A geostatistical methodology to simulate the transmissivity in a highly heterogeneous rock body based on borehole data and pumping tests
Sofia Barbosa (1) (*), José Almeida (1), António Chambel (2)
(1) Departamento de Ciências da Terra e GeoBioTec, FCT Universidade NOVA de Lisboa, Campus da Caparica, 2829-516 Caparica, Portugal, [email protected]; [email protected]
(2) Departamento de Geociências e Instituto de Ciências da Terra (ICT),Universidade de Évora, Rua Romão Ramalho, 59, 7000-671 Évora, Portugal, [email protected]
(*) corresponding author; e-mail: [email protected]; telephone: +351 212 948 573; fax: +351 212 948 556
Abstract
In this study we develop a geostatistical methodology with the aim of simulating 3D grids of transmissivity. The case study is a highly heterogeneous massif rock body mainly composed by granites and schists with distinct weathering and fracture conditions that surrounds and is part of a former uranium mine. Contrasting hydraulic behaviour is given by fractured rock and a pervasively weathered rock matrix composed mainly by clay minerals. Lithology, weathering, and fracture were the geological attributes selected for simulation. Data and information were analysed in detail for their respective integration into a sequential geostatistical modelling approach. Simulation process was conditioned to local data histograms. Simulated images of transmissivity show high levels of heterogeneity both laterally and vertically. High potential areas for flow propagation are restricted in number and interconnectivity is not particularly evident. Small channels, the main structures responsible for groundwater flow propagation, can be identified within the rock body.
Key words
Heterogeneity, sequential indicator simulation, direct sequential simulation, 3D fracturing models, double-porosity 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
1 Introduction
A geological conceptual model is a simplified version of reality and is the first tool for further developments regarding hydrogeological model framing. The accuracy of a model of a given hydrogeological attribute depends directly on the adequacy of the geological models that are previously constructed according to possible formulations and conceptualisations. Conversion of geological data into its possible correspondent hydrogeological model is a complex, time-consuming and demanding in terms of the quantity and quality of the information that must be available for this purpose. For heterogeneous fractured rocks, several studies related to fracture modeling and double-porosity flow modeling applied to oil industry case-studies represents very significant contributes that constituted a basis of inspiration to the present work (Ngo et al., 2017; Berrone at al. 2016; Noetinger et al. 2016; Berrone at al., 2015; Benedetto et al., 2014; Delorme et al., 2014; Fourno et al., 2013; Verscheure et al., 2012; Dreuzy et al., 2012; Noetinger and Jarrige, 2012; Pichot at al., 2012; Pichot at al., 2010; Lemonnier at Bourbiaux, 2010; Jenni et al., 2007; Bourbiaux et al., 2005; Fourno at al., 2004; Landereau at al., 2001; Noetinger et al., 2001; Bourbiaux at al., 1998).One of the most significant stages in the development of a groundwater model of a heterogeneous medium is the characterization and understanding of the spatial distribution of lithology and its related attributes, weathering and fracture patterns and frequency (Almeida et al., 1993; Marsily et al., 2005; Srivastava, 2005; Chambel, 2006; Eaton, 2006; Telles, 2006; Caumon et al., 2009; Almeida, 2010; Matias, 2010; Michael et al., 2010; Carlotto et al, 2018). For this purpose, the interpretation of geological parameters both laterally and vertically is essential. In a first step, a lithological model and its corresponding hydrological facies must be developed from information obtained from boreholes, geological maps, and geological interpretation of profiles of the site (Wu et al., 2005; Kaufmann et al., 2008). This lithological (and its corresponding hydro facies) model is a morphological type and can be represented by limiting the top and basal surfaces or by a block model where a certain lithology is attributed to each grid block. If storage and groundwater flow are of the “porosity” type, then the hydrogeological properties needed for groundwater modelling and dynamic simulation are calculated according to the original geological model. Upscaling adaptation may be needed to produce a more accurate and 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52
adequate final hydrogeological model. If groundwater flow is of the “fracture” type, then grids of equivalent hydraulic properties may be generated conditional to the spatial variability in the number and types of fractures. For “fracture” type models, a fracture network may be developed for which equivalent hydraulic properties are estimated (Adler & Thovert, 1999; Bogdanov et al., 2003; Molson et al., 2012). In all cases, the final result is a grid block model in which each cell block is attributed certain hydrogeological properties (such as transmissivity or storage coefficient) that are crucial for groundwater flow simulation.
Model realism, that is to say, the way in which a model reproduces reality, depends on the degree of lithological discretization, cell block dimensions, and the degree of interpretational simplification that may be needed. It should be noted that a certain model may be very detailed in terms of lithology and spatial resolution but may also be less realistic if interpretations of information or conditional statistics are not adequate (Dimitrakopoulos, 1998; Kanevski et al., 2004; Arpat et al., 2007; Michael et al., 2010; Shafer, 2014).
For geological modelling, either object-based (deterministic or stochastic) models or geostatistical (estimation or simulation) models may be developed (Kanevski et al., 2004; Wu et al, 2005; Kaufmann et al, 2008; Michael et al, 2010; Comunian et al., 2011; Castilla-Rho et al., 2014). The resolution, variability and spatial representation of the different types of information that are needed (e.g., geological maps, logging, geophysics, hydrochemistry, and pumping tests) may vary significantly. For this reason, most of the time, to integrate as much information as possible, combinations of object-based and geostatistical models are used.
Hydrogeological models may be extremely simple with all the grid cells having the same average value for a certain hydrogeological attribute but may also be extremely complex with each grid cell having different values. In addition to this diversity, different equiprobable scenarios may be considered to represent an unknown by essence reality and therefore it is necessary to conceive the uncertainty of the model. Equiprobable scenarios may be very different (high uncertainty), moderately different (medium uncertainty), or equal or very similar (low uncertainty). A good model, in addition to realistically reproducing the information known in all variables (e.g., in situ tests, log information 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79
and cartographic and profile information), must carry the uncertainty based on the variability of the various scenarios generated.
Object-based modelling is a technique traditionally used in geological modelling to reconstructed channel structures and fracture networks. Such modelling is also used to generate geological objects such as discontinuities. Deterministic (geological modelling and interpolation) or stochastic strategies (fracture simulation) can be used to generate geological objects (Chilès et al., 1992; Soares and Brusco, 1997; Castilla-Rho et al., 2014).
Deterministic object-based modelling generates a unique average scenario and is strongly dependent on the interpretational capabilities of experts. Such modelling does not quantify uncertainty and therefore its utilization is limited. Conversely, stochastic object modelling, however, allows the generation of high-density (spatial frequency) geological objects such as fractures. A deterministic model may be used as a base model for the development of a stochastic model.
Geostatistical models are adequate for generating continuous and categorical attributes (morphological models) in regular cell block grids. Either an estimation or a simulation modelling strategy may be considered. A substantial advantage may be achieved when geostatistical models are generated conditional to geological information (Dimitrakopoulos, 1998; Matheron et al., 1998; Kanevski et al., 2004; Arpat and Caers, 2007; Michael et al., 2010; Shafer, 2014).
In the case study presented (Barbosa, 2013), we model a highly heterogeneous fractured aquifer system. Zones with high fracture densities are found interspersed with zones of low fracture densities. Zones of weathering also occur at different depths and sometimes in unexpected locations and contribute barrier effects to flow propagation. In such circumstances, it is to be expected that a transmissivity model reproduces these circumstances by generating (by simulation) several equiprobable scenarios. In each equiprobable scenario, we expect to observe small groups of cell blocks of high transmissivity that are included in a highly contrasting rock matrix of medium to low transmissibility. High level of spatial heterogeneity must be well evidenced in simulation results (Marsily et al., 2005; Eaton, 2006; Michael et al., 2010, Carlotto et al., 2018). There are several main determinants of groundwater flow propagation in a rock matrix. One is the rock matrix porosity (Pereira and Almeida, 1997; Fialho et al., 1998). A second is the fracture porosity, whereby flow 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107
transmissivity is a function of the fractures in the rock matrix and their 3D spatial interconnectivity. In fractured rock, flow circulates through intergranular pores of the more weathered parts of the rock body, microscopic interstitial pores, micro-fractures, and macroscopic features such as fractures, schistosity, and other structural discontinuities. It has become common to consider double porosity models (Moench; 1984; Röhrich, 2005; Chambel, 2006; Telles, 2006) when interpreting the flow conditions of fractured rock, whereby the total porosity of the rock system is calculated as the sum of rock matrix porosity and fracture porosity. The relationship between storage capacity (i.e., porosity) and hydraulic conductivity for rock matrix and fracture systems vary significantly according to lithology, the degree of alteration, the intrinsic characteristics of fractures, and the interconnectivity of the fracture network.
The main scope of our work was the development of a geostatistical methodology where distinct types of data, achieved from distinct characterization site investigation campaigns, are associated in order to produce a unique final result: 3D models of transmissivity of a massif rock with high heterogeneity. Our main objective was that the simulated 3D models of transmissivity could respect simultaneously distinct types of data and information, deterministic or randomly generated, directly or indirectly related to the local intrinsic and structural properties of the rock massif and its corresponding transmissivity conditions. Also, it was objective of our work that the data available for the site and related with distinct sampling campaigns and surveys could be integrated and have its own influence on the final generated models. First, an exhaustive study considering different drilling and logging surveys, geophysical site investigations and hydrochemical results from a groundwater monitoring system was performed. Secondly, data sets of the main variables considered relevant for the development of the transmissivity models - “lithology”, “weathering” and “fracture” – were prepared. In this article we present the geostatistical methodology that was conceived in order to conciliate data obtained from well drilling surveys with data generated from in situ pumping tests that were executed in distinct campaigns. For this, and in order to be possible to aggregate the maximum data information possible that influence the transmissivity of the rock massif, a sequential geostatistical procedure step was developed. Data from distinct variables were sequentially introduced in the simulated models according with its geological hierarchical relevance on the resulting transmissivity of the rock matrix. 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135
The order who data was introduced in the sequential geostatistical step procedure was: (1) “Lithology”, (2) Weathering (3) “Fracture” and (4) “Transmissivity” from pumping tests. Simulation procedures and results relatively to each step are presented and discussed. For the categorical variables “Lithology” and “Weathering”, Sequential Indicator Simulation (SIS) was the geostatistical methodology considered, and for numerical (continuos) variables “Fracture” and “Transmissivity” it was considered the Direct Sequential Simulation (DSS) algorithm. The models were simulatesd with conditioning to local data histograms. In a final step, Transmissivity results from pumping tests were integrated in 3D simulated models through a crossing histogram procedure that is explained in detail. This procedure establish correlations between the determinist parameter “Transmissivity” and the intrinsic and structural properties of the rock massif that were numerically simulated in previous steps. 2 Materials and Methods
2.1 Study area
The studied massif rock body is located at Central Region of Portugal and is composed of Hercynian granites and a schist–greywacke metasediment complex of the Iberian Hesperian Massif. In general, these granite rock have very low hydraulic conductivity and storage coefficient and a natural fracture system usually characterized by poor storativity but high hydraulic conductivity. The rock body surrounds a former uranium mine located in central Portugal. Groundwater exploitation is restricted to local use, such as for subsistence agriculture and private domestic use, or even small industries. Uranium mineralization has been exploited from a rocky enclave of ferriferous mica schist bounded to the north by a to coarse-grained porphyritic granite and to the south by a fine- to medium-grained biotitic granite. Mining activity has resulted in the formation of an open pit, and rock waste dumps still remain in the area (Barbosa, 2013). This mining site therefore has potential for contamination by radionuclide dispersion both through surface waters and groundwater flow. The rock matrix is highly heterogeneous, and hydrothermal and supergenic alteration has led to the generation of variations in mineralogy with consequent high levels of local variability in lithological facies. Fractures and veins with different mineralogical compositions also add to the heterogeneity in conditions that affect hydraulic behaviour.
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Two granite facies occur in the study area, porphyritic granite and biotitic granite, and exhibit distinct hydraulic behaviours. Porphyritic granite has a more intensive weathered rock matrix whereas biotitic granite has a greater tendency to exhibit an open fracture network and is much less weathered. For this reason, porphyritic granite has some zones that have primary permeability. This primary permeability is low to very low as a consequence of clay-rich minerals derived from alteration of the host rock. Locally, some fractures have a clay-rich infill, and therefore barrier effects are present. In contrast, biotitic granite has secondary permeability. Open fractures, clay-mineral-free quartz veins, and granitic breccia are the main features responsible for controlling local groundwater flow propagation. The rock massif shows high levels of variability with respect to its weathering and fracture spatial patterns and is in direct contact with an old pit related to uranium exploitation (Barbosa, 2013). Groundwater flow modelling for this case study is therefore a very environmentally sensitive and relevant issue (Pereira et al., 2004; Pereira et al., 2005).
The granitic rock bodies studied have a very high level of heterogeneity with respect to its
major geological characteristics such as lithology, weathering degree and fracture network
(Barbosa, 2013).
A general view of the study area and borehole location is present in Figure 1a). In our study, lithology was considered to be the key variable with respect to its influence on local and regional hydraulic properties, with both weathering and fractures being partial functions of lithology (i.e., each is, in part, statistically and spatially dependent on lithology). In the study area, a two-mica porphyritic granite and a grey biotitic low- to medium-grained granite are the main lithologies as judged by rock volume. The porphyritic granite dominates in the northern part of the open pit whereas the biotitic granite predominates in the southern part (Fig. 1b). The remaining of a mica schist rock body that has been exploited for uranium production is found in the central area of the open pit. 2.2 .Geostatistical methodology and frameworkThe main objective of the developed methodology was to generate 3D equally probable transmissivity models for a highly heterogeneous, fractured, and weathered massif rock body mainly composed of granites and mica schist. As part of the methodology, data and information were inspected and analysed concerning their respective integration into 3D geostatistical modelling. For this, a detailed 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189
and systematic analysis of borehole data was performed with the aim of identifying local geological conditions as well as the relevant geological properties and their spatial variability. Three variables (geological properties) were selected because of their direct influence on groundwater flow conditions: lithology, weathering level, and the intensity (and orientation) of open fractures. Data for these three variables were processed along with direct information from boreholes and other indirect information such as geophysical surveys (Carvalho et al., 2005), gamma radiation from well logging (Lukes, 2005), and physical and chemical results of groundwater monitoring.
The results of pre-processing and interpretations of pumping test were used to establish relationships between the selected geological variables and their corresponding transmissivities and storage coefficients. This was a crucial methodological step where cumulative histograms of the geological and hydrogeological variables were crossed making possible to find correlations between geological attributes and transmissivity.
The grids of the geological attributes were generated in a sequential step approach with the aim of integrating, to the greatest possible extent, local heterogeneity in the final hydraulic 3D grids.
With the applied crossing histogram procedure, the value of transmissivity of each cell grid in the final 3D models is expressed in the direct dependence of its spatial correspondent geological attributes of lithology, weathering, and fracture characteristics.
According to Figure 2, the developed geostatistical methodology involved the following sequential steps:
Step 1. Build a 3D geological model of the lithologies and weathering levels using sequential indicator simulation (SIS) conditional to borehole data and regional proportions of most spatial representative lithologies.
Step 2. Analyse the intensity and orientations of fractures encountered in the boreholes and their correlation with lithologies and weathering levels.
Step 3. Build a 3D object-based model of fractures conditional to the number of open fracture of each block; fractures are simulated as polygons, in this particular case as squares, and fracture sizes and openings follow a power law and a uniform distribution law, respectively.
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Step 4. Simulate the number of fractures (NF) in a 3D grid of blocks conditional to the borehole data and to the previous geological model using direct sequential simulation (DSS) with local histograms conditional to lithologies and weathering.
Step 5. Analysis and interpretation of pumping tests results, compute the local transmissivity near the tested piezometers or along directional section.
Step 6. Build distribution functions of transmissivity conditional to each lithology and weathering level by associating transmissivity values with the corresponding local geological conditions.
Step 7. Establish a bivariate relationship between fracture intensity and transmissivity according to corresponding local geological conditions.
Step 8. Cross histograms for converting the number of fractures to transmissivity according to lithology and weathering classes.
Step 9. Simulate equally probable 3D transmissivity scenarios considering and distinguishing zones of rock matrix with primary permeability from those with secondary permeability.
Specific codes for conditional sequential indicator simulation (SIS) and conditional direct sequential simulation (DSS) were developed in Visual C++ language with the aid of Visual Basic v. 6.0. Object-based models of fractures were developed with the computer program application FTRIAN (Almeida and Barbosa, 2008; Barbosa, 2013) and analysis of pumping tests were made with the software AquiferTest v.4®. Map results are presented using the Geostatistical Modelling Software geoMs copyright CMRP-IST 2000 and the software MOVE ® 2012.
As will be evidenced, with this geostatistical workflow, the final model of transmissivity is simultaneously conditional to the geological model (equally-probable simulated images of lithology and weathering), to the well data (equally-probable local fracture intensity), and to the pumping tests (regional fracture intensity).
Simulation geostatistical models allow to evaluate the spatial uncertainty associated to the simultaneous behaviour of the starting data. This evaluation is obtained through several equiprobable images, which together are understood as representative of the spatial phenomenon under study and, thereby, of uncertainty (Soares, 2006; Almeida 2010; Nunes e Almeida, 2010). This is not verified through estimation models like, for example, those that results from kriging.
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In stochastic simulation, if we denote by
Z
c(
x )
the set of simulated values and byZ
(
xα)
,=1 , … , n , the n experimental values, the simulated image fulfil the following conditions:First) for any value of z
z : prob
{
(
x
α)
<
z
}
=prob
{
Z
c(
x)<z
}
(1)Second)
γ (h)=γ
c(h) ,
(2)being γ
(
h)
andγ
c(h )
the variograms, respectively, of the experimental values and the simulated values;Third) Conditioning at experimental values is respected. At any experimental point xα , the value
Z
(
xα)
and the simulated valueZ
c(
x
α)
are coincident, that is,Z
(
xα)
=Zc(xα) (3)This means that, in addition to having the same variability, the simulated images passes through the experimental points and, therefore, sample values and simulated values are spatially coincident. This means that sample values influence the simulated maps according to the variograms models which determined the greater or lesser structural continuity of data.
For these reasons, simulation models, rather than reproduce the average characteristics of reality, intend to transmit the "impression" resulting from the variability of this same reality (Soares, 2006). This situation fits within the scope and objectives of the work.
2.3 3D lithology and weathering geological models (Step 1)
Lithology was processed as a dichotomous categorical variable, and the 3D grids of lithology were generated using a Sequential Indicator Simulation (SIS) algorithm with local probability correction (Ruben and Almeida, 2010; Soares, 2006).
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When the categorical variable to be simulated displays several phases (or classes), multiphase variogram can be used for the simulation process, that is, instead of using one variogram by class, a single variogram is used. This variogram results from the sum of the individual variograms of each phase (or class) (Soares 1992). A vector of categorical variables
I
k(
x)
of a multiphase system can be expressed by:{
1 if x
∈ X
k, k =1,… K
0if x
∈ X
k, j≠ k
(4)For any phase
X
k we can define the meanm
k=
E
{
I
k(
x )
}
, which is the proportion ofX
k inA . The spatial continuity of the set
I
k( x ) , k=1, … , K
maybe measured by covarianceC (h)
or multiphase variogram(h)
, that is:C (h)=E
{
∑
k =1 K[
I
k( x ) I
k(x +h )
]
}
(5) or(h)=
1
2
E
{
∑
k=1 K[
I
k(x )−I
k( x +h)
]
2}
(6)Sequential conditioning has the practical drawback of generating dependencies in the results between the first points when there are still few simulated nodes. In fact, if the first simulated nodes are in the vicinity of the experimental values of a given phase, the proportion of values of that phase tends to increase rapidly at the beginning and will hardly return to the values that were set as objective. This effect is all the more significant as the amplitudes of the variograms are used, affecting mainly the classes of lower overall proportions (Soares, 2006). Correction of local probabilities, iteration to iteration, as proposed by Soares (2006) has the advantage of approaching the proportions of each phase in the final image to the objective that is intended to be reached. This procedure can be developed globally or by regions, especially in case of less abundant phases.
Considering the field of study
A ,
divided into a set of disjoint multiphase bodies,X
k, k=1, … , K
, the SIS algorithm for multiphase sets with correction of the local probabilities follows the following steps:267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290
First) For a given node to simulate
x
1 in a random locationA ,
the estimation by kriging of theindicator of the probability vector is
[
I
k(
x
1)
]
¿
=
prob
{
x
1∈ X
k}
¿
, k=1, … K
(7)Estimated values
[
Ik(x1)]
¿, k=1, … , K can be obtained based on the individual covariance ofeach phase or on the multiphase covariance. In this case, only one kriging system is required for the estimation of the multiphase structures.
Second) Correction of possible alteration of order relations is performed. The sum of the estimated values of local probabilities must necessarily be unitary, that is,
∑
k=1 K
[
I
k(
x
1)
]
¿=1
(8)It may happen that the estimated values are not contained in the range
[
0,1
]
,
that is,I
k(
x
1)
¿>
1
orI
k(
x
1)
<
0
, due to the existence of negative weights that may result from the individual phase ormultiphase kriging process. In this situation, the usual methods of correction of the kriging order relations of the indicator variable should be used. In general, however, it should be expected that multiphasic kriging ensures this relationship, since it uses a single model to estimate the k values
[
I
k(
x
1)
]
¿
.
Third) Computation of the cumulative function in
x
1 is expressed by[
F
1(
x
1)
]
¿=
[
I
1(
x
1)
]
¿ (9)[
F
2(
x
1)
]
¿¿
[
I
1(
x
1)
]
¿+
[
I
2(
x
1)
]
¿ (10) (…)[
F
l(
x
1)
]
¿=
∑
k=1 lI
1(
x
1)
¿, l=1, … , K
, (11)Fourth) Generation of a uniformly distributed p-value between 0 and 1.Then, x1 belongs to the
phase k if p∈
[
[
Fk−1(x1)]
¿,[
Fk 1(x1)]
¿]
, that is, the simulated value in x1 becomesI1
(
x1)
=1 and Ij(
x1)
=0 with j≠ k . 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313Fifth) The simulated values
I
k(
x
1)
, k=1, …, K
, are integrated into the simulation conditioningset.
Sixth) The sequence from 1 to 5 is repeated until the total set of points in A is simulated.
Soares (2006) proposes a correction performed by conditioning the overall proportions (or objectives)
of each class after the calculation of the multiphase kriging estimator of the probabilities
[
I
k(
x
1)
]
¿ , as indicated in second step.Considering pk as the objective proportion of each class K , a certain deviation
e
KS may be
calculated between
p
k and the correspondent proportion of a certain given simulation steps
e
K S=
p
k−
p
K S (12) wherep
K Sis the proportion of class K in the simulation step s
p
k s=
1
N
s∑
i=1 Ns[
I
k(
x
i)
]
¿, k=1, … , K
(13)e
N
s is the number of estimated points until steps
is achieved.Estimated local probability
[
I
k(
x
i)
]
¿ are corrected by the deviation eKS, k =1, … , K as follows:[
I
ks(
x
1)
]
¿=
[
I
k(
x
1)
]
¿+
e
ks, k=1, …, K
(14) If the deviations are null, that is,∑
i=1 K
e
ks=
0 ,
and the sum of estimated values is equal to 1,∑
k=1 K
[
I
k(
x
1)
]
¿=1
, then, the sum of the corrected values will also be equal to 1 in any steps
of the simulation, that is:∑
k=1 K
[
I
is(
x
1)
]
¿=1
(15)With this algorithm of local probabilities correction, the best possible reproduction of global probabilities of each class in each simulated scenario is guaranteed. This is one of the main objectives that needs to be insured so that the simulated scenarios are reliable.
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Simulations of lithology were run conditional to borehole data to achieve a better de-clustering effect for the simulation results (Soares, 1998). 11 conceptual models for lithology were designed once the study area is complex in terms of its geology. From these, one model with six classes of lithology porphyritic granite, biotitic granite, mica schist, migmatite granite, breccia granite, and pegmatite -was selected for simulation. Simulation conditional to regional proportions of the two main spatially prevalent lithologies was also performed because the two main lithologies in the area (the porphyritic granite and the grey biotitic granite) have nonstationary patterns and a 3D spatially variable location with two main domination areas: porphyritic granite in the Northern and biotitic granite in the southern, respectively. To ensure a high degree of spatial continuity of the two major lithologies of porphyritic granite and biotitic granite, and to respect the main boundary between these two lithologies, simulations were performed conditional to the spatial extents of the two types of granite. These geometric regions were established according to geological cartography of the site at a scale of 1:5000 (Fig. 1b). Subsequently all categorical data for lithology were re-scaled in the dependency of porphyritic granite and grey biotitic granite 3D location. Twenty equally probable simulated scenarios for lithology were generated using SIS with regional conditioning to the main lithologies, porphyritic granite, and, grey biotitic granite. In our case study, multiphase variogram models, that is, variographic interpretations considering simultaneously the distinct lithology classes (usually called “phases”) were not constructed because of the high spatial variation detected for different lithological classes. In fact, differences between the amplitudes of the variograms for distinct classes were found and, in such conditions, variographic interpretations considering simultaneously the distinct classes (or phases of the multiphasic variogram models) leads to inadequate results. In such conditions, and in order to obtain a better declustering effect, simulations were performed conditional to the local proportions of feach lithology class according to previous indicator kriging of experimental data. Similar to lithology, weathering was also treated as a categorical variable. Four classes were established ranging from less weathered (W1) to most weathered (W4) rock matrix. Weathering classes were compared with lithology classes with respect to two depth intervals of investigation (0–15 m and 16–95 m). The stationarity of weathering was analysed, and dependencies on lithology and depth interval were found. Weathering was therefore simulated according to its classification into 12 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362
distinct modalities that are the result of the combination of 6 lithology classes and 2 depth interval classes.
The methodology developed to simulate lithology and weathering 3D models is shown in Figure 3. In our study case, 20 equally probable simulated scenarios for lithology were performed using SIS and, subsequently, a total of 100 equally probable scenarios of weathering were simulated, comprising 5 simulations of weathering for each of the 20 scenarios of lithology previously simulated.
2.4 3D fracturing models (Steps 2, 3 and 4)
In rock massifs, equivalent hydraulic parameters are conditional to the local fracture network and its intrinsic characteristics (Chilès et al., 1992; Grossman, 1988; Arpat et al., 2007; Almeida and Barbosa, 2008; Chesnaux et al., 2009, Michael et al., 2010; Molson et al, 2012; Shafer, 2014). For 3D models of fracturing (Soares and Brusco, 1997; Caumon et al., 2009), fracture characteristics, intensity and orientation (Sausse and Genter, 2005; Lukes, 2005; Lin et al., 2007), encountered in the boreholes and their correlation with lithology and weathering levels were processed and coded with the intent of using geostatistical treatment procedures. However, fracture is equally as crucial to understand as to quantify, especially in those cases where data is collected and interpreted from boreholes that lacked core orientation procedures during drilling operations (Lukes, 2005). The linear density of fracturing (LDF) is a common index for quantifying fractures. In the present study, because all the boreholes are vertical, this index tends to underestimate the density of vertical fractures whereas horizontal to subhorizontal fractures are generally better detected compared with other possible existent fractures with different orientations. The relationship between LDF and 3D fracture number (NF) was established with FTRIAN. Following an object-based modelling procedure, FTRIAN generates a certain desired total number of fracture as square polygons. Through successive iterations, it uses a unit volume in which randomly generate fractures according to the orientations and dimensions previously measured. Fracture lengths and openings follow a power law and a uniform distribution law, respectively (Handy, 1989). The final result is a lookup table with two columns that give the correspondence between LDF (1D information) and 3D fracture number.
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A detailed fracture pattern study was developed from the interpretation of the recovered testimonies in drilling works. In the performed study, the relative frequencies of “fractures per linear meter” according to the characteristics "inclination", "fill" and "aperture" and according to the lithology and weathering degree were measured considering 1D scanlines of 5 meters. However, as drilling works were not carried out with core orientation, the conclusions and representativeness of fracture data interpretations and simulations must be considered with caution and with the necessary reservations. The performed measurements allowed the establishment of the cumulative histograms of Linear Density of Fracture (LDF) in function of the predominant lithologies (Porphyry Granite and Biotite Granite), weathering degree and depth (relative to the reference level of 15 meters). For the simulations in FTRIAN, some assumptions and adaptations had to be considered. Taking into consideration the limitations of the fracture pattern recognition derived from non-oriented testimonies, the simulations of the variable “Number of fractures” in FTRIAN were performed considering the sum of all existing fracturing data with full rotations, that is, using the total spectrum of possible attitudes (directions from de 0 to 360 and inclinations from 0 to 90), and the sum of all fracture families (sub-horizontal, sub-vertical and oblique) intersected by 1D scanlines of 5 meters. The treatment for the sum of all facture families classified according to its inclination was considered because it was impossible to determine its real attitude. Also, the treatment was performed considering the sum of the three families of fractures (sub-horizontal, sub-vertical and oblique) since the proportions between the number of horizontal and oblique families was very close, being possible to consider the possibility of proximity also relatively to the quantity of the number of vertical fractures (although the measured quantities of this family are always in lower number). Closed or filled fractures were treated as “non-existent” being only considered for subsequent simulation the number of fractures apparently "without fills" (that is, the fractures with more potentiality on fluid flow performance). Also, it was considered appropriate to simulate the variable “Number of fractures” from the conditional cumulative histograms established by lithology and degree of weathering. Fracture dimension in FTRIAN varied from 10 cm (the smallest dimension captured in drilling testimonies considered to have sufficient representativeness for accounting) and 5 meters (the model reference dimension).
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The number of fractures per unit of volume (3D fracture number, NF) was simulated in the dependence of LDF and used for subsequent simulation of 3D models of fracture. 3D models of fracture number were simulated using Direct Sequential Simulation (DSS) algorithm (Soares, 2001; Soares, 2006; Nunes and Almeida, 2010; Durão et al., 2010; Barbosa, 2013).
Direct sequential simulation (DSS) is a simulation algorithm which uses directly the original numeric number of the variable (continuous variable). Previous data transformation, like Gaussian transformation for example, it is not needed which is an obvious advantage (Soares, 2006). According to Soares (2006), the idea of SSD arose from the following postulate: if the local distribution laws of
Z ( x )
are centered on the simple kriging estimator (eq.16) with local conditional variance equal tothe kriging variance
σ
ks2
(
x
0)
,
then spatial covariates or variograms are necessarily reproduced insimulated final maps (Journel, 1994).
[
z
(
x
0)
]
¿
−
m=
∑
α❑
α(
z
(
x
α)
−
m
)
(16)The problem with this principle is that the local distribution laws cannot be characterized only by the local means and variances of the original variables and it is necessary to ensure the reproduction of the histogram of the local distributions Z
(
x)
.For this, the idea associated to the DSS variant proposed by Soares (2006) is the use of local averages and variances to re-sample the global distribution law. More specifically, Z
(
x)
intervals of the global distribution lawF
Z(
x)
are chosen to construct a new function F'Z(
z)
which isconsidered to simulate the values zS(x0) . The intervals of the new function F'Z
(
z)
are centredin local mean which is estimated by simple kriging
[
z (x0)]
¿ . These intervals have a proportional amplitude to local conditional variance which estimated by from variance of simple krigingσ
ks2
(
x
0)
. DSS of a variable Z(
x)
inx
0 ,z
(
x
0)
, can be explain according the nextfollowing steps:
First) Establish a random path that traverses in a unique step all mesh nodes to be simulate; 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439
Second) In each node to be simulated,
x
0,
estimate the local meanz (x
0)
¿
by simple kriging and
the variance
σ
2sk(
x
u)
conditional on experimental data and previously simulated nodes;Third) Definition of the intervals to be considerate for the histogram of the global distribution function of the variable to be simulated
F
z(
z )
;Fourth) Generation of the simulated value
z
s(
x
0)
, according to the following procedure: (1st)Generation of a random p -value with uniform distribution law
U (0,1) ,
(2nd) Generation of thesimulated value
y
s through the relationshipG
(
y
(
x
0
)
¿
, σ
sk2(
x
0)
)
,
(3rd) Inverse transformation toobtain the transformed value of the original distribution law
z
s(
x
0)
=
φ
−1(
y
s)
and add the value as conditioning information to the set of the remaining nodes;Fifth) Repeat steps 2 to 5 until all nodes are simulated.
If instead of a global distribution law,
F
Z(
z)
, local distribution laws are knownF
Zi( z ) ,i=1 … n
R , that is, regional distribution laws are known, then, the simulation can be better locally conditioned with this simulation procedure, being respected the formalism and algorithmic principles (Roxo et al, 2011).In the sequential simulation procedure, the 3D models of NF (which is a continuous variable) were generated by Direct Sequential Simulation (SSD) with conditioning to the local histograms of the number of fractures defined per degree of weathering W1, W2 and W3 (which indirectly includes depth and lithology).
To determine the correlations between open fracture and the other variables, cumulative histograms conditional to lithology and to the three weathering classes W1, W2, and W3 were established. According with this local histograms (Roxo, 2011; Barbosa, 2013; Roxo et al 2016), the variables were simulated in a chain, with weathering conditioned to lithology and fracture number conditioned to weathering (Figure 4). Following this simulation procedure, three simulated scenarios were generated for each aforementioned “weathering” simulated scenarios (total of 100). For each previous group of 5 simulations of weathering developed for each of the 20 simulations of lithology, 3 equally 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464
probable scenarios of the number of fractures were generated using DSS. Therefore, a total of 300 equally probable scenarios of 3D NF were generated. Correspondence between the FTRIAN “3D fracture number” and local geological conditions was possible to be established once cumulative histograms established by lithology and degree of weathering were considered in the DSS simulation procedure.
In synthesis, 3D simulated models of number of fracture were develop according with the next three steps:
(1) Conduct a detailed analysis of fracturing intensities and orientations in boreholes and their correlation with lithology and weathering level;
(2) Establish a lookup table of correspondence between LDF (1D fracture information) and 3D fracture number (NF) following an object-based modelling procedure; and
(3) Simulate the intensity of fractures in a 3D grid of blocks conditional to borehole data and to the geological model established previously using DSS with local histograms conditional to lithology and weathering.
2.5 Analysis and interpretation of pumping tests (Steps 5 and 6)
Pumping test results were analysed in order to quantify transmissivity and storage coefficient for different monitoring piezometers which corresponds to differing lithology and weathering local conditions. Transmissivity is the parameter considered in the present discussion for model simulation. The study area may be characterized as a complex weathered and fractured geological media that have been subjected to hydrothermal and supergenic alteration processes. Geological materials with high levels of heterogeneity generally give rise to pumping tests with more complex shapes and with different drawdown stages that may have different meanings compared with less heterogeneous materials. The interpretations made considered that each pumping test is divided into different sectors, each one representing distinct hydrodynamic conditions. Distinct hydrodynamic behaviours are defined according to the main sectors identified. These behaviours are typical on fractured rock massif were distinct pumping stages are present and represent flow of distinct origins (for example: main fractures, secondary permeability of fracture network systems, and, primary permeability derived from 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491
the porosity of the rock matrix) A schematic example of a general pattern were it is possible to distinct different behavioural sectors is presented in Figure 5a). In this case study, individual sectors of each pumping test were adjusted to the standardized curves of “Double porosity” and “Theis with Jacob’s correction” models. Interpretations were made with software AquiferTest v.4®, which includes a specific module for “Double-Porosity” models (Röhrich, 2005). In general, three main interpretative sections were possible to be distinguished (Figure 5a):
(1) An initial step, with pronounced drawdown effects derived from pumping. In fractured rock matrix, the pumped water usually comes directly from groundwater flow productive fractures. (2) An intermediate step, where flow starts to involve water that is present in the rock matrix.
Drawdown is therefore significantly lower when compared with that which originates in the fracture network.
(3) A final step, where drawdown from the rock matrix tends to equalize drawdown from fractures. Globally, this represents a system that produces a response similar to that of a simple porous-equivalent media. This response results from the combination of flows that propagate in fractures and in the rock matrix. It is an equilibrium stage in which it is possible to establish approximations to the “Theis” model.
This type of behaviour is completely different from what typically happens in primary permeability rock matrix (example of drawdown line in grey in Figure 5a), like it is the case of totally weathered rock matrix (W4 class). The interpretations of the pumping test results allowed to determinate values of transmissivity and storage coefficient. Distribution functions of transmissivity conditional to each lithology and weathering level class were built by the association of transmissivity experimental data with their corresponding in situ geological conditions (Figure 5b). At this stage, secondary information, especially drill logging information, was taken into account for the interpretation of the results. This secondary information was very useful for establishing relationships between the hydraulic parameters and lithology and weathering, ns. Distribution functions of transmissivity conditional to each lithology and weathering class were built allowing, subsequently, to find correspondence between transmissivity and the number of fractures through the respective lithology 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518
and weathering classes as it is explain in next modeling steps relative to “3D models of transmissivity” (crossing histograms procedure).
2.6 3D models of transmissivity (Steps 7, 8 and 9)
Modelling the hydraulic parameters of the fractured and weathered rock body implies that zones with contrasting behaviours regarding the mechanisms of flow propagation need to be taken into account. In our study, the most intensive (pervasive) weathering zones (W4) with primary porosity were considered differently from the less weathered zones (W1, W2, and W3), where fracture patterns prevail over weathering level and act as determinant elements for flow propagation. Hydraulic parameters for these distinct media conditions (W4, and W1-W2-W3) have marked differences. For this reason, the hydraulic parameters for W4 were discriminated from other zones that are dominated by a fractured rock matrix (W1-W2-W3). 3D modelling of hydraulic parameters of zones with primary permeability were discriminated from the 3D modelling of hydraulic parameters of zones with secondary permeability. The specific procedure for simulating transmissivity models was developed in order to generate equally probable scenarios of a rock matrix that has contrasting behaviour with respect to primary permeability and secondary permeability. The developed methodology take into consideration that the final 3D results must be coherent, with no spatial overlap of these two contrasting hydraulic behaviour conditions. First, histograms were crossed to convert number of fractures to transmissivity according to lithology and weathering classes (Figure 6). Then, 3D transmissivity of the fractured rock matrix of secondary permeability zones (W1, W2, and W3) and of pervasively weathered rock (W4) were simulated. 3D transmissivity of the fractured rock matrix of W1, W2, and W3 (secondary permeability zones) was simulated by crossing histograms of experimental data for transmissivity with histograms derived from simulated scenarios of the number of fractures and lithology. On the other hand, 3D transmissivity of pervasively weathered rock (W4) was simulated by probability field simulation (PFS) (Srivastava, 1992; Froidevaux, 1993; Barbosa, 2013) according to the previously simulated 3D matrix of weathering and conditional to the experimental data for transmissivity for W4 zones.
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PFS algorithm aims to generate values of a property in a known study area, the histogram of this variable and its spatial continuity model (Srivastava, 1992, Froidevaux, 1993). If we do not consider a variogram model, PFS coincides with Monte Carlo Simulation. In fact, PFS is a simulation algorithm similar to Monte Carlo algorithm where a spatial continuity model is imposed by a variogram. PFS is not conditional to data, however, errors can be estimated for later conditioning. With the W4 zones and as it was intended to simulate W4 zones randomly but following a previous spatial continuity model, PFS algorithm was considerate to be adequate for simulation.
PFS algorithm is simple. Firstly, an image of probability values with a uniform law between 0 and 1 is generated. This image describes the desired continuity model. This mesh of probability values can be obtained from a mesh of Gaussian values simulated by SSG and transformed to probability after its respective ordering. In each localization to be simulated,
x
u , the cumulative distribution function of the variableZ
is known and may be the same for the whole area A or per region (eq. 17)F
Z(
x
u)
=
Prob
{
Z
(
x
u)
<
z
}
(17)After generating a mesh of probability values
(
x)
with uniform law for the entire areaA
and following a variogram model, the simulated value in each localization is equal to:Z
s(
x
u)
=
F
−1z(
x
u, p
)
,x
up=
¿
) (18)In order to generate several images, the PFS method starts from several simulated probability images independently of each other.
The non-overlapping of zones with differing hydraulic behaviour (of primary permeability and of secondary permeability) is ensured because of the following: (Frist) W4 3D locations are established in the initial steps of the procedure in the weathering simulation stage, (Second) the simulation of a variable number of fractures is conditional to local histograms of lithology and weathering (and the depth of investigation), and (Third) pervasive weathering conditions (W4) are randomly and independently simulated using PFS conditional to the experimental data for transmissivity only for the W4 simulated zones resulting from step (4 ), where 3D transmissivity of pervasively weathered rock (W4) is simulate by PFS. 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570
This methodology ensures that the hydraulic parameters of each grid cell of the 3D models are simulated conditional to experimental data for pumping tests and weighted according to local conditions of lithology, weathering (and depth), and (when applicable) to a certain open fracture intensity. This conditional simulation sequence is crucial because the variables of lithology, weathering (and depth), and the number of open fractures were identified as the most relevant variables for the propagation of local groundwater flow in the studied rock body.
For the groundwater modelling of fractured rock, the degree of weathering needs to be considered because it affects the flow and propagation conditions of fluids in underground environments. In matrix zones with weathering levels of W1, W2, and W3, flow propagation is highly dependent on the characteristics and network of fractures. In contrast, the flow in pervasively weathered matrix (W4) has a different behaviour essentially conditioned by primary permeability. In granites, clay minerals resulting from host rock alteration in zones of pervasive weathering will decrease the primary permeability of weathered rock matrix. As described, W4 experimental transmissivity data were considered and introduced in a later procedure step following a particular PFS geostatistical procedure. The number of fractures was not considered when performing simulations for W4. W4 transmissivity values were randomly simulated using PFS directly from the corresponding cumulative curve of transmissivity experimental data.
As discussed above, the modelling of hydraulic parameters of fractured and weathered rock implies the consideration of zones with contrasting behaviour regarding the mechanisms of flow propagation. Zones of the rock body characterized by intensive (pervasive) weathering (W4) with primary porosity must be considered differently from less weathered zones (W1, W2, and W3), where fracture patterns prevail as conditioners of flow propagation. For this reason, the developed methodology includes two main steps for the 3D modelling of “transmissivity” (1) First, to simulate the 3D transmissivity of fractured rock matrix according to weathering levels W1, W2, and W3. This was achieved by crossing histograms of experimental data on transmissivity with histograms derived from simulated scenarios of NF. For this (Figure 6), cumulative frequency of transmissivity and cumulative frequency of NF are expressed by the same classes of lithology and weathering level. W4 class (primary porosity behaviour) is considered independently and for all lithological classes. A certain quantile of NF is, 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598
therefore, transformed into a correspondent quantile of transmissivity across the possible (union) classes present in the case-study (porphyritic granite W1, W2 and W3; biotitic granite W1, W2 and W3, breccia granite W1, W2 and W3; micaschist and migmatite W1, W2 and W3, and, W4 for all lithology classes with intensive to pervasive weathered rock matrix). This histogram cross procedure allows the establishment of correlations between hydrogeological parameters deterministically obtained, such as porosity and transmissivity values that are calculated from in situ tests, and intrinsic and structural properties of the massif, whose matrices can be generated by using 3D geostatistical modeling techniques. (2) Second, simulate the 3D transmissivity of pervasively weathered rock (W4) using PFS according to previously simulated 3D matrix weathering (W1, W2, W3, and W4) and conditional to the transmissivity experimental data for W4 zones.
3 Results
3.1 3D geological models of lithology and weathering
Mineralization and supergenic alteration phenomena in the study area have led to spatial variation in mineralogy, with a consequent high level of local variability in lithological facies. As a consequence of such a high level of variability, 11 distinct conceptual models for lithology based on possible different interpretations of data and information were developed. The consistency of the 11 models was tested regarding their geological and statistical representativeness and spatial continuity. Comparisons with conceptual interpretations and geophysical interpretations (Carvalho et al., 2005) for the area were made and a final model with six classes of lithology was selected for further processing and stochastic simulation (Soares, 1992; Almeida et al. 1993; Journel, 1994; Srivastava, 2005; Almeida, 2010; Matias, 2010; Nunes and Almeida, 2010; Quental at al., 2012, Barbosa, 2013). Data for each borehole was coded at regular depth intervals of 5 m (the block grid dimension) according to an indicator vector classified using six distinct modalities that reflect the six aforementioned classes of lithology: porphyritic granite (class1), biotitic granite (class 2), mica schist (class 3), migmatite granite (class 4), breccia granite (class 5), and pegmatite (class 6). The models generated by SIS have a total number of cells of 388700, having each cell a volume of 5x5x5 cubic meters. Global proportions of each lithology class that results from the conditionality of simulations to 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625
the two granite main regions are presented in Table 1. As it can be seen, global proportion of each lithology class is respected in relation to input data proportions.
Variogram results (Figure 7) show zonal effects as a consequence of the conditional simulation process. Each lithology class has two structures: one with lower spatial continuity derived from input data, and another with higher spatial continuity that arises from simulation conditional to lithological region (porphyritic granite facies and biotitic granite facies). The presence of two main structures is well patent in variograms of Figure 7. One, with lower spatial continuity derived from input data (amplitude< 300 meters) is according with the dimensions of the case study and the major distances between logs and piezometers. The other second structure, with amplitude higher than approximately 300 meters (with higher spatial continuity), is present in consequence of the first step of simulation (that is, the simulation conditional to the two main lithological regions -porphyritic granite and biotitic granite facies). In this case, a more continuous structure tendency in XY direction (higher amplitude in simulated results) is achieved with the first step of simulation. Without this step, the simulated results will be much more circumscribed to the surrounding areas of the vertical logs and will not respect so well the general regional lithology. With the adopted simulation procedure, the variograms of the simulated results present a more continuous structure tendency. An average scenario resulting from the 20 equally probable simulated scenarios for lithology is shown in Figure 8.
For weathering, four categorical classes were established ranging from less weathered (W1) to most weathered (W4) rock matrix: W4 (phase 4: rock with pervasively weathered matrix); W3 (class 3: rock with weathered matrix); W2 (class 2: rock with slightly weathered matrix); and W1 (class 1: rock with no weathered matrix). This variable was also regularized with a dimension of 5 m (i.e., the block dimension of 3D grids). Categorical variables “lithology” and “weathering” are not dichotomous. For each variable, regularization is expressed according to the percentage of occurrence of each class (which ranges from 0% or “0” to 100% or “1”). The sum of all classes by each measurement interval of 5 meters is always 100% (1). This percentage of occurrence is, in fact, the numerical value of the proportion of each of the classes per interval of 5 meters.). Weathering classes were compared with lithology classes with respect to the depth intervals of investigation 0–15 m and 16–95 m. A specific pattern regarding the relationship between different lithologies and weathering classes is evident 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653
(Table 2). Differences of weathering classes between porphyritic granite and biotitic granite are pronounced. Porphyritic granite is substantially more weathered at higher depths than biotitic granite (Table 2). The effects of more intense argillic and sericitic alteration can be expected in porphyritic granite facies. Because of this fact, 3D models of weathering were simulated conditional to lithology and to depth interval (0–15 m and 16–95 m, Table 3).
Considering the detected relationships between lithology, weathering, and depth of investigation, the equally probable scenarios of weathering were simulated with SIS conditional to lithology, to regional binary zoning for lithology (porphyritic granite and biotitic granite), and to regional binary zoning of depth interval (0–15 m and 16–95 m). For 3D models of weathering, 5 simulations were run for each of the 20 previously developed 3D lithology models, resulting in a total of 100 equally probable scenarios. The high variability of data created difficulties in variogram modelling, but a multiphase variogram with exponential and isotropic models and with a range of 60 m was considered adequate. The highest structure tendency in classes W3 and W4 is found in the northern part of the study area, where porphyritic granite dominates, whereas classes W1 and W2 have higher frequencies in the southern part (Figs. 9 and 10).
3.2 3D fracturing models
In FTRIAN simulations, the starting unit volume is a block with the same dimensions as those of the unit grid block of the 3D models, in this case, a 5m×5m×5m block. Considering the general characteristics of the media - a granite rock matrix with differing degrees of weathering and fracturing – only open fracture (Sausse and Genter, 2005) identified in borehole data were considered was considered for the LDF counts. In a first step, open fracture networks were simulated using FTRIAN in a unit reference block with the same dimensions as those of models’ grid cells (5m×5m×5m). Fracture dimensions follow a “-2” power law and fracture openings follow a uniform distribution law. The 3D object-based models of number of fractures is conditional to the open fracture densities of each block. In this object-based simulation procedure, the following assumptions were made: (i) The total sum of all fracture families previously classified according to the degree of inclination was used because drilling was not performed with respect to the spatial orientation of rock core; (ii) The sum of 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680
