Volume Averaging

Porous Media 17

Figure 2.6: An example of a porous medium where two fluid phases are co-located with a solid.

for which phaseαis present,i.e.,

Ωα :={x∈Ω|Positionxis occupied by the phaseα} .

• We define thecharacteristic functionχαof a phaseαas the function that is one for the part of the domain where the fluid phaseαis present, and zero elsewhere,i.e.,

χα(x) :=

1 ifx∈Ω_α 0 else.

• Theaveraging volumeVfor any pointx∈Ωis the sphere of points which are closer tox than an arbitrary but fixed radiusr,i.e.,

V(x) :={y∈Ω| kx−yk< r} .

• Theintrinsic averaging volumeV_αof a phaseαfor any pointx∈Ωis the set of points which are inV(x)and in the characteristic set ofα,i.e.,

V_α(x) :=V(x)∩Ω_α.

• Theinterior boundary∂V_sαof the averaging volumeV_αof a fluid phaseαis the set of points that is on the boundary ofV_αbut not on the boundary ofV,i.e.,

∂V_sα :=∂V_α\∂V , where the lower indexh·i_sindicates the solid phase.

• Theporosityφof the porous medium is defined as φ:= 1−kV_sk

kVk ,

and represents volume fraction of the pores,i.e., the fraction of space which can be occupied by fluids.

• ThesaturationS_α of a fluid phaseαis the fraction of the pore space occupied by the fluid phase,i.e.,

S_α:= kV_αk φkVk .

• Thephase averagehb_αiof a quantityb_αwhich is associated with the phaseαis defined as hb_αi(x) := 1

kV(x)k Z

Vα(x)

bα(y) dy. (2.24)

Porous Media 19

Note, that we can consider Equation 2.24 as a convolution ofb·χαwith the kernel m(y) =

( ₁

kVk ifkyk< r 0 else.

This means that we can think of the procedures discussed in this section as analogous to the upscaling procedure from molecular to continuum-scale which we sketched in Section 2.1.

• Theintrinsic phase averagehb_αi^αof the quantitybαis the average ofbαconsidering only the points where the phaseαis present,i.e.,

hb_αi^α(x) := 1 kV_α(x)k

Z

Vα(x)

bα(y) dy. (2.25)

The phase average and the intrinsic phase average are thus connected by the relation hb_αi^α= kVk

kV_αkhb_αi= hb_αi φS_α . The Spatial Averaging Theorem

Using these tools, we can now average the micro-scale conservation Equations 2.7, 2.21, and 2.23. One problem which occurs during this procedure is that we will get averages of gradients in the results but that we would rather like to express the equations in terms of gradients of averages.

We can overcome this issue using theSpatial Averaging Theorem[89, 67]

hgradb_αi=gradhb_αi+ 1 kVk

Z

∂Vsα

b_α ndy, (2.26)

wherenis the outer unit normal vector ofV_αat positiony.

We will also get an average of the divergence instead of the divergence of averaged quantities.

In this case, we can substitute the gradients in Equation 2.26 by divergences [89, 67], and get hdivb_αi= divhb_αi+ 1

kVk Z

∂V_sα

b_α·ndy. (2.27)

Conservation of Mass

Now, let us have a closer look at the mass conservation Equation 2.7. Applying Equations 2.26 and 2.24 to it, we get

∂ρα

∂t + div(ραvα)

= ∂hρ_αi

∂t + div(hρ_αvαi) + 1

kVk Z

∂Vsα

n·ραvαdy

=hq_mass,αi.

If we now assume a no-slip condition for the internal boundaries of the averaging volume, i.e.,v_α = 0on∂V_sα, we get

∂hρ_αi

∂t + div(hρ_αvαi) =hq_mass,αi as the macro-scale equation for the conservation of mass.

Still, the term inside the divergence is the average of density times velocity, but we would like to formulate the mass conservation equation in terms of the product of the two averages.

To achieve this, we use the GRAYdecomposition [35] which expresses any quantityb_αas the sum of its intrinsic phase averagehb_αi^α, and of its deviation˜bα:

b_α =hb_αi^α+ ˜b_α.

We now observe that the average of the deviation term˜bαis zero, and we can thus write the mass conservation equation as

∂φS_αhρ_αi^α

∂t + div(hρ_αi^αhv_αi) =hq_mass,αi.

If we repeat this procedure for Equation 2.10, the compositional form of mass conservation, we get

∂φSαhx^κ_αi^αhρ_mol,αi^α

∂t + div hx^κ_αi^αhρ_mol,αi^αhv_αi −D_pm,α^κ gradhx^κ_αi^α

=hq_α^κi

for the conservation of each component in fluid phaseα. In this case, we need to adapt the molecular diffusion coefficientsD_α^κto account for the presence of the solid phase.

If there is more than a single component, we have to preserve each in the whole porous medium. This implies that we need to sum up the component conservation equations over all phases in order to get an equation that describes the conservation of the component for

Porous Media 21

the whole multi-phase system. This leads to X

α

∂φS_αhx^κ_αi^αhρ_mol,αi^α

∂t +X

α

div hx^κ_αi^αhρ_mol,αi^αhv_αi −D^κ_pm,αgradhx^κ_αi^α

=X

α

hq^κ_αi. (2.28)

Conservation of Momentum

The procedure for conservation of momentum is much more elaborate than the one for the conservation of mass. Thus, we will only sketch it here and refer the interested reader to WHITAKER[89] for further details.

First, we neglect compressibility and the inertia terms of the NAVIER-STOKESequations, and start with the STOKESEquations 2.21. Averaging those leads us to

0 =−hgradp_αi+hµ_αdivgradv_αi+hρ_αgi+

q_mom,α .

We now use the GRAY decomposition for the viscosity µα and the mass densityρα, and assume a sufficiently small averaging volume, so that we may neglect the deviation terms.

Further assuming that the gravitational acceleration g is constant within the averaging volume, and that the momentum source termq_mom,αis zero, we get

0 =−hgradp_αi+hµ_αi^αhdivgradv_αi+hρ_αi^αg. (2.29) Taking advantage of the Spatial Averaging Theorem 2.26, we may write the pressure term as

hgradpαi=gradhp_αi+ 1 kVk

Z

∂Vsα

pαndy.

Now, we apply the GRAYdecomposition onpα, and rewritepα as the sum of its intrinsic phase average and the deviation, and get

hgradpαi=hp_αi^αgrad(φSα) +φSαgradhp_αi^α + 1

kVk Z

∂Vsα

nhp_αi^αdy+ 1 kVk

Z

∂Vsα

np˜αdy (2.30) after applying the product rule to the first term. Sincehp_αi^αis constant within an averaging volume,

1 kVk

Z

∂Vsα

nhp_αi^αdy= hp_αi^α kVk

Z

∂Vsα

ndy

holds. Now we use the spatial averaging theorem withbα≡1to obtain 1

kVk Z

∂V_sα

ndy=−grad(φS_α) . (2.31)

Taking advantage of Equations 2.31 and 2.30 in Equation 2.29, and using the same procedure to the phase velocities, we get

0 =−φS_αgradhp_αi^α− 1 kVk

Z

∂Vsα

np˜_αdy+hµ_αi^α kVk

Z

∂Vsα

n·gradv˜_αdy+hρ_αi^αg. (2.32) The problem we now face is that we need to definep˜_αandv˜_α. WHITAKER[89] argues that

˜

vα=Bαhv_αi^α and (2.33)

˜

pα=bα· hµ_αi^αhv_αi^α (2.34) hold, whereBαandbαare the solutions of the following boundary value problem:

−gradb_α+ divgradB_α = 1 kV_αk

Z

V_α

−gradb_α+ divgradB_αdV

Bα =−I onV_α

Bα =Gα on∂V_α\∂V_s hB_αi^α =hb_αi^α = 0.

Inserting Equations 2.33 and 2.34 into Equation 2.32, we get 0 =−gradhp_αi^α+

hµ_αi^α kVk

Z

∂Vsα

n·gradBα−n⊗bαdy

hv_αi^α+hρ_αi^αg. (2.35) Using the abbreviation

C_α =− 1 kVk

Z

∂V_sα

n·gradB−n⊗b_αdy

we get a relation for the intrinsic phase velocity of fluid phaseα:

hv_αi^α=− C⁻¹_α

hµ_αi^α (gradhp_αi^α− hρ_αi^αg) . (2.36) We can convert Equation 2.36 from the intrinsic phase velocity to the more common phase velocity by introducing

K_α=φS_αC⁻¹_α which yields

hv_αi=− K_α

hµ_αi^α(gradhp_αi^α− hρ_αi^αg) . (2.37) Conservation of Energy

To obtain an equation for the conservation of energy in a fluid phase, we neglect potential and kinetic energy, and start with Equation 2.23. After applying the phase average (2.24) and

Porous Media 23

GRAY’s decomposition, we get

∂ φS_αhu_αi^α hρ_αi^α

∂t + div(hh_αi^αhρ_αi^α hv_αi − hλ_αi^αgradhT_αi) =hq_energyi.

Unlike mass and momentum, energy can be transported by the solid. Since we assumed a rigid solid phase, the advective part of the divergence term is zero for the solid, and we get

∂(1−φ)hu_si^s hρ_si^s

∂t −div(hλ_si^sgradhT_si) =hq_energyi.

In this context we usually assume the internal energyusof the solid to be proportional to temperature,i.e.,

us=cp,sTs

holds with a constantheat capacityc_p,s. 2.4.2 Applicability

To derive the macro-scale conservation equations, and especially the momentum conservation equation (2.37), we had to make several assumptions:

• These equations are only valid for creeping flows, and we required an incompressible fluid with constant dynamic viscosity when simplifying the NAVIER-STOKES equa-tions (2.20) to the STOKESequations (2.21). At this point, we note that we do not need to consider these simplifications in practice, since in order to derive Equation 2.37, we only used the STOKESequations within the averaging volumes where compressibility and variations in viscosity can usually be neglected.

• The averaging volume must be large enough for the continuum assumption to be appli-cable for the macro-scale system. This means that slightly increasing or decreasing the size of the averaging volume only leads to small changes of the averaged quantities [89].

• The averaging volume must be small enough so that we may neglect the deviation of all physical quantities from their average within an averaging volume. This assumption is potentially problematic for systems featuring low absolute pressures in conjunction with highly compressible fluids.

Of these assumptions, we usually have to pay most attention to the constraint of the flow velocity. To get a feeling about this, we first need to specify a REYNOLDSnumber,i.e., we define a characteristic lengthLcand a characteristic kinematic viscosityνc=^µc/ρc. In this case, we choose the diameter of a typical pore of the porous medium as the characteristic length;

for the characteristic kinematic viscosityν_cwe choose the value of the fluid at a representative temperature and pressure. Allowing REYNOLDSnumbers smaller than1, Equation 2.37 is valid up to relatively high velocities³: For example, if the typical pore diameter of a porous medium is the one of a typical sand with100·10⁻⁶ m, and the characteristic kinematic

3At least “high” velocities in the context of fluid flows in porous media.

viscosity is20·10^{−5 m2}/s(roughly the value of air at20^◦C), the velocity constraints implied by Equation 2.37 are met for intrinsic phase velocities smaller than about2^m/s.

Simplification of Notation

In order to simplify notation, we will not write the averaging operator henceforth, so we will use the conservation equations

∂ραφSα

∂t + div(ραvα) =qmass,α for mass, (2.38) v_α =−K_α

µα

(gradp_α−ρ_αg) for momentum, (2.39)

∂u_αρ_αφS_α

∂t + div(h_αρ_αv_α−λ_αgradT_α) =q_energy,α for energy within a fluid, and (2.40)

∂(1−φ)ρ_sc_p,sT_s

∂t −div(λ_sgradT_s) =q_energy,s for energy within the solid. (2.41) Also, since the momentum conservation equation (2.39) determines the phase velocities explicitly, we can also insert it into the mass and energy conservation equations. In this case momentum is conserved implicitly, and we get

∂ραφSα

∂t −div

ρ_αKα

µ_α (gradp_α−ρ_αg)

=q_mass,α (2.42)

for mass conservation as well as

∂ραuαφSα

∂t −div

h_αρ_αKα

µ_α (gradp_α−ρ_αg) +λ_αgradT_α

=q_energy,α (2.43) for energy conservation within a fluid.

Im Dokument Theory and numerical applications of compositional multi-phase flow in porous media (Seite 47-54)