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# Glossary

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## A Solute that has transferred (adsorbed) from the fluid onto the solid.

Adsorption (or sorption) is the transfer of solute from the fluid onto the solid. Solute adsorbed onto the solid is called adsorbate. SUTRA offers a choice of three equilibrium adsorption models: linear, Freundlich, and Langmuir. See Section 2.4 of the SUTRA documentation for details.

anisotropic

Direction-dependent. The permeability tensor is anisotropic if the effective permeability varies with the direction of groundwater flow (see Section 2.2 of the SUTRA documentation for details). The dispersion tensor is anisotropic if the longitudinal dispersivity or either of the two transverse dispersivities varies with the direction of groundwater flow (see Section 2.5 of the SUTRA documentation for details). A property that is independent of the flow direction is said to be isotropic.

areal simulation

A 2D simulation in which flow and transport are modeled in the horizontal plane; the system is considered to be vertically homogeneous. Compare cross-sectional simulation.

## B basis function

A function used to describe the variation of a variable (such as pressure or concentration) between nodes in a finite-element discretization. SUTRA uses bilinear basis functions in 2D and trilinear basis functions in 3D. SUTRA basis functions vary linearly along element edges (straight lines that connect neighboring nodes). See Sections 3.2 and 4.1 of the SUTRA documentation for details.

boundary conditions

In SUTRA, "boundary condition" refers to a specification of pressure, concentration/temperature, or mass/energy flux made at a node. Boundary conditions are not restricted to the physical boundary of the model; they can be assigned to any node. By default, there is no flux in or out of the model domain at nodes for which a boundary condition has not been explicitly assigned. Boundary conditions are assigned through datasets 17, 18, 19, and 20 of the main input (".inp") file. In SUTRA Version 2.1 and earlier, time-dependent boundary conditions must be programmed by the user in subroutine BCTIME. In later versions, time-dependent boundary conditions can be specified using one or more input files; no programming is required. For details, see Section 7.5 and Appendix B of the SUTRA documentation, and "Time-dependent boundary conditions" on the "Special topics" page.

budget

An accounting of the input/output, production/decay, and accumulation/depletion of mass or energy in the system. In a perfect simulation, (net accumulation/depletion) = (net input/output) + (net production/decay). SUTRA reports separate budgets for fluid mass and solute mass or thermal energy.

## C cell

The region of space associated with a node.

CG

An iterative matrix solver. The name stands for "Conjugate Gradient".

cold start

One of two methods for initializing a SUTRA run; compare warm start. A cold start is used to initialize a simulation that is being run for the first time, or a continuation run in which input parameters or boundary conditions have changed since the previous run. See "Q: What is the difference between a "cold" and a "warm" start, and which one should I use?" on the "Frequently asked questions" page.

compressibility

For a fluid, the fractional change in density per unit change in pressure. The compressibility of pure water at 20ºC is approximately 4.47x10-10 kg/(m·s2)-1. For a solid, minus the fractional change in volume per unit change in intergranular stress. Compressibilities range from about 10-10 kg/(m·s2)-1 for sound bedrock to about 10-7 kg/(m·s2)-1 for clay. See also storativity.

concentration

In SUTRA, the mass fraction of solute in the fluid, or the mass fraction of adsorbate on the solid.

conductance

In SUTRA, either of two parameters, GNUP and GNUU, used to enforce specified pressure and specified concentration/temperature boundary conditions. These parameters are set in dataset 5 of the main input (".inp") file. See Sections 3.5 and 7.7 of the SUTRA documentation, and "Q: What do the boundary conductances GNUP and GNUU do, and why are they important?" on the "Frequently asked questions" page.

conductivity

constant-density flow

Groundwater flow in which the fluid density is uniform in space and constant in time. Compare variable-density flow.

convergence tolerance

The value to which a measure of convergence must fall for an iterative solution to be considered acceptable. A SUTRA run can involve two different types of iteration: non-linearity iteration, and matrix solver iteration. For non-linearity iteration, the convergence tolerances specify the maximum changes in computed pressures and concentrations/temperatures allowed from one iteration to the next; iterations continue until the changes are sufficiently small or the maximum number of iterations allowed has been reached.  For matrix solver iteration, the convergence tolerances specify the maximum acceptable estimated errors in the pressure and concentration/temperature solutions; for each solution, iterations continue until the estimated error is sufficiently small or the maximum number of iterations allowed has been reached.

cross-sectional simulation

A 2D simulation in which flow and transport in a vertical plane are modeled; the system is considered to be homogeneous in the direction perpendicular to the plane. Compare areal simulation.

## D density

The mass of substance per unit volume. In SUTRA, fluid density can vary linearly with solute concentration or fluid temperature. The solid grain density is uniform and constant and is relevant only in simulations involving energy transport or solute transport with sorption.

density-dependent flow

direct matrix solver

A matrix solver that computes the solution to a matrix problem directly (not by iteration) using an algebraic elimination procedure. Such a solver is available in SUTRA. Compare iterative matrix solver.

discretization

In SUTRA, discretization is the process of or result of approximating the physical system (spatially) as an assemblage of discrete elements and (temporally) as a series of discrete time steps. Smaller elements or steps result in finer discretization; larger elements or steps result in coarser discretization.

dispersion

In SUTRA, "dispersion" refers specifically to mechanical dispersion, as distinguished from molecular diffusion. Dispersion describes the spreading of solute mass or thermal energy relative to the average advective motion of the fluid; it is the macroscopic (model-scale) representation of the effect of fluid mixing at the microscopic (pore) scale. In SUTRA, the dispersive flux depends linearly on both the concentration/temperature gradient (it is "Fickian") and the magnitude of the fluid velocity. To learn more about the SUTRA dispersion model, see Section 2.5 of the SUTRA documentation and "Visualizing the SUTRA dispersion model in 3D" on the "Special topics" page.

dispersivity

In 3D, dispersion of solute or energy is characterized by three dispersivities: the longitudinal dispersivity, aL, and two transverse dispersivities, aT1 and aT2. The longitudinal dispersivity characterizes the tendency of solute or energy to disperse along the direction of groundwater flow. The transverse dispersivities characterize the tendency of solute or energy to disperse in two directions perpendicular to the direction of groundwater flow. In 3D, SUTRA requires the user to input six parameters (in dataset 15) that are used to compute aL, aT1, and aT2 and their associated spreading directions as functions of the groundwater flow magnitude and direction. In 2D, the situation is similar, except that there is only one transverse dispersivity and one transverse spreading direction (which is perpendicular to the direction of groundwater flow), and only four input parameters are required. To learn more about the SUTRA dispersion model, see Section 2.5 of the SUTRA documentation and "Visualizing the SUTRA dispersion model in 3D" on the "Special topics" page.

## E element

The basic "building block" used to model physical systems using the finite-element method. In SUTRA, each element is a quadrilateral (2D) or a generalized hexahedron (3D).

elementwise

Element-by-element. Elementwise properties (such as permeability) are assigned a value at each element in the mesh. Compare nodewise.

energy

In SUTRA, "energy" refers specifically to thermal energy, which is the energy associated with changes in temperature.

## F finite-element method

A numerical method used to approximate a physical system as an assemblage of discrete elements. SUTRA uses a hybridization of the Galerkin finite-element and integrated finite-difference methods. See Section 1.5 and Chapters 3 and 4 of the SUTRA documentation for details.

flow equation

The equation that embodies the physical laws governing groundwater flow. The flow equation used in SUTRA results from the combination of the fluid mass balance with Darcy's Law. It is formulated in terms of fluid pressure. The discretized form of the flow equation is a matrix equation that is solved for the pressure at each node. See Section 2.2 of the SUTRA documentation for details.

## G Gaussian elimination

An algebraic procedure used to solve sets of linear equations. This is the procedure used by the direct matrix solver available in SUTRA.

generalized hexahedron

In SUTRA, a closed 3D figure having six (not necessarily planar) faces, with each face defined by four straight edges. Examples include (but are not limited to) cubes and parallelepipeds. 3D SUTRA elements are generalized hexahedra, the eight corners of which are called nodes.

global coordinates

The physical (x, y, z) coordinate system in which a SUTRA problem is formulated by the user. Compare local coordinates. See Sections 4.1 and 4.2 of the SUTRA documentation for details.

GMRES

An iterative matrix solver. The name stands for "Generalized Minimum Residual".

GNUP

A "conductance" used to enforce specified pressure boundary conditions. This parameter is set in dataset 5 of the main input (".inp") file. See Section 3.5 and Appendix B of the SUTRA documentation, and "Q: What do the boundary conductances GNUP and GNUU do, and why are they important?" on the "Frequently asked questions" page.

GNUU

A "conductance" used to enforce specified concentration/temperature boundary conditions. This parameter is set in dataset 5 of the main input (".inp") file. GNUU works analogously to GNUP; see Section 3.5 and Appendix B of the SUTRA documentation, and "Q: What do the boundary conductances GNUP and GNUU do, and why are they important?" on the "Frequently asked questions" page.

## H Hydraulic head is defined as the sum of the pressure head and the elevation. Pressure head is, in turn, defined as p/(rg), where p is the fluid pressure, r is the fluid density, and g is the acceleration of gravity.

hexahedron

A closed 3D figure having 6 planar faces. 3D SUTRA elements are generalized hexahedra.

hydraulic conductivity

Hydraulic conductivity is a measure of the ease with which fluid flows through a porous medium under the influence of a hydraulic head gradient. Conductivity depends on the properties of both the fluid and the solid. It is most useful in the context of isothermal, constant-density flow. See Section 2.2 of the SUTRA documentation for details. Compare permeability.

## I incidence list

A list of the nodes that are associated with each element in a SUTRA model. The incidence list is entered in dataset 22 of the main input (".inp") file.

isothermal

Having uniform, constant temperature.

isotropic

Independent of direction. The permeability tensor is isotropic if the effective permeability does not vary with the direction of groundwater flow. The dispersion tensor is anisotropic if the longitudinal and transverse dispersivities do not vary with the direction of groundwater flow. A property that depends on the flow direction is said to be anisotropic.

iteration

The process of arriving at a solution through a series of steps that, ideally, computes solutions that are progressively closer to the correct answer. Also, a single step in such a process. A SUTRA run can involve two different types of iteration: non-linearity iteration, and matrix solver iteration.

iterative matrix solver

A matrix solver that computes the solution to a matrix problem by iteration. Three such solvers, CG, GMRES, and ORTHOMIN, are available in SUTRA. Compare direct matrix solver.

## J Jacobian matrix

The matrix that defines the linear transformation between global and local coordinates.

## L local coordinates

A coordinate system used to simplify and carry out calculations within an element. The local coordinate system is unique to each element. The user formulates the SUTRA problem in terms of the global coordinates (x, y, z), and SUTRA automatically performs transformations between local and global coordinates as needed. See Sections 4.1 and 4.2 of the SUTRA documentation for details.

logically rectangular

Having the same logical organization as a regular array of squares (2D) or cubes (3D). In SUTRA 2.0 (2D3D.1), 3D SUTRA meshes had to be logically rectangular, although their geometry could be deformed. This restriction was removed beginning with SUTRA 2.1. See Section 3.1 of the SUTRA documentation for details.

longitudinal

Along the direction of groundwater flow.

longitudinal dispersivity

See dispersivity.

## M matrix solver

A numerical algorithm designed to solve matrix problems. In SUTRA, matrix solvers are used to solve the flow equation for pressure and the transport equation for concentration or temperature.

matrix solver iteration

The scheme employed by an iterative matrix solver to converge to the correct solution to a matrix problem to within a specified convergence tolerance. Also, a single step in such a scheme. Not to be confused with non-linearity iteration.

max direction

The direction of groundwater flow for which the effective permeability is at its maximum value; a.k.a. the maximum permeability direction. The max direction is perpendicular to the min direction and (in 3D) the mid direction. See Section 2.2 of the SUTRA documentation for details.

mesh

The assemblage of nodes and elements used to model a physical system using the finite-element method.

mesh spacing

The distance between opposite sides of an element in a SUTRA mesh. The mesh spacing can vary from place to place within a model, and can depend on the direction in which it is measured. The mesh spacing can be important in determining the accuracy and numerical stability of SUTRA simulations; see Section 7.2 of the SUTRA documentation for details.

mid direction

In 3D, the direction that is perpendicular to both the max direction and the min direction, and for which the effective permeability assumes a value between the maximum and minimum values; a.k.a. the middle permeability direction. See Section 2.2 of the SUTRA documentation for details.

min direction

The direction of groundwater flow for which the effective permeability is at its minimum value; a.k.a. the minimum permeability direction. The min direction is perpendicular to the max direction and (in 3D) the mid direction. See Section 2.2 of the SUTRA documentation for details.

## N node

A corner of an element.

nodewise

Node-by-node. Nodewise properties (such as porosity) are assigned a value at each node in the mesh. Compare elementwise.

non-linearity iteration

Variable-density and/or unsaturated problems yield a non-linear set of equations that must be solved for pressure (p) and concentration or temperature (U). However, the SUTRA flow equation is linear with respect to p alone, and the SUTRA transport equation is linear with respect to U alone. Thus, the non-linear set of equations can be solved by iterating between the flow and transport equations, solving two linear matrix equations in succession on each iteration. First, the latest U solution is used in setting up the flow equation, which is then solved for p using a matrix solver. Then, the new p solution is used in setting up the transport equation, which is then solved for U. This so-called non-linearity (or "Picard") iteration cycle continues until a user-specified convergence tolerance has been satisfied or the maximum allowable number of iterations has been reached. Non-linearity iteration is not to be confused with matrix solver iteration.

numbering directions

The three indexing directions, I, J, and K, in a logically rectangular, 3D SUTRA mesh, ordered to indicate the order in which the nodes are numbered.

ORTHOMIN

## P permeability

Permeability is a measure of the ease with which fluid flows through a porous medium under the influence of a pressure gradient and/or gravity. Permeability is, in most situations, essentially independent of pressure, temperature, and concentration and depends only on the nature of the porous medium. SUTRA simulations are formulated in terms of permeability (rather than hydraulic conductivity). The effective permeability at any point in the system can depend on the direction of groundwater flow. See Section 2.2 of the SUTRA documentation for details.

Picard iteration

An iterative scheme used to solve a set of non-linear equations simultaneously. See non-linearity iteration.

pinch node

A type of node used to effect large changes in mesh spacing over relatively short distances. The pinch node option was discontinued in SUTRA Version 2.0 (2D3D.1).

porosity

The fractional volume of pore space in a porous medium. In SUTRA, the porosity is assumed to remain constant with time.

## Q A closed plane (2D) figure having four straight sides. Examples include (but are not limited to) squares, rectangles, parallelograms, and trapezoids. 2D SUTRA elements are quadrilaterals, the four corners of which are called nodes.

## S saturated flow

Groundwater flow in which all available pore space is filled with water. Compare unsaturated flow.

saturation

The fraction of the total available pore space that is filled with water. When the saturation equals unity (Sw = 1), the groundwater flow is said to be saturated.

sorption

storativity

The specific pressure storativity is the volume of water released from saturated pore storage, per volume of porous medium, due to a unit drop in fluid pressure. SUTRA computes the specific pressure storativity from the fluid and solid compressibilites under the assumption that individual solid grains are incompressible and that the total stress (the sum of the fluid pore pressure and the intergranular stress) remains constant. See Section 2.1 of the SUTRA documentation for details.

## T time step size

The size of a time increment in a transient simulation. Time step size can be important in determining the accuracy and numerical stability of the solution. See Section 7.2 of the SUTRA documentation for details.

transport equation

The equation that embodies the physical laws governing the transport of solute mass or thermal energy. The transport equation used in SUTRA results from the combination of the solute mass or energy balance with the dispersion and adsorption models. It is formulated in terms of the solute concentration or temperature. The discretized form of the transport equation is a matrix equation that is solved for the solute concentration or temperature at each node. The SUTRA transport equation is formulated such that it can represent either solute mass transport or thermal energy transport, depending on how its coefficients are defined. See Sections 2.3, 2.4, and 2.6 of the SUTRA documentation for details.

transverse

Perpendicular to the direction of groundwater flow.

transverse dispersivity

See dispersivity.

## U unsaturated flow

Groundwater flow in which the available pore space is only partially filled with water. SUTRA is capable of simulating both saturated and unsaturated flow. See Sections 2.1, 2.2, and 7.5 of the SUTRA documentation, and "Unsaturated flow functions" on the "Special topics" page.

## V variable-density flow

Groundwater flow in which the fluid density is variable in space and/or in time as a result of its dependence on solute concentration or fluid temperature. Compare constant-density flow.

velocity

In SUTRA, velocity refers to the average fluid velocity relative to the stationary solid matrix. It is an average velocity in the sense that it represents fluid movement as viewed at the model scale, rather than at the microscopic (pore) scale. The average fluid velocity defined here is not to be confused with the Darcy velocity, which is a measure of the volumetric flux of fluid and is equal to eSwv, where e is porosity, Sw is saturation, and v is average fluid velocity. See Section 2.2 of the SUTRA documentation for details.

viscosity

A measure of a fluid's resistance to shearing, which in part determines its resistance to flowing through a porous medium. In SUTRA, viscosity is a function of temperature in energy transport simulations and is uniform and constant in solute transport simulations. See Section 2.1 of the SUTRA documentation for details.

## W warm start

One of two methods for initializing a SUTRA run; compare cold start. A warm start is used to continue an earlier simulation as though it had never been interrupted. See "Q: What is the difference between a "cold" and a "warm" start, and which one should I use?" on the "Frequently asked questions" page.

water table

In saturated-unsaturated flow, the surface along which the fluid pressure is equal to atmospheric pressure. SUTRA does not explicitly track the position of the water table. Rather, the transition from saturated to unsaturated flow is modeled using unsaturated flow functions provided by the user. See Sections 2.1, 2.2, and 7.5 of the SUTRA documentation, and "Unsaturated flow functions" on the "Special topics" page.

## X x-coordinate

The x-component of the physical (x, y, z) coordinate system in which the user has formulated the SUTRA problem. See also global coordinates.

## Y y-coordinate

The y-component of the physical (x, y, z) coordinate system in which the user has formulated the SUTRA problem. See also global coordinates.

## Z z-coordinate

The z-component of the physical (x, y, z) coordinate system in which the user has formulated the SUTRA problem. See also global coordinates.  U.S. Department of the Interior | U.S. Geological Survey
URL: https://water.usgs.gov/nrp/gwsoftware/sour/glossary/glossary.htm
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