1 / 15

Mathematical Model of Toth Problem: Equations and Solutions

Explore the Mathematical Model of the Toth Problem, including governing equations, analytical and numerical solutions, and types of solutions in 2D steady state. Learn about boundary conditions, initial conditions, and various methods in solving mathematical models.

mackensie-acosta
Télécharger la présentation

Mathematical Model of Toth Problem: Equations and Solutions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Mathematical Model a set of equations that describes the physical and/or chemical processes occurring in a system.

  2. Components of a Mathematical Model • Governing Equation • Boundary Conditions • Initial conditions (for transient conditions) problems)

  3. Mathematical Model of the Toth Problem h = c x + zo Laplace Equation 2D, steady state

  4. Types of Solutions of Mathematical Models • Analytical Solutions: h= f(x,y,z,t) • (example: Theis eqn., Toth 1962) • Numerical Solutions • Finite difference methods • Finite element methods • Analytic Element Methods (AEM)

  5. Toth Problem h = c x + zo z Math Model x Analytical Solution Numerical Solution continuous solution discrete solution

  6. Toth Problem h = c x + zo z Math Model x Analytical Solution Numerical Solution h(x,z) = zo + cs/2 – 4cs/2 … (eqn. 2.1 in W&A) z x continuous solution discrete solution

  7. Toth Problem h = c x + zo z Math Model x Analytical Solution Numerical Solution h(x,z) = zo + cs/2 – 4cs/2 … (eqn. 2.1 in W&A) z hi,j =(hi+1,j +hi-1,j +hi,j+1 +hi,j-1)/4 x continuous solution discrete solution

  8. div q = 0 Steady state mass balance q = - Kgrad h Darcy’s law

  9. div q = 0 Steady state mass balance q = - Kgrad h Darcy’s law q equipotential line grad h q grad h Isotropic Anisotropic

  10. div q = 0 Steady state mass balance q = - Kgrad h Darcy’s law Assume that K = a constant Laplace Equation

  11. Governing Eqn. for TopoDrive Steady-state, heterogeneous, anisotropic

  12. q = - Kgrad h K is a tensor with 9 components

  13. Kxx Kxy Kxz KyxKyy Kyz Kzx KzyKzz K = Principal components of K

More Related