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WATER RESOURCES RESEARCH, VOL. 43, W08430, doi:10.1029/2006WR005018, 2007
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Interpretation of macroscale variables in Darcy’s law J. M. Nordbotten,1,2 M. A. Celia,2 H. K. Dahle,1 and S. M. Hassanizadeh3 Received 7 March 2006; revised 2 March 2007; accepted 12 March 2007; published 29 August 2007.
[1] The pursuit of a theoretical foundation of Darcy’s law based on volume averaging of
equations at the scale of flow in pores has a long history. While theories are well established for homogeneous systems, more complex systems exhibit inconsistencies in the resulting equations. The difficulties often lie in the treatment of surface integral terms arising from the classical averaging theorems used to transform averages of derivatives into derivatives of averages. In this work we extend the intrinsic phase average as a macroscale variable to a family of more general macroscale variables, which take into account systematic dependencies of averaging volume size on the macroscale. Comparison to Darcy’s law gives new insight into the relationship between variables at the microscale and macroscale. Citation: Nordbotten, J. M., M. A. Celia, H. K. Dahle, and S. M. Hassanizadeh (2007), Interpretation of macroscale variables in Darcy’s law, Water Resour. Res., 43, W08430, doi:10.1029/2006WR005018.
1. Introduction [2] Flow of fluids in porous media is traditionally modeled by the empirical relationship known as Darcy’s law, which relates the pressure gradient to the volumetric flux vector as follows: 1 qa ¼ K ðr½ Pa rgrzÞ: m
ð1Þ
Here a denotes a fluid phase, qa is the volumetric flux vector, K is permeability tensor, [P]a is macroscopic pressure, r is the mass density, m is the viscosity, g is the gravitational constant, and z is the vertical coordinate increasing downward. [3] Many authors have studied the derivation of Darcy’s law through the use of averaging methods. Commonly, governing pore-scale equations, such as the Navier-Stokes equation, are averaged over a representative elementary volume (REV) in order to obtain macroscale governing equations (see, for example, Hubbert [1940, 1956], Whitaker [1969], Gray and O’Neill [1976], Neuman [1977], Bachmat and Bear [1986], Quintard and Whitaker [1994a, 1994b], and He and Sykes [1996] among many others). [4] Hubbert [1940] was the first to show that an equation similar to Darcy’s law can be obtained from averaging the equations of flow on the microscale. In that formulation, he observed that the macroscopic pressure gradient is equal to the average of the microscopic pressure gradient. Thus he assumes the existence of a macroscale pressure, but he does not give an explicit definition to relate it directly to the microscale pressure.
1
Department of Mathematics, University of Bergen, Bergen, Norway. Department of Civil and Environmental Engineering, Princeton University, Princeton, New Jersey, USA. 3 Department of Earth Sciences, Utrecht University, Utrecht, Netherlands. 2
Copyright 2007 by the American Geophysical Union. 0043-1397/07/2006WR005018$09.00
[ 5 ] Whitaker [1969] and Gray and O’Neill [1976] addressed this question by using averaging theorems to manipulate the average of microscopic pressure gradients to obtain the gradient of an average pressure. This allows Darcy’s equation to be written with all the variables defined directly in terms of averages of microscale quantities. However, as was pointed out by Quintard and Whitaker [1994a], in the presence of gravity, results from the work of Whitaker [1969] and Gray and O’Neill [1976] are limited to media with essentially constant porosity in space. Otherwise, nonphysical gravitational effects appear in the resulting equations. This point was elaborated in the work of Goyeau et al. [1997], where they considered evolving heterogeneities. They found that both porosity and permeability depended on the size of the averaging volume, indicating conceptual problems with the notion of an REV. [6] Quintard and Whitaker [1994a, 1994b, 1994c] circumvented the problems in the earlier formulations by introducing a generalized average. This is equivalent to smoothing the problem before averaging, such that the effect of heterogeneities on the scale of the averaging volume could be neglected. While this allows for spatially periodic porosity, it has the drawback that the averaging volume required must be larger than the scale of the wavelength of the spatial variance in porosity, such that the porosity at the scale of the averaging volume is essentially constant. [7] In this paper, we focus on averaging the microscale flow equation for cases where there is a porosity gradient and show that the classical definition of average pressure does not lead to the classical Darcy’s law. Consequently, the classical average pressure does not correspond to the macroscopic pressure that appears in Darcy’s equation (1). In our development, we will average the microscale momentum balance equation. Our derivation will differ from that of previous works in allowing for a more general porosity variation within the averaging volume. As such, we seek to overcome limitations of previous approaches and provide a new view on the definition of averaged variables in Darcy’s law.
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[8] Here we will only discuss the flow of a single fluid in a rigid medium. This is part of a broader research effort aimed at increasing our understanding of multifluid flow in the presence of significant spatial and temporal variations in geometrical quantities such as porosity and fluid saturations.
generalized divergence theorem allows us to obtain the averaging theorem of Whitaker [1967],
2. Averaging Microscale Equations
Here Aa,s is the interfacial area between fluid a and solid s, and na is the unit normal vector along this interface. For the intrinsic averages we have the associated averaging theorem obtained by applying the product rule to rhwia [Gray, 1975],
[9] For a microscale function w and an averaging volume Vx centered at a point in N-dimensional space denoted by the vector x, a general definition of the average of w is hwiðxÞ ¼
Z
1 Vx
wdV:
g a ðx; tÞ ¼
:
hrwia ¼ rhwia þ
ð2Þ
Vx
where w may be nonzero over the whole or part of volume Vx. A special case of interest is when w is defined only over the parts of Vx that are occupied by a phase a. This and other definitions are facilitated by the definition of an indicator function g a, defined by 8 0. Let S: (2‘REV , 2‘REV + c). Then, by the definition of an REV for porosity, we have that 1 @Vx ð pÞ
Z @Vx ð pÞ
hg a if ‘ REV dS ¼ hg a i1 ¼
1 @Vx ð pÞ
Z @Vx ð pÞ
hg a i1 dS
ðA8Þ
for all p 2 S. Note the slight abuse in notation when denoting both the sphere and its area by @Vx(p). Equivalently, Z @Vx ð pÞ
Appendix A
Z
‘2 hg a i1 N þ2
5. Conclusions [56] In this paper we have shown how volume averaging can lead to nonphysical results for nonhomogeneous systems. For the case of linear spatial variations, such as the gravitational component term, this can lead to unconventional macroscale gravitational terms in Darcy’s law. We circumvent this problem by introducing a polynomial correction to the intrinsic average, and we show that this higher-order macroscale function still satisfies Darcy’s law. The application of the new macroscale function is demonstrated through an example with a porous medium where a strong gradient in porosity is present. [57] The analysis herein is particularly relevant for the upscaling of heterogeneous porous media. For common upscaling techniques, such as homogenization, scale separation or statistical uniformity is required at the REV scale. The results herein allow this assumption to be relaxed. In particular, when the medium has a linearly varying porosity, an REV according to the definition herein will exist, and the macroscale definitions presented can be applied. As such, the restriction of constant porosity across an REV has been removed, and linear variations can be accommodated. [58] We believe these results can have important implications for upscaling studies involving more general spatial variability and for upscaling studies involving multiple fluid phases. This new definition of average pressure can also provide new insights into the meaning of pressures calculated routinely in numerical simulators and in the interpretation of what is actually measured by a measuring device. We expect some of these topics to be the focus of further studies.
1 Vx ð‘Þ
hg a if ‘ REV hg a i1 dS ¼ 0;
ðA9Þ
for all x and p 2 S. This is the classical Pompeiu problem, which for c > 0 implies (for noncompact domains, Berentsen and Zalcman [1980]):
rVx ðrÞða ðrÞ a Þdr:
hg a if ‘ REV ¼ hg a i1 :
0
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ðA10Þ
NORDBOTTEN ET AL.: MACROSCALE VARIABLES IN DARCY’S LAW
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[60] Combining equations (A5) and (A10) we obtain the desired result, 1 Vx ð‘Þ
Z
constant c, the difference between the weighted averages must equal zero. We thus have that
‘
rVx ðrÞða ðrÞ a Þdr ¼ 0;
n X n rk hx i x ¼ k a Vx k¼1
ðA11Þ
n a
0
þ for all ‘ such that ‘ REV ‘ < ‘REV =3. [61] We note that our definition of an REV implies that the porosity is a harmonic function. This follows from the second equality in equation (A8) [see, e.g., Courant and Hilbert, 1962]. These results should therefore be interpreted locally; if the porosity can be represented locally as a harmonic function, we can define an REV. With this definition of an REV, we are assured that equation (A11) holds.
n
1 Vx
a hxn ia ¼
Z
‘
0
Z @V ðr1 ;xÞ
g a ðx þ x Þxn1 dSdr1
a 1 ¼ a x xn1 þ Vx
a 1 ¼ a x xn1 þ Vx
Z
Z
‘
r1 @V ðr1 ;xÞ
0
Z
‘
g a nxn1 dSdr1
a
r1 Vx ðr1 Þr a ðr1 Þ xn1 ðr1 Þ dr1
0
ðB3Þ
We can apply this equation recursively to obtain n a
a hx i ¼
n k X n r
k Vx Z ‘ Z r1 Z xnk r1 r2 k¼0
0
0
rk Vx ðrk Þa ðrk Þdrk dr2 dr1 :
rk1
0
ðB4Þ
The integration limits are independent of space (since we have assumed ‘ independent of space), and we have therefore brought the kth-order gradient outside the integrals. Using the substitution a(rn) = a + (a(rn) a), we then obtain after integration by parts a hxn ia ¼
n k X n r k¼0
k
Vx
!! hg a if hg a i1 a þ Qk k ; j¼1 ð N þ 2jÞ j¼1 ð N þ 2jÞ
xnk Qk
‘2k
ðB5Þ
where the weighing functions fk are 2kth-order polynomials. The same arguments can be applied, as in Appendix A, to show that, if 3REV + c < REV+, for some positive
x
!
‘2k
Qk
j¼1
m X
Pm ¼
ð N þ 2jÞ
a :
ðB6Þ
ðiÞ
Bi : x i ;
ðB7Þ
i¼0
ðiÞ
where the coefficient Bi immediately have that a
hPm i Pm ¼
m X i¼0
ðiÞ Bi
is an ith-order tensor, we
i X rk i : k a Vx k¼1
ik
x
Qk
j¼1
‘2k ð N þ 2jÞ
! a : ðB8Þ
Since this is a 2mth polynomial in ‘, it follows that ½Pm an ¼ Pm
ðB1Þ
ðB2Þ
nk
Defining an arbitrary mth-order polynomial Pm as
Appendix B [62] The main part of the paper and Appendix A showed that ½wan = w for linear functions, when n 2. This argument will be generalized in this appendix. [63] Consider the nth power of a coordinate, hxnia. We then have as in section 3.3
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8 n 2m:
ðB9Þ
[64] Acknowledgments. The authors thank Arne Stray and Sabine Manthey for interesting discussions. This work was supported in part by BP and Ford Motor Company through funding to the Carbon Mitigation Initiative at Princeton University and by the National Science Foundation under grant EAR-0309607.
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Whitaker, S. (1969), Advances in theory of fluid motion in porous media, Ind. Eng. Chem., 61(12), 14 – 28. Whitaker, S. (1986), Flow in porous media: I. A theoretical derivation of Darcy’s law, Transp. Porous Media, 1(1), 3 – 26. Woods, L. C. (1986), The Thermodynamics of Fluid Systems, Oxford Univ. Press, Oxford Univ. Press.
M. A. Celia, Department of Civil and Environmental Engineering, Princeton University, 08544 Princeton, NJ, USA. H. K. Dahle and J. M. Nordbotten, Department of Mathematics, University of Bergen, 5089 Bergen, Norway. (
[email protected]) S. M. Hassanizadeh, Department of Earth Sciences, Utrecht University, 3508 TA Utrecht, Netherlands.
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