ground water flow modeling

2. Ground Water Flow Modeling A Powerful Tool for furthering our understanding of hydrogeological systems I want to show you how to use the power of hydrogeological modeling You should be able to do more than just go through the motions of hydrogeological modeling, you should to be able to use the modeling process to further your understanding of the hydrogeological system that you are investigating. This will hinge on the development of a sound conceptual model, a concept in your mind of how the plumbing works and how it relates to the problem to be addressed We will use mathematical models (analytical and numerical) as tools to address these problems The next step is to learn how to convert your conceptual model into a mathematical model. This could be as simple as applying 1-D Darcy�s Law and as complex as setting up and calibrating a 3-D, transient numerical model. In any case the procedure is the same: 1) Define the problem in lay-terms (demonstrate the significance to your audience), 2) define the specific objectives in technical (hydrogeological) terms, 3) Develop a conceptual model [site description and general hydrogeology], 4) convert the conceptual model into mathematical models that will address the objectives [methodology] 5) determine specifically where you will get the information from to set up your model [more methodology], 6) set up your model, calibrate and use it to address the objective [results] This will also help write the documentation which you should be writing all along I want to show you how to use the power of hydrogeological modeling You should be able to do more than just go through the motions of hydrogeological modeling, you should to be able to use the modeling process to further your understanding of the hydrogeological system that you are investigating. This will hinge on the development of a sound conceptual model, a concept in your mind of how the plumbing works and how it relates to the problem to be addressed We will use mathematical models (analytical and numerical) as tools to address these problems The next step is to learn how to convert your conceptual model into a mathematical model. This could be as simple as applying 1-D Darcy�s Law and as complex as setting up and calibrating a 3-D, transient numerical model. In any case the procedure is the same: 1) Define the problem in lay-terms (demonstrate the significance to your audience), 2) define the specific objectives in technical (hydrogeological) terms, 3) Develop a conceptual model [site description and general hydrogeology], 4) convert the conceptual model into mathematical models that will address the objectives [methodology] 5) determine specifically where you will get the information from to set up your model [more methodology], 6) set up your model, calibrate and use it to address the objective [results] This will also help write the documentation which you should be writing all along

3. Ground Water Flow Modeling Avoid the �Black Box� Approach The temptation is to plug in numbers and have a model spit out answers, especially with fancy preprocessors and postprocessors This will not provide a sound appreciation for the accuracy of the results or how reliable the predictions will be Modeling is intimidating: equations, vectors, partial differential equations, huge computer code� However, it is surprising how straightforward the finite difference modeling process is

4. Ground Water Flow Modeling Goals Gain an appreciation of the Modeling Process Develop finite difference equations from first principles with a minimum of higher level mathematics Implement the finite difference models with a spreadsheet Learn what is involved in setting up a finite difference model In general: make the modeling process transparent

5. The Two Fundamental Equationsof Ground Water Flow Basic Form Full Form

6. Darcy�s Law Darcy�s Experiment (1856) Controlled laboratory experiment designed to investigate what controls the flow rate (e.g. through a dam) Experiment A saturated porous medium is placed in a �column� (this could just as easily be vertical and was in Darcy�s experiment but this orientation, horizontal flow, is more familiar to us hydrogeologists) A method of measuring the dependent variable, volumetric flow rate, Q A method of individually changing the variables that were hypothesized to control flow (the dependant variable) Manometers can be thought of as wells (but you know that you need more than two wells to calculate groundwater flow rates and directions) discuss proportionalities Introduce hydraulic gradient as the slope of head with distance, Slope at a point is the derivative of the functional relationship between h and x In reality we use delta h/delta x Negative sign results from the fact that head decreases in the direction of flow, I.e. when the slope (hydraulic gradient is negative flow id in the positive x direction) Dimensional analysis shows that the units of K are [L/T] (this is not a velocity but can be thought as a quantitative measure of how easily water flows through the porous media Controlled laboratory experiment designed to investigate what controls the flow rate (e.g. through a dam) Experiment A saturated porous medium is placed in a �column� (this could just as easily be vertical and was in Darcy�s experiment but this orientation, horizontal flow, is more familiar to us hydrogeologists) A method of measuring the dependent variable, volumetric flow rate, Q A method of individually changing the variables that were hypothesized to control flow (the dependant variable) Manometers can be thought of as wells (but you know that you need more than two wells to calculate groundwater flow rates and directions) discuss proportionalities Introduce hydraulic gradient as the slope of head with distance, Slope at a point is the derivative of the functional relationship between h and x In reality we use delta h/delta x Negative sign results from the fact that head decreases in the direction of flow, I.e. when the slope (hydraulic gradient is negative flow id in the positive x direction) Dimensional analysis shows that the units of K are [L/T] (this is not a velocity but can be thought as a quantitative measure of how easily water flows through the porous media

7. Darcy�s Law (cont.) Other useful forms of Darcy�s Law Volumetric Flux: Volumetric flow per unit area (e.g. Rainfall depth is a volumetric flux, R*A=Q) q*A The volumetric flux is easy to calculate once you have measured (or approximated) the hydraulic gradient and measured or estimated q In order to calculate velocity you need to account for porosity because the same flow is squeezed through the percentage of pore space represented by the porosity therefor the velocity is greater than the volumetric flux by a factor of 1/n Once you�ve calculated q you cal calculate Q by multiplying by the cross sectional area (A: the area perpendicular to the flow vector) and you can calculate v by dividing by n (measured or approximated) Volumetric Flux: Volumetric flow per unit area (e.g. Rainfall depth is a volumetric flux, R*A=Q) q*A The volumetric flux is easy to calculate once you have measured (or approximated) the hydraulic gradient and measured or estimated q In order to calculate velocity you need to account for porosity because the same flow is squeezed through the percentage of pore space represented by the porosity therefor the velocity is greater than the volumetric flux by a factor of 1/n Once you�ve calculated q you cal calculate Q by multiplying by the cross sectional area (A: the area perpendicular to the flow vector) and you can calculate v by dividing by n (measured or approximated)

8. 3-D Darcy�s Law Hydrogeology is Three Dimensional

9. 3-D Darcy�s Law Velocity* Vectors and Components It is easy to think of velocity as a vector, choose coordinate axes, decompose the velocity vector into its components parallel to those axes Those coordinates are usually x and y are perpendicular horizontal coordinates (e.g., east and north) and z is the vertical (e.g., positive downwards) Usually the horizontal components are represented as one vector and If sliced parallel to the flow direction the velocity vector can be seen as the vector addition of the horizontal component and the vertical componentIt is easy to think of velocity as a vector, choose coordinate axes, decompose the velocity vector into its components parallel to those axes Those coordinates are usually x and y are perpendicular horizontal coordinates (e.g., east and north) and z is the vertical (e.g., positive downwards) Usually the horizontal components are represented as one vector and If sliced parallel to the flow direction the velocity vector can be seen as the vector addition of the horizontal component and the vertical component

10. 3-D Darcy�s Law (cont.) Volumetric Flux* Vector ? The Velocity Vector It is easy to think of velocity as a vector, choose coordinate axes, decompose the velocity vector into its components parallel to those axes Those coordinates are usually x and y are perpendicular horizontal coordinates (e.g., east and north) and z is the vertical (e.g., positive downwards) Usually the horizontal components are represented as one vector and If sliced parallel to the flow direction the velocity vector can be seen as the vector addition of the horizontal component and the vertical componentIt is easy to think of velocity as a vector, choose coordinate axes, decompose the velocity vector into its components parallel to those axes Those coordinates are usually x and y are perpendicular horizontal coordinates (e.g., east and north) and z is the vertical (e.g., positive downwards) Usually the horizontal components are represented as one vector and If sliced parallel to the flow direction the velocity vector can be seen as the vector addition of the horizontal component and the vertical component

11. 3-D Darcy�s Law (cont.) Hydraulic Gradient and Components Hydraulic gradient vector Describe partial derivitivesDescribe partial derivitives

12. 3-D Darcy�s Law (cont.) Horizontal and Vertical Hydraulic Gradients Usually the horizontal components are represented as one vector and approximated with a 3pt prob or from a piezometric surface map, only vert comp Usually the horizontal components are represented as one vector and approximated with a 3pt prob or from a piezometric surface map, only vert comp

13. 3-D Darcy�s Law (cont.) Vector representation of Darcy�s Law We know how to calculate the 1-D velocity and flux How do we calculate the horizontal components? We need to introduce the assumption that hydraulic conductivity does not depend on horizontal direction and groundwater flow most easily in the horizontal direction (horizontal isotropy and aligned anisotropy)We know how to calculate the 1-D velocity and flux How do we calculate the horizontal components? We need to introduce the assumption that hydraulic conductivity does not depend on horizontal direction and groundwater flow most easily in the horizontal direction (horizontal isotropy and aligned anisotropy)

14. 3-D Darcy�s Law (cont.) Generalization of Darcy�s Law to 3-D Anisotropic K Anisotropy aligned with coordinate axes

15. 3-D Darcy�s Law (cont.) 3-D Velocity Calculation Divide flux by porosity (n), a scalar.

16. Alternative form of 3-D Darcy�s Law U: velocity (in some texts) T: Transmissivity (T=K*b)

17. Horizontal (2-D) Darcy�s Law Horizontal ground water flow Common characteristics of aquifers and resulting assumption Aquifers tend to be much more extensive in the horizontal direction than in the vertical Hydraulic conductivity tends to be higher in the horizontal direction

18. The Ground Water Flow Equation Mass Balance ?Objective of modeling: represent h=f(x,y,z,t) A common method of analysis in sciences For a �system�, during a period of time (e.g., a unit of time), Assumption: Water is incompressible Mass per unit volume (density, r) does not change significantly Volume is directly related to mass by density V=m/r In this case water balance models are essentially mass balance models divided by density

19. The Flow Equation (cont.) Example 1: Storage in a reservoir If Qi = Qo, dVw/dt = 0 ? no change in level, i.e., steady state If Qi > Qo, dVw/dt > 0 ?+filling If Qi < Qo, dVw/dt < 0 ? -emptying E.g., Change in storage due to linearly varying flows

20. The Flow Equation (cont.) Example 1: Storage in a reservoir If Qi = Qo, dVw/dt = 0 ? no change in level, i.e., steady state If Qi > Qo, dVw/dt > 0 ?+filling If Qi < Qo, dVw/dt < 0 ? -emptying

21. The Flow Equation (cont.) Example 1: Storage in a reservoir If Qi = Qo, dVw/dt = 0 ? no change in level, i.e., steady state If Qi > Qo, dVw/dt > 0 ?+filling If Qi < Qo, dVw/dt < 0 ? -emptying Example 2: Storage in a REV (Representative Elementary Volume) REV: The smallest parcel of a unit that has properties (n, K, r�) that are representative of the formation The same water balance can be used to examine the saturated (or unsaturated) REV

22. The Flow Equation (cont.) Mass Balance for the REV (or any volume of a flow system) aka, �The Ground Water Flow Equation�

23. The Flow Equation (cont.) External Sources and Sinks (Qs)

24. The Flow Equation (cont.) 3-D, flow equation Summing the mass balance equation for each coordinate direction gives the total net inflow per unit volume into the REV Add a source term Qs/V volumetric flow rate per unit volume injected into REV

25. The Flow Equation (cont.) Specific storage and homogeneity Due to aquifer compressibility: change in porosity is proportional to a change in head (over a infinitesimally small range, dh)*

26. The Flow Equation (cont.) Flow Equation Simplifications

27. Introduction to Ground Water Flow Modeling Predicting heads (and flows) and Approximating parameters

28. Flow Modeling (cont.) Analytical models (a.k.a., closed form models) The previous model is an example of an analytical model

29. Flow Modeling (cont.) Common Analytical Models Thiem Equation: steady state flow to a well within a confined aquifer Analytic solution to the radial (1-D), steady-state, homogeneous K flow equation Gives head as a function of radial distance

30. 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 transport Capture zones for ground water contaminant plume capture Travel time zones for wellhead protection Velocity distributions and flow paths are then used in contaminant transport modeling

31. Flow Modeling (cont.) Inverse Modeling: Aquifer Characterization Use of forward modeling requires estimates of aquifer parameters Simple models can be solved for these parameters e.g., 1-D Steady State: This inverse model can be used to �characterize� K This estimate of K can then be used in a forward model to predict what will happen when other variables are changed

32. Flow Modeling (cont.) Inverse Modeling: Aquifer Characterization The Thiem Equation can also be solved for K Pump Test: This inverse model allows measurement of K using a steady state pump test A 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 r2 The values are �plugged into� the inverse model to calculate K (a bulk measure of K over the area stressed by pumping)

33. Flow Modeling (cont.) Inverse Modeling: Aquifer Characterization Indirect solution of flow models More 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 solution gives storage parameters S (or Ss) as well as T (or K) Pump test does not have to be continued to steady state Modifications allow the calculation of many other parameters e.g., Specific yield, aquitard leakage, anisotropy�

34. Flow Modeling (cont.) Limitations of Analytical Models Closed form models are well suited to the characterization of bulk parameters However, the flexibility of forward modeling is limited due to simplifying assumptions: Homogeneity, Isotropy, simple geometry, simple initial conditions� Geology is inherently complex: Heterogeneous, anisotropic, complex geometry, complex conditions�

35. Numerical Modeling in a Nutshell A solution of flow equation is approximated on a discrete grid (or mesh) of points, cells or elements Flow Modeling (cont.)

36. An Introduction to Finite Difference Modeling Approximate Solutions to the Flow Equation

37. Finite Difference Modeling (cont.) Approximation of the second derivative The second derivative of head with respect to x represents the change of the first derivative with respect to x The second derivative can be approximated using two finite differences centered around x2 This is known as a central difference

38. Finite Difference Modeling (cont.) Finite Difference Approximation of 1-D, Steady State Flow Equation

39. Finite Difference Modeling (cont.) Physical basis for finite difference approximation

40. Finite Difference Modeling (cont.) Discretization of the Domain Divide the 1-D domain into equal cells of heterogeneous K

41. Finite Difference Modeling (cont.) 2-D, Steady State, Uniform Grid Spacing, Finite Difference Scheme Divide the 2-D domain into equally spaced rows and columns of heterogeneous K

42. Finite Difference Modeling (cont.) Incorporate Transmissivity: Confined Aquifers multiply by b (aquifer thickness)

43. Finite Difference Modeling (cont.) Incorporate Transmissivity: Unconfined Aquifers b depends on saturated thickness which is head measured relative to the aquifer bottom

44. Finite Difference Modeling (cont.) 2-D, Steady State, Isotropic, Homogeneous Finite Difference Scheme

45. Finite Difference Modeling (cont.) Spreadsheet Implementation Spreadsheets provide all you need to do basic finite difference modeling Interdependent calculations among grids of cells Iteration control Multiple sheets for multiple layers, 3-D, or heterogeneous parameter input Built in graphics: x-y scatter plots and basic surface plots

46. Finite Difference Modeling (cont.) Spreadsheet Implementation of 2-D, Steady State, Isotropic, Homogeneous Finite Difference Type the formula into a computational cell Copy that cell into all other interior computational cell and the references will automatically adjust to calculate value for that cell Note: Boundary cells will be treated differently

47. Finite Difference Modeling (cont.) A simple example This will give a circular reference error Set? Tools:Options� Calculation to Manual Select? Tools:Options� Itteration Set?Maximum Iteration and Maximum Change Press F9 to iteratively calculate

48. Basic Finite Difference Design Discretization and Boundary Conditions

49. Basic Finite Difference Design (cont.) Discretization: Grid orientation Grid rows and columns should line up with as many rivers, shorelines, valley walls and other major boundaries as much as possible

50. Basic Finite Difference Design (cont.) Discretization: Variable Grid Spacing Rules of Thumb Refine grid around areas of interest Adjacent rows or columns should be no more than twice (or less than half) as wide as each other Expand spacing smoothly Many implementations of Numerical models allow Onscreen manipulation of Grids relative to an imported Base map

51. Basic Finite Difference Design (cont.) Boundary Conditions Any numerical model must be bounded on all sides of the domain (including bottom and top) The types of boundaries and mathematical representation depends on your conceptual model Types of Boundary Conditions Specified Head Boundaries Specified Flux Boundaries Head Dependant Flux Boundaries

52. Basic Finite Difference Design (cont.) Specified Head Boundaries Boundaries along which the heads have been measured and can be specified in the model e.g., surface water bodies They must be in good hydraulic connection with the aquifer Must influence heads throughout layer being modeled Large streams and lakes in unconfined aquifers with highly permeable beds Uniform Head Boundaries: Head is uniform in space, e.g., Lakes Spatially Varying Head Boundaries: e.g., River heads can be picked of of a topo map if: Hydraulic connection with and unconfined aquifer the streambed materials are more permeable than the aquifer materials

53. Basic Finite Difference Design (cont.) Specified Flux Boundaries Boundaries along which, or cells within which, inflows or outflows are set Recharge due to infiltration (R) Pumping wells (Qp) Induced infiltration Underflow No flow boundaries Valley wall of low permeable sediment or rock Fault

54. Finite Difference Modeling (cont.) No Flow Boundary Implementation A special type of specified flux boundary Because there are no nodes outside the domain, the perpendicular node is reflected across the boundary as and �image node�

55. Finite Difference Modeling (cont.) No Flow Boundary Implementation Corner nodes have two image nodes

56. Finite Difference Modeling (cont.) No Flow Boundary Implementation Combinations of edge and corner points are used to approximate irregular boundaries

57. Finite Difference Modeling (cont.) Head Dependent Flux Boundaries Flow into or out of cell depends on the difference between the head in the cell and the head on the other side of a conductive boundary e.g. 1, Streambed conductance hs: stage of the stream ho: Head within the cell Ksb: K of streambed materials bsb: Thickness of streambed w: width of stream L: length of reach within cell Csb: Streambed conductance Based on Darcys law

58. Finite Difference Modeling (cont.) Head Dependent Flux Boundaries e.g. 2, Flow through aquitard hc: Head within confined aquifer Ho: Head within the cell Kc: K of aquitard bc: Thickness of aquitard Dx2: Area of cell Cc: aquitard conductance Based on Darcys law

59. Case Study The Layered Modeling Approach

60. Finite Difference Modeling (cont.) 3-D Finite Difference Models Approximate solution to the 3-D flow equation e.g., 3-D, Steady State, Homogeneous Finite Difference approximation 3-D Computational Cell

61. Finite Difference Modeling (cont.) 3-D Finite Difference Models Requires vertical discretization (or layering) of model

62. Implementing Finite Difference Modeling Model Set-Up, Sensitivity Analysis, Calibration and Prediction

63. Implementation Anatomy of a Hydrogeological Investigation and accompanying report Significance Define the problem in lay-terms Highlight the importance of the problem being addressed Objectives Define the specific objectives in technical terms Description of site and general hydrogeology This is a presentation of your conceptual model Understanding of the hydrogeological system will hinge on the development of a sound conceptual model, a concept in your mind of how the plumbing works and how it relates to the problem to be addressed We will use mathematical models (analytical and numerical) as tools to address these problems The next step is to learn how to convert your conceptual model into a mathematical model. This could be as simple as applying 1-D Darcy�s Law and as complex as setting up and calibrating a 3-D, transient numerical model. In any case the procedure is the same: 1) Define the problem in lay-terms (demonstrate the significance to your audience), 2) define the specific objectives in technical (hydrogeological) terms, 3) Develop a conceptual model [site description and general hydrogeology], 4) convert the conceptual model into mathematical models that will address the objectives [methodology] 5) determine specifically where you will get the information from to set up your model [more methodology], 6) set up your model, calibrate and use it to address the objective [results] This will also help write the documentation which you should be writing all along Understanding of the hydrogeological system will hinge on the development of a sound conceptual model, a concept in your mind of how the plumbing works and how it relates to the problem to be addressed We will use mathematical models (analytical and numerical) as tools to address these problems The next step is to learn how to convert your conceptual model into a mathematical model. This could be as simple as applying 1-D Darcy�s Law and as complex as setting up and calibrating a 3-D, transient numerical model. In any case the procedure is the same: 1) Define the problem in lay-terms (demonstrate the significance to your audience), 2) define the specific objectives in technical (hydrogeological) terms, 3) Develop a conceptual model [site description and general hydrogeology], 4) convert the conceptual model into mathematical models that will address the objectives [methodology] 5) determine specifically where you will get the information from to set up your model [more methodology], 6) set up your model, calibrate and use it to address the objective [results] This will also help write the documentation which you should be writing all along

64. Implementation Anatomy of a Hydrogeological Investigation (cont.) Methodology Convert your conceptual model into mathematical models that will specifically address the Objectives Determine specifically where you will get the information to set-up the models Results Set up the models, calibrate, and use them to address the objectives Conclusions Discuss specifically, and concisely, how your results achieved the Objectives (or not) If not, discuss improvements on the conceptual model and mathematical representations 4) convert the conceptual model into mathematical models that will address the objectives [methodology] 5) determine specifically where you will get the information from to set up your model [more methodology], 6) set up your model, calibrate and use it to address the objective [results] This will also help write the documentation which you should be writing all along 4) convert the conceptual model into mathematical models that will address the objectives [methodology] 5) determine specifically where you will get the information from to set up your model [more methodology], 6) set up your model, calibrate and use it to address the objective [results] This will also help write the documentation which you should be writing all along

65. Developing a Conceptual Model Settling Pond Example* The company wanted to discharge sediment laden water into Pond 1 and have the water filter through a sand dike before it discharges into the stream This brings up the first question: How much water could they discharge without overflowing Pond 2? The upper settling pond is separated from the lower pond by 186 ft of sand The maximum elevation of Pond 1 is 658 ft and the outlet of Pond 2 is at an elevation of 652 If Pond 1 becomes contaminated with a dissolved contaminant it will flow towards Pond 2 at a rate roughly equivalent to the average groundwater velocity Once we figure out this velocity (given the distance of travel) it can be used to get a worst case calculation of the time of arrival of the advective front Once the contamination reaches Pond 2 the volumetric flow rate of the groundwater The company wanted to discharge sediment laden water into Pond 1 and have the water filter through a sand dike before it discharges into the stream This brings up the first question: How much water could they discharge without overflowing Pond 2? The upper settling pond is separated from the lower pond by 186 ft of sand The maximum elevation of Pond 1 is 658 ft and the outlet of Pond 2 is at an elevation of 652 If Pond 1 becomes contaminated with a dissolved contaminant it will flow towards Pond 2 at a rate roughly equivalent to the average groundwater velocity Once we figure out this velocity (given the distance of travel) it can be used to get a worst case calculation of the time of arrival of the advective front Once the contamination reaches Pond 2 the volumetric flow rate of the groundwater

66. Conceptual Model (cont.) Develop your conceptual model Two settling ponds were dug in 10 ft of sand, bottomed on low permeability clay. Two settling ponds, different levels, separated by an earthen barrier Start with conceptual model Two settling ponds were dug in 10 ft of sand, bottomed on low permeability clay. Two settling ponds, different levels, separated by an earthen barrier Start with conceptual model

67. Conceptual Model (cont.) Develop your mathematical representation (model) (i.e., convert your conceptual model into a mathematical model) Formulate reasonable assumptions Saturated flow (constant hydraulic conductivity) Laminar flow (a fundamental Darcy�s Law assumption) Parallel flow (so you can use 1-D Darcy�s law) Formulate a mathematical representation of your conceptual model that: Meets the assumptions and Addresses the objectives Any mathimatical model requires assumptions in order to represent an infinitely complex natural system with a set of mathematical equations This is a bit of an art because technically you need to develop a more complex model and demonstrate that the simplifications do not significantly effect the results If you understand the limits of the model you can make more appropriate assumptionsAny mathimatical model requires assumptions in order to represent an infinitely complex natural system with a set of mathematical equations This is a bit of an art because technically you need to develop a more complex model and demonstrate that the simplifications do not significantly effect the results If you understand the limits of the model you can make more appropriate assumptions

68. Conceptual Model (cont.) Collect data to complete your Conceptual Model and to Set up your Mathematical Model The model determines the data to be collected Cross sectional area (A = w b) w: length perpendicular to flow b: thickness of the permeable unit Hydraulic gradient (Dh/Dx) Dh: difference in water level in ponds Dx: flow path length, width of barrier Hydraulic Parameters K: hydraulic tests and/or laboratory tests n: estimated from grainsize and/or laboratory tests Sensitivity analysis Which parameters influence the results most strongly? Which parameter uncertainty lead to the most uncertainty in the results? The mathematical model is helpful in that it will dictate what information you need to collect Cross sectional area (perpendicular to flow): A=b*w Hydraulic gradient: Dh: head difference between ponds, Dx flow pathlength Discuss sensitivity analysis The accuracy of the model results depends on 1) how well your assumptions represent reality and 2) how accurately you have determined the parameters of the model.The mathematical model is helpful in that it will dictate what information you need to collect Cross sectional area (perpendicular to flow): A=b*w Hydraulic gradient: Dh: head difference between ponds, Dx flow pathlength Discuss sensitivity analysis The accuracy of the model results depends on 1) how well your assumptions represent reality and 2) how accurately you have determined the parameters of the model.

69. Implementing Finite Difference Modeling Testing and Sensitivity Analysis Adjust parameters and boundary conditions to get realistic results Test each parameter to learn how the model reacts Gain an appreciation for interdependence of parameters Document how each change effected the head distribution (and heads at key points in the model)

70. Implementation (cont.) Calibration �Fine tune� the model by minimizing the error Quantify the difference between the calculated and the measured heads (and flows) Mean Absolute Error Minimize? Calibration Plot Allows identification of trouble spots Calebration of a transient model requires that the model be calibrated over time steps to a transient event e.g., pump test or rainfall episode Automatic Calibration allows parameter estimation e.g., ModflowP

71. Implementation (cont.) Prediction A well calibrated model can be used to perform reliable �what if� investigations Effects of pumping on Regional heads Induced infiltration Inter aquifer flow Flow paths Effects of urbanization Reduced infiltration Regional use of ground water Addition and diversion of drainage

72. Case Study An unconfined sand aquifer in northwest Ohio Conceptual Model

73. Case Study An unconfined sand aquifer in northwest Ohio Surface water hydrology and topography

74. Boundary Conditions