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Dr. James M. Martin-Hayden Associate Professor Dr. James M. Martin-Hayden Associate Professor Analytical and Numerical Ground Water Flow Modeling An Introduction (419) 530-2634 Jhayden@Geology.UToledo.edu Slide 2 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, ) does not change significantly Volume is directly related to mass by density V=m/ In this case water balance models are essentially mass balance models divided by density Mass In Mass Out = Change in Mass Stored Volume In Volume Out = Change in Volume Stored Q i Q o = V w / t Dividing by unit time gives: If Q is a continuous function of time Q(t) then dv/dt is at any instant in time Q i (t) Q o (t) = dV w /dt Slide 3 The Flow Equation (cont.) Example 1: Storage in a reservoir If Q i = Q o, dV w /dt = 0 no change in level, i.e., steady state If Q i > Q o, dV w /dt > 0 +filling If Q i < Q o, dV w /dt < 0 -emptying E.g., Change in storage due to linearly varying flows QiQi QoQo dV w /dt Q i (t) Q o (t) = dV w /dt Q1Q1 Q2Q2 Q i =m 1 t+Q 1 Q o =m 2 t+Q 2 t Q 0 dV/dt = 0 dV/dt < 0 dV/dt > 0 Slide 4 The Flow Equation (cont.) Example 1: Storage in a reservoir If Q i = Q o, dV w /dt = 0 no change in level, i.e., steady state If Q i > Q o, dV w /dt > 0 +filling If Q i < Q o, dV w /dt < 0 -emptying QiQi QoQo dV w /dt Q i (t) Q o (t) = dV w /dt Q1Q1 Q2Q2 Q i =m 1 t+Q 1 Q o =m 2 t+Q 2 t Q 0 dV/dt = 0 dV/dt < 0 dV/dt > 0 t V dV/dt = 0 dV/dt < 0 dV/dt > 0 Slide 5 The Flow Equation (cont.) Example 1: Storage in a reservoir If Q i = Q o, dV w /dt = 0 no change in level, i.e., steady state If Q i > Q o, dV w /dt > 0 +filling If Q i < Q o, dV w /dt < 0 -emptying Example 2: Storage in a REV (Representative Elementary Volume) REV: The smallest parcel of a unit that has properties (n, K, ) that are representative of the formation The same water balance can be used to examine the saturated (or unsaturated) REV QiQi QoQo dV w /dt Q i (t) Q o (t) = dV w /dt Slide 6 Q o = q o y z The Flow Equation (cont.) Mass Balance for the REV (or any volume of a flow system) aka, The Ground Water Flow Equation xx yy zz Vw/tVw/t Q i Q o = dV w /dt y z x = V (q i q o ) y z =dV w /dt Q i Q o = (q o q i ) = q= ( q/ x) x Take change in q with x at a point (the derivative of q rwt x) For saturated, incompressible, 1-D flow Q i =q i y z Slide 7 The Flow Equation (cont.) External Sources and Sinks ( Q s ) y z x = V Differential Form Slide 8 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 Q s /V volumetric flow rate per unit volume injected into REV Net inflow* = Change in volume stored* *per unit volume per unit time qxqx qxqx qzqz qzqz qyqy qyqy Substitute components of q from 3-D Darcys law Slide 9 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)* Assumption: K is homogeneous over small distances, i.e., K f(x,y,z) Compressibility of water is much less than aquifer compressibility This gives the equation on which ground water flow models are based: h=f(x,y,z,t) S s : Specific Storage, a proportionality constant Slide 10 The Flow Equation (cont.) Flow Equation Simplifications 0 Our job is not yet finished, h=f(x,y,z,t)? Isotropic, 2-D, steady state flow equation ( without source term), a.k.a.: The Laplace Equation 2-D, horizontally isotropic flow equation K x =K y =K h T=K h b units: [L 2 /T], S=S s b A h : horizontal area of recharge S s /K h =S/T hydraulic diffusivity Two dimensional flow equation Horizontal flow (Dupuit assumption) Thus: dh/dz = 0 Steady State Flow Equation If inflow = out flow, Net inflow = 0 Change in storage = 0 0 00 Slide 11 Introduction to Ground Water Flow Modeling 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 Potentiometric Surface x x x hoho x 0 h(x) x K q e.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) h(x,y,z,t)? Darcys LawIntegrated Slide 12 Flow Modeling (cont.) Analytical models (a.k.a., closed form models) The previous model is an example of an analytical model is a solution to the 1-D Laplace equation i.e., the second derivative of h(x) is zero With 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 equations Thiem Equation: Steady state flow to a well in a confined aquifer The Theis Equation: Transient flow to a well in a confined aquifer Slug test solutions: Transient response of head within a well to a pressure pulse Slide 13 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 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 Slide 14 Pump Tests an Groundwater Modeling h as a function of r (radial distance) and t (time) Aquitard Aquifer Aquitard r1r1 r2r2 h1h1 h2h2 Q T?, S? Slide 15 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 1-D, SS ThiemTheis Slide 16 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 hoho h1h1 Clay b x hoho h1h1 Q Q Slide 17 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, h 1 and h 2, are measured in two observation wells at different radial distances from the pumping well r 1 and r 2 The values are plugged into the inverse model to calculate K (a bulk measure of K over the area stressed by pumping) Slide 18 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 S s ) 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 Slide 19 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 This complexity calls for a more powerful solution to the flow equation Numerical modeling Slide 20 Numerical Modeling in a Nutshell A solution of flow equation is approximated on a discrete grid (or mesh) of points, cells or elements Within this discretized domain: 1)Aquifer parameters can be set at each cell within the grid 2)Complex aquifer geometry can be modeled 3)Complex boundary conditions can be accounted for Requires 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 characterization Flow Modeling (cont.) The parameters and variables are specified over the boundary of the domain (region) being modeled Slide 21 An Introduction to Finite Difference Modeling Approximate Solutions to the Flow Equation Partial derivatives of head represent the change in head with respect to a coordinate direction (or time) at a point.e.g., h y hh yy h1h1 h2h2 y1y1 y2y2 These derivatives can be approximated as the change in head ( h) over a finite distance in the coordinate direction ( y) that traverses the point i.e., The component of the hydraulic gradient in the y direction can be approximated by the finite difference h/ y The Finite Difference Approximation of Derivatives Slide 22 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 x 2 This is known as a central difference h x xx haha hoho xoxo xbxb xaxa xx hbhb ho-haho-ha hb-hohb-ho xx Slide 23 Finite Difference Modeling (cont.) Finite Difference Approximation of 1-D, Steady State Flow Equation With external source (Q s ) Q s /A h = R Slide 24 Finite Difference Modeling (cont.) Physical basis for finite difference approximation h xx haha hoho xoxo xbxb xaxa xx hbhb ho-haho-ha hb-hohb-ho xx K oa : average K of cell and K of cell to the left; K ob : average K of cell and K of cell to the right yy zz xx KaKa KbKb KoKo Slide 25 Finite Difference Modeling (cont.) Inclusion of and external source K oa : average K of cell and K of cell to the left; K ob : average K of cell and K of cell to the right yy zz xx KaKa KbKb KoKo QsQs QiQi QoQo Slide 26 Finite Difference Modeling (cont.) Discretization of the Domain Divide the 1-D domain into equal cells of heterogeneous K h1h1 h2h2 h3h3 h i-1 hnhn xx xx xx xx xx xx xx hihi h i+1 xx xx Solve for the head at each node gives n equations and n unknowns The head at each node is an average of the head at adjacent cells weighted by the Ks hoho h n+1 Specified Head Specified Head Slide 27 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 haha hoho hbhb hdhd hchc xx xx xx KaKa KcKc KbKb KdKd KdKd xx xx xx Solve for h o Slide 28 Finite Difference Modeling (cont.) 2-D, Steady State, Uniform Grid Spacing, Finite Difference Scheme With Source Term Heterogeneous K haha hoho hbhb hdhd hchc xx xx xx KaKa KcKc KbKb KdKd KdKd xx xx Solve for h o V= x y z= x 2 b Slide 29 haha hoho hbhb hchc KaKa KcKc KbKb KdKd Finite Difference Modeling (cont.) Incorporate Transmissivity: Confined Aquifers multiply by b (aquifer thickness) xx xx xx xx xx KoKo KbKb KaKa baba bobob Solve for h o R or Q s Slide 30 haha hoho hbhb hchc KaKa KcKc KbKb KdKd xx xx xx xx xx KoKo KbKb KaKa haha hoho hbhb Finite Difference Modeling (cont.) Incorporate Transmissivity: Unconfined Aquifers b depends on saturated thickness which is head ( h ) measured relative to the aquifer bottom Solve for h o Slide 31 haha hoho hbhb hchc KaKa KcKc KbKb KdKd xx xx xx xx xx KoKo KbKb KaKa haha hoho hbhb Finite Difference Modeling (cont.) Incorporate Transmissivity: Unconfined Aquifers Homogeneous K Solve for h o Slide 32 Finite Difference Modeling (cont.) 2-D, Steady State, Isotropic, Homogeneous Finite Difference Scheme haha hoho hbhb hdhd hchc xx xx xx xx xx xx Solve for h o Slide 33 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 ABCDE 1 2 3 4 Slide 34 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 h B3 h C3 h D3 h C4 h C2 ABCDE 1 2 3 4 5 = (C2+D3+C4+B3)/4 Slide 35 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 10 ABCDE 2 3 4 5 10 752 1 1 1 Lake 1 Lake 2 River Slide 36 Basic Finite Difference Design Discretization and Boundary Conditions Grids should be oriented and spaced to maximize the efficiency of the model Boundary conditions should represent reality as closely as possible Slide 37 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 Slide 38 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 Slide 39 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 Slide 40 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 Slide 41 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 (Q p ) Induced infiltration Underflow No flow boundaries Valley wall of low permeable sediment or rock Fault Slide 42 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 h B3 h C4 h A2 AB C DE 1 2 4 5 = (A2 +2*B3 +C4)/4 = (B1+D1+2*C2)/4 A3 C1 E3 = (2*D3+E2 +E4)/4 = (B5+2*C4+D5)/4 C5 h A3 h B3 Slide 43 Finite Difference Modeling (cont.) No Flow Boundary Implementation Corner nodes have two image nodes BCDE 2 4 5 = (2*B1 +2*A2)/4 = (2*D1+2*E2)/4 A1 E1 E5 = (2*D5+2*E4)/4 = (2*A4+2*B5)/4 A5 3 Slide 44 Finite Difference Modeling (cont.) No Flow Boundary Implementation Combinations of edge and corner points are used to approximate irregular boundaries Slide 45 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 h s : stage of the stream h o : Head within the cell K sb : K of streambed materials b sb : Thickness of streambed w: width of stream L: length of reach within cell C sb : Streambed conductance Based on Darcys law 1-D Flow through streambed hshs hoho w L b sb Q sb Slide 46 Finite Difference Modeling (cont.) Head Dependent Flux Boundaries e.g. 2, Flow through aquitard h c : Head within confined aquifer H o : Head within the cell K c : K of aquitard b c : Thickness of aquitard x 2 : Area of cell C c : aquitard conductance Based on Darcys law 1-D Flow through aquitard hoho hchc xx bcbc QcQc QcQc xx Slide 47 Case Study The Layered Modeling Approach Slide 48 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 haha hoho hbhb hdhd hchc hfhf Slide 49 Finite Difference Modeling (cont.) 3-D Finite Difference Models Requires vertical discretization (or layering) of model K1K1 K2K2 K3K3 K4K4 Slide 50 Implementing Finite Difference Modeling Model Set-Up, Sensitivity Analysis, Calibration and Prediction Model Set-Up Develop a Conceptual Model Collect Data Develop Mathematical Representation of your System Model set-up is an Iterative process Start simple and make sure the model runs after every added complexity Make Back-ups Slide 51 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 Slide 52 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 Slide 53 Developing a Conceptual Model Settling Pond Example* Questions to be addressed: (Objectives) How much flow can Pond 1 receive without overflowing? Q? How long will water (contamination) take to reach Pond 2 on average? v? How much contaminant mass will enter Pond 2 (per unit time)? M? A company has installed two settling ponds to: (Significance) Settle suspended solids from effluent Filter water before it discharges to stream Damp flow surges *This is a hypothetical example based on a composite of a few real cases 5000 ft 652 658 0 N Pond 1 Pond 2 Slide 54 Conceptual Model (cont.) Develop your conceptual model W 1510 ft x =186 Pond 1Pond 2 Outfall Elev.= 658.74 ft Elev.= 652.23 ft Q? v? M? K x =186 ft b=8.56 ft Water flows between ponds through the saturated fine sand barrier driven by the head difference Sand Clay h =6.51 ft Contaminated Pond b xx Not to scale Overflow Slide 55 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 Darcys Law assumption) Parallel flow (so you can use 1-D Darcys law) Formulate a mathematical representation of your conceptual model that: Meets the assumptions and Addresses the objectives M = Q C Q?Q?v?v?M?M? Slide 56 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 ( h/ x) h: difference in water level in ponds x: 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? M = Q C Q?Q? v?v? M?M? Slide 57 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) Slide 58 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 ErrorMinimize 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 Measured Head Calculated Head x x o o x x x x MW28d x Calculated= Measured Slide 59 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 Slide 60 Case Study An unconfined sand aquifer in northwest Ohio Conceptual Model Slide 61 Case Study An unconfined sand aquifer in northwest Ohio Surface water hydrology and topography Slide 62 Boundary Conditions Slide 63