Page 6 - Antonellini_2013

6 M. Antonellini et al. / Marine and Petroleum Geology xxx (2013) 1e16
between slip surfaces. In this work, we have assumed that the slip
T
k zdb ¼ a k 0zdb a ij (6) surfaces permeability is included in the measurements made in
ij
ij ij
zone I with the field permeameter. The permeability of the struc-
T
where a is the transpose of the matrix of director cosines. tures has been measured in the different zones (zones I, II, and III)
ij
As the formulation above may be complete for structures in and implemented according to the following rules: (1) If the fault
porous rocks just containing cataclastic material and no apparent contains a slip surface (DF), the permeability of the fault has been
diagenetic zonation, it should be re-formulated for cases where computed according to a harmonic mean of zones I and II for the
diagenesis affects differently the areas next to the shear structures. direction normal to the bands and an arithmetic mean of zones I
A form of the above equations for the studied carbonate rocks and II for the direction parallel to the fault. (2) If the shear structure
should include the different diagenetic zones (I, II, and III) in a way did not contain slip surfaces (CSB or ZB), the permeability of the
that a complete formulation for the up-scaled permeability in the structure was computed according to a harmonic mean of zones II
cell would be: and III for the direction normal to the bands and an arithmetic
mean of zones II and III for the direction parallel to the bands.
d cell
k r_ncell ¼ T r db zone 1 N r db zone 1 T r db zone 2 N r db zone 2 T r db zone 3 N r db zone 3
þ þ 3.5. DFN model construction
k r ndb zone 1 k r ndb zone 2 k r ndb zone 3
d cell
þ The DFN model was built using the Fracture Modeling module
ðd cell T r db zone 1 N r db zone 1 T r db zone 2 N r db zone 2 T r db zone 3 N r db zone 3Þ
within the commercial MOVEÔ software package from Midland
k m
Valley Exploration Ltd. Like many other commercial packages used
(7)
and to generate DFN models, MOVEÔ is designed for tight rocks
T r db zone 1 N r db zone 1 k r pdb zone 1 þ T r db zone 2 N r db zone 2 k r pdb zone 2 T r db zone 3 N r db zone 3 k r pdb zone 3
k r_pcell ¼ þ
d cell d cell

d cell T r db zone 1 N r db zone 1 T r db zone 2 N r db zone 2 T r db zone 3 N r db zone 3 k m
þ (8)
d cell

Where T r db zone 1 , T r db zone 2 and T r db zone 3 are the thicknesses of the (negligible matrix porosity) where porosity and permeability are
different zones described by Tondi (2007).N r db zone 1 , N r db zone 2 ,and provided only by the fractures. The software assigns null values of
N r db zone 3 are the numbers of zones in the cell, k r ndb zone 1 , k r ndb zone 2 , porosity to those portions of the volume that are not crosscut by
and k r ndb zone 3 are the permeability normal and k r pdb zone 1 , k r pdb zone 2 , any structure (i.e. the host rock). It then computes the porosity of a
and k r pdb zone 3 are the permeabilities parallel to the zones as given cell as the ratio of total fracture volume in a cell per cell
mentioned above. volume. The volume of an individual fracture polygon is equal to its
The permeability tensor for a ZB with associated diagenetic surface area multiplied by its aperture size. In the study rocks, both
zones ðk 0zdb diag Þ in the local reference system and within a single cell host rock porosity and permeability are not negligible. Thus we did
ij
of the model is given by not apply a standard workflow to obtain a porosity/permeability
map but we defined a workaround, which provided us with the
k r_pcell 0 0 stochastic representation of the structures within the geo-cellular
2 3
k 0zdb diag 0 k 0 5 (9) volume.
ij ¼ 4 r_pcell
0 0 k r_ncell A rectangular surface of the same size of the map shown in
Figure 3a (45 m easting and 32 m northing) was generated and geo-
The permeability tensor in the geographic reference system of referenced. The surface was then converted into a geo-cellular
the model (UTM) is according to the transformation of coordinates volume with a thickness (height) of 1 m. Squared (map view)
relationship cells with 0.2 m side (total number 7.2 10 ) made up the geo-
4
cellular volume. Two 0.5-m thick layers were present in the verti-
T
k zd b diag ¼ a k 0zd b diag (10) cal dimension of the volume. The generated geo-cellular volume
ij ij ij a ij
was then populated with six different sets of structures: right-
By using the properties of the tensors, we can add r permeability lateral DF, left-lateral DF, right-lateral ZB, left-lateral ZB, right-
tensors (where r ¼ 1, 2, 3,...n) obtained from r ZB (with or without lateral CSB, and left-lateral CSB. Specific parameters were
associated slip surfaces) having different orientations and hydraulic assigned to each set of structures: intensity (P 32 ), length, orienta-
attributes (thickness, opening of the slip surface, diagenetic zones, tion (dip azimuth, angle of dip, and Fisher K value), and aspect ratio
etc.) to obtain the total up-scaled permeability tensor of a single (length over height). The P 32 represents fracture area per unit
cell. volume expressed in units per meter [m /m ]. The Fisher K is a
3
2
statistical parameter used to define if a data cloud is clustered or
k cell ¼ k zd b diag r¼1 þ k zd b diag r¼2 þ k zd b diag r¼3 þ . (11) not (Fisher et al., 1987). Some of these parameters were obtained
ij
ij
ij
ij
from literature: orientation-related information was taken from
This procedure allows to up-scale the permeability measured Tondi (2007; Fig. 1c), and the length of the structures was defined
with the field permeameter to the cell of the numerical model in using the power laws distributions shown in Figure 5b. The in-
MODFLOW 2005. A few remarks need to be added. One of the most tensity (P 32 ) we assigned to each set of structures was calculated
difficult parameters to measure in the field is the opening in using the workflow proposed by Golder Associates Ltd. (2009). The
Please cite this article in press as: Antonellini, M., et al., Fluid flow numerical experiments of faulted porous carbonates, Northwest Sicily (Italy),
Marine and Petroleum Geology (2013), http://dx.doi.org/10.1016/j.marpetgeo.2013.12.003