From ce6acea9a9e49abf5b9f4fefced10b7f0dd1b405 Mon Sep 17 00:00:00 2001 From: bangerth Date: Thu, 19 Oct 2006 23:03:17 +0000 Subject: [PATCH] More text git-svn-id: https://svn.dealii.org/trunk@14030 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-21/doc/intro.dox | 154 +++++++++++++++++-------- 1 file changed, 109 insertions(+), 45 deletions(-) diff --git a/deal.II/examples/step-21/doc/intro.dox b/deal.II/examples/step-21/doc/intro.dox index c232b6bcef..77af722896 100644 --- a/deal.II/examples/step-21/doc/intro.dox +++ b/deal.II/examples/step-21/doc/intro.dox @@ -4,61 +4,125 @@ This program grew out of a student project by Yan Li at Texas A&M University. Most of the work for this program is by her. In this project, we propose a numerical simulation for two phase -flow problem in porous media. The two phase flow system includes one -elliptic equation and one nonlinear transport equation. We apply -mixed finite element method and Discontinuous Galerkin method for -this system. Some numerical results for two dimensional case are -given by $RT_{0}\times DQ_{0}\times DQ_{0}$. - -The numerical computation is based on $dealII$. We use vector shape -functions from step9, DG method from step12, mixed method and Schur -complement from step20 and many many useful tools from the library. - -

Two Phase Flow Problem

- -The modeling of two phase flow in porous media is important for both -environmental rededication and the management of petroleum -reservoirs. Practical situations involving two phase flow include -the dispersal of a nonaqueous phase liquid in an aquifer or the -displacement of a non-aqueous heterogeneity on the flow and -transport. Simulation models, if they are to provide realistic +flow problems in porous media. This problem includes one +elliptic equation and one nonlinear, time dependent transport +equation. This is therefore also the first time-dependent tutorial +program (besides the somewhat strange time-dependence of step-18). + + +

The two phase flow problem

+ +Modeling of two phase flow in porous media is important for both +environmental remediation and the management of petroleum +reservoirs. Practical situations involving two phase flow include the +dispersal of a nonaqueous phase liquid in an aquifer, or the joint +movement of a mixture of fluids such as oil and water in a +reservoir. Simulation models, if they are to provide realistic predictions, must accurately account for these effects. -In our project,we consider a kind of periodic permeability,our -numerical result shows that the heterogeneity effects are simulated -accurately. +To derive the governing equations, consider two phase flow in a +reservoir $\Omega$ under the assumption that the movement of fluids is +dominated by viscous effects; i.e. we neglect the effects of gravity, +compressibility, and capillary pressure. Porosity will be considered +to be constant. We will denote variables refering to either of the two +phases using subscripts $w$ and $o$, short for water and oil. The +derivation of the equations holds for other pairs of fluids as well, +however. -Consider two phase flow in a reservoir $\Omega$ under the assumption -that the displacement is dominated by viscous effects; i.e. we -neglect the effects of gravity, compressibility, and capillary -pressure. Porosity will be considered to be constant. The two phase -will be referred to as water and oil, designated by subscripts $w$ -and $o$, respectively. We write Darcy's for each phase as follows: +The velocity with which molecules of each of the two phases move is +determined by Darcy's law that states that the velocity is +proportional to the pressure gradient: @f{eqnarray*} -\mathbf{u}_{j} = \frac{k_{rj}(S)}{\mu_{j}} \mathbf{K} \cdot \nabla p + \mathbf{u}_{j} + = + -\frac{k_{rj}(S)}{\mu_{j}} \mathbf{K} \cdot \nabla p @f} - where, $\mathbf{u}_{j}$ is the phase velocity, $K$ is the -permeability tensor, $k_{rj}$ is the relative permeability to phase -$j$($j=o,w$),$S$ is the water saturation(volume fraction), $P$ is -pressure and $\mu_{j}$ is the viscosity of phase $j$($j=o,w$). - -Combining Darcy's law with a statement of conservation of mass -allows us to express the governing equations in terms of the -so-called pressure and saturation equations: +where $\mathbf{u}_{j}$ is the velocity of phase $j=o,w$, $K$ is the +permeability tensor, $k_{rj}$ is the relative permeability of phase +$j$, $p$ is the +pressure and $\mu_{j}$ is the viscosity of phase $j$. Finally, $S$ is +the saturation (volume fraction), i.e. a function with values between +0 and 1 indicating the composition of the mixture of fluids. In +general, the coefficients $K, k_{rj}, \mu$ may be spatially dependent +variables, and we will always treat them as non-constant functions in +the following. + +We combine Darcy's law with the statement of conservation of mass for +each phase, +@f[ + \textrm{div}\ \mathbf{u}_{j} = q_j, +@f] +with a source term for each phase. By summing over the two phases, +we can express the governing equations in terms of the +so-called pressure equation: @f{eqnarray*} -\nabla \cdot (\mathbf{K}(x,y)\lambda(S) \nabla p)= q(x,y) && \forall(x,y)\in\Omega\\ - S_{t} + \mathbf{u} \cdot \nabla F(S) = 0&& \forall(x,y)\in\Omega +- \nabla \cdot (\mathbf{K}\lambda(S) \nabla p)= q. @f} +Here, $q$ is th sum source term, and +@f[ + \lambda(S) = \frac{k_{rw}(S)}{\mu_{w}}+\frac{k_{ro}(S)}{\mu_{o}} +@f] +is the total mobility. +So far, this looks like an ordinary stationary, Poisson-like equation +that we can solve right away with the techniques of the first few +tutorial programs (take a look at step-6, for example, for something +very similar). However, we have not said anything yet about the +saturation, which of course is going to change as the fluids move around. - where, $\lambda$ is the total mobility, f is the fractional -flow of water, $q$ is a source term and $\mathbf{u}$ is the total -velocity, which are respectively given by: +The second part of the equations is a therefore description of the +dynamics of the saturation. We model this as an advected quantity: +@f{eqnarray*} + S_{t} + \mathbf{u} \cdot \nabla F(S) = 0. +@f} +where $\mathbf u$ is the total velocity @f[\mathbf{u} = -\mathbf{u}_{o} + \mathbf{u}_{w} = -\lambda(S) \mathbf{K}\cdot\nabla -p@f] -@f[\lambda(S) = \frac{k_{rw}(S)}{\mu_{w}}+\frac{k_{ro}(S)}{\mu_{o}}@f] -@f[F(S) = \frac{k_{rw}(S)/\mu_{w}}{k_{rw}(S)/\mu_{w} + k_{ro}(S)/\mu_{o}}@f] + \mathbf{u}_{o} + \mathbf{u}_{w} = -\lambda(S) \mathbf{K}\cdot\nabla p. +@f] +In addition, +@f[ + F(S) + = + \frac{k_{rw}(S)/\mu_{w}}{k_{rw}(S)/\mu_{w} + k_{ro}(S)/\mu_{o}} +@f] + +In summary, what we get are the following two equations: +@f{eqnarray*} + - \nabla \cdot (\mathbf{K}\lambda(S) \nabla p) &=& q + \qquad \textrm{in}\ \Omega\times[0,T], + \\ + S_{t} + \mathbf{u} \cdot \nabla F(S) &=& 0 + \qquad \textrm{in}\ \Omega\times[0,T]. +@f} +Here, $p=p(\mathbf x, t), S=S(\mathbf x, t)$ are now time dependent +functions: while at every time instant the flow field is in +equilibrium with the pressure (i.e. we neglect dynamic +accellerations), the saturation is transported along with the flow and +therefore changes over time, in turn affected the flow field again +through the dependence of the first equation on $S$. + +This set of equations has a peculiar character: one of the two +equations has a time derivative, the other one doesn't. This +corresponds to the character that the pressure and velocities are +coupled through an instantaneous constraint, whereas the saturation +evolves over finite time scales. + +Such systems of equations are called Differential Algebraic Equations +(DAEs), since one of the equations is a differential equation, the +other is not (at least not with respect to the time variable) and is +therefore an "algebraic" equation. (The notation comes from the field +of ordinary differential equations, where everything that does not +have derivatives with respect to the time variable is necessarily an +algebraic equation.) This class of equations contains pretty +well-known cases: for example, the time dependent Stokes and +Navier-Stokes equations (where the algebraic constraint is that the +divergence of the flow field, $\textrm{div}\ \mathbf u$, must be zero) +as well as the time dependent Maxwell equations (here, the algebraic +constraint is that the divergence of the electric displacement field +equals the charge density, $\textrm{div}\ \mathbf D = \rho$ and that the +divergence of the magnetic flux density is zero: $\textrm{div}\ \mathbf +B = 0$). We will see that the different character of the two equations +will inform our discretization strategy for the two equations.

Discretization

-- 2.39.5