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Understanding Transient and Steady-State Mass Balance Equations in Groundwater Flow

This overview presents a comprehensive examination of Darcy's law, the transient mass balance equation, and steady-state conditions in groundwater flow. Emphasizing anisotropic and heterogeneous media, we analyze the characteristics of hydraulic conductivity (K) in various dimensional frameworks. The text details the implementation of the Gauss-Seidel iteration method to solve the Toth problem in 2D groundwater scenarios. Special attention is given to specified head boundary conditions, showcasing examples with initial guesses and iteration solutions to illustrate concepts effectively.

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Understanding Transient and Steady-State Mass Balance Equations in Groundwater Flow

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  1. q = - Kgrad h Darcy’s law grad is a vector

  2. Transient mass balance equation: q is a vector

  3. Steady State Mass Balance Equation with W = 0 gradq = 0   q = 0 (del notation) The dot product of grad and q is div q = 0

  4. Anisotropic medium: K at a point varies with direction Kx Ky  Kz or Kx= Ky  Kz Heterogeneous medium: K varies in space

  5. Characteristics of K in two dimensions A  Kx A B  B Kz Kx Kz Kx =Kz = K Homogeneous, isotropic Homogeneous, anisotropic B A A B Heterogeneous, anisotropic Heterogeneous, isotropic

  6. Characteristics of K in two dimensions A  Kx A B  B Kz Kx =Kz = K Kx Kz Homogeneous, isotropic Homogeneous, anisotropic B A A B Heterogeneous, anisotropic Heterogeneous, isotropic

  7. homogeneous, isotropic porous medium: Kx = Ky = Kz = K steady state conditions: W = 0 Laplace equation h = 0 2

  8. Toth Problem (2D) Groundwater divide Groundwater divide Impermeable Rock 2D, steady state

  9. i i, j-1 The 5 point star in a regular grid j i, j i-1, j i+1, j y i, j+1 x = y x

  10. Iteration indices Solution by iteration Gauss-Seidel Iteration

  11. Gauss-Seidel Iteration solution m+3 Iteration planes m+2 m+1 initial guesses m

  12. 1 2 3 4 i 1 2 3 4 5 6 10 9 8 7 10 6 2D example with specified head boundary conditions 10 5 10 5 (Assume an initial guess of h =10 for all interior nodes) 6 10 9 8 10 7 j

  13. 1 2 3 4 i 1 2 3 4 5 6 10 9 8 7 10 6 9.75 Example with specified head boundary conditions 10 5 10 5 6 10 9 8 10 7 j

  14. 1 2 3 4 i 1 2 3 4 5 6 10 9 8 7 9.75 10 6 Example with specified head boundary conditions 8.44 10 5 10 5 6 10 9 8 10 7 j

  15. 1 2 3 4 i 1 2 3 4 5 6 10 9 8 7 8.44 9.75 10 6 Example with specified head boundary conditions 9.94 10 5 10 5 6 10 9 8 10 7 j

  16. 1 2 3 4 i 1 2 3 4 5 6 10 9 8 7 10 6 Example with specified head boundary conditions 9.94 10 5 10 5 6 10 9 8 10 7 j

  17. i j i, j Gauss-Seidel Iteration

  18. C4=(C3+C5+B4+D4)/4

  19. Example of spreadsheet formula Toth Problem

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