representative values. The models can also be used for the if-then analysis: if the value of a parameter is A, then what is the result, and if the value of the parameter is B instead, what is the influence? This analysis may be sufficient to obtain a rough impression of the groundwater behavior, but it can also serve to do a ''sensitivity analysis'' to answer the question: which factors have a great influence and which have less influence. With such information one may direct the efforts of investigation more to the influential factors. When sufficient data have been assembled, it is possible to determine some of missing information by calibration. This implies that one assumes a range of values for the unknown or doubtful value of a certain parameter and one runs the model repeatedly while comparing results with known corresponding data. For example, if

salinity Salinity () is the saltiness or amount of salt dissolved in a body of water, called saline water (see also soil salinity). It is usually measured in g/L or g/kg (grams of salt per liter/kilogram of water; the latter is dimensionless and equal t ...

figures of the groundwater are available and the value of

is uncertain, one assumes a range of conductivities and the selects that value of conductivity as "true" that yields salinity results close to the observed values, meaning that the groundwater flow as governed by the hydraulic conductivity is in agreement with the salinity conditions. This procedure is similar to the measurement of the flow in a river or canal by letting very saline water of a known salt concentration drip into the channel and measuring the resulting salt concentration downstream.

Dimensions

Groundwater models can be one-dimensional, two-dimensional, three-dimensional and semi-three-dimensional. Two and three-dimensional models can take into account the

anisotropy Anisotropy () is the property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, in a material's physi ...

of the aquifer with respect to the

, i.e. this property may vary in different directions.

One-, two- and three-dimensional

#One-dimensional models can be used for the vertical flow in a system of parallel horizontal layers. #

Two-dimensional In mathematics, a plane is a Euclidean ( flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as ...

models apply to a vertical plane while it is assumed that the groundwater conditions repeat themselves in other parallel vertical planes (Fig. 4). Spacing equations of subsurface drains and the

groundwater energy balance The groundwater energy balance is the energy balance of a groundwater body in terms of incoming hydraulic energy associated with groundwater inflow into the body, energy associated with the outflow, energy conversion into heat due to friction of fl ...

applied to drainage equations are examples of two-dimensional groundwater models. #

Three-dimensional Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informal ...

models like

Modflow MODFLOW is the U.S. Geological Survey modular finite-difference flow model, which is a computer code that solves the groundwater flow equation. The program is used by hydrogeologists to simulate the flow of groundwater through aquifers. The s ...

require

discretization In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerica ...

of the entire flow domain. To that end the flow region must be subdivided into smaller elements (or cells), in both horizontal and vertical sense. Within each cell the parameters are maintained constant, but they may vary between the cells (Fig. 5). Using numerical solutions of groundwater flow equations, the flow of groundwater may be found as horizontal, vertical and, more often, as intermediate.

Semi three-dimensional

In semi 3-dimensional models the horizontal flow is described by 2-dimensional flow equations (i. e. in horizontal x and y direction).

Vertical Vertical is a geometric term of location which may refer to: * Vertical direction, the direction aligned with the direction of the force of gravity, up or down * Vertical (angles), a pair of angles opposite each other, formed by two intersecting s ...

flows (in z-direction) are described (a) with a 1-dimensional flow equation, or (b) derived from a

water balance The law of water balance states that the inflows to any water system or area is equal to its outflows plus change in storage during a time interval. In hydrology, a water balance equation can be used to describe the flow of water in and out of ...

of horizontal flows converting the excess of horizontally incoming over the horizontally outgoing groundwater into vertical flow under the assumption that water is

incompressible In fluid mechanics or more generally continuum mechanics, incompressible flow ( isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An e ...

. There are two classes of semi 3-dimensional models: *''Continuous models'' or ''radial models'' consisting of 2 dimensional submodels in vertical

radial Radial is a geometric term of location which may refer to: Mathematics and Direction * Vector (geometric), a line * Radius, adjective form of * Radial distance, a directional coordinate in a polar coordinate system * Radial set * A bearing from ...

planes intersecting each other in one single axis. The flow pattern is repeated in each vertical plane fanning out from the central axis. *''Discretized models'' or ''prismatic models'' consisting of submodels formed by vertical blocks or prisms for the horizontal flow combined with one or more methods of superposition of the vertical flow.

Continuous radial model

An example of a non-discretized radial model is the description of groundwater flow moving radially towards a deep well in a network of wells from which water is abstracted. The radial flow passes through a vertical, cylindrical, cross-section representing the hydraulic

equipotential In mathematics and physics, an equipotential or isopotential refers to a region in space where every point is at the same potential. This usually refers to a scalar potential (in that case it is a level set of the potential), although it can als ...

of which the

surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...

diminishes in the direction of the axis of intersection of the radial planes where the well is located. Garmsar

Prismatically discretized model

Prismatically discretized models like SahysModILRI, 1995. ''SahysMod: Spatial Agro-Hydro-Salinity Model. Description of Principles, User Manual, and Case Studies''. International Institute for Land Reclamation and Improvement (ILRI), Wageningen, Netherlands. On line

/ref> have a grid over the land surface only. The 2-dimensional grid network consists of triangles, squares, rectangles or

polygons In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two to ...

. Hence, the flow domain is subdivided into vertical blocks or prisms. The prisms can be discretized into ''horizontal'' layers with different characteristics that may also vary between the prisms. The groundwater flow between neighboring prisms is calculated using 2-dimensional horizontal groundwater flow equations. Vertical flows are found by applying one-dimensional flow equations in a vertical sense, or they can be derived from the water balance: excess of horizontal inflow over horizontal outflow (or vice versa) is translated into vertical flow, as demonstrated in the article

Hydrology (agriculture) Agricultural hydrology is the study of water balance components intervening in agricultural water management, especially in irrigation and drainage. Water balance components The water balance components can be grouped into components correspo ...

. In semi 3-dimensional models, intermediate flow between horizontal and vertical is not modelled like in truly 3-dimensional models. Yet, like the truly 3-dimensional models, such models do permit the introduction of horizontal and vertical subsurface drainage systems. Semiconfined aquifers with a slowly permeable layer overlying the aquifer (the

aquitard An aquifer is an underground layer of water-bearing, permeable rock, rock fractures, or unconsolidated materials (gravel, sand, or silt). Groundwater from aquifers can be extracted using a water well. Aquifers vary greatly in their character ...

) can be included in the model by simulating vertical flow through it under influence of an overpressure in the aquifer proper relative to the level of the watertable inside or above the aquitard.

Groundwater modeling software and references

* Analytic Element Method *

FEFLOW FEFLOW (Finite Element subsurface FLOW system) is a computer program for simulating groundwater flow, mass transfer and heat transfer in porous media and fractured media. The program uses finite element analysis to solve the groundwater flow eq ...

* SVFlux * FEHM *

MicroFEM
*