# Fast Upscaling of the Hydraulic Conductivity of Three-Dimensional Fractured Porous Rock for Reservoir Modeling

## Abstract

A fast upscaling procedure for determining the equivalent hydraulic conductivity of a three-dimensional fractured rock is presented in this paper. A modified semi-analytical superposition method is developed to take into account, at the same time, the hydraulic conductivity of the porous matrix (KM) and the fractures (KF). The connectivity of the conductive fracture network is also taken into account. The upscaling approach has been validated by comparison with the hydraulic conductivity of synthetic samples calculated with full numerical procedures (flow simulations and averaging). The extended superposition approach is in good agreement with numerical results for infinite size fractures. For finite size fractures, an improved model that takes into account the connectivity of the fracture network through multiplicative connectivity indexes determined empirically is proposed. This improved model is also in good agreement with the numerical results obtained for different configurations of fracture networks.

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## Notes

1. 1.

The terms “narrow bounds” and “tight bounds” both refer to the case where the lower and upper bounds (KLOW and KUP) are relatively close to each other; obviously, the inequality becomes an equality in the ideal case of equal bounds (KLOW  =  KUP).

2. 2.

Homogeneity, ergodicity, isotropy. A random field, such as K(x) or lnK(x), is a random function of position (x). The moments of a statistically homogeneous, or “stationary”, random field are invariant by translation. Ergodicity refers to the convergence of spatial averages (spatial moments) to ensemble averages (ensemble moments), in the limit of infinite spatial domain. In practice, only the 1st and 2nd order moments are considered (2nd order stationarity and ergodicity). Finally, statistical isotropy refers to the case where the moments of the random field are invariant by rotation.

3. 3.

Probability distribution of a random field F(x). One should distinguish the single-point probability law of F(x), from its general multipoint probability law. In the case of a Gaussian random field, the one-point and two-point moments suffice to entirely define its multipoint probability law. In the two-dimensional case at hand, if F  =  lnK(x,y) is assumed Gaussian (in the sense “multi-Gaussian”), then K(x,y) is by definition a “log-normal” random field. In that case, it can be shown that K/KG and its inverse KG/K both have well defined multipoint laws (log-normal) with the same moments. Furthermore, the mean, variance and two-point covariance of the log-normal K(x,y) can be explicitly related to those of the Gaussian lnK(x,y).

4. 4.

Infinite size fractures. It is understood in this paper that the term “infinite size fracture” or “infinite fracture” refers to any planar fracture that entirely crosses the upscaling domain. More precisely, if the upscaling domain is a convex region of finite volume (e.g., a three-dimensional parallelepiped), then an “infinite fracture” is a planar object that completely crosses the domain: it separates the domain in two subdomains, and its trace on the domain boundary is a closed curve. If the upscaling domain is infinite, then an “infinite fracture” is simply a planar fracture of infinite diameter.

5. 5.

The consequence in three-dimensional space is that the longitude angle Ө must be distributed uniformly in [0, 2π], and (independently) the latitude angle φ has its cosine distributed uniformly in [− 1, + 1].

6. 6.

A finite size” fracture is any planar fracture which does not entirely cross the upscaling domain: the opposite of “infinite sire” fracture mentioned in Sect. 2.

7. 7.

The planar disc fractures studied in this work are a special limit form of planar convex polygons as the number of edges goes to infinity.

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## Acknowledgements

The first three authors wish to acknowledge the financial support of ADEME (France) in the framework of the GeotRef project “Géothermie haute énergie en Reservoirs fracturés” (www.geotref.org). We thank the anonymous reviewers for their careful reading of the manuscript and for their insightful comments and suggestions.

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Correspondence to Tawfik Rajeh.

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Rajeh, T., Ababou, R., Marcoux, M. et al. Fast Upscaling of the Hydraulic Conductivity of Three-Dimensional Fractured Porous Rock for Reservoir Modeling. Math Geosci 51, 1037–1074 (2019). https://doi.org/10.1007/s11004-019-09785-w

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### Keywords

• Fractured porous medium
• Three-dimensional upscaling
• Equivalent hydraulic conductivity
• Numerical simulations
• Darcy
• Random fracture sets
• Network connectivity