A new approach for the prediction of unexposed fractured reservoirs: a case study in Millas Granite (French Pyrenees)

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1 Hydrological Sciences -Journal- des Sciences Hydrologiques,40,3, June A new approach for the prediction of unexposed fractured reservoirs: a case study in Millas Granite (French Pyrenees) S. PISTRE & L. M. BANGOY Laboratoire d'hydrogéologie, Place Eugène Bataillon, F Montpellier Cedex 05, France T. RIVES ELF Aquitaine, CSTJF, Avenue Larribeau, F Pau Cedex, France Abstract A fracture map is an essential aid in the study of fissured aquifers even though it often includes covered areas where the fractures are not visible. Is it possible to predict fracturing or permeable zones under the cover? This problem is dealt with using a stochastic approach based on fracture density. Classical methods of fracture density assessment involve several biases and are not adapted to hydrogeology. A proposed "fracture density index" (FDI) eliminates these biases and gives weight to two important parameters influencing the hydraulic conductivity of a zone: length and proximity of long fractures. The FDI is a regionalized variable and allows one to make predictions by kriging over covered zones. The proposed method has been applied to the Campoussy site (Pyrénées Orientales, France). In contrast to previous methods, the FDI predictions were in good agreement with the apparent hydraulic transmissivity and electrical anisotropy. The results were promising but the method needs to be tested on other sites before being considered a useful and efficient hydrogeological tool. Une nouvelle approche pour la prévision des réservoirs fissurés sous couverture: exemple du granite de Millas (Pyrénées françaises) Résumé Les cartes de fracturation sont des documents indispensables pour l'étude des aquifères fissurés, mais elles présentent souvent des zones couvertes où les fractures ne sont pas visibles. Est-il possible de prévoir la fracturation ou les zones perméables sous cette couverture? Ce problème est discuté grâce à une approche stochastique, à partir de la densité de fracturation. Les méthodes classiques de calcul de la densité de fracturation induisent plusieurs biais et ne sont pas adaptées à Fhydrogéologie. L'"index de densité de fracturation" (FDI) proposé élimine ces biais et donne beaucoup de poids à deux paramètres importants pour la perméabilité d'une zone: longueur et proximité des grandes fractures. Le FDI correspond à une variable régionalisée et permet des prévisions par krigeage sur des zones couvertes. La méthode est appliquée au site de Campoussy (Pyrénées Orientales, France). Contrairement aux méthodes existantes, les prédictions montrent une bonne corrélation avec la transmissivité hydraulique apparente et 1'anisotropic électrique. Ces résultats sont prometteurs mais la méthode doit être testée sur d'autres sites pour être considérée utile et efficace en hydrogéologie. Open for discussion until I December 1995

2 352 S. Pistre et al. INTRODUCTION Crystalline rocks have low matrix porosity so their hydraulic behaviour is mainly controlled by their fracture network geometry. A hydrogeologieal study in this type of environment provides a fracture network analysis. Fracture maps derived from direct observations (field studies and aerial photographs) are often the only reliable data for the characterization of the fracture pattern in a groundwater reservoir. However, fractures are not always visible at the ground surface, in particular where they are buried under natural cover (farmland, wooded areas) or artificial cover (inhabited areas). The characterization of fracture networks in such unexposed areas may be of economic importance and so extrapolations are then necessary. Previous studies have shown that geostatistics (Matheron, 1965, 1973), which analyses regionalized phenomena using a probabilistic approach, is a mathematical tool well adapted to rock fracture characterization at the scale of outcrops (Rossier, 1986), aerial views (Razack, 1986) or SPOT photos (Castaing et al, 1989). A fracture map is transformed into a map of fracture density values, and kriging then allows the estimation of this density parameter in unexposed areas from values in the surrounding zones. The fracture density value is assumed to be the total fracture length in each square unit (Brière, 1982) or circle unit (Kiraly, 1969) on the fracture map. The classical statistical methods involve several geometric biases and do not account for water circulation data, so a new method is proposed herein for the estimation of fracture density values specifically adapted for hydrogeology. THE FRACTURE DENSITY INDEX METHOD Fracture density maps are spatially gridded fracture maps; the fracture density value assigned to the centre of each grid unit corresponds to the total fracture length in that unit. In order to produce a continuous field of fracture density values, this gridding technique involves several biases: in particular, where a long fracture crosses a unit but not an adjacent unit, then two neighbouring fracture density values may be very different. This could involve a discontinuity of data corresponding to an accentuation of the local variability of the fracturing phenomenon, difficult to consider from a geostatistic point of view, and which could be conditioned by the accuracy of the map. Furthermore, for hydrogeologieal applications, such a discontinuity is inconsistent with the spatial variability of the hydrodynamic parameters which are rather continuous at this scale, owing to the small fracture network. This bias may be partially avoided by shifting the grid a half unit length successively in two orthogonal directions and then recalculating the fracture density values. Nevertheless, the parameters of most influence on productivity in drilling are the proximity and length of important fractures which are not taken into account in these methods.

3 Prediction of unexposed fractured reservoirs 353 The fracture density index method (Pistre, 1992a), specifically developed for hydrogeology, aims to predict the fracturing of unexposed zones without any geometric bias. It involves three successive stages: Fracture map In order to consider the entire aquifer, the fracture map must cover a much wider area than the zone of interest. The scale is determined from the minimum fracture length influencing global circulations in the reservoir (generally ten to several tens of metres). Aerial views are generally the means most adopted. Fractures are drawn as linear segments and covered zones, where fractures are unexposed, are shaded. Fracture density index map The fracture map is then transformed into a fracture density index map. A regular grid is placed on the map and the fracture density index is calculated for the centre of each unit. The size of unit is sufficient to allow accurate kriging approximately ten times the size of the unexposed zone, but this has to be limited by time considerations. The fracture density index FDI(x,y) at a point P(x,y), for a single fracture i, is defined as: FDIjfXy) = [fir) dr (1) where r u and r 2i are the distances between P and the two extremities of the fracture. The index FDI depends on the fracture length and on the relative position of P(x,y) and of the fracture (Fig. 1(a)). Fig. 1 Fracture density index estimation for one fracture, at one point.

4 354 S. Pistre et al. In order to adapt the FDI to hydrogeology, one needs to give a large weight to long or proximal fractures and a small weight to short or distal fractures whose hydrodynamic influence on P(x,y) is negligible. This weighting can be made by assuming/(r) to be a function derived from the Gaussian law fir) = e" r2/2t2 where T has the dimension of length. The graph of this function (Fig. 1(b)) shows the influence of fracture proximity and length on FDI ; (x,y), which can be modified by varying T. This latter parameter is derived from the estimated average distance for which a fracture will have only a minor influence on the hydrogeological characteristics at P(x,y), i.e. for which exp( r 2 /!-! 2 ) becomes insignificant (Fig. 2(a)). The effect of a fracture is considered to be negligible at r = AT (Fig. 1(b)). For a set of N f fractures FDI(x,y) is defined as: N f r n FDI(x,y) = f e^^dr ( 2 ) where FDI(J,)>) is in metres of fractures. The method obviously implies an edge effect at map boundaries or close to shaded zones where the "visible" fracture field is reduced. This bias remains restricted (Fig. 2(b)) but is likely to alter the validity of interpolation calculations close to hidden areas. A correcting factor A(x,y) is therefore introduced which takes this bias into account: A(x,y) = ^fir)ds/ N, J>)d 5 -ÇJ/(r)d5 (3)

5 Prediction of unexposed fractured reservoirs 355 where N z is the number of hidden areas, r represents the distance between P(x,y) and the surface differential ds, 5, represents the area of each hidden zone (Fig. 1) and S is the whole area of the field considered. The corrected fracture density index FDI corr (x,y) becomes: N f hi FDI corr (x,y) = J2 f e~ r2,2t2 dr.a(x,y) (4) Calculation of FDI corr (x,y) at each point of the grid leads to the fracture density index map. Prediction of fracture networks by kriging The fracture density index map is an incomplete field of numerical values corresponding to the fracture field of the aquifer. The value FDI corr (x,y) may be considered as a regionalized variable z(x) = FDI corr (x,y) (X represents simultaneously the two coordinates of a point studied) which corresponds to the realization of a random function at each point of the zone studied (Delhomme, 1978). The spatial variability of z(x) = FDI corr (x,y) on the map can be described by an experimental variogram and characterized in numerical terms by adjustment to a theoretical variogram (De Marsily, 1986). The knowledge of the values FDI corr (x,y) at points close to hidden areas and the spatial variability of this parameter allows estimation of the fracture density index FDI(x,y) in unexposed zones by interpolation (point kriging). The more the variable is structured and regular, the more this interpolation is relevant or accurate (deviation estimations are very small). The kriging results in a complete fracture density index map. APPLICATION TO THE MILLAS GRANITE The Campoussy experimental site is located in the late Variscan calc-alkaline granitoids of the Quérigut-Millas massif (Pyrénées Orientales, France, Fig. 3). This granitic massif, located along the North Pyrenean Fault, has a complex geological history (Choukroune, 1976). The development of the observed fracture network can be related to two important tectonic events: the brittle pyrenean north-south compression and the NW-SE Oligo-Miocene extension (Soliva et al., 1992; Arthaud & Pistre, 1993). Intensely fractured, the massif is, in a general way, an aquifer. Moreover, the relief and large faults create a pattern of reservoirs with an extent of several square kilometres which are often in elevated positions with free, semi-captive or captive characters according to the presence or not of an alluvial cover. The Campoussy site is an area where the granitic massif is covered by alluvial deposits. The problem is to characterize its fracturing.

6 356 S. Pistre et al. Fig. 3 Location of the Querigut-Millas massif in the Pyrenees Axial Zone, and of the area studied (black square): 1- gneissic massif; 2- granite massifs; and 3- sedimentary formations. Data collection and corresponding fracture density index map The field selected corresponds to a 5 km 2 rectangle (the entire aquifer) roughly centred on the experimental site. Fracture traces longer than 50 m were drawn from high quality aerial photographs at 1: scale, complemented with field observations (Pistre, 1989). A map of the fractures was obtained with shaded areas representing the unexposed zones (Fig. 4). The fracture network comprised two dominant sets: N170 to N010, N030 to N050, and two subordinate sets: N080 to N100, N120 to N135 (Fig. 5). These four fracture sets occurred over the entire massif (Pistre, 1992b; Arthaud & Pistre, 1993). The values of FDI corr (jc,y) were calculated for each point on a regular grid (50 m unit length) and contours were drawn for different r values (25, 50, 100 and 200 m) to cover the range of the possible average distances ( m) for which the influence of a fracture is not negligible (see above) (Bangoy, 1992). Figure 6 shows the map for T = 50 m. Prediction of fracture network by kriging Maps of the selected field were calculated for the four T values. With increasing T, the map contours became smoother and showed less detail (e.g. alignments, fracture intersections). The selection of r, which reflects the influence of fracture closeness and length, therefore determines the behaviour

7 Prediction of unexposed fractured reservoirs 357 Fig. 4 Fracture map drawn from aerial views: location of the hydrogeological site at Campoussy (star), Fig. 5 Fractures directional histogram, from the fractures cumulated lengths, with azimuthal classes of 10. of the variogram which indicates the spatial variability of FDI and which guides kriging. Thus, this selection is based not only on hydrodynamic considerations (see above), but also, for technical reasons (kriging), on the variograms. They show oscillations (with more instabilities when T decreases) at approximately identical distances for different T values (Fig. 7), which correspond to "hole effects" ("rich" areas surrounded by "poor" ones) except at 1400 m (not visible for T = 25 m) where the regionalization of the phenomenon is proved. The T chosen needs to lead to a reliable map in total

8 358 S. Pistre et al. Ill Recent deposits mm Campoussy site Fig. 6 Map of the fracture density index deduced from Fig. 2, with r = 50 m. T=25m 1=50 m Spherical + Linear Spherical + Linear 0 -,MII.. I. 1 I 1 I.. I MI 1 I.... IM! 0- I...1.1". L h, m h, m 60i m 2 : 40- T= =100m /? so, 2 : m T=200m 20-2 x Spherical i «Se ibbo"" iebb h, m h, m Fig. 7 Experimental and adjusted variograms for different values of T. accordance with the preferential alignments of the fracture map. The value T = 25 m provides the most contrasted map, but is not representative because the distinctness is only illusory according to the resolution of the aerial views. All the variograms were adjusted (Fig. 7) to spherical and linear models as follows:

9 Prediction of unexposed fractured reservoirs 359 u> 1 h + u) 2 [3f2(hfa)-l/2(.h/a) 3 ] + C Q h < a a)j/z + co 2 + C() h > a where a is range and co 1; w 2 are positive coefficients. The adjustments with linear and spherical models were satisfactory for all the variograms (the average difference between the experimental and analytical variograms was less than 3% of the variance values) with the prospect of further calculations during interpolation. The value T = 50 m was selected considering its variogram: - FDI was structured and regular for distances below 300 m. The presence of this 300 m large regionalization was not visible for higher r, whereas it was visible for T = 25 m. For distances beyond 300 m the variogram was quite stable but quite flat (more than for r = 25 m): FDI was no more regular. This structure was important for interpolation under the cover because its dimension was about that of the covered zones: the 300 m distance values, which were barely correlated, had a weak influence on the kriging. - The behaviour at the origin did not show any nugget-effect, T = 50 m seemed to be well adapted to this fracture map, and the nearest values were correlated (the "erratic" component of the FDI was small). This would allow a more stable or more "accurate" kriged map to be obtained, reducing the standard deviation of the estimation. The kriging allowed a prediction of the fracture density index below the Campoussy alluvial parcel. Maps of predicted FDI values and associated standard deviations (Fig. 8) were prepared. The FDI values obtained were larger in the eastern and central parts of the site than in the western part. Fig. 8 (a) kriged map of FDI at the Campoussy site; and (b) map of the standard deviation of estimation (m). PREDICTION VALIDITY The fracture density index method is one of the methods providing a regionalized variable which shows fracture density on a fracture map, and is then able

10 360 S. Pistre et al. to predict fracture densities of unexposed zones. In order to test such predictions, the coherence of the FDI method with other approaches is investigated below. Comparison with previous methods The classical grid methods differ from the fracture index method in the construction of fracture density maps. In the simple grid method (Kulatilake & Wu, 1984) a regular grid is superimposed on the fracture map and the total j 1 Recent deposits ^M Campoussy site Fig. 9 Maps of fracture density, with a 100 m x 100 m unit (isodensities in m per unit), deduced from Fig. 2: estimated from (a) the simple grid method; and (b) the multiple grid method.

11 Prediction of unexposed fractured reservoirs 361 a ;' t ;' ; ' ' / / / / / / // ; / : : i ' / /' / ; /, - 4 y ','' / ^ 60 /,, 0 ' t,--' -r 20m Fig. 10 Kriged map of fracture density (m), with a 50 m x 50 m unit, on the Campoussy site for (a) the simple grid method; and (b) the multiple grid method. fracture length of each unit square is measured. The calculated value of the fracture density d(x,y) is expressed in metres of fracture for the chosen unit and is assigned to the centre of each unit. The unit size is determined by the aerial photograph resolution (100 m x 100 m was chosen). The resulting map (Fig. 9(a)) was quite similar to the fracture density index map (Fig. 6), the only differences being the amplitudes of spatial variations and many details close to the unexposed zones. Nevertheless, the kriged map of the Campoussy site (Fig. 10(a)) was very different from that produced by using the fracture density index method (Fig. 8). In the multiple grid method, the grid is translated half a unit length in N-S and E-W directions in order to multiply the points of determination of the fracture density. This overlapping of squares makes the edge effect disappear. The map obtained (Fig. 9(b)) presented the same characteristics as the fracture density maps drawn with the previous methods. However, it was more contrasted and appeared to point out more details than Fig. 9(a), but it needed a longer time of calculation and bulkier files (x4 for a similar fractures field). The interpolated map of the Campoussy site (Fig. 10(b)) was also very different from Fig. 8. The three methods led to predictions of the fracture density of the site. In spite of biases inherent to the grid methods - edge effect (linked to the position of each square unit) and shape effect (influence of fracture orientations) one could not determine which predicted map was correct. Onlycomparison with geophysical and hydrological data from the Campoussy site could allow the best method to be determined. Validation by other approaches A series of short pumping tests in the seven wells of the Campoussy site (F1-F7 on Fig. 11) was made in order to investigate the hydraulic properties of the captive aquifer located in the deep fissured granite. The hydraulic

12 362 S. Pistre et al. behaviour of the fissured granite appeared to be equivalent to a porous medium (Pistre, 1992b): among the classical models for pumping-test interpretation, the Theis model gave the best results (except for the beginning of pumping) for each borehole. The presence of many small-sized fractures, as observed at the surface (Arthaud & Pistre, 1993), can explain this behaviour. Assuming the reservoir as a porous medium without any leakage, a map of "apparent transmissivities" may be drawn (Fig. 11). The apparent transmissivity is the experimental value which represents the conductive properties of a volume of granite functioning at the time of pumping. Because the circulations in fissured media occur in fractures, the apparent transmissivity is an increasing function of fracture density. An east-west profile across the Campoussy site showed an apparent transmissivity curve (Fig. 12), which suggested a greater fracture density in the centre of the site. A series of electrical surveys were made at the Campoussy site in order to determine the electric properties of the medium and to characterize its structure (Aslanian, 1989). Schlumberger soundings across the Campoussy site showed three successive strata interpreted as alterites, altered granite and unaltered \ \ '. / F2 / F3 # / F4 \ 1 * C1 V \ D C4 0 20m \ / Fig. 11 Map of the apparent transmissivity ( x l(r 5 m 2 s" 1 ) at the Campoussy site. Drill locations (Fl to F7); Schlumberger locations (SI to S4) and square electrical soundings locations (CI to C4). \ ' /CM*?'' / F V\V>\ \ F6 S3 C2y ; *^.\ \ \ t 3-i 1 i i i i i i i i i i i i i i i i i i i " " m Fig. 12 Comparisons at the site between electrical anisotropy coefficient (\: squares), apparent transmissivity (Ta: stars = data; dashed curve = interpolation) and interpolated fracture densities computed from simple grid (iil), multiple grids (dz) and fracture density index (FDI) methods.

13 Prediction of unexposed fractured reservoirs 363 granite with an apparent resistivity (p am ) of about 700 ohm m. Schlumberger drag soundings revealed increasing thickness of the altered zone to the west (from 15 to 25 m). Three multidirectional square soundings (C1-C3 on Fig. 11) showed an anisotropy of the fresh granite along the NE-SW direction. This anisotropy could be interpreted as indicating the presence of N050 fractures at depth. The values of p am and X, the elongation coefficient (i. e. the anisotropy coefficient), are indicative of fracture density (Haberjam, 1972; Darboux-Afouda, 1985). The value of p am was constant across the site and X was greater in the centre (Fig. 12), suggesting a higher fracture density in the centre of the Campoussy site. The hydraulic and electrical methods showed a variation in the east-west structure of the Campoussy site; in particular, a probably more densely fractured zone in its central part. The east-west profiles of the estimated fracture densities obtained by the three methods (Fig. 12) showed a constant fracture density in the simple grid method, a regularly increasing fracture density from west to east in the multiple grid method and a maximum fracture density in the centre of the site in the fracture density index method. The fracture density index profile was the most coherent with the direct hydraulic measurements and electrical surveys, which indicated that the fracture density index method was the most valid fracture density approach, at least for hydrogeological considerations. In the western part of the profile, the apparent transmissivity decreased more quickly than the FDI probably because the connectivity decreased more quickly than the density of fractures (Robinson, 1984). DISCUSSION AND CONCLUSIONS The fracture density index (FDI) is a new variable with a hydrodynamic connotation. It can be considered as a realization of a regionalized variable of a studied area and so allows predictions by kriging in unexposed zones. The fracture density index method avoids geometric biases (effects of the position and the orientation of each unit) and hydrodynamic bias (cutting of long fractures) which are encountered in the simple and multiple grid methods: thus it corrects hidden area effects. Like the other methods, it implies partial interpretations to select the parameter values, in particular for the selection of the fractures to be taken into account. The minimum fracture length of the fracture map corresponds to the minimum fracture length which could significantly influence the hydrodynamic behaviour of groundwater reservoirs. At the Campoussy site, this was assumed to be fractures of tens of metres. As shown by hydraulic tests, the FDI variable seemed to be linked with the hydraulic transmissivity of the reservoir, so the kriged maps indicated the poorly and highly fractured zones. The reliability of predictions depends on the parameter r which reflects the distance (AT) from which a fracture will have only a minor hydraulic influence on a possible well. To provide useful results, kriging preferably necessitates a continuous variable. So, for this influence distance, several r values need to be tested. The behaviours at the origin of the corresponding

14 364 S. Pistre et al. variograms permit selection of the best r value (in the example given herein T = 50 m). The fracture density index method necessitates more complex and longer calculations than those of the other methods. Moreover, in order to validate the FDI method, in particular the link between FDI and hydraulic transmissivity, other applications will have to be carried out at other sites. FDI allows characterization of three dimensional hydraulic behaviour of a groundwater reservoir from two dimensional surface (fracture) data. A remaining problem is the choice of the fracture sets to be taken into account since in a fissured network not all fractures contribute equally to fluid circulation. For example, at the Campoussy site the N170-N010 and N030-N050 fractures, which were the most represented, could only be used following a tectonic analysis of the domain (Arthaud & Pistre, 1993) which attributed the N030-N050 set formation and opening of the N170-N010 fractures to the Oligo-Miocene extension. Moreover, once may also consider that all the fractures are potential drains. The N080-N100 and N120-N135 fracture sets insure local drainage through the main fractures and could not be neglected. The fracture density index method constitutes a promising new predictive technique for hydrogeology. It can be improved by other applications on other sites and by taking into account fracture organization and hierarchy. It is necessary, therefore, to understand fracture development mechanisms via field analysis in well exposed areas (Rawnsley et al., 1992), analogue modelling (Rives & Petit, 1990a, 1990b) and numerical simulations (Rives et al., 1992) to guide further extrapolations. REFERENCES Arthaud, F. & Pistre, S. (1993) Les fractures et les paléocontraintes du granite hercynien de Millas (Zone Axiale des pyrénées). Un cas d'étude de la tectonique cassante d'un aquifère de socle. Geodin. Acta, 6(3), Aslanian, D. (1989) Approche électrique de la géométrie d'un magasin aquifère en milieu cristallin. DEA, Montpellier II, France. Bangoy, L. M. (1992) Hydrodynamique d'un site expérimental en aquifère de socle fissuré; nouvelle méthode d'interprétation des essais hydrauliques. Thèse Doctorat es Sciences, Université de Montpellier II, France. Brière, G. (1982) Introduction à l'analyse géostatistique sur clichés aériens de la fracturation des magasins aquifères en roches fissurées. Thèse 3 Cycle, Montpellier II, France. Castaing, C, Dutartre, P., Gouyet, J.F., Loiseau, P., Martin, P. & Pointet, T. (1989) Etude pluridisciplinaire d'un réseau de discontinuités-images SPOT en milieu granitique couvert. Implication en hydrogéologie des milieux fissurés. Bull. BRGM, no. 1, Choukroune, P. (1976) Structure et évolution tectonique de la zone nord-pyrénéenne. Mém. Soc. Géol. France, no. 127, Darboux-Afouda, 8. (1985) Recherches hydrogéologiques par les méthodes géophysiques dans les régions de socle en Afrique Occidentale. Thèse 3 Cycle, Montpellier II, France. Delhomme, J. P. (1978) Application de la théorie des variables régionalisées dans les sciences de l'eau. Bull. BRGM, série 2, section III, De Marsily, G. (1986) Quantitative Hydrogeology. Groundwater Hydrogeology for Engineers. Academic Press, Orlando, USA. Haberjam, G. M. (1972) The effects of anisotropy on square array resistivity measurements. Geophys. Prosp. 20, Kiraly, L. (1969) Statistical analysis of fractures (orientation and densities). Geol. Rundschau, 59(2),

15 Prediction of unexposed fractured reservoirs 365 Kulatilake, P. H. S. W. & WU, T. H. (1984) The density of discontinuity traces in sampling windows. J. Rock Mech. & Geomech. Abstr. 21(6), Matheron, G. (1965) Les Variables Régionalisées et Leur Estimation. Masson, Paris, France. Matheron, G. (1973) The intrinsic random functions and their applications. Adv. in Appl. Prob. 5, Pistre, S. (1989) Approche tectonique et microtectonique de la géométrie d'un aquifère fissuré en zone de socle: exemple du site expérimental de Campoussy dans le massif de Millas. DEA, Montpellier II, France. Pistre, S. (1992a) Méthode d'évaluation d'un indice de densité de fracturation en hydrogéologie et prévision des zones perméables par une approche probabiliste. C. R. Acad. Se. Paris, 314(11), Pistre, S. (1992b) Structure fissurale et modélisation hydrodynamique. Thèse Doctorat es Sciences, Université de Montpellier II, France. Rawnsley, K. D., Rives, T., Petit, J. P., Hencher, S. R. & Lumsden, A. C. (1992) Joint development in perturbed stress fields near faults. /. Struct. Geol. 14, Razack, M. (1986) Approche probabiliste de l'étude en subsurface de la géométrie des réservoirs fissurés. Effet de l'échelle d'investigation. Bull. BRGM Hydrogéologie no. 2, Rives, T. & Petit, J. P. (1990a) Diaclases et plissement: une approche expérimentale. C. R. Acad. Se. Paris, 310(11), Rives, T. & Petit, J. P. (1990b) Experimental study of jointing during cylindrical and non-cylindrical folding. In: Mechanics of Jointed and Faulted Rock, ed. H. P. Rossmanith, Balkema, Rotterdam, The Netherlands. Rives, T., Razack, M., Petit, J. P. & Rawnsley, K. D. (1992) Joint spacing: analogue and numerical simulations. /. Struct. Geol. 14, Robinson, P. C. (1984) Connectivity, flow and transport in network models of fracture media. PhD dissertation, St Catherine's College, Oxford, UK. Rossier, Y. (1986) Exemple d'applications d'une méthode d'analyse de la fissuration. Illustration en pays granitique et calcaire. Bull. BRGM Hydrogéologie no. 2, Soliva, J., Pistre, S., Brunei, M. & Arthaud, F. (1992) Les zones de cisaillement mylonitiques dans le granite hercynien de Millas (Pyrénées Orientales): déformation ductile d'âge hercynien et/ou pyrénéen. C. R. Acad. Se. Paris 314(11), Received 24 January 1994; accepted 9 December 1994

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