From 407d4d112ede74b3440f9a662c933705715efdbf Mon Sep 17 00:00:00 2001 From: bangerth Date: Tue, 4 Jan 2011 04:33:20 +0000 Subject: [PATCH] Some more text. git-svn-id: https://svn.dealii.org/trunk@23119 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-43/doc/intro.dox | 60 ++++++++++++++++++-------- 1 file changed, 42 insertions(+), 18 deletions(-) diff --git a/deal.II/examples/step-43/doc/intro.dox b/deal.II/examples/step-43/doc/intro.dox index e806a79d28..6b19ba258b 100644 --- a/deal.II/examples/step-43/doc/intro.dox +++ b/deal.II/examples/step-43/doc/intro.dox @@ -24,27 +24,40 @@ Research Fellowship.

Introduction

-The simulation of multiphase flow in porous media is a ubiquitous problem, but it faces two major difficulties: numerical accuracy and efficiency. In this tutorial, in order to overcome these two problems, there are five areas which we are trying to improve for a high performance simulator: +The simulation of multiphase flow in porous media is a ubiquitous problem, and +we have previously addressed it already in some form in step-20 and +step-21. However, as was easy to see there, +it faces two major difficulties: numerical accuracy and efficiency. The first +is easy to see in step-20: using lowest order Raviart-Thomas elements can not +be expected to yield highly accurate solutions. We need more accurate +methods. The second reason is apparent from step-21: that program is +exceedingly slow, and there is no hope to get highly accurate solutions in 3d +within reasonable time frames. + +In this +program, in order to overcome these two problems, there are five areas which +we are trying to improve for a high performance simulator: +

Advection-dominated two-phase flow mathematical model.

We consider the flow of a two-phase immiscible, incompressible -fluid. Capillary and gravity effects are neglected, and that viscous +fluid. Capillary and gravity effects are neglected, and viscous effects are assumed dominant. The governing equations for such a -flow that are identical to those used in step-21 are then +flow that are identical to those used in step-21 and are @f{align*} \mathbf{u}_t &= - \mathbf{K} \lambda_t \left(S\right) \nabla p, \\ \nabla \cdot \mathbf{u}_t &= q, \\ \epsilon \frac{\partial S}{\partial t} + \nabla \cdot \left( \mathbf{u}_t F\left( S \right) \right)&=0, @f} -where $S$ is the saturation (volume fraction) of the second (wetting) phase, $p$ is the pressure, $\mathbf{K}$ is the permeability tensor, $\lambda_t$ is the total mobility, $\epsilon$ is the porosity, $F$ is the fractional flow of the wetting phase, $q$ is the source term and $\mathbf{u}_t$ is the total velocity. The total mobility, fractional flow of the wetting phase and total velocity are respectively given by +where $S$ is the saturation (volume fraction between zero and one) of the second (wetting) phase, $p$ is the pressure, $\mathbf{K}$ is the permeability tensor, $\lambda_t$ is the total mobility, $\epsilon$ is the porosity, $F$ is the fractional flow of the wetting phase, $q$ is the source term and $\mathbf{u}_t$ is the total velocity. The total mobility, fractional flow of the wetting phase and total velocity are respectively given by @f{align*} \lambda_t(S)&= \lambda_w + \lambda_{nw} = \frac{k_{rw}(S)}{\mu_w} + \frac{k_{rnw}(S)}{\mu_{nw}}, \\ F(S) &= \frac{\lambda_w}{\lambda_t} = \frac{\lambda_w}{\lambda_w + \lambda_{nw}} = \frac{k_{rw}(S)/\mu_w}{k_{rw}(S)/\mu_w + k_{rnw}(S)/\mu_{nw}}, \\ @@ -59,7 +72,7 @@ scaling factor for the time variable, is set to one. Following a commonly used prescription for the dependence of the relative permeabilities $k_{rw}$ and $k_{rnw}$ on saturation, we use @f{align*} - k_{rw} &= S^2, \qquad\qquad + k_{rw} &= S^2, \qquad&\qquad k_{rnw} &= \left( 1-S \right)^2. @f} @@ -76,15 +89,21 @@ parts. Specifically, and we arrive at a complete model by also imposing boundary values for the saturation variable on the inflow boundary $\mathbf{\Gamma}_{in}$. +

Adaptive operator splitting and time stepping.

-Based on facts that an implicit pressure-velocity part is computationally -costly and the pressure and velocity depend only weakly on saturation, we -don't have to solve it every saturation time step. Therefore, operator -splitting is employed. However, with operator splitting that is not enough, we -still needs an objective criterion to tell us when the pressure-velocity part -needs to be solved at every saturation time step to control the amount of -splitting error in a reasonable frame. Here, we use an a poteriori criterion -(which detailed derivations and descriptions can be found in [Chueh, Bangerth + +As seen in step-21, solving the flow equations for velocity and pressure are +the parts of the program that take far longer than the (explicit) updating +step for the saturation variable once we know the flow variables. On the other +hand, the pressure and velocity depend only weakly on saturation, so one may +think about only solving for pressure and velocity every few time steps while +updating the saturation in every step. If we can find a criterion for when the +flow variables need to be updated, we call this splitting an "adaptive +operator splitting" scheme. + +Here, we use the following a posteriori criterion to decide when to re-compute +pressure and velocity variables: +(detailed derivations and descriptions can be found in [Chueh, Bangerth and Djilali 2010]): @f{align*} \theta(n,n_p) @@ -93,11 +112,16 @@ and Djilali 2010]): \left( \left\| \frac 1{\lambda_t\left(S^{(n-1)}\right)} - - \frac 1{\lambda_t\left(S^{(n_p-1)}\right)} \right\|_{L^\infty(\kappa)} + - \frac 1{\lambda_t\left(S^{(n_p)}\right)} \right\|_{L^\infty(\kappa)} \left\|\|\mathbf{K}^{-1}\|_1\right\|_{L^\infty(\kappa)} \right). @f} -where superscripts in parentheses denote the number of the saturation time step at which any quantity is defined and $n_p