## Environmental Modeling: Using MATLAB®“Environmental Modeling using MATLAB R ” by Ekkehard Holzbecher is an excellent publication and a novel approach covering the intersection of two important, growing worlds – the world of environmental modeling and of mathematical software. Environmental modeling is a science that uses mathematics and comp- ers to simulate physical and chemical phenomena in the environment (e.g., environmental pollution). This science was initially based on pen-and-paper calculations using simple equations. In the last 50 years, with the devel- mentofdigitalcomputers,environmentalmodelshavebecomemoreandmore complex, requiring often numerical solutions for systems of partial di?erential equations. Mathematical software, such as MATLAB R , has been developed in the lasttwo decades. Thesepackageshavebeen particularlysuccessfulfor usersof personal computers. Mathematical software provides a set of tools for solving equations both analytically and numerically. This is a major improvement in comparison to the programming tools (e.g., FORTRAN) previously used by scientists. Mathematical software o?ers extremely valuable and cost-e?ective tools that improve the productivity of the programmer by at least an order of magnitude. The use of these tools also minimizes the risk of programming errors. In addition, mathematical software o?ers unique visualization tools that allow the user to immediately visualize and often animate simulation results. Scientists who become familiar with a tool like MATLAB R will never go back to previous ways of computer programming. |

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### Contents

3 | |

7 | |

20 | |

14 MATLAB Graphics The Figure Editor | 24 |

15MATLB Help System | 25 |

References | 27 |

Fundamentals of Modeling Principles and MATLABR 21 Model Types | 29 |

22 Modeling Steps | 30 |

Advanced Modeling using MATLAB | 205 |

Flow Modeling | 206 |

111 The NavierStokes Equations for Free Fluids | 208 |

112 The Euler Equations and the Bernoulli Theorem | 213 |

113 Darcys Law for Flow in Porous Media | 217 |

114 Flow in Unsaturated Porous Media | 222 |

References | 227 |

Groundwater Drawdown by Pumping | 229 |

23 Fundamental Laws | 34 |

24 Continuity Equation for Mass | 36 |

25 MATLAB Mﬁles | 40 |

26 Ifs and Loops in MATLAB | 42 |

27 Debugging of Mﬁles | 44 |

Reference | 46 |

Transport 31 The Conservation Principle | 47 |

32 Ficks Law and Generalizations | 49 |

33 The Transport Equation Mass Transport | 55 |

34 Dimensionless Formulation | 60 |

35 Boundary and Initial Conditions | 61 |

References | 63 |

Transport Solutions | 65 |

42 A Simple Numerical Model | 69 |

43 Comparison between Analytical and Numerical Solution | 77 |

44 Numerical Solution using MATLAB pdepe | 79 |

1D Inﬂow Front | 83 |

References | 85 |

Transport with Decay and Degradation | 86 |

52 1D Steady State Solution | 90 |

53 Dimensionless Formulation | 92 |

54 Transient Solutions | 98 |

References | 100 |

Transport and Sorption | 101 |

62 Retardation | 107 |

63 Analytical Solution | 109 |

64 Numerical Solutions | 111 |

65 Slow Sorption | 115 |

66 MATLAB Animations | 119 |

References | 121 |

Transport and Kinetics | 123 |

72 Law of Mass Action for Kinetic Reactions | 125 |

73 Monod MichaelisMenten and Blackwell Kinetics | 126 |

74 Bacteria Populations | 128 |

75 Steady States | 130 |

References | 134 |

Transport and Equilibrium Reactions | 137 |

82 The Law of Mass Action for Equilibrium Reactions | 141 |

83 Speciation Calculations | 143 |

84 Sorption and the Law of Mass Action | 147 |

85 Transport and Speciation | 150 |

References | 156 |

Ordinary Differential Equations Dynamical Systems | 158 |

91 StreeterPhelps Model for River Puriﬁcation | 160 |

92 Details of MichaelisMenten or Monod Kinetics | 163 |

93 1D Steady State Analytical Solution | 165 |

94 Redox Sequences | 173 |

References | 178 |

Parameter Estimation | 181 |

102 Polynomial Curve Fitting | 182 |

103 Exponential Curve Fitting | 185 |

104 Parameter Estimation with Derivatives | 189 |

105 Transport Parameter Fitting | 196 |

106 General Procedure | 199 |

References | 204 |

121 Conﬁned Aquifer | 230 |

122 Unconﬁned Aquifer | 232 |

123 Halfconﬁned Aquifer | 235 |

124 Unsteady Drawdown and Well Function | 237 |

125 Automatic Transmissivity Estimation | 238 |

References | 241 |

Aquifer Baseﬂow and 2D Meshing | 243 |

132 1D Implementation | 245 |

133 2D Implementation | 246 |

134 Meshs and Grids | 250 |

Reference | 254 |

Potential and Flow Visualization | 255 |

142 Potential and Real World Variables | 258 |

Groundwater Baseﬂow and Well | 260 |

144 MATLAB 2D Graphics | 264 |

145 MATLAB 3D Graphics | 268 |

References | 270 |

Streamfunction and Complex Potential | 271 |

152 The Principle of Superposition | 275 |

153 Complex Analysis and Complex Potential | 282 |

Vortices or Wells Systems | 286 |

References | 291 |

2D and 3D Transport Solutions Gaussian Puffs and Plumes 161 Introduction | 293 |

162 2D Instantaneous Line Source | 298 |

163 2D Constant Line Source | 299 |

165 3D Constant Source | 301 |

References | 305 |

Image Processing and Georeferencing | 307 |

172 Reading and Display | 308 |

173 GeoReferencing | 310 |

174 Digitizing | 312 |

175 MATLABR Functions | 314 |

References | 316 |

Compartment Graphs and Linear Systems | 317 |

182 Linear Systems | 321 |

183 Eigenvalues and Phase Space | 331 |

References | 336 |

Nonlinear Systems | 338 |

192 Competing Species | 342 |

193 PredatorPrey Models | 348 |

194 Chaos Lorenz Attractor | 353 |

References | 355 |

Graphical User Interfaces | 357 |

202 The Transport GUI | 366 |

References | 369 |

MATLAB Data Import | 370 |

Data Export | 374 |

Data Presentation in a Histogram | 375 |

Epilogue | 379 |

References | 380 |

381 | |

384 | |

385 | |

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### Common terms and phrases

advection algorithm analytical solution approach baseflow boundary conditions calculated cfit Chap chemical coefficient column command sequence command window compartment complex components computed confined aquifer constant contours corresponding curve decay degradation denotes depends depicted derived diffusion dimensionless dispersion drawdown eigenvalues environmental modeling equilibrium example exponential factor figure editor flux following command formula given gradient graphical graphical user interface groundwater Holzbecher hydraulic hydraulic conductivity inflow input isotherm kinetics lambda linear loop M-file mathematical MATLABR MATLABR command matrix mesh obtained options ordinary differential equations output parameter pdepe Péclet phase space physical unit piezometric piezometric head plot porosity porous media position potential processes pumping reaction redox represents result sediments Sidebar simulation situation solid phase sorption species specified steady streamfunction streamline surface term tion transport equation unconfined valid values variable vector velocity xlabel zero

### Popular passages

Page 15 - ... the element in the ith row and jth column of the matrix Q3.

Page 15 - Matrix multiplication is only possible when the first matrix has the same number of columns as the second matrix has rows. If this requirement isn't met, the matrices are said to be "incompatible" and matrix multiplication cannot be performed.