This section contains a brief glossary of terms specifically relevant to mathematical modelling of ground water systems. The terms have been selected somewhat arbitrarily for inclusion in the glossary which is thus not intended to be comprehensive. The terms are arranged, for the sake of convenience, in alphabetical order. The object of this glossary is to provide unambiguous definitions of relevant terms, concerning mathematical modelling of complex ground water systems, that have been employed in documents introduced by experts in this case. It may also be of some assistance to the court in adopting a consistent understanding of terms associated with expert investigations undertaken in this case and with expert testimony based upon them.

Aquiclude

A saturated, but weakly permeable bed, unit, formation, or group of sub-surface formations that does not yield water freely to a well or spring. However, an aquiclude may transmit appreciable water to or from adjacent aquifers. Its properties are usually established through appropriate field tests.

Aquifer

A formation that contains saturated and permeable water-bearing materials such as the Upper South Park formation which transmit water and is subject to recharges and discharges. An aquifer formation may be composed of one or more geologic units. Its properties are usually established through appropriate field tests.

Aquifer Test

A man-made test involving the withdrawal of measured quantities of ground water from, or addition of water to, a well and the measurement of resulting changes in potentiometric head within the aquifer both during and after the period of discharge or addition. The purpose of aquifer tests is to obtain quantitative estimates of aquifer property values. Such field tests should be conducted with sufficient spatial completeness to allow results to be used for modelling.

Aquitard

A geologic formation, group of formations, or part of a formation known as a unit through which virtually no ground water flows. Its properties are usually established through appropriate field tests.

Boundary Conditions

Mathematical representations of either aquifer recharge or aquifer discharge conditions which relate hydrologic connections between the chosen model domain and portions of the ground water system which lie external to it. A specific form is the so-called general-head boundary condition. Boundary conditions are based upon the premise that hydrogeologic conditions outside the domain remain unaffected by changes occurring inside the domain.

Calibration, [of a Model]

The activity of systematically adjusting values assigned to model parameters, that have been incompletely quantified in the field, in successive model runs until predicted variables such as potentiometric heads and ground water fluxes, compare well with observed, i.e. measured, values in a plausible way and within preset tolerances. Valid and accurate data on such variables, set aside specifically for model calibration, are thus important prerequisites. Calibration is a necessary part of model development for any given site or ground water system. This is so because spatially and temporally complete data sets for purposes of modelling the variety sub-surface flow mechanisms, in both spatial and temporal senses, will rarely be available. Great care is needed in selecting and in implementing proper procedures of model calibration, for, upon its validity and success will depend the model's predictive capability.

Coefficient of Storage

The volume of water which an aquifer either releases from, or takes into, its pore-space volume per unit surface area of the aquifer, per unit change in its potentiometric head. It affects the rate of drawdown and the shape of the cone of depression predicted by a model for a given pumping rate. Values of this property are usually established through appropriate field tests.

Cone of Depression

A depression that forms in a ground water table or potentiometric surface and which usually assumes the shape of an inverted cone. It usually develops around a well or cluster of wells from which water is being withdrawn by well pumping and defines the zone of influence of the pumping well. Cones of depression are invariably dynamic and influenced by other natural and man-made phenomena. Accurate predictions of the size, shape and location of cones of depression require the application of a valid, well-calibrated and accurate model to represent the location and rates of well pumping.

Confined Aquifer

A formation in which the ground water is isolated from local atmospheric conditions at the point of discharge by impermeable geologic formations or units. Ground water in confined aquifers is generally subject to pressures greater than the local atmospheric value. Aquifers are said to be under artesian pressure or head under such circumstances. Ground water recharge to a confined aquifer usually occurs in complex ways.

Confining Pressure [Upon a Model Layer]

This variable denotes the mathematical difference between the elevation of potentiometric head within a model layer and the elevation of the top surface of this layer. It is clearly a spatial distribution within the domain covered by the specific model layer, that may or may not be constant. Confining pressure identifies that amount of excess pressure which will be influenced by pumping ground water from the model layer under consideration. It is positive when confining pressure indeed exists and negative when the potentiometric head in the layer has been reduced to water table conditions through well pumping.

Darcy's Law

A derived algebraic expression for the flow of fluids within porous media based upon the postulation that such flows are relatively slow, laminar and that influences of inertia forces can be neglected. The derived expression expresses a linear relationship between ground water flow rate in any chosen direction, per unit perpendicular area, and the gradient of potentiometric head or water level in that direction. The constant of proportionality in this relationship is called the hydraulic conductivity. This "law" is one of the foundations of ground water models.

Discharge [Rate]

Rate at which ground water is removed from aquifers either by natural mechanisms such as flows to low-lying rivers, springs, wetlands, fens, swamps, natural ponds, or man-made features such as well pumping. The mechanisms by which ground water may be naturally removed from rocks with fractures and fault zones are more complex than those associated with unconsolidated soils.

Domain, [or Model Domain]

The region or spatial extent of an aquifer which is represented in a mathematical model of ground water flow systems. In the case of finite-difference computer programs used for modelling such as MODFLOW, this domain is usually rectangular when viewed as a plane. However, such rectangular domains may in general contain inactive numerical grid cells representing areas that are external to irregularly shaped aquifers. The selection of the modelling domain influences the spatial extent of zones upon which impacts may be sought to be predicted by a model.

Drawdown

The distance (usually in feet) between the static water level and the local surface of the cone of depression when pumping from an aquifer takes place. Care is needed to ensure that this distance is calculated from a true static water level. Either measured values or values calculated by a model may be used for this purpose.

Equipotential Line

A contour line on the water table or potentiometric surface along which the potentiometric head (or head) of ground water within an aquifer is invariant. It is also known as a potentiometric head contour line. Water flow takes place perpendicular to these lines and is positive in the direction of decreasing fluid potential or potentiometric head.

Evapotranspiration [or ET or Consumptive Use by Vegetation]

A mechanism defining consumptive use of water by vegetation. It includes both evaporation and transpiration of water to support plant growth. Beneficial ET is a term that refers to consumption by crops; non-beneficial ET refers to consumption by natural vegetation. ET affects the availability of water for development and its timing. Representation of ET in models requires understanding of the specific types of vegetation involved, their demand for water, their opportunistic tendency to capture available precipitation and soil moisture. A proper representation involves the specification of several variables which characterize such behavior in the manner appropriate to vegetation in a given ground water system.

Faults [or Fault Zones]

Fissures or openings within geologic formations, caused by relative motions of the earth's crust within geologic time frames, that may be filled with a variety of solid materials. When such fissures transect water-bearing units at any angle to the vertical, they can cause fluid motions to depart from those that occur in porous formations and obey Darcy's Law. In many practical situations, faults occur within zones of finite spatial extent which may or may not be rectilinear and within which hydrologic property values may vary substantially. Fault zones may or may not be associated with more recent movements of geologic formations that can give rise to variations in transmissive properties.

Flow lines

Imaginary lines drawn to indicate the direction followed by ground water flows toward points of discharge from an aquifer, usually under presumed or postulated steady-state conditions. Flow lines are, by definition, perpendicular to the equipotential lines which are also known as potentiometric head contour lines. Flow lines within complex ground water systems are invariably curved.

General-Head Boundary

A boundary of an aquifer that is represented in a ground water flow model in a specific way which allows that boundary to serve either as a recharge or as a discharge surface. This representation is based upon a prescribed, i.e. presumed known, potentiometric-head value outside the aquifer and a conductivity value at the boundary itself. Depending upon whether calculated potentiometric heads within a model domain just inside the boundary are lower or higher than the external value, the boundary can serve as either a recharge or a discharge surface.

Grid Cells [or Numerical Grid Cells]

The smallest individual segments of a domain used in a mathematical model to represent ground water flow mechanisms. It is usually rectangular in shape for finite-difference methods and may be irregular in shape for finite-element methods. Both of these methods are available for use in computer programs upon which mathematical models of ground water flows can be based. The chosen size of a grid cell governs the precision and accuracy with which mechanisms such as stream-aquifer interactions can be represented.

Grid Lines

Straight lines drawn to connect grid nodes in any given direction. These lines are themselves rectilinear in finite difference models and orthogonal to those aligned with different coordinate directions.

Grid Nodes

Points which represent the geometric centers of grid cells or intersections of grid lines in finite-difference models or apexes of triangular elements in finite-element models which employ them. They represent the finite number of discrete locations at which the equation governing ground water flows is solved by the modelling program.

Ground Water Table

The surface between the zone of saturation and the zone of aeration; it is thus the potentiometric head surface of an unconfined aquifer. Interpretations of ground water table are usually made from water-elevation measurements made in wells located in such an aquifer, when they are sufficient and spatially distributed. Such interpretations may be used to depict ground water flow directions, to prescribe initial conditions for modelling or for purposes of model calibrations.

Head [or Potentiometric Head]

A quantity denoting energy possessed by water within an aquifer, by virtue of its elevation and/or fluid pressure. Other contributions to such energy are usually negligible. For instance, the influences of velocity upon head in sub-surface formations are negligible. It is the level of water (in feet) attained by a piezometer inserted into an aquifer. For an unconfined aquifer the head is the water-level elevation and is usually measured in feet.

Head Loss

That part of potentiometric head energy (in feet) which is lost due to friction as water flows through either pipes or other conduits or from wells.

Homogeneous

In the context of aquifer property values that are material to ground water systems, and to mathematical modelling thereof, this term refers to spatial uniformity in material property values. Thus a homogeneous aquifer is one composed of such materials that they can be assigned spatially uniform property values. Such a representation is of course an idealization leading to simplifications.

Hydraulic Conductivity

An aquifer property representing the rate of flow of water in gallons per day through a cross section of one square foot under a unit hydraulic gradient, at the prevailing temperature. In common terms the unit of this aquifer property is ft³/day per ft², i.e. ft/day. Its values are usually established through appropriate field pump tests.

Hydraulic Gradient

It is the rate of change in potentiometric head per unit distance in the direction of ground water flow. When this direction is unknown or unclear, the term may be used to express the rate of change in any chosen coordinate direction.

Hydrogeologic Factors

Those factors dealing with sub-surface water flows, stream-aquifer interactions and related geologic features of any chosen aquifer site. A mathematical model is said to be valid when it accounts properly for hydrogeologic factors that are relevant in an aquifer being simulated.

Initial Conditions

Mathematical representations of aquifer recharge and discharge conditions which identify the hydrogeologic conditions within the chosen model domain at the commencement of time dependent model calculations. Such conditions prescribe the potentiometric-head state from which changes are intended to be evaluated by a model. Initial conditions generally represent spatial interpretations based upon field measurements, although they may also be based upon other model calculations. Care must be taken to ensure that initial conditions supplied to a model are self-consistent.

Input [Data or File]

Numerical values, i.e. of data, assigned to variables and intended to be supplied to a mathematical model. Such values are usually contained in computer files which are read by the computer program embodying the mathematical model. Reading of input files usually takes place at the start of the program before arithmetic operations which use the data are begun. Sometimes data representing time-dependent variations are read by programs using the same read instructions that are executed repetitiously.

Isotropic

Also in the context of aquifer property values that are material to ground water systems, and to mathematical modelling thereof, this term refers to independence of material property values upon coordinate directions. Thus, an isotropic aquifer is composed of materials that can be assigned property values, like hydraulic conductivity as identical in all coordinate directions on a plane. In contrast, an anisotropic aquifer is one in which material property values depend upon coordinate directions. This results in values in one direction being larger than those in another. In general, aquifer systems are anisotropic in one sense: property values along bedding planes are larger than those in a direction perpendicular to such planes.

Iterations

Repetitious numerical calculations employed in order to obtain consistent solutions to equations such as those representing ground water flows in an aquifer. Iterations are usually required, within each time step taken, when numerical algorithms are employed in mathematical models based upon grid cells. This is either due to the large matrix of discrete, or algebraic, equations which are required to be solved, or because these equations are non-linear, or both. Iterations are said to converge to a solution when further numerical calculations do not produce appreciable changes in predicted results. In contrast, iterations can also lead to oscillating or divergent solutions that are not meaningful.

Iterative Calculations

This term is generally used in the context of representing complex ground water systems which interact with flowing streams. When different models are used to represent streams and ground water systems, iterative calculations are usually performed to ensure that the quantitative relationship between streams and aquifers is represented correctly, compatibly and consistently by both models. For instance, a surface water model may presume ground water conditions as known to begin calculations of available stream flow. The ground water model, in sequence would utilize this estimate of stream flow to calculate ground water flow conditions. In turn, these revised conditions would be used in the surface water model to revise its stream flow calculations, passing information on stream flows back to the ground water model. This process of exchanging revised estimates or iterative model calculations would be repeated until both stream flows and ground water flows calculated and used by both models, for a given set of conditions, are substantially the same and variations due to calculational differences have been removed.

Layer

This term has two meanings which can differ significantly. The first refers to geologic formations or units or groups of units which act as layers for purposes of describing the lateral movement of ground water. The second is a strict definition, as used in modelling, referring to those geologic materials chosen for inclusion in a geometric grouping of model grid cells and to which property values are assigned. While these two definitions may occasionally coincide, they are in general distinctly different and should be considered when evaluating modelling descriptions.

Mathematical Model, [or Model]

A theoretical representation of physical and chemical mechanisms that may be used to obtain predictions under either static or time-dependent conditions within a particular ground water system. A model of ground water flows within such a system permits predictions of potentiometric-head distributions and ground water recharges and discharges to be obtained under a variety of postulated stimuli. Models are usually embodied in computer programs which can be run on a variety of computers with appropriate input data sets so as to generate corresponding output sets. A computer program such as MODFLOW itself does not constitute a model; only when it is supplied with site or basin specific information and material properties does it become a model.

Non-Linear [Equation, Model or Response]

Equations governing fluid flows are referred to as non-linear because the coefficients which appear in the equations are themselves a function of the dependent variable which in the case of ground water flows is potentiometric head. Unconfined aquifer flows are represented by a flow equation which is essentially non-linear. Non-linearity stems from the fact that results predicted by solutions of such equations for scenarios involving a single parameter value may not be added together in order to obtain a solution for a scenario involving the combination of parameters. The opposite of non-linear equations is linear equations; the equation governing confined aquifer flows is essentially linear.

Observation Well [or Monitor Well]

A well drilled at a selected location within an aquifer, at selected depths and screened-intervals, for the purpose of observing or measuring variables such as water level changes caused by either natural or man-made conditions. In the context of man-made actions such as well pumping from an aquifer, observation wells are usually placed at nearby locations.

Output [Results or File]

Numerical values, i.e. predicted results, of calculated variables produced by a mathematical model. Such values are usually contained in computer files which are created by the computer program embodying the mathematical model. Writing of output files usually takes place within designated segments of the program after arithmetic operations are completed. When calculations representing time-dependent variations are performed by model programs, such output is written to a file or a device by executing the identical write instructions repetitiously at user-selected time instants.

Permeability

The property or capacity of a porous rock, sediment, or soil for transmitting a fluid; it is a measure of the relative ease of fluid flow under pressure gradients. Commonly expressed in units of cm/sec. A unit more frequently used in mathematical models of ground water flow is (ft/day). Such properties are usually established through appropriate tests.

Porosity

The percentage of the bulk volume of a rock or soil that is occupied by interstices, whether isolated or connected, giving rise to pores. A property that is usually established through appropriate tests.

Post-Processing

The preparation of results predicted by mathematical models into tabular and graphical forms intended to be more meaningful than large numbers of individual numerical values. Post-processing activities usually occur after an appropriate model run has been completed.

Potentiometric Surface

An imaginary surface representing the total head of ground water in a confined aquifer that is defined by levels, at various spatial locations, to which water will rise in a well. This surface is derived by interpretations of potentiometric-head measurements made at wells that are spatially distributed in an aquifer.

Pre-Processing

The preparation of site-specific field data into forms which are designed to be supplied as input parameters for a mathematical model. The levels of manipulations involved in pre-processing depend upon the physical mechanisms involved, operational features of the site, the geometric complexity, etc. Pre-processing activities usually take place before the appropriate model run is undertaken.

Quasi-Steady

A loose term used occasionally by modellers to designate modelling calculations which are meant to reach true steady-state conditions, but are never really achieved by their model runs. There is no physical condition described by this term other than an interpretation of hydrologic conditions when time-dependent changes are perceived as being relatively small.

Radius of Influence

The radial distance (in feet) from the center of a well bore to the point where there is no significant lowering of the water table or potentiometric surface, i.e. the edge of its cone of depression.

Recharge [Rate]

The addition of water to an aquifer or water-bearing formation by natural means, such as precipitation, or by artificial means such as the deep percolation of applied irrigation water.

Restart/Plot [File]

Numerical values, i.e. predicted results, of variables calculated by a mathematical model and converted into a compact, i.e. binary, form readable only by machines used to perform the calculations. Such values are usually contained in computer files which are created by the computer program embodying the mathematical model. Writing of restart/plot files usually takes place within designated segments of the program after arithmetic operations are completed. When calculations that represent time-dependent variations are performed by programs, such files are written by executing the identical write instructions repetitiously at user-selected time instants. The contents of restart/plot files serve one or both of two specific purposes: to re-start model runs using the results of previous runs as starting conditions; to enable plots of predicted results to be obtained by allowing graphics packages to read their contents.

Saturated Thickness

Saturated thickness (in feet) of an aquifer is the vertical extent which is saturated with water and perpendicular to which ground water flows within the aquifer. In an unconfined aquifer it is the difference between elevation of water level and elevation of the impermeable aquifer base. The product of saturated thickness and hydraulic conductivity yields transmissivity.

Sensitivity Analysis

Test runs made with a mathematical model in order to evaluate the variability in predicted results to systematic variations in values assigned to model parameters. Sensitivity analysis may be conducted with a model at any stage of its development. When referring however to the predictive capability of a model, sensitivity analysis refers only to tests performed upon its final calibrated state, not those made during its development.

Specific Yield

The ratio of the volume of water that a given mass of saturated rock or soil will yield by gravity to the volume of rock or soil. This ratio is usually also stated as a percentage value. Specific yield is only an approximate measure of the relation between storage and head in unconfined aquifers. A property that is usually established through appropriate field tests.

Static Water Level

The level of water in a well, located in an aquifer, that is not affected by withdrawals of ground water through pumping from it. It is important to recognize that this water level will not, in general, be invariant with time but will be altered by natural changes or other man-made effects such as pumping from neighboring wells. It is usually measured soon after a well is drilled and prior to its use.

Steady-State [Conditions or Run]

The application or use of a mathematical model of ground water flow under the specific assumption that complete equilibrium exists between all recharge and discharge rates. Such an assumed condition, when time-dependent variations either in aquifer stimuli or aquifer responses are precisely zero or minute enough to be ignored, rarely if ever occurs in nature. It simply allows ground water calculations to be simplified. Separate analysis of available aquifer data, not just a model's postulates, is generally conducted to demonstrate that such is indeed the case in an aquifer under investigation.

Stream-Aquifer Interaction

This term refers to the quantitative relationship between a stream and an aquifer in contact with it. It may refer to conditions when the stream is actually gaining water or when it is losing it. The relationship is generally expressed as an algebraic one involving the stage or elevation of water in the stream, the elevation of water in the ground water system at the contact location with the stream, the width and length of the wetted perimeter of the contact, the hydraulic conductivity and thickness of the stream beds that influence the interactions. When properly represented with appropriate measurement data, the quantitative relationship allows the gains and/or losses by streams to be reliably calculated with a model.

Stress Period

A time interval used in modelling to prescribe recharge or discharge values, designated to be aquifer stresses, to one or more grid cells. The prescribed values are intended to be constant during the stress period but may vary from one stress period to the next. When involving such variations, the smaller the size of the stress periods the more variable aquifer stresses are prescribed to be with time.

Time Step

A time interval used in mathematical models to calculate new values of dependent variables, such as potentiometric head, based upon previous-time values and changes which are intended to occur during the interval. A number of time steps, usually of varying sizes, is generally taken when making model runs in order to traverse a chosen simulation time period. The sizes of time steps are generally smaller than, but may be assigned equal to, the sizes of stress periods.

Transient [Conditions or Run]

The application or use of a mathematical model of ground water flow when dynamic equilibrium does not exist between recharge and discharge rates. Under such conditions, time-dependent variations in aquifer stimuli or aquifer responses are significant and cannot be ignored. This is particularly true when considering variations in stream flows and, thus, of aquifer gains and losses during seasons of a year.

Transmissivity

The rate at which ground water is transmitted through a unit width of an aquifer under a unit hydraulic gradient. It represents the product of aquifer hydraulic conductivity and saturated thickness. Transmissivity values are occasionally expressed in gallons per day through a vertical section of an aquifer one foot wide and extending the full saturated height of an aquifer under a unit hydraulic gradient. A unit more frequently used in mathematical models is (ft²/day). A property that is usually established through appropriate field tests.

Verification, [or Validation or Model Verification]

This is a necessary step of testing a mathematical model, after it has been calibrated to conditions within a given ground water system. A model is said to be verified or validated with respect to data set aside for this purpose in order to demonstrate the reliability with which it allows predictions to be obtained of mechanisms or conditions, other than those used for calibration. Only a model that has been so verified can be used reliably to make predictions of aquifer and stream responses to postulated water-development actions.

Volumetric Budget

This is a term employed in the computer program MODFLOW to denote the quantitative accounting for all ground water recharge and discharge rates, predicted by a model either at defined intervals of time or over the entire model simulation time period. The volumetric budget is usually presented as a table of recharge, i.e. positive, and discharge, i.e. negative, values corresponding to each relevant mechanism represented in, and predicted by, a model. It may also be presented in graphical forms which improve its clarity.