Page:4SIGHT manual- a computer program for modelling degradation of underground low level waste concrete vaults (IA 4sightmanualcomp5612snyd).pdf/141

65. UPDATE PRESSURES '''65. Update Pressures.'''

The pressures are updated using the continuity equation: $$\frac{\partial{p}}{\partial{t}}=\nabla\cdot \rho\mathbf{v} $$

where $$\mathbf{v} $$ the intrinisic pore fluid velocity and $$\rho$$ is the pore fluid density. An equation for the bulk material can be obtained from a volume average over a represetative volume V

For these equations, assume $$\rho$$ is a constant since water is virtually incompressible. Rearranging and simplifying the above equation gives:

$$ \frac{\partial{}}{\partial{t}}\frac{1}{V}\int_v \frac{\partial{p}}{\partial{t}}d^3\mathbf{x} = -\frac{1}{V}\int_v \nabla\cdot\rho\mathbf{v}d^3\mathbf{x}$$

Since $$\rho$$ is zero outside the pore volume, if $$V_p$$ represents only the pore volume then the above equation simplifies to

$$ \frac{\partial{\phi}}{\partial{t}}-\nabla\cdot\frac{\phi}{V_p} \int_{V_p}\mathbf{v}d^3\mathbf{x} $$

which finally gives $$ \frac{\partial{\phi}}{\partial{t}}=-\nabla\cdot\phi \mathbf{u}=-\nabla\cdot\phi \mathbf{v}_D$$

where $$\mathbf{v}_D$$ is the Darcy velocity. Substituting for the Darcy velocity gives

$$\frac{\partial{\phi}}{\partial{t}}=-\nabla\cdot\frac{k}{\mu}\nabla p$$

⟨Function declarations 23⟩ $$+\equiv$$ void update.pressures (void);

66. update.pressures void update.pressures {   int k, iteration; real perm [NUM_CELLS], imp [NUM.SURFACES], tol = 0.0005, $$\varepsilon$$; for (k = FIRST.CELL; k < NUM.CELLS; k++) perm[k]=$$\kappa$$[k] * CUB($$\xi$$[ke]/$$\vartheta_0$$ ); $$\Phi$$[FIRST.CELL] = Pout + (NUM.CELLS - 1 + A A [FIRST.CELL]) * dP   $$\Phi$$[NUM.SURFACES-1]=0; iteration = 0; do { for (k= FIRST.CELL + 1; k < NUM.SURFACES - 1; k++) { imp[k]= perm * $$\Psi$$[k + 1]/$$\Delta X$$[k] + perm[k -1]*$$\Psi$$[k-1]/$$\Delta X$$[k-1]; /* **** tmp[k]-=0.5*(DX[k-1]_DX[k])*(PHIn[k]-PHIn[k-1])/DT; *****/ imp[k]*=1.0/(perm[k]/$$\Delta X$$[k] = perm[k-1]/$$\Delta X$$[k-1]); } $$\varepsilon$$=0.0; for (k= FIRST_CELL+1; K0.0 $$\epsilon$$=MAX($$\epsilon$$,$$\Psi$$[k] -imp[k])/$$\Psi$$[k]);