Results

After a visual inspection of the different test cases (Figs. 2 and 4), we ranked them by their apparent channeling degree (Table 2).

Table 2

Porous and fractured test cases ranked visually by increasing order of channeling.

Field type Properties Short name

Porous Rearranged with the C-method,r²_y¼1 or 3,k¼L=16 orL/2 PC

Porous Gaussian correlation,r²y¼1 or 3,k¼L=16 orL/2 PG

Porous Exponential correlation,r²y¼1 or 3,k¼L=16 orL/2 PE

Porous Rearranged with the C+ method,r²_y¼1 or 3,k¼L=16 orL/2 PC+

Porous Rearranged with the F method,r²_y¼1 or 3,k¼L=16 orL/2 PF

Porous Rearranged with the F2 method,r²y¼1 or 3,k¼L=16 orL/2 PF2

Fractured Denseðd¼3p_cÞ, dominated by short fracturesða¼3:5Þ, uniform fracture transmissivityðr²y¼0Þ FDS0 Fractured Denseðd¼3p_cÞ, dominated by short fracturesða¼3:5Þ, distributed fracture transmissivityðr²_ylogT¼1Þ FDS1 Fractured Denseðd¼3p_cÞ, dominated by short fracturesða¼3:5Þ, distributed fracture transmissivityðr²_ylogT¼2Þ FDS2 Fractured Sparseðd¼p_cÞ, dominated by short fracturesða¼3:5Þ, constant fracture transmissivityðr²_ylogT¼0Þ FTS0 Fractured Sparseðd¼p_cÞ, dominated by long fracturesða¼2:0Þ, constant fracture transmissivityðr²_ylogT¼0Þ FTL0

Table 3

Mean and variance on 500 realizations for the different indicators and test cases. N/A stands for indicators that cannot be computed in the corresponding cases.

CASE r²y k D_ic=L r²ðDic=LÞ D_cc=L⁰ r²ðDcc=L⁰Þ CF₁ r²ðCF1Þ CT₁ r²ðCT1Þ

PC 1 8 0.063 0.0040 0.4 0.0820 -0.20 0.17 1.4 0.22

3 0.073 0.011 0.38 0.066 0.21 0.19 1.6 0.38

1 64 0.061 0.0080 0.40 0.18 0.10 0.50 1.4 0.35

3 0.070 0.022 0.44 0.20 0.10 0.51 1.4 0.34

PG 1 8 0.065 0.0027 0.41 0.032 0.0042 0.076 1.6 0.25

3 0.092 0.0082 0.44 0.037 0.0014 0.071 2.3 0.64

1 64 0.064 0.014 0.29 0.088 0.00023 0.43 1.3 0.28

3 0.096 0.049 0.31 0.13 0.020 0.41 1.6 0.55

PE 1 8 0.063 0.0031 0.40 0.016 0.035 0.075 1.5 0.25

3 0.088 0.010 0.45 0.022 0.026 0.073 2.2 0.60

1 64 0.062 0.0087 0.37 0.030 0.015 0.27 1.4 0.26

3 0.089 0.032 0.39 0.065 0.00018 0.26 1.7 0.55

PC+ 1 8 0.071 0.0064 0.64 0.066 0.17 0.15 1.7 0.42

3 0.094 0.020 0.60 0.075 0.23 0.16 2.1 0.67

1 64 0.064 0.016 0.57 0.21 0.039 0.53 1.5 0.50

3 0.077 0.035 0.51 0.22 0.065 0.53 1.7 0.73

PF 1 8 0.066 0.0030 0.74 0.082 0.23 0.12 1.9 0.54

3 0.10 0.011 0.86 0.067 0.29 0.09 3.4 1.8

1 64 0.057 0.0053 0.70 0.12 0.31 0.31 1.4 0.25

3 0.071 0.017 0.70 0.15 0.31 0.31 1.8 0.68

PF2 1 8 0.067 0.0037 0.78 0.076 0.42 0.12 3.5 2.4

3 0.12 0.0030 0.92 0.052 0.64 0.078 15 9.6

1 64 0.056 0.060 0.71 0.11 0.33 0.32 1.5 0.55

3 0.069 0.019 0.72 0.14 0.35 0.33 2.6 2.6

FDS0 N/A 0.11 0.10 0.59 0.036 N/A 1.7 0.30

FDS1 0.24 0.15 0.86 0.050 7.2 8.4

FDS2 0.29 0.14 0.91 0.048 26 26

FTS0 0.64 0.18 0.65 0.089 2.2 5.3

FTL0 0.82 0.21 0.90 0.045 1.2 6.9

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The order was derived separately in the porous and fracture cases according to the flow-tube widths and regularity. In porous media, this order is consistent with CF₁ values (Table 3). All results discussed in the following paragraph are given inTable 3.

4.1. Relation between Dicand the number of considered flow tubes

The interchannel distanceDic(5)depends on the proportion of flow used to define a channel. Ifnis the number of flow tubes, each flow tube carries 1/npart of the total flow.Fig. 5displays the rela-

tion betweenDic and 1/n. When 1/n tends to 1, Dic tends to L, meaning that no channel contains all the flow by itself. For the smallest values of 1/n, the fracture cases reach a plateau character- istic of the smallest distance between flowing fractures. 1/ncan be interpreted as characteristic of the smallest flow channel that can be identified. InFig. 5, we chose a value ofn= 20, for which all test cases have a characteristic interchannel distance larger than the interfracture distance. The value of Dic remains dependant of n but the relative order for the different test cases remains the same whenevern620. We will thus compareDicvalues between test cases rather than their absolute values in single test cases. The cho- sen value ofnis the flow-tube resolution. IfDic<L=n, the medium will be considered as homogeneous in the sense that interchannel distances are smaller thanL/n. The value ofnshould be increased for distinguishing closer channels.

4.2. Channeling characteristics Dicand Dcc

Fig. 6a and b display the two new indicatorsDic=LandDcc=L⁰in highly-heterogeneous porous casesðr²y¼3Þand fracture cases. PG and PE configurations have very close values ofDicandDccdespite the visual ranking, meaning that the short-range correlations do not affect the channeling degree.Fig. 6c and d display the two other indicatorsCF1andCT1.CT1performs poorly while CF1cap- tures the channeling increase of porous configurations (PC+, PF and PF2). However,CF1can be used in porous cases but is not avail- able in fractured media. Moreover,CT1is not discriminating in por- ous cases and does not account for the apparent ranking of channeling in the fracture test cases.Dic=LandDcc=L⁰are thus more adapted to characterize channeling consistently in porous and frac- tured cases.

Fig. 6.Values of (a)D_ic/L, (b)D_cc=L⁰, (c)CF₁and (d)CT₁. For the different test cases ranked by their increasing intuitive rating of channeling (Table 2), Error bars are the standard deviations of the underlying distributions. Parameters of porous cases arer²y¼3 andk¼8. The vertical double bar separates porous and fracture test cases.

Fig. 5.Dic=Lversus 1/nin various porous and fractured test cases. The vertical black line represents the values used in the current study.D_ic=L¼1=nis the lower limit representing a homogeneous field whileD_ic=L¼1 is the upper limit representing a field with all flows concentrated in a unique channel.

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The characteristic interchannel distance Dic=Lconsistently in- creases with more visual channeling. This increase is much smaller in the porous cases than in the fracture cases.D_cc=L⁰also increases in porous media and has significantly larger values in all channeled fracture and porous fracture cases (FDS1, FDS2, FTL0, PF and PF2).

In fact, in fractured media, flow is focused within the fractures and the variations of flow rates are more restricted than in porous med- ia.Dcc=L⁰reaches values close to one equivalent to little variation of flow rates within flow lines in FTL0, FDS2, PF and PF2. The compar- ison of the variations inDic=LandDcc=L⁰shows that in the porous cases,Dcc=L⁰increases over a range twice as large as that ofDic=L from PCto PC+. In the fracture case, however,Dic=Lis more con- sistent with the visual ranking of channeling thanDcc=L⁰. These re- sults indicate that a flow organization indicator (Dcc) better characterizes porous flow channeling while a flow localization indicator (D_ic) better characterizes fracture flow channeling.

The variability ofDic/Lis much larger in fractured media than in porous media (Error bars inFig. 6a), which means thatDicdoes not vary much in porous configurations where channels are distributed over the field. However,Dicis highly variable in fracture configura- tions where channels can be either very clustered or spread.

The absence of systematic correlation betweenDic=LandDcc=L⁰ shown inFig. 7confirms thatDic=LandDcc=L⁰characterize two dif- ferent channeling properties.Fig. 7also shows thatDic=LandDcc=L⁰ consistently characterize channeling in respectively the fracture and the porous cases. First, the visually ranked non-channeled frac- ture case FDS0 is in fact close to a highly-correlated porous case ðPCþ;ry¼3;k¼8Þ. Second, the porous fracture cases PF and PF2 located at the top left corner ofFig. 7have largerDcc=L⁰values than the porous cases and smallerDcc=L⁰values than the fracture chan- neled cases.

Based onFig. 7, we distinguish three types of flow structures.

First, weakly-channeled flow structures are characteristic of mul- ti-Gaussian fields (PG, PE and PC) and lead to small Dic=Land Dcc=L⁰ values. Second, the mildly-channeled flow structures were obtained for high-k connected patterns (PC+, PF and PF2 with

r²y¼1) and have smallDic=Lvalues and largeDcc=L⁰values. Third, the highly-channeled media have largeDic=LandDcc=L⁰values, like FTL0 or PF2 withr²_y¼3. The latter case corresponds to extreme channeling for which flow is both highly localized and highly con- tinuous in a very small number of channels.

4.3. Relation between channeling characteristics and k-field parameters

In this section, we look for a finer understanding of indicators Dic=LandDcc=L⁰ by analyzing their dependence on the structures of the porous (Figs. 8 and 9) and fractured test cases (Figs. 10–

12). We then comment on the variation trends and amplitudes.

Dic=L systematically increases with more heterogeneity. In fact, Dic=Lincreases withr²yin porous media (Fig. 8) and withr²yin frac- ture networks (Fig. 12). Largerr²_y values imply that flows focus in sparser transmissivity zones.Dic=Lalso systematically decreases in denser fracture networks, i.e. when increasing the number of con- nected parallel fracture paths (Fig. 11). Increasing the probability of occurrence of long fractures with smalleravalues yields similar causes and effects (Fig. 10). Similarly, increasing the correlation length k in porous media from small to median values induces more channeling. The sole non-obvious Dic=L variation is its de- crease from intermediary to large correlation lengths, approxi-

Fig. 7.Dic=LversusDcc=L⁰in porous and fractured fields. Porous cases are represented by squares, fractured cases by triangles and porous fractured cases by hexagons. The combination of the two indicators give an estimation of the channeling degree with weakly-channeled configurations (smallD_ic, smallD_cc), mildly-channeled configurations (highD_cc, smallD_ic) and highly-channeled configurations (highD_ic, highD_cc).

Fig. 8.Variation ofD_cc=L⁰versusD_ic=Lin porous test cases PG and PF for varying permeability standard deviationsry.ryvalues are given next to the corresponding symbols. Whenry tends to zero,Dictends to 1/nandDcctends to zero as the permeability values tend to be homogeneous.

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mately fromkL=8 tok¼L=2 (Fig. 9). This may be due to two rea- sons. First, the standard deviation ofDic=Lsteeply increases withk

to the point where its variations become smaller than its variabil- ity. Second, when the large correlation lengthkis comparable to the system sizeL, for example fork¼L=2, the system becomes more homogeneous and channels are more regularly distributed, explaining the smaller value ofDic=L. Apart from these side effects, Dic=Lis first determined by the density of potential channels given by the correlation length in porous media and by fracture density and length distributions. Among the potential channels, only those made up of the higher permeabilities lead to effective channels.

This is confirmed by the variation ranges ofDic=Lpresented inTa- ble 4. The largest variation ranges are due to the fracture density first and to the difference ofDic=Lvalues between porous and frac- ture cases (Fig. 6). They cover at least two thirds of the full varia- tion ofDic=L. The variation ranges according tor²_yare smaller but not negligible and account on average for less than half of the full variation range of porous and fracture cases.

Dcc=L⁰is less variable thanDic=L. It varies significantly only in PF as a function ofr²y(Fig. 8), in FDS as a function ofr²y(Fig. 12), in PG as a function ofk(Fig. 9) and in FD0 as a function ofa(Fig. 10). We argue in the following that the sole genuine variation is the last one. The two first variations are due to the transition from porous to fractured cases. Beyond the transitionðr²y>1Þ;Dcc=L⁰is almost constant. In porous-fracture fields, smallr²y values correspond to almost pure porous cases without fractures whereas highr²_yvalues lead to fracture-like cases. In dense fracture networks withr²y¼0, the fracture network looks like a porous medium. Increasingr²y

triggers channeling while keepingDcc=L⁰almost constant.Dcc=L⁰de- creases with the correlation length. This counter-intuitive result is Fig. 10.D_cc=L⁰versusD_ic=Lin fractured test cases with varying power-law length

exponentsa. Values ofaare given next to the corresponding symbols.

Fig. 9.Variation ofD_cc=L⁰versusD_ic=Lin porous test cases PG and PF for varying permeability correlation lengthsk.kvalues are given next to the corresponding symbols.

Fig. 11.D_cc=L⁰versusD_ic=Lin fractured test cases with varying fracture densitiesd.

Values ofdare given next to the corresponding symbols. Density is measured as the percentage of fractures above percolation threshold. It is 0 at percolation threshold and 100 in networks having twice as much fractures as in networks at percolation threshold.

Fig. 12.Dcc=L⁰versusDic=Lin fractured test cases with varying variances of the transmissivity distributionr²y. Values ofr²y are given next to the corresponding symbols.

Table 4

Variation range (maximal minus minimal values) ofDic=LandDcc=L⁰according to porous and fracture parameters. High values in bold are due to transitions from porous-like to fracture-like structures rather than to variations withr²y.

Porous cases Fractured cases D_ic=L Dcc=L⁰ D_ic=L Dcc=L⁰ Full range of

variation

0.12 0.66 0.73 0.40

r²y 0.08 (PG) 0.1 (PG) 0:42ða¼3:5Þ 0.31(a= 3.5) 0.10 (PF) 0.55(PF) 0:24ða¼2:0Þ 0:07ða¼2:0Þ

k 0.06 (PG) 0.09 (PG)

0.04 (PF) 0.14 (PF)

a 0:35ðr²_y¼0Þ 0:30ðr²_y¼0Þ

0:22ðr²y¼3Þ 0:05ðr²y¼3Þ

d 0:55ða¼3:5Þ 0:1ða¼3:5Þ

0:55ða¼2:0Þ 0:12ða¼2:0Þ

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only apparent and mostly due to the simultaneous variations ofL and k in our simulation settings. In fact, we have found that Dcc=L⁰depends more onL=kthan onk. The range of variations of Dcc=L⁰is reduced from 0.25 to 0.09 when decreasing the range of variations ofL=kfrom 64 to 8. Disregarding these dummy varia- tions, the sole genuine variation that is not a transition is the in- crease inDcc=L⁰ with smalleravalues for dense networks (FD0).

What fundamentally changes in the latter case is the nature of the correlations.Dcc=L⁰ seems to be more affected by the nature of the correlation than by the variability of permeability. Aside from the large variation ranges due to a transition from porous-like to fracture-like media marked in bold inTable 4, it is the sole case where the variation range ofDcc=L⁰is significant. More precisely, it is of the order of three quarters of the full range of variations in all fracture networks, where all other cases are restricted to one quar- ter. The channel continuity measured byDcc=L⁰is thus much more influenced by the nature of the correlation structure than by the other parameters including the permeability variability, the frac- ture density and the correlation length.Dcc=L⁰ can be considered as an indicator of the nature of correlation. Finally, the absence of correlation between variability andDcc=L⁰ expresses that there is a fundamental limit in channeling related to the local permeabil- ity structure rather than to the permeability variability.