Stochastic Flow Through Multifractal Porous Media

Stochastic Flow Through Multifractal Porous Media

Abstract We study the flow through a random porous medium when the hydraulic conductivity is assumed an isotropic lognormal multifractal field. We find that the resulting hydraulic gradient H  x  and specific flow q  x  are also multifractal with renormalization properties and marginal distributions obtained analytically as functions of the space dimension D and a multifractal parameter of K (the codimension CK. The analytical results are confirmed with two-dimensional simulations.

1.

Introduction

We study the flow of a homogenous liquid through a saturated random porous medium when the hydraulic conductivity K is a multifractal. We find that the hydraulic gradient H and specific flow q fields are also multifractal. When a scalar quantity such as K is multifractal, it possesses a scale invariance property such that the distribution of K at a smaller scale is equivalent to K at a larger scale that is isotropically contracted and multiplied by a random number. Following [3 and 6], we say that a scalar homogeneous random measure V(S) in ¡

D

is

isotropically multifractal if there exists a sequence of non-negative random variables

A r ,

r  1 independent of V such that the measure density v  S  V  S / S is

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statistically invariant under the scale transformations x  x r, v  A r v , i.e. if, for any r  1 , d

v  S  A r v  rS

(1)

d

where  denotes equality of all finite dimensional distributions. In the case when Ar has lognormal distribution, the spectral density of the log hydraulic conductivity F=ln(K) has the form  2  SF  k    C K  k  D  SD 

(2)

where k=|k| and SD is the area of the unit sphere in RD; hence S1=2, S2 = 2 and S3 = 4 [7]. Vector quantities such as H and q exhibit a more general form of multifractality. Their renormalization properties involves not only rescaling of the amplitude and support of the original process but also involves rotation of the support, as shown in Figure 1. Hence, their measure density v and amplitude v have the following renormalization properties:

i 



d

v  S  A r R r v R r  rS T



for any r  1 (3)

 ii 

d



v rr1  S  A r v r1 R r  rS T



for any r, r1  1

 1  T where R r  rS   x : R r x  S is the set S isotropically expanded by r and randomly  r  T

rotated by R r . Details on multifractal properties are given in [6 and 7].

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2.

Methodology

We examine the hydraulic head and flow inside the unit cube   x : 0  x i  1,..., D in a medium with an isotropic lognormal field K with

multifractal renormalization invariance over a finite range of scales. Specifically, F=ln(K) is modeled as an isotropic Gaussian field with spectral density

2 D  CK k , SF  k    SD 0, 

k o  k  k max

(4)

otherwise

where k = |k|, k max  k o ? 1 and CK = 1 . The spectral range is finite because for k max / k o   , the variance 2F  2CK ln  k max / k o  diverges and the medium becomes

non-conductive. We apply an average hydraulic gradient -Jo in the x1 direction and examine whether the resulting hydraulic gradient H and specific flow q satisfy multifractal conditions. We find that they do and examine their scaling properties and obtain the marginal distributions of H and q. To study the multifractality of H and q, we consider a cascade of bare hydraulic gradient and flow fields at different resolutions r  1. The fields at resolution r are obtained by using a log-conductivity Fr in which all Fourier components at wavenumbers k  rk o have been filtered out; hence the spectral density of Fr except that it is zero for k  rk o . Using a subscript r to denote quantities derived under F = Fr, the hydraulic gradient Hr and specific flow q r satisfy

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i

q r   K r H r

 ii 

 H r  Fr .H r  0

(5) 2

Using Eqs (5i) and (5ii) the scaling relations are derived in [7] and the renormalization properties of H and q are obtained under the following assumptions: 1. In spaces of dimension D > 1, zero-mean high frequency fluctuations of the head and flow along the boundary affect the hydraulic gradient and flow only in a narrow region close to the boundary. 2.

Consider a sub-region  ' of  and split F into a low-frequency component FLF and a high-frequency component FHF such that the hydraulic gradient HLF due to FLF may be considered constant inside  ' , say with value H LF   '  . We assume that the hydraulic head field in  ' can be obtained accurately by replacing F with FHF while subjecting  to the large-scale hydraulic gradient H o  H LF   ' .

Details of our results are presented below.

3.

Multifractal Scaling of Hydraulic Gradient and Specific Flow

We found the hydraulic gradient to satisfy the following renormalization equation, d



H rr1  x    J r R r  .H r1 rR r x T



x  r

(6)

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where R r is an orthogonal random matrix, and  J r R r  is independent of the random field

Hr1 and Jr is a random variable distributed like H r  x  and has a lognormal distribution with parameters

D4  E ln  J r    D(D  2) CK ln  r    6 Var ln  J    C K ln  r  r    D(D  2)

(7)

where D is the space dimension and CK is the parameter that controls the level of the spectral density of F. Notice that for D =1 there is no rotation and for D=2 the matrix

R r is completely defined by the vector er , which has the distribution of Brownian motion (Bm) on the D-dimensional unit sphere, with log-resolution r '  ln  r  as time parameter. Details on the derivation of the distributional properties of H and J can be found in [7]. We investigated the scaling behavior of the specific flow q by comparing the fields q r in Eq (1) when F=ln(K) is developed to different resolutions r. First, we found the specific flows q r to satisfy the scaling relation d



q rr  x    Br R r  .q r rR r x 1

1

T



x r

(8)

A direct implication of Eq (10) is that the flow amplitudes qr scale as d



q rr  x   Br .q r rR r x 1

1

T



x r

(9)

where Br is a random scaling parameter that has lognormal distribution with parameters

  D4   1 C K ln  r  E ln  Br      D(D  2)    2  D 2  1  Var ln B      r   D(D  2) CK ln  r   

(10)

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and initial moments

1   E  Bsr   exp E ln  Br   s  Var ln  Br   s 2  2   =r

 D 5 D2 1 2 CK   s s s  D(D  2) D(D  2)





  

(11)

Eq (11) shows that lim E  Br   0 which implies through Eq (10) that the expected flow r 

amplitude vanishes asymptotically as the hydraulic conductivity is developed to infinite resolution. This behavior is caused by the negative correlation between the log conductivity Fr and the log hydraulic gradient amplitude ln  J r  x   . In fact the correlation coefficient  between these two variables does not depend on the resolution r and varies with the space dimension as

  Corr  Fr , ln  J r    

D2 3D

(12)

To validate the analytical results, we perform two-dimensional simulations on 512x512 grid and present the results of the specific flow field for illustration. Figure 1 shows the flow field for a K field with CK  0.1. Figure 2 shows the scaling of the specific flow field. Next, we used the scaling properties of H and q to derive their spectral density tensors. Equations for these spectral densities are provided in [7] and provide two important differences between the spectra of the linear theory [1] and those of the nonlinear theory: 1.

The spectral density from the linear theory decay at a faster rate.

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2.

The spectral contour lines from the linear theory have the same non-circular shape at all scales, whereas those from nonlinear theory exhibit anisotropy at large scales and isotropy at small scales.

The above features can be seen in Figure 3, which numerically compares the linear and nonlinear transverse velocity fields. Using the spectral density tensors of q, we obtained the macrodispersivities for the nonlinear case. Reflecting the differences in the spectral densities, we found the macrodispersivities from the nonlinear theory to be isotropic at small travel distances and anisotropic at large travel distances. Moreover, the power law dependence of the macrodispersivities at small travel distances, have different exponents ( = 1 for the linear case and  = 1-CK for the nonlinear case, see [7] for details) for the linear and nonlinear case. Figure 4 shows the macrodispersivities from the linear and nonlinear theories.

4.

Summary and Conclusions

When flow occurs in field with a multifractal K, the resulting hydraulic gradient H and specific discharge q fields are also multifractal, with scale invariance properties related to those of K. The marginal distributions of H and q are not Gaussian as typically assumed. Rather, their amplitudes have lognormal distributions and their common direction has the distribution of Brownian motion on a sphere. The multifractality of H and q produces isotropy in the spectral density tensors of H and q at small scales and anisotropy at large scales. The transition from

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isotropy to anisotropy with increasing scales is also observed in the resultant macrodispersivities.

References [1]

Gelhar, L. W., Stochastic Subsurface Hydrology, Prentice Hall, New Jersey, 1993

[2]

Dagan, G., Flow and Transport in Porous Formations, Springer-Verlag, New York, 1989.

[3]

Deng, F. W. and J. H. Cushman, Higher-order corrections to the flow velocity covariance tensor revisited, Water Resour. Res., 34(4): 103-106, 1998.

[4]

Gupta, V. K. and E. C. Waymire, Multiscaling properties of spatial rainfall and river flow distributions, J. Geophys. Res., 95:1999-2009, 1993.

[6]

Veneziano, D., Basic Properties and Characterization of Stochastically SelfSimilar Processes in ¡

[7]

D

, Fractals, 7(1):59-78, 1999.

Veneziano, D and A. K. Essiam, Flow Through Random Media with Lognormal Multifractal Hydraulic Conductivity 2. Spectral Tensors, Macrodispersivity and Effective Hydraulic Conductivity, Submitted to Water Resources Research, 2000.

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