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Introduction to Groundwater Modelling

Presentation OutlineGroundwater in Hydrologic CycleWhy Groundwater Modeling is needed?Mathematical ModelsGroundwater Flow Models

Groundwater in Hydrologic Cycle

Types of Terrestrial WaterGround waterSoilMoistureSurfaceWater

Unsaturated Zone / Zone of Aeration / Vadose (Soil Water)Pores Full of Combination of Air and WaterZone of Saturation (Ground water)Pores Full Completely with Water

GroundwaterImportant source of clean waterMore abundant than SWLinked to SW systems

Sustains flows in streamsBaseflow

pollutionGroundwater Concerns?groundwater miningsubsidence

Problems with groundwater

Groundwater overdraft / mining / subsidence

Waterlogging

Seawater intrusion

Groundwater pollution

Why Groundwater Modelling is needed?

Groundwater

An important component of water resource systems.

Extracted from aquifers through pumping wells and supplied for domestic use, industry and agriculture.

With increased withdrawal of groundwater, the quality of groundwater has been continuously deteriorating.

Water can be injected into aquifers for storage and/or quality control purposes.

GROUND WATER MODELING

WHY MODEL?

To make predictions about a ground-water systems response to a stress

To understand the system

To design field studies

Use as a thinking tool

Use of Groundwater modelsCan be used for three general purposes:To predict or forecast expected artificial or natural changes in the system. Predictive is more applied to deterministic models since it carries higher degree of certainty, while forecasting is used with probabilistic (stochastic) models.

Use of Groundwater modelsTo describe the system in order to analyse various assumptionsTo generate a hypothetical system that will be used to study principles of groundwater flow associated with various general or specific problems.

Processes we might want to model

Groundwater flowcalculate both heads and flowSolute transport requires information on flow (velocities) calculate concentrations

TYPES OF MODELS CONCEPTUAL MODEL QUALITATIVE DESCRIPTION OF SYSTEM "a cartoon of the system in your mind" MATHEMATICAL MODEL MATHEMATICAL DESCRIPTION OF SYSTEM

SIMPLE - ANALYTICAL (provides a continuous solution over the model domain)

COMPLEX - NUMERICAL (provides a discrete solution - i.e. values are calculated at only a few points)

ANALOG MODEL e.g. ELECTRICAL CURRENT FLOW through a circuit board with resistors to represent hydraulic conductivity and capacitors to represent storage coefficient PHYSICAL MODEL e.g. SAND TANK which poses scaling problems

Mathematical Models

Mathematical model: simulates ground-water flow and/or solute fate and transport indirectly by means of a set of governing equations thought to represent the physical processes that occur in the system.

(Anderson and Woessner, 1992)

Components of a Mathematical Model

Governing Equation (Darcys law + water balance equation) with head (h) as the dependent variable Boundary Conditions Initial conditions (for transient problems)

R x yQyxzConsider flux (q) through REVOUT IN = - StorageCombine with: q = -K grad hqDerivation of the Governing Equation

Numerical MethodsAll numerical methods involve representing the flow domain by a limited number of discrete points called nodes.A set of equations are then derived to relate the nodal values of the dependent variable such that they satisfy the governing PDE, either approximately or exactly.

Numerical Solutions

Discrete solution of head at selected nodal points. Involves numerical solution of a set of algebraic equations. Finite difference models (e.g., MODFLOW)Finite element models (e.g., SUTRA)

Finite difference modelsmay be solved using:

a computer program (e.g., a FORTRAN program)

a spreadsheet (e.g., EXCEL)

Groundwater Flow Models

The Two Fundamental Equationsof Ground Water FlowBasic FormFirst Law of HydrogeologySecond Law of HydrogeologyBasic Form

Darcys LawDarcys Experiment (1856)DxQQ: Volumetric flow rate [L3/T]DhhxSlope = Dh/Dx ~ dh/dx Darcy investigated ground water flow under controlled conditionsDhDxh1h2h1h2x1x2K: The proportionality constant is added to form the following equation:K units [L/T] : Hydraulic GradientAA: Cross Sectional Area (Perp. to flow)

Darcys Law (cont.)Other useful forms of Darcys LawQA=QA.n=qn= Volumetric Flux Ave. Linear VelocityUsed for calculating Q given AUsed for calculating average velocity of contaminant transport Assumptions: Laminar, saturated flow

Introduction to Ground Water Flow ModelingPredicting heads (and flows) and Approximating parametersSolutions to the flow equationsMost ground water flow models are solutions of some form of the ground water flow equatione.g., unidirectional, steady-state flow within a confined aquifer

The partial differential equation needs to be solved to calculate head as a function of position and time, i.e., h=f(x,y,z,t)

Flow Modeling (cont.)Analytical models (a.k.a., closed form models)The previous model is an example of an analytical modelis a solution to the 1-D Laplace equationi.e., the second derivative of h(x) is zeroWith this analytical model, head can be calculated at any position (x) Analytical solutions to the 3-D transient flow equation would give head at any position and at any time, i.e., the continuous function h(x,y,z,t)Examples of analytical models:1-D solutions to steady state and transient flow equationsThiem Equation: Steady state flow to a well in a confined aquiferThe Theis Equation: Transient flow to a well in a confined aquiferSlug test solutions: Transient response of head within a well to apressure pulse

Flow Modeling (cont.)Common Analytical ModelsThiem Equation: steady state flow to a well within a confined aquiferAnalytic solution to the radial (1-D), steady-state, homogeneous K flow equationGives head as a function of radial distance Theis Equation: Transient flow to a well within a confined aquifer Analytic solution of radial, transient, homogeneous K flow equation Gives head as a function of radial distance and time

Flow Modeling (cont.)Forward Modeling: Prediction

Models can be used to predict h(x,y,z,t) if the parameters are known, K, T, Ss, S, n, b

Heads are used to predict flow rates,velocity distributions, flow paths, travel times. For example:Velocities for average contaminant transportCapture zones for ground water contaminant plume captureTravel time zones for wellhead protection

Velocity distributions and flow paths are then used in contaminant transport modeling

Flow Modeling (cont.)Inverse Modeling: Aquifer CharacterizationUse of forward modeling requires estimates of aquifer parametersSimple models can be solved for these parameterse.g., 1-D Steady State:

This inverse model can be used to characterize KThis estimate of K can then be used in a forward model to predict what will happen when other variables are changed

hoh1Claybxhoh1QQ

Flow Modeling (cont.)Inverse Modeling: Aquifer CharacterizationThe Thiem Equation can also be solved for K

Pump Test: This inverse model allows measurement of K using a steady state pump testA pumping well is pumped at a constant rate of Q until heads come to steady state, i.e., The steady-state heads, h1 and h2, are measured in two observation wells at different radial distances from the pumping well r1 and r2The values are plugged into the inverse model to calculate K (a bulk measure of K over the area stressed by pumping)

Flow Modeling (cont.)Inverse Modeling: Aquifer CharacterizationIndirect solution of flow modelsMore complex analytical flow models cannot be solved for the parameters Curve Matching or Iteration

This calls for curve matching or iteration in order to calculate the aquifer parameters Advantages over steady state solutiongives storage parameters S (or Ss) as well as T (or K)Pump test does not have to be continued to steady stateModifications allow the calculation of many other parameterse.g., Specific yield, aquitard leakage, anisotropy

Flow Modeling (cont.)Limitations of Analytical ModelsClosed form models are well suited to the characterization of bulk parametersHowever, the flexibility of forward modeling is limited due to simplifying assumptions:Homogeneity, Isotropy, simple geometry, simple initial conditionsGeology is inherently complex:Heterogeneous, anisotropic, complex geometry, complex conditionsThis complexity calls for a more powerful solution to the flow equation Numerical modeling

Numerical Modeling in a NutshellA solution of flow equation is approximated on a discrete grid (or mesh) of points, cells or elementsWithin this discretized domain: Aquifer parameters can be set at each cell within the gridComplex aquifer geometry can be modeledComplex boundary conditions can be accounted forRequires detailed knowledge of 1), 2) , and 3)As compared to analytical modeling, numerical modeling is:Well suited to prediction but More difficult to use for aquifer characterizationFlow Modeling (cont.)The parameters and variables are specified over the boundary of the domain (region) being modeled

An Introduction to Finite Difference ModelingApproximate Solutions to the Flow EquationPartial derivatives of head represent the change in head with respect to a coordinate di