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Groundwater Modeling - 1 Groundwater Hydraulics Daene C. McKinney

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Models …? Input (Explanatory Variable) Input (Explanatory Variable) Model (Represents the Phenomena) Model (Represents the Phenomena) Output (Results – Response variable) Output (Results – Response variable) Run off Infiltration Evaporation ET PrecipitationSoil Characteristics

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Models and more models … Input (Explanatory Variable) Input (Explanatory Variable) Model (Phenomena) Model (Phenomena) Output (Results) Output (Results) Inflow Data Basin Water Allocation Policy Response to the Policy Inflow Data Basin Objectives and Constraints Optimum Policy Precip. & Soil Charact. Mimic Physics of the Basin Runoff Simulation Model Optimization Model Hydrologic Simulation Predict Response to given design/policy Identify optimal design/policy Source for Input data of other models

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Modeling Process Problem identification (1) – Important elements to be modeled – Relations and interactions between them – Degree of accuracy Conceptualization and development (2 – 3) – Mathematical description – Type of model – Numerical method - computer code – Grid, boundary & initial conditions Calibration (4) – Estimate model parameters – Model outputs compared with actual outputs – Parameters adjusted until the values agree Verification (4) – Independent set of input data used – Results compared with measured outputs Problem identification and description Model verification & sensitivity analysis Model Documentation Model application Model calibration & parameter estimation Model conceptualization Model development Data Present results 1 1 2 2 3 3 4 4 5 5 6 6 7 7

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Tools to Solve Groundwater Problems Physical and analog methods – Some of the first methods used. Analytical methods – What we have been discussing so far – Difficult for irregular boundaries, different boundary conditions, heterogeneous and anisotropic properties, multiple phases, nonlinearities Numerical methods – Transform PDEs governing flow of groundwater into a system of ODEs or algebraic equations for solution

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Conceptual Model Descriptive representation of groundwater system incorporating interpretation of geological & hydrological conditions What processes are important to model? What are the boundaries? What parameter values are available? What parameter values must be collected?

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What Do We Really Want To Solve? Horizontal flow in a leaky confined aquifer Governing Equations Boundary Conditions Initial conditions Ground surface Bedrock Confined aquifer QxQx K x y z h Head in confined aquifer Confining Layer b FluxLeakageSource/SinkStorage

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Finite Difference Method Finite-difference method – Replace derivatives in governing equations with Taylor series approximations – Generates set of algebraic equations to solve 1 st derivatives

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Taylor Series Taylor series expansion of h(x) at a point x+ x close to x If we truncate the series after the n th term, the error will be

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First Derivative - Forward Consider the forward Taylor series expansion of a function h(x) near a point x Solve for 1 st derivative

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First Derivative - Backward Consider the backward Taylor series expansion of a function f(x) near a point x Solve for 1 st derivative

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Finite Difference Approximations 1 st Derivative ( Backward) 1 st Derivative ( Forward) i

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Grids and Discretrization Discretization process Grid defined to cover domain Goal - predict values of head at node points of mesh – Determine effects of pumping – Flow from a river, etc Finite Difference method – Popular due to simplicity – Attractive for simple geometry i,j i,j+1 i+1,j i-1,j i,j-1 x, i y, j Domain Mesh Node point x y Grid cell

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Three-Dimensional Grids An aquifer system is divided into rectangular blocks by a grid. The grid is organized by rows (i), columns (j), and layers (k), and each block is called a "cell" Types of Layers – Confined – Unconfined – Convertible Layers can be different materials i, rows j, columns k, layers

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1-D Confined Aquifer Flow Homogeneous, isotropic, 1-D, confined flow Governing equation Initial Condition Boundary Conditions Ground surface Aquifer x y z hBhB Confining Layer b hAhA xx i =012345678910 Node Grid Cell

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Derivative Approximations Need 2nd derivative WRT x

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Derivative Approximations Governing Equation 2nd derivative WRT x Need 1st derivative WRT t ForwardBackward Which one to use?

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Time Derivative Explicit – Use all the information at the previous time step to compute the value at this time step. – Proceed point by point through the domain. Implicit – Use information from one point at the previous time step to compute the value at all points of this time step. – Solve for all points in domain simultaneously.

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Explicit Method Use all the information at the previous time step to compute the value at this time step. Proceed point by point through the domain. Can be unstable for large time steps. FD Approx. Forward

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Explicit Method l +1 time level unknown l time level known

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1-D Confined Aquifer Flow Initial Condition Boundary Conditions Ground surface Aquifer x y z hBhB Confining Layer b hAhA xx i =012345678910 Node Grid Cell L x = 1 m L = 10 m T=bK = 0.75 m 2 /d S = 0.02

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Explicit Method Ground surface Aquifer hBhB Confining Layer b hAhA xx i =012345678910 Node Grid Cell Consider: r = 0.48 r = 0.52 x = 1 m L = 10 m T = 0.75 m 2 /d S = 0.02

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Explicit Results ( t = 18.5 min; r = 0.48 < 0.5)

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Explicit Results ( t = 20 min; r = 0.52 > 0.5)

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What’s Going On Here? At time t = 0 no flow At time t > 0 flow Water released from storage in a cell over time t Water flowing out of cell over interval t Ground surface Aquifer hBhB Confining Layer b hAhA xx i =012…i-1ii+1…8910 xx Grid Cell i r > 0.5 Tme interval is too large Cell doesn’t contain enough water Causes instability

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Implicit Method Use information from one point at the previous time step to compute the value at all points of this time step. Solve for all points in domain simultaneously. Inherently stable FD Approx. Backward

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Implicit Method l +1 time level unknown l time level known

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2-D Steady-State Flow General Equation Homogeneous, isotropic aquifer, no well Equal spacing (average of surrounding cells) Node No. Unknown heads Known heads

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2-D Heterogeneous Anisotropic Flow T x and T y are transmissivities in the x and y directions

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2-D Heterogeneous Anisotropic Flow Harmonic average transmissivity

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Transient Problems

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MODFLOW USGS supported mathematical model Uses finite-difference method Several versions available – MODFLOW 88, 96, 2000, 2005 (water.usgs.gov/nrp/gwsoftware/modflow.html) Graphical user interfaces for MODFLOW: – GWV (www.groundwater-vistas.com)www.groundwater-vistas.com – GMS (www.ems-i.com)www.ems-i.com – PMWIN (www.ifu.ethz.ch/publications/software/pmwin/index_EN)www.ifu.ethz.ch/publications/software/pmwin/index_EN – Each includes MODFLOW code

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What Can MODFLOW Simulate? 1.Unconfined and confined aquifers 2.Faults and other barriers 3.Fine-grained confining units and interbeds 4.Confining unit - Ground-water flow and storage changes 5.River – aquifer water exchange 6.Discharge of water from drains and springs 7.Ephemeral stream - aquifer water exchange 8.Reservoir - aquifer water exchange 9.Recharge from precipitation and irrigation 10.Evapotranspiration 11.Withdrawal or recharge wells 12.Seawater intrusion

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