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VI.C Macroscopic Description

VI.C.1 Macroscopic Equations of Motion

The accepted large scale equations of motion for two phase flow involve a generalization of Darcy’s law to relative permeabilities including offdiagonal viscous coupling terms [349, 270, 322, 321, 350, 351, 352]. The importance of viscous coupling terms has been recognized relatively late [353, 354, 355, 356, 357]. The equations which are generally believed to describe multiphase flow on the reservoir scale as well as on the laboratory scale may be written as [322, 350]

ϕ¯S¯𝕎t¯=¯v¯𝕎ϕ¯S¯𝕆t¯=¯v¯𝕆 (6.38)
v¯𝕎=-K𝕎𝕎rKμ𝕎¯P¯𝕎-ρ𝕎g¯z¯+K𝕎𝕆rKμ𝕆¯P¯𝕆-ρ𝕆g¯z¯v¯𝕆=-K𝕆𝕎rKμ𝕎¯P¯𝕎-ρ𝕎g¯z¯+K𝕆𝕆rKμ𝕆¯P¯𝕆-ρ𝕆g¯z¯ (6.39)
S¯𝕎+S¯𝕆=1 (6.40)
P¯𝕆-P¯𝕎=P¯cS¯𝕎 (6.41)

where A¯ denotes the macroscopic volume averaged equivalent of the pore scale quantity A. In the equations above K stands for the absolute (single phase flow) permeability tensor, K𝕎𝕎r is the relative permeability tensor for water, K𝕆𝕆r the oil relative permeability tensor, and K𝕎𝕆r,K𝕆𝕎r denote the possibly anisotropic coupling terms. The relative permeabilities are matrix valued functions of saturation. The saturations are denoted as S¯𝕎,S¯𝕆 and they depend on the macroscopic space and time variables x¯,t¯. The capillary pressure curve P¯cS¯𝕎 and the relative permeability tensors KijrS¯𝕎,i,j=𝕎,𝕆 must be known either from solving the pore scale equations of motion, or from experiment. KijrS¯𝕎 and P¯cS¯𝕎 are conventionally assumed to be independent of v¯ and P¯ and this convention is followed here, although it is conceivable that this is not generally correct [354].

Eliminating v¯ and choosing P¯𝕎x¯,t¯ and S¯𝕎x¯,t¯ as the principal unknowns one arrives at the large scale two-phase flow equations

ϕ¯S¯𝕎t¯=¯{K𝕎𝕎r(S¯𝕎)Kμ𝕎(¯P¯𝕎-ρ𝕎g¯z¯) (6.42)
ϕ¯1-S¯𝕎t¯=¯{K𝕆𝕎r(S¯𝕎)Kμ𝕎(¯P¯𝕎-ρ𝕎g¯z¯) (6.43)

for these two unknowns. Equations (6.42) and (6.43) are coupled nonlinear partial differential equations for the large scale pressure and saturation field of the water phase.

These equations must be complemented with large scale boundary conditions. For core experiments these are typically given by a surface source on one side of the core, a surface sink on the opposite face, and impermeable walls on the other faces. For a reservoir the boundary conditions depend upon the drive configuration and the geological modeling of the reservoir environment, so that Dirichlet as well as von Neumann problems arise in practice [339, 351, 352].

VI.C.2 Macroscopic Dimensional Analysis

The large scale equations of motion can be cast in dimensionless form using the definitions

x¯=l¯x¯^ (6.44)
¯=¯^l¯ (6.45)
v¯=u¯v¯^ (6.46)
t¯=l¯t¯^u¯ (6.47)
P¯=P¯bP¯^ (6.48)

where as before A¯^ denotes the dimensionless equivalent of the macroscopic quantity A¯. The length l¯ is now a macroscopic length, and u¯ a macroscopic (seepage or Darcy) velocity. The pressure P¯b denotes the “breakthrough” pressure from the capillary pressure curve P¯cS¯𝕎. It is defined as

P¯b=P¯cS¯b (6.49)

where S¯b is the breakthrough saturation defined as the solution of the equation

d2P¯cS¯𝕎dS¯𝕎2=0. (6.50)

Thus the dimensionless pressure is defined in terms of the inflection point P¯b,S¯b on the capillary pressure curve, and it gives a measure of the macroscopic capillary pressure. Note that P¯b is process dependent, i.e. it will in general differ between imbibition and drainage. This dependence reflects the influence of microscopic wetting properties [348] and flow mechanisms on the macroscale [358].

The definition (6.48) differs from the traditional analysis [49, 329, 330, 331]. In the traditional analysis the normalized pressure field is defined as

P¯=μ𝕎u¯l¯kP¯^ (6.51)

which immediately gives rise to three problems. Firstly the permeablity is a tensor, and thus a certain nonuniqueness results in anisotropic situations [339]. Secondly equation (6.51) neglects the importance of microscopic wetting and saturation history dependence. The main problem however is that Eq. (6.51) is not based on macroscopic capillary pressures but on Darcy’s law which describes macroscopic viscous pressure effects. On the other hand the normalization (6.48) is free from these problems and it includes macroscopic capillarity in the same way as the microscopic normalization (6.15) includes microscopic capillarity.

With the normalizations introduced above the dimensionless form of the macroscopic two-phase flow equations (6.42),(6.43) becomes

ϕ¯S¯𝕎t¯^=¯^{K𝕎𝕎r(S¯𝕎)(Ca¯𝕎-1¯^P¯^𝕎-Gr¯𝕎-1¯^z¯^) (6.52)
ϕ¯1-S¯𝕎t¯^=¯^{K𝕆𝕎r(S¯𝕎)(Ca¯𝕎-1¯^P¯^𝕎-Gr¯𝕎-1¯^z¯^) (6.53)

In these equations the dimensionless tensor

Ca¯𝕎=μ𝕎u¯l¯P¯bK-1=macroscopic viscous pressure dropmacroscopic capillary pressure (6.54)

plays the role of a macroscopic or large scale capillary number. Similarly the tensor

Gr¯𝕎=μ𝕎u¯ρ𝕎gK-1=macroscopic viscous pressure dropmacroscopic gravitational pressure (6.55)

corresponds to the macroscopic gravity number.

If the traditional normalization (6.51) is used instead of the normalization (6.48), and isotropy is assumed then the same dimensionless equations are obtained with

Ca¯𝕎=1 (6.56)

where Ca¯𝕎 is the macroscopic capillary number. Thus the traditional normalization is equivalent to the assumption that the macroscopic viscous pressure drop always equals the macroscopic capillary pressure. While this assumption is not generally valid, it sometimes is a reasonable approximation as will be illustrated below. First, however, the consequences of the traditional assumption (6.56) for the measurement of relative permeabilities will be discussed.

VI.C.3 Measurement of Relative Permeabilities

For simplicity only the isotropic case will be considered from now on, i.e. let K=k 1 where 1 is the identity matrix. The tensors Ca¯𝕎 and Gr¯𝕎 then become Ca¯𝕎=Ca¯𝕎 1 and Gr¯𝕎=Gr¯𝕎 1 where Ca¯𝕎 and Gr¯𝕎 are the macroscopic capillary and gravity numbers.

The unsteady state or displacement method of measuring relative permeabilities consists of monitoring the production history and pressure drop across the sample during a laboratory displacement process [2, 359, 337]. The relative permeability is obtained as the solution of an inverse problem. The inverse problem consists in matching the measured production history and pressure drop to the solutions of the multiphase flow equations (6.52) and (6.53) using the Buckley-Leverett approximation.

In the present formulation the Buckley-Leverett approximation comprises several independent assumptions. Firstly it is assumed that gravity effects are absent, which amounts to the assumption

Ca¯𝕎Gr¯𝕎. (6.57)

Secondly the viscous coupling terms are neglected, i.e.

k𝕎𝕆rμ𝕎μ𝕆Ca¯𝕎andk𝕆𝕎rCa¯𝕎. (6.58)

Finally the resulting equations

ϕ¯S¯𝕎t¯^=¯^k𝕎𝕎rS¯𝕎¯^P¯^𝕎Ca¯𝕎 (6.59)
ϕ¯1-S¯𝕎t¯^=¯^k𝕆𝕆rS¯𝕎μ𝕎μ𝕆¯^P¯^𝕎Ca¯𝕎+¯^P¯^cS¯𝕎Ca¯𝕎. (6.60)

are further simplified by assuming that the term involving P¯^cS¯𝕎 in equation (6.60) may be neglected [332].

Combining (6.57) with the traditional normalization (6.56) yields the consistency condition

Gr¯𝕎1 (6.61)

for the application of Buckley-Leverett theory in the determination of relative permeabilities. It is now clear from the definition of the macroscopic gravity number, see Eq. (6.55), that the consistent use of Buckley-Leverett theory for the unsteady state measurement of relative permeabilities depends strongly on the flow regime. This is valid whether or not the capillary pressure term P¯^cS¯𝕎 in (6.60) is neglected. In addition to these consistency problems the Buckley-Leverett theory is also plagued with stability problems [360].

VI.C.4 Pore Scale to Large Scale Comparison

The comparison between the macroscopic and the microscopic dimensional analysis is carried out by relating the microscopic and macroscopic velocities and length scales. The macroscopic velocity is taken to be a Darcy velocity defined as (see discussion following equation (5.79))

u¯=ϕ¯u (6.62)

where ϕ¯ is the bulk porosity and u denotes the average microscopic flow velocity introduced in the microscopic analysis (Eq. (6.12)). The length scales l and l¯ are identical (l¯=l).

Using these relations between microscopic and macroscopic length and time scales together with the assumption of isotropy yields

Ca¯𝕎=μ𝕎ϕ¯ulkP¯b=ulν*¯𝕎=σ𝕆𝕎ϕ¯lkP¯bCa𝕎 (6.63)

as the relationship between microscopic and macroscopic capillary numbers. Similarly one obtains

Gr¯𝕎=μ𝕎ϕ¯uρ𝕎gk=uu*¯𝕎=ϕ¯l2kGr𝕎 (6.64)

for the gravity numbers. Taking the quotient gives

Gl¯𝕎=Ca¯𝕎Gr¯𝕎=ρ𝕎glP¯b=ll*¯𝕎=σ𝕆𝕎lP¯bGl𝕎 (6.65)

for the macroscopic gravillary number. Note that the ratio σ𝕆𝕎/lP¯b is the ratio of the microscopic to the macroscopic capillary pressures. The characteristic numbers

ν*¯𝕎=kP¯bϕ¯μ𝕎 (6.66)
u*¯𝕎=ρ𝕎gkμ𝕎ϕ¯ (6.67)
l*¯𝕎=P¯bρ𝕎g (6.68)

are the macroscopic counterparts of the microscopic numbers defined in equations (6.22), (6.30) and (6.34).

An interesting way of rewriting these relationships arises from interpreting the permeability as an effective microscopic cross sectional area of flow, combined with the Leverett J-function. More precisely, let

Λ=kϕ¯ (6.69)

denote a microscopic length which is characteristic for the pore space transport properties. Then equations (6.63), (6.64) and (6.65) may be rewritten as

Ca¯𝕎=l¯ΛCa𝕎JS¯bcosθ (6.70)
Gr¯𝕎=l¯2Λ2Gr𝕎 (6.71)
Gl¯𝕎=Λl¯Gl𝕎JS¯bcosθ (6.72)

where JS¯b=P¯bk/ϕ¯/σ𝕆𝕎cosθ is the value of the Leverett-J-function [28, 2] at the saturation corresponding to breakthrough, and θ is the wetting angle.

The capillary number scales as l¯/Λ while the gravity number scales as l¯/Λ2. Inserting (6.71) and (6.72) into (6.57) this implies that the Buckley-Leverett approximation (6.57) becomes invalid whenever l¯<ΛGl𝕎/JS¯bcosθ.

VI.C.5 Macroscopic Estimates

The present section gives order of magnitude estimates for the relative importance of capillary, viscous and gravity effects at different scales in representative categories of porous media. These estimates illustrate the usefulness of the macroscopic dimensionless ratios for the problem of upscaling.

Three types of porous media are considered: high permeability unconsolidated sand, intermediate permeability sandstone and low permeability limestone. Representative values for ϕ¯,k and P¯b are shown in Table VI.

Table VI: Representative values for porosity, permeability and breakthrough capillary pressure in unconsolidated sand, sandstone and low permeability limestone.
Quantity Sand Sandstone Limestone
ϕ¯ 0.36 0.22 0.20  
k 10000 mD 400 mD 3 mD  
P¯b 2000 Pa 104 Pa 105 Pa  
Table VII: Definition and representative values for macroscopic dimensionless numbers in different porous media on laboratory (llab0.1m) and reservoir scale (lres100m) under uniform flow conditions (u3×10-6ms-1).
Quantity Definition Unconsolidated Sand Sandstone Limestone
Laboratory Reservoir Laboratory Reservoir Laboratory Reservoir
Gr¯𝕎 μ𝕎ϕ¯uρ𝕎gk 0.01 0.13 18.6 
Ca¯𝕎 μ𝕎ϕ¯ulkP¯b 0.005 4.9 0.015 15.0 0.19 187.5 
Gl¯𝕎 ρ𝕎glP¯b 0.5 492 0.1 115 0.01 10 
Λ kϕ¯ 5.2μm 1.3μm 0.1μm 
Ca𝕎/Ca¯𝕎 ΛlJS¯bcosθ 1.510-4 1.510-7 4.810-6 4.810-9 2.810-7 2.810-10 
CacCa¯𝕎Ca𝕎 ΛlJS¯bcosθ 0.67 --- 0.63 --- 0.71 --- 

To estimate the dimensionless numbers the same microscopic velocity u3×10-6ms-1 as for the microscopic estimates will be used. The length scale l, however, differs between a laboratory displacement and a reservoir process. llab0.1m and lres100m are used as representative values. Combining these values with those in Table IV and VI yields the results shown in Table VII.

The first row in Table VII can be used to check the consistency of the Buckley-Leverett approximation with the traditional normalization. The consistency condition (Eq. (6.61)) is violated for unconsolidated sand and sandstones. Such a conclusion, of course, assumes that the values given in Table VI are representative for these media.

The fifth row in Table VII gives the ratio between macroscopic and microscopic capillary numbers which according to Eq. (6.63) is length scale dependent. The last row in Table VII compares this ratio to the typical critical capillary number Cac reported for laboratory desaturation curves in the different porous media. Using the Cac10-4 for sand, Cac310-6 for sandstone, and Cac210-7 for limestone [28] as before one finds that the corresponding critical macroscopic capillary number is close to 1. This indicates that the macroscopic capillary number is indeed an appropriate measure of the relative strength of viscous and capillary forces.

As a consequence one expects differences between residual oil saturation S𝕆r in laboratory and reservoir floods. Given a laboratory measured capillary desaturation curve S𝕆rCa𝕎 as a function of the microscopic capillary number Ca𝕎 the analysis predicts that the residual oil saturation in a reservoir flood can be estimated from the laboratory curve as S𝕆rCacCa¯𝕎 [47, 48]. For Ca¯𝕎>1 the S𝕆r value based on macroscopic capillary numbers will in general be lower than the value S𝕆rCa𝕎 expected from using microscopic capillary numbers. Such differences have been frequently observed, and Morrow [361] has recently raised the question why field recoveries are sometimes significantly higher than those observed in the laboratory. The revised macroscopic analysis of [47, 48] suggests a possible answer to this question.

The values of the dimensionless numbers in Table VII allow an assessment of the relative importance of the different forces for a displacement. To illustrate this consider the values Gr¯𝕎=0.01, Ca¯𝕎=0.005 and Gl¯𝕎=0.5 for unconsolidated sand on the laboratory scale. A moments reflection shows that this implies VGC where V stands for macroscopic viscous forces, C for macroscopic capillary forces, and G for gravity forces. The notation AB indicates that A/B<10-2 while A<B means 10-2<A/B<0.5 and AB stands for 0.5<A/B<2. Repeating this for all cases in Table VII yields the results shown in Table VIII. Table VIII contains also the results from the microscopic dimensional analysis, as well as the results one would obtain from a traditional macroscopic dimensional analysis which assumes Ca¯=1 (see Eq. (6.56)).

Table VIII: Relative importance of viscous (V), gravity (G) and capillary (C) forces in unconsolidated sand, sandstone and limestone. The notation AB (with A,BV,G,C) indicates that A/B<10-2 while A<B means 10-2<A/B<0.5 and AB stands for 0.5<A/B<2.
Sand Sandstone Limestone
pore scale VGC 
traditional analysis V=CG V=C<G G<V=C 
large scale [47] laboratory scale VGC V<G<C G<V<C 
[48] field scale C<VG C<V<G C<G<V 

Obviously, the relative importance of the different forces may change depending on the type of medium, the characteristic fluid velocities and the length scale. Perhaps this explains part of the general difficulty of scaling up from the laboratory to the reservoir scale for immiscible displacement.

VI.C.6 Applications

The characteristic macroscopic velocities, length scales and kinematic viscosities defined respectively in equations (6.66), (6.67) and (6.68) are intrinsic physical characteristics of the porous media and the fluid displacement processes. These characteristics can be useful in applications such as estimating the width of a gravitational segregation front, the energy input required to mobilize residual oil or gravitational relaxation times.

The macroscopic gravillary number Gl¯𝕎 defines an intrinsic length scale l*¯𝕎 (see Eq. (6.65)). Because Gl¯𝕎 gives the ratio of the gravity to the capillary forces the length l*¯𝕎 directly gives the width of a gravitational segregation front when the fluids are at rest and in gravitational equilibrium, i.e. when viscous forces are negligible or absent. Using the same estimates for ϕ¯,k and P¯b as those used for Table VII one obtains a characteristic front width of 20cm for unconsolidated sand, 1m for sandstone, and roughly 10m for a low permeability limestone.

Similarly, the macroscopic capillary number defines an intrinsic specific action (or energy input) ν*¯𝕎 via Eq. (6.63) which is the energy input required to mobilize residual oil if gravity forces may be considered negligible or absent. Represenative estimates are given in Table IX below.

The gravitational relaxation time is the time needed to return to gravitational equilibrium after its disturbance. This may be defined from the balance of gravitational forces versus the combined effect of viscous and capillary forces. Analogous to equation (6.35) for the microscopic case the dimensionless ratio becomes

Gl¯𝕎Gr¯𝕎=macr. gravitational pressure2macr. capillary pressure×macr. viscous pressure drop (6.73)

which defines the gravitational relaxation time t*¯𝕎 as

t*¯𝕎=l*¯𝕎u*¯𝕎=μ𝕎ϕ¯P¯bρ𝕎2g2k. (6.74)

Estimated values are given in Table IX. They correspond to gravitational relaxation times of roughly 10 minutes for unconsolidated sand, 13 hours for a sandstone and 736 days for a low permeability limestone.

Another interesting intrinsic number arises from comparing the strength of macroscopic capillary forces versus the combined effect of viscous and gravity forces

Gl¯𝕎Ca¯𝕎-1=macr. capillary pressure2macr. grav. pressure×macr. viscous pressure drop (6.75)

where Q denotes the volumetric flow rate. Thus Q*¯𝕎 defined as

Q*¯𝕎=l*¯𝕎2u*¯𝕎=kP¯b2ϕ¯μ𝕎ρ𝕎g (6.76)

is an intrinsic system specific characteristic flow rate. The estimates for ν*¯𝕎,u*¯𝕎,l*¯𝕎,t*¯𝕎 and Q*¯𝕎 are summarized in Table IX.

Table IX: Characteristic macroscopic energies, velocities, length scales, time scales and volumetric flow rates for oil-water flow under reservoir conditions in unconsolidated sand, sandstone and low permeability limestone.
Quantity Sand Sandstone Limestone
ν*¯𝕎 6.110-5m2s-1 2.010-5m2s-1 1.610-6m2s-1 
u*¯𝕎 2.9910-4ms-1 2.1710-5ms-1 1.6110-7ms-1 
l*¯𝕎 0.2m 1.02m 10.2m 
t*¯𝕎 669s 4.7104 s 6.36107 s  
Q*¯ 1.2210-6m3s-1 2.0410-5m3s-1 1.6310-5m3s-1 

In summary, the dimensional analysis of the upscaling problem for two phase immiscible displacement suggests to normalize the macroscopic pressure field in a way which differs from the traditional normalization. This gives rise to a macroscopic capillary number Ca¯ which differs from the traditional microscopic capillary number Ca in that it depends on length scale and the breakthrough capillary pressure P¯b. The traditional normalization corresponds to the tacit assumption that viscous and capillary forces are of equal magnitude. With the new macroscopic capillary number Ca¯ the breakpoint Cac in capillary desaturation curves seems to occur at Ca¯1 for all types of porous media. Representative estimates of Ca¯ for unconsolidated sand, sandstones and limestones suggest that the residual oil saturation after a field flood will in general differ from that after a laboratory flood performed under the same conditions. Order of magnitude estimates of gravitational relaxation times and segregation front widths for different media are consistent with experiment.