Transcript Lesson 6
Introduction to MT3DMS All equations & illustrations taken from the MT3DMS manual Refer to the document on the course homepage entitled “MT3DMS Solution Methods and Parameter Options” (Look under the MT3DMS tab on the homepage) General form of the ADE: Expands to 9 terms Expands to 3 terms (See eqn. 3.48 in Z&B) 9 Dispersion Coefficients This schematic assumes that MODFLOW MT3DMS MT3DMS time steps are selected by the code considering stability constraints, if any, and Courant numbers. Dispersion, sink/source, chemical reactions Advection MT3DMS Solution Options 1 2 3 4 x j-1 j-1/2 j j+1 j+1/2 Upstream weighting Central differences MT3DMS Solution Options Explicit Approximation Courant Number Stability constraints for explicit solutions Courant Number v t Cr x 6 Courant Numbers One for each face of the cell block Cr < 1 MT3DMS Solution Options Use GCG Solver Use GCG Solver Use GCG Solver Implicit Approximation for advection term MT3DMS Solution Options TVD ULTIMATE METHOD a higher order FD method Conventional FD methods use 3 nodes in the FD approximation. The TVD method uses 4 nodes with upstream weighting. This essentially eliminates numerical dispersion. Steps in the TVD Method Check for oscillation errors Correction for oscillation errors oscillation TVD ULTIMATE METHOD In one dimension Compare with an equation for a lower order explicit approximation c j n 1 vt (c j n c j 1n ) c j n x MT3DMS Solution Options Eulerian vs Lagrangian Methods • Eulerian: fixed coordinate system with mass flux through an REV • Lagrangian: moving particles; each particle carries mass. The Random Walk method is a Lagrangian method. • Mixed Eulerian-Lagrangian methods use particles to solve the advection portion of the ADE and an Eulerian method to solve the rest of the equation. Method of Characteristics (MOC) 2 where is a weighting factor to weight concentration between time level n and an intermediate time level n*, normally = 0.5 3 1 n 1 n 1 Step 1 is a Lagrangian method; Cm Cm 4 Cm Step 3 is a Eulerian method. Also update concentration of each particle. For example, n 1 n n 1 C C C for particles in cell m: p p m n* • MOC uses multiple particles per cell. • MMOC uses one particle per cell. • HMOC uses multiple particles in high concentration regions and one particle per cell elsewhere. Dynamic Particle Allocation Breakthrough curve for example problem in the MT3DMS manual Compare with Fig. 7.26 in Z&B 1.20 Central FD TVD 1.00 HMOC TVD Upstream weighting Central FD Concentration 0.80 Upstream FD 0.60 0.40 0.20 0.00 0.00 0.20 0.40 0.60 Time (years) 0.80 1.00 1.20 MT3DMS Solution Options PS#2 1 3 4 2