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437 lines
20 KiB
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437 lines
20 KiB
TeX
\chapter{The \eWoms Fluid Framework}
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\label{sec:fluidframework}
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This chapter discusses the \eWoms fluid framework. \eWoms users who
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do not want to write new models and who do not need new fluid
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configurations may skip this chapter.
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In the chapter, a high level overview over the the principle concepts
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is provided first, then some implementation details follow.
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\section{Overview of the Fluid Framework}
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The \eWoms fluid framework currently features the following concepts
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(listed roughly in their order of importance):
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\begin{description}
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\item[Fluid state:] Fluid states are responsible for representing the
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complete thermodynamic configuration of a system at a given spatial
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and temporal position. A fluid state always provides access methods
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to \textbf{all} thermodynamic quantities, but the concept of a fluid state does not
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mandate what assumptions are made to store these thermodynamic
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quantities. What fluid states also do \textbf{not} do is to make sure
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that the thermodynamic state which they represent is physically
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possible.
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\item[Fluid system:] Fluid systems express the thermodynamic \textbf{
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relations}\footnote{Strictly speaking, these relations are
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functions, mathematically.} between quantities. Since functions do
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not exhibit any internal state, fluid systems are stateless classes,
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i.e. all member functions are \texttt{static}. This is a conscious
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decision since the thermodynamic state of the system is expressed by
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a fluid state!
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\item[Parameter cache:] Fluid systems sometimes require
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computationally expensive parameters for multiple relations. Such
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parameters can be cached using a so-called parameter
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cache. Parameter cache objects are specific for each fluid system
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but they must provide a common interface to update the internal
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parameters depending on the quantities which changed since the last
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update.
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\item[Constraint solver:] Constraint solvers are auxiliary tools to
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make sure that a fluid state is consistent with some thermodynamic
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constraints. All constraint solvers specify a well defined set of
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input variables and make sure that the resulting fluid state is
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consistent with a given set of thermodynamic equations. See section
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\ref{sec:constraint_solvers} for a detailed description of the
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constraint solvers which are currently available in \eWoms.
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\item[Equation of state:] Equations of state (EOS) are auxiliary
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classes which provide relations between a fluid phase's temperature,
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pressure, composition and density. Since these classes are only used
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internally in fluid systems, their programming interface is
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currently ad-hoc.
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\item[Component:] Components are fluid systems which provide the
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thermodynamic relations for the liquid and gas phase of a single
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chemical species or a fixed mixture of species. Their main purpose
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is to provide a convenient way to access these quantities from
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full-fledged fluid systems. Components are not supposed to be used
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by models directly.
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\item[Binary coefficient:] Binary coefficients describe the relations
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of a mixture of two components. Typical binary coefficients are
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\textsc{Henry} coefficients or binary molecular diffusion
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coefficients. So far, the programming interface for accessing binary
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coefficients has not been standardized in \eWoms.
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\end{description}
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\section{Fluid States}
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Fluid state objects express the complete thermodynamic state of a
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system at a given spatial and temporal position.
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\subsection{Exported Constants}
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\textbf{All} fluid states \textbf{must} export the following constants:
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\begin{description}
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\item[numPhases:] The number of fluid phases considered.
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\item[numComponents:] The number of considered chemical
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species or pseudo-species.
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\end{description}
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\subsection{Accessible Thermodynamic Quantities}
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Also, \textbf{all} fluid states \textbf{must} provide the following methods:
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\begin{description}
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\item[temperature():] The absolute temperature $T_\alpha$ of
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a fluid phase $\alpha$.
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\item[pressure():] The absolute pressure $p_\alpha$ of a
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fluid phase $\alpha$.
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\item[saturation():] The saturation $S_\alpha$ of a fluid phase
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$\alpha$. The saturation is defined as the pore space occupied by
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the fluid divided by the total pore space:
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\[
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\saturation_\alpha := \frac{\porosity \mathcal{V}_\alpha}{\porosity \mathcal{V}}
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\]
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\item[moleFraction():] Returns the molar fraction $x^\kappa_\alpha$ of
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the component $\kappa$ in fluid phase $\alpha$. The molar fraction
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$x^\kappa_\alpha$ is defined as the ratio of the number of molecules
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of component $\kappa$ and the total number of molecules of the phase
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$\alpha$.
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\item[massFraction():] Returns the mass fraction $X^\kappa_\alpha$ of
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component $\kappa$ in fluid phase $\alpha$. The mass fraction
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$X^\kappa_\alpha$ is defined as the weight of all molecules of a
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component divided by the total mass of the fluid phase. It is
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related with the component's mole fraction by means of the relation
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\[
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X^\kappa_\alpha = x^\kappa_\alpha \frac{M^\kappa}{\overline M_\alpha}\;,
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\]
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where $M^\kappa$ is the molar mass of component $\kappa$ and
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$\overline M_\alpha$ is the mean molar mass of a molecule of phase
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$\alpha$.
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\item[averageMolarMass():] Returns $\overline M_\alpha$, the mean
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molar mass of a molecule of phase $\alpha$. For a mixture of $N > 0$
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components, $\overline M_\alpha$ is defined as
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\[
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\overline M_\alpha = \sum_{\kappa=1}^{N} x^\kappa_\alpha M^\kappa
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\]
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\item[density():] Returns the density $\rho_\alpha$ of the fluid phase
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$\alpha$.
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\item[molarDensity():] Returns the molar density $\rho_{mol,\alpha}$
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of a fluid phase $\alpha$. The molar density is defined by the mass
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density $\rho_\alpha$ and the mean molar mass $\overline M_\alpha$:
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\[
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\rho_{mol,\alpha} = \frac{\rho_\alpha}{\overline M_\alpha} \;.
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\]
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\item[molarVolume():] Returns the molar volume $v_{mol,\alpha}$ of a
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fluid phase $\alpha$. This quantity is the inverse of the molar
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density.
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\item[molarity():] Returns the molar concentration $c^\kappa_\alpha$
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of component $\kappa$ in fluid phase $\alpha$.
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\item[fugacity():] Returns the fugacity $f^\kappa_\alpha$ of component
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$\kappa$ in fluid phase $\alpha$. The fugacity is defined as
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\[
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f_\alpha^\kappa := \Phi^\kappa_\alpha x^\kappa_\alpha p_\alpha \;,
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\]
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where $\Phi^\kappa_\alpha$ is the {\em fugacity
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coefficient}~\cite{reid1987}. The physical meaning of fugacity
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becomes clear from the equation
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\[
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f_\alpha^\kappa = f_\alpha^{\kappa,0} \exp\left\{\frac{\zeta^\kappa_\alpha - \zeta^{\kappa,0}_\alpha}{R T_\alpha} \right\} \;,
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\]
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where $\zeta^\kappa_\alpha$ represents the $\kappa$'s chemical
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potential in phase $\alpha$, $R$ stands for the ideal gas constant,
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$\zeta^{\kappa,0}_\alpha$ is the chemical potential in a reference
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state, $f_\alpha^{\kappa,0}$ is the fugacity in the reference state
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and $T_\alpha$ for the absolute temperature of phase $\alpha$. The
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fugacity in the reference state $f_\alpha^{\kappa,0}$ is in princle
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arbitrary, but in the context of the \eWoms fluid framework, we
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assume that it is the same for all fluid phases, i.e.
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$f_\alpha^{\kappa,0} = f_\beta^{\kappa,0}$.
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Assuming thermal equilibrium, there is a one-to-one mapping between
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a component's chemical potential $\zeta^\kappa_\alpha$ and its
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fugacity $f^\kappa_\alpha$. With the above assumptions, chemical
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equilibrium can thus be expressed by
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\[
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f^\kappa := f^\kappa_\alpha = f^\kappa_\beta\quad\forall \alpha, \beta
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\]
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\item[fugacityCoefficient():] Returns the fugacity coefficient
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$\Phi^\kappa_\alpha$ of component $\kappa$ in fluid phase $\alpha$.
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\item[enthalpy():] Returns specific enthalpy $h_\alpha$ of a fluid
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phase $\alpha$.
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\item[internalEnergy():] Returns specific internal energy $u_\alpha$
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of a fluid phase $\alpha$. The specific internal energy is defined
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by the relation
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\[
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u_\alpha = h_\alpha - \frac{p_\alpha}{\rho_\alpha}
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\]
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\item[viscosity():] Returns the dynamic viscosity
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$\mu_\alpha$ of fluid phase $\alpha$.
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\end{description}
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\subsection{Available Fluid States}
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Currently, the following fluid states are provided by \eWoms:
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\begin{description}
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\item[NonEquilibriumFluidState:] This is the most general fluid state
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supplied. It does not assume thermodynamic equilibrium and thus
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stores all phase compositions (using mole fractions), fugacity
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coefficients, phase temperatures, phase pressures, saturations and
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specific enthalpies.
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\item[CompositionalFluidState:] This fluid state is very similar to
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the \texttt{Non\-Equilibrium\-Fluid\-State} with the difference that
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the \texttt{Compositional\-Fluid\-State} assumes thermodynamic
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equilibrium. In the context of multi-phase flow in porous media,
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this means that only a single temperature needs to be stored.
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\item[ImmisicibleFluidState:] This fluid state assumes that the fluid
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phases are immiscible, which implies that the phase compositions and
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the fugacity coefficients do not need to be stored explicitly.
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\item[PressureOverlayFluidState:] This is a so-called {\em overlay}
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fluid state. It allows to set the pressure of all fluid phases but
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forwards everything else to another fluid state.
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\item[SaturationOverlayFluidState:] This fluid state is like the
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\texttt{PressureOverlayFluidState}, except that the phase
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saturations are settable instead of the phase pressures.
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\item[TempeatureOverlayFluidState:] This fluid state is like the
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\texttt{PressureOverlayFluidState}, except that the temperature is
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settable instead of the phase pressures. Note that this overlay
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state assumes thermal equilibrium regardless of underlying fluid
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state.
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\item[CompositionOverlayFluidState:] This fluid state is like the
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\texttt{PressureOverlayFluidState}, except that the phase
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composition is settable (in terms of mole fractions) instead of the
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phase pressures.
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\end{description}
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\section{Fluid Systems}
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Fluid systems express the thermodynamic relations between the
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quantities of a fluid state.
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\subsection{Parameter Caches}
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All fluid systems must export a type for their \texttt{ParameterCache}
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objects. Parameter caches can be used to cache parameter that are
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expensive to compute and are required in multiple thermodynamic
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relations. For fluid systems which do need to cache parameters,
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\eWoms provides a \texttt{NullParameterCache} class.
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The actual quantities stored by parameter cache objects are specific
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to the fluid system and no assumptions on what they provide should be
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made outside of their fluid system. Parameter cache objects provide a
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well-defined set of methods to make them coherent with a given fluid
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state, though. These update are:
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\begin{description}
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\item[updateAll(fluidState, except):] Update all cached quantities for
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all phases. The \texttt{except} argument contains a bit field of the
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quantities which have not been modified since the last call to a
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\texttt{update()} method.
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\item[updateAllPresures(fluidState):] Update all cached quantities
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which depend on the pressure of any fluid phase.
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\item[updateAllTemperatures(fluidState):] Update all cached quantities
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which depend on temperature of any fluid phase.
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\item[updatePhase(fluidState, phaseIdx, except):] Update all cached
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quantities for a given phase. The quantities specified by the
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\texttt{except} bit field have not been modified since the last call
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to an \texttt{update()} method.
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\item[updateTemperature(fluidState, phaseIdx):] Update all cached
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quantities which depend on the temperature of a given phase.
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\item[updatePressure(fluidState, phaseIdx):] Update all cached
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quantities which depend on the pressure of a given phase.
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\item[updateComposition(fluidState, phaseIdx):] Update all cached
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quantities which depend on the composition of a given phase.
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\item[updateSingleMoleFraction(fluidState, phaseIdx, compIdx):] Update
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all cached quantities which depend on the value of the mole fraction
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of a component in a phase.
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\end{description}
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Note, that the parameter cache interface only guarantees that if a
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more specialized \texttt{update()} method is called, it is not slower
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than the next more-general method (e.g. calling
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\texttt{updateSingleMoleFraction()} may be as expensive as
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\texttt{updateAll()}). It is thus advisable to rather use a more
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general \texttt{update()} method once than multiple calls to
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specialized \texttt{update()} methods.
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To make usage of parameter caches easier for the case where all cached
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quantities ought to be re-calculated if a quantity of a phase was
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changed, it is possible to only define the \texttt{updatePhase()}
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method and derive the parameter cache from
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\texttt{Ewoms::ParameterCacheBase}.
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\subsection{Exported Constants and Capabilities}
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Besides providing the type of their \texttt{ParameterCache} objects,
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fluid systems need to export the following constants and auxiliary
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methods:
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\begin{description}
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\item[numPhases:] The number of considered fluid phases.
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\item[numComponents:] The number of considered chemical (pseudo-)
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species.
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\item[init():] Initialize the fluid system. This is usually used to
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tabulate some quantities
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\item[phaseName():] Given the index of a fluid phase, return its name
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as human-readable string.
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\item[componentName():] Given the index of a component, return its
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name as human-readable string.
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\item[isLiquid():] Return whether the phase is a liquid, given the
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index of a phase.
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\item[isIdealMixture():] Return whether the phase is an ideal mixture,
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given the phase index. In the context of the \eWoms fluid
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framework a phase $\alpha$ is an ideal mixture if, and only if, all
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its fugacity coefficients $\Phi^\kappa_\alpha$ do not depend on the
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phase composition. (Although they might very well depend on
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temperature and pressure.)
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\item[isIdealGas():] Return whether a phase $\alpha$ is an ideal gas,
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i.e. it adheres to the relation
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\[
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p_\alpha v_{mol,\alpha} = R T_\alpha \;,
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\]
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with $R$ being the ideal gas constant.
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\item[isCompressible():] Return whether a phase $\alpha$ is
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compressible, i.e. its density depends on pressure $p_\alpha$.
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\item[molarMass():] Given a component index, return the molar mass of
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the corresponding component.
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\end{description}
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\subsection{Thermodynamic Relations}
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Fluid systems have been explicitly designed to provide as few
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thermodynamic relations as possible. A full-fledged fluid system thus
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only needs to provide the following thermodynamic relations:
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\begin{description}
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\item[density():] Given a fluid state, an up-to-date parameter cache
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and a phase index, return the mass density $\rho_\alpha$ of the
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phase.
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\item[fugacityCoefficient():] Given a fluid state, an up-to-date
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parameter cache as well as a phase and a component index, return the
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fugacity coefficient $\Phi^\kappa_\alpha$ of a the component for the
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phase.
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\item[viscosity():] Given a fluid state, an up-to-date parameter cache
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and a phase index, return the dynamic viscosity $\mu_\alpha$ of the
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phase.
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\item[diffusionCoefficient():] Given a fluid state, an up-to-date
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parameter cache, a phase and a component index, return the calculate
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the molecular diffusion coefficient for the component in the fluid
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phase.
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Molecular diffusion of a component $\kappa$ in phase $\alpha$ is
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caused by a gradient of the chemical potential. Using some
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simplifying assumptions~\cite{reid1987}, they can be also expressed
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in terms of mole fraction gradients, i.e. the equation used for mass
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fluxes due to molecular diffusion is
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\[
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J^\kappa_\alpha = - \rho_{mol,\alpha} D^\kappa_\alpha\ \mathbf{grad} x^\kappa_\alpha\;,
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\]
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where $\rho_{mol,\alpha}$ is the molar density of phase $\alpha$,
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$x^\kappa_\alpha$ is the mole fraction of component $\kappa$ in
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phase $\alpha$, $D^\kappa_\alpha$ is the diffusion coefficient and
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$J^\kappa_\alpha$ is the diffusive flux.
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\item[enthalpy():] Given a fluid state, an up-to-date parameter cache
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and a phase index, this method calulates the specific enthalpy
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$h_\alpha$ of the phase.
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\item[thermalConductivity:] Given a fluid state, an up-to-date
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parameter cache and a phase index, this method returns the thermal
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conductivity $\lambda_\alpha$ of the fluid phase. The thermal
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conductivity is defined by means of the relation
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\[
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\dot Q = \lambda_\alpha \mathbf{grad}\;T_\alpha \;,
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\]
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where $\dot Q$ is the heat flux caused by the temperature gradient
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$\mathbf{grad}\;T_\alpha$.
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\item[heatCapacity():] Given a fluid state, an up-to-date parameter
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cache and a phase index, this method computes the isobaric heat
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capacity $c_{p,\alpha}$ of the fluid phase. The isobaric heat
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capacity is defined as the partial derivative of the specific
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enthalpy $h_\alpha$ to the fluid pressure:
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\[
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c_{p,\alpha} = \frac{\partial h_\alpha}{\partial p_\alpha}
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\]
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% TODO: remove the heatCapacity() method??
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\end{description}
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Fluid systems may chose not to implement some of these methods and
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throw an exception of type \lstinline{Dune::NotImplemented} instead. Obviously,
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such fluid systems cannot be used for models that depend on those
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methods.
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\section{Constraint Solvers}
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\label{sec:constraint_solvers}
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Constraint solvers connect the thermodynamic relations expressed by
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fluid systems with the thermodynamic quantities stored by fluid
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states. Using them is not mandatory for models, but given the fact
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that some thermodynamic constraints can be quite complex to solve,
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sharing this code between models makes sense. Currently, \eWoms
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provides the following constraint solvers:
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\begin{description}
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\item[CompositionFromFugacities:] This constraint solver takes all
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component fugacities, the temperature and pressure of a phase as
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input and calculates the composition of the fluid phase. This means
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that the thermodynamic constraints used by this solver are
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\[
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f^\kappa = \Phi^\kappa_\alpha(\{x^\beta_\alpha \}, T_\alpha, p_\alpha) p_\alpha x^\kappa_\alpha\;,
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\]
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where ${f^\kappa}$, $T_\alpha$ and $p_\alpha$ are fixed values.
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\item[ComputeFromReferencePhase:] This solver brings all
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fluid phases into thermodynamic equilibrium with a reference phase
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$\beta$, assuming that all phase temperatures and saturations have
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already been set. The constraints used by this solver are thus
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\begin{eqnarray*}
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f^\kappa_\beta = f^\kappa_\alpha = \Phi^\kappa_\alpha(\{x^\beta_\alpha \}, T_\alpha, p_\alpha) p_\alpha x^\kappa_\alpha\;, \\
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p_\alpha = p_\beta + p_{c\beta\alpha} \;,
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\end{eqnarray*}
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where $p_{c\beta\alpha}$ is the capillary pressure between the
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fluid phases $\beta$ and $\alpha$.
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\item[NcpFlash:] This is a so-called flash solver. A flash solver
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takes the total mass of all components per volume unit and the phase
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temperatures as input and calculates all phase pressures,
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saturations and compositions. This flash solver works for an
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arbitrary number of phases $M > 0$ and components $N \geq M - 1$. In
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this case, the unknown quantities are the following:
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\begin{itemize}
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\item $M$ pressures $p_\alpha$
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\item $M$ saturations $\saturation_\alpha$
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\item $M\cdot N$ mole fractions $x^\kappa_\alpha$
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\end{itemize}
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This sums up to $M\cdot(N + 2)$. The equations side of things
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provides:
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\begin{itemize}
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\item $(M - 1)\cdot N$ equations stemming from the fact that the
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fugacity of any component is the same in all phases, i.e.
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\[
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f^\kappa_\alpha = f^\kappa_\beta
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\]
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holds for all phases $\alpha, \beta$ and all components $\kappa$.
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\item $1$ equation comes from the fact that the whole pore space is
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filled by some fluid, i.e.
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\[
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\sum_{\alpha=1}^M \saturation_\alpha = 1
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\]
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\item $M - 1$ constraints are given by the capillary pressures:
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\[
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p_\beta = p_\alpha + p_{c\beta\alpha} \;,
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\]
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for all phases $\alpha$, $\beta$
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\item $N$ constraints come the fact that the total mass of each
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component is given:
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\[
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c^\kappa_{tot} = \sum_{\alpha=1}^M x_\alpha^\kappa\;\rho_{mol,\alpha} = const
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\]
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\item And finally $M$ model assumptions are used. This solver uses
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the NCP constraints proposed in~\cite{LHHW2011}:
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\[
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0 = \mathrm{min}\{\saturation_\alpha, 1 - \sum_{\kappa=1}^N x_\alpha^\kappa\}
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\]
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\end{itemize}
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The number of equations also sums up to $M\cdot(N + 2)$. Thus, the
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system of equations is closed.
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\item[ImmiscibleFlash:] This is a flash solver assuming immiscibility
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of the phases. It is similar to the \texttt{NcpFlash} solver but a
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lot simpler.
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\item[MiscibleMultiphaseComposition:] This solver calculates the
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composition of all phases provided that each of the phases is
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potentially present. Currently, this solver does not support
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non-ideal mixtures.
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\end{description}
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