Introduction to Groundwater Modelling C. P. Kumar Scientist ‘F’ National Institute of Hydrology Roorkee – 247667 (Uttaranchal) India Email: firstname.lastname@example.org Webpage: http://www.angelfire.com/nh/cpkumar/
Presentation Outline • Groundwater in Hydrologic Cycle • Why Groundwater Modelling is needed? • Mathematical Models • Modelling Protocol • Model Design • Calibration and Validation • Groundwater Flow Models • Groundwater Modelling Resources
Types of Terrestrial Water Surface Water Soil Moisture Ground water
Pores Full of Combination of Air and Water Unsaturated Zone / Zone of Aeration / Vadose (Soil Water) Zone of Saturation (Ground water) Pores Full Completely with Water
Groundwater Important source of clean water More abundant than SW Baseflow Linked to SW systems Sustains flows in streams
Groundwater Concerns? pollution groundwater mining subsidence
Problems with groundwater • Groundwater overdraft / mining / subsidence • Waterlogging • Seawater intrusion • Groundwater pollution
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.
Management of a groundwater system, means making such decisionsas: • The total volume that may be withdrawn annually from the aquifer. • The location of pumping and artificial recharge wells, and their rates. • Decisions related to groundwater quality. • Groundwater contamination by: • Hazardous industrial wastes • Leachate from landfills • Agricultural activities such as the use of fertilizers and pesticides
MANAGEMENT means making decisions to achieve goals without violating specified constraints. • Good management requires information on the response of the managed system to the proposed activities. • This information enables the decision-maker, to compare alternative actions and to ensure that constraints are not violated. • Any planning of mitigation or control measures, once contamination has been detected in the saturated or unsaturated zones, requires the prediction of the path and the fate of the contaminants, in response to the planned activities. • Any monitoring or observation network must be based on the anticipated behavior of the system.
A tool is needed that will provide this information. • The tool for understanding the system and its behavior and for predicting this response is the model. • Usually, the model takes the form of a set of mathematical equations, involving one or more partial differential equations. We refer to such model as a mathematical model. • The preferred method of solution of the mathematical model of a given problem is the analytical solution.
The advantage of the analytical solution is that the same solution can be applied to various numerical values of model coefficients and parameters. • Unfortunately, for most practical problems, because of the heterogeneity of the considered domain, the irregular shape of its boundaries, and the non-analytic form of the various functions, solving the mathematical models analytically is not possible. • Instead, we transform the mathematical model into a numerical one, solving it by means of computer programs.
Prior to determining the management scheme for any aquifer: We should have a CALIBRATED MODEL of the aquifer, especially, we should know the aquifer’s natural replenishment (from precipitation and through aquifer boundaries). The model will provide the response of the aquifer (water levels, concentrations, etc.) to the implementation of any management alternative. We should have a POLICY that dictates management objectives and constraints. Obviously, we also need information about the water demand (quantity and quality, current and future), interaction with other parts of the water resources system, economic information, sources of pollution, effect of changes on the environment---springs, rivers,...
GROUND WATER MODELING • WHY MODEL? • To make predictions about a ground-water • system’s response to a stress • To understand the system • To design field studies • Use as a thinking tool
Use of Groundwater models • Can 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 models • To describe the system in order to analyse various assumptions • To generate a hypothetical system that will be used to study principles of groundwater flow associated with various general or specific problems.
ALL GROUND-WATER HYDROLOGY WORK IS MODELING A Model is a representation of a system. Modeling begins when one formulates a concept of a hydrologic system, continues with application of, for example, Darcy's Law to the problem, and may culminate in a complex numerical simulation.
Ground Water Flow Modelling A Powerful Tool for furthering our understanding of hydrogeological systems • Importance of understanding ground water flow models • Construct accurate representations of hydrogeological systems • Understand the interrelationships between elements of systems • Efficiently develop a sound mathematical representation • Make reasonable assumptions and simplifications • Understand the limitations of the mathematical representation • Understand the limitations of the interpretation of the results
h(x,y,z,t)? Potentiometric Surface x q K ho x h(x) x x x Darcy’s Law Integrated 0 Introduction to Ground Water Flow Modelling Predicting heads (and flows) and Approximating parameters • Solutions to the flow equations • Most ground water flow models are solutions of some form of the ground water flow equation • 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) • “e.g., unidirectional, steady-state flow within a confined aquifer
Groundwater Modeling • The only effective way to test effects of groundwater management strategies • Takes time, money to make model • Conceptual model Steady state model Transient model • The model is only as good as its calibration
Processes we might want to model • Groundwater flow • calculate both heads and flow • Solute transport – requires information on flow (velocities) • calculate concentrations
MODELING PROCESS ALL IMPORTANT MECHANISMS & PROCESSES MUST BE INCLUDED IN THE MODEL, OR RESULTS WILL BE INVALID.
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 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 • (Darcy’s law + water balance equation) with head (h) as the dependent variable • Boundary Conditions • Initial conditions (for transient problems)
Derivation of the Governing Equation Q R x y q z x y • Consider flux (q) through REV • OUT – IN = - Storage • Combine with: q = -Kgrad h
Law of Mass Balance + Darcy’s Law = Governing Equation for Groundwater Flow --------------------------------------------------------------- div q = - Ss (h t) (Law of Mass Balance) q = - Kgrad h (Darcy’s Law) div (K grad h) = Ss (h t) (Ss = S / z)
General governing equation for steady-state, heterogeneous, anisotropic conditions, without a source/sink term with a source/sink term
General governing equation for transient, heterogeneous, and anisotropic conditions Specific Storage Ss = V / (x y z h)
h h b S = V / A h S = Ss b Confinedaquifer Unconfinedaquifer Storativity Specific yield Figures taken from Hornberger et al. (1998)
General 3D equation 2D confined: 2D unconfined: Storage coefficient (S) is either storativity or specific yield. S = Ss b & T = K b
Types of Solutions of Mathematical Models • Analytical Solutions: h= f(x,y,z,t) • (example: Theis equation) • Numerical Solutions • Finite difference methods • Finite element methods • Analytic Element Methods (AEM)
Limitations of Analytical Models • The flexibility of analytical modeling is limited due to simplifying assumptions: • Homogeneity, Isotropy, simple geometry, simple initial conditions… • Geology is inherently complex: • Heterogeneous, anisotropic, complex geometry, complex conditions… This complexity calls for a more powerful solution to the flow equation Numerical modeling
Numerical Methods • All 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 Modelling • 3-D Finite Difference Models • Requires vertical discretization (or layering) of model K1 K2 K3 K4
Finite difference models • may be solved using: • a computer program • (e.g., a FORTRAN program) • a spreadsheet (e.g., EXCEL)
Finite Elements: basis functions, variational principle, Galerkin’s method, weighted residuals • Nodes plus elements; elements defined by nodes • Properties (K, S) assigned to elements • Nodes located on flux boundaries • Able to simulate point sources/sinks at nodes • Flexibility in grid design: • elements shaped to boundaries • elements fitted to capture detail • Easier to accommodate anisotropy that occurs at an • angle to the coordinate axis
Hybrid Analytic Element Method (AEM) Involves superposition of analytic solutions. Heads are calculated in continuous space using a computer to do the mathematics involved in superposition. The AE Method was introduced by Otto Strack. A general purpose code, GFLOW, was developed by Strack’s student Henk Haitjema, who also wrote a textbook on the AE Method: Analytic Element Modeling of Groundwater Flow, Academic Press, 1995. Currently the method is limited to steady-state, two-dimensional, horizontal flow.
What is a “model”? • Any “device” that represents approximation to field system • Physical Models • Mathematical Models • Analytical • Numerical
Modelling Protocol • Establish the Purpose of the Model • Develop Conceptual Model of the System • Select Governing Equations and Computer Code • Model Design • Calibration • Calibration Sensitivity Analysis • Model Verification • Prediction • Predictive Sensitivity Analysis • Presentation of Modeling Design and Results • Post Audit • Model Redesign
Purpose - What questions do you want the model to answer? • Prediction; System Interpretation; Generic Modeling • What do you want to learn from the model? • Is a modeling exercise the best way to answer the question? Historical data? • Can an analytical model provide the answer? System Interpretation: Inverse Modeling: Sensitivity Analysis Generic: Used in a hypothetical sense, not necessarily for a real site
Model “Overkill”? • Is the vast labor of characterizing the system, combined with the vast labor of analyzing it, disproportionate to the benefits that follow?