Effects of linking up of discontinuities on fracture growth and groundwater transport
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- Gudmundsson, A., Gjesdal, O., Brenner, S.L. et al. Hydrogeology Journal (2003) 11: 84. doi:10.1007/s10040-002-0238-0
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It is proposed that the growth of fractures is the basic process for generating and maintaining permeability in solid rock (bedrock). Many extension fractures grow as hydrofractures, whereas many shear (and extension) fractures grow through the formation of transverse fractures that connect the adjacent tips of existing fractures. In a boundary-element analysis, the hydrofractures are modeled as being driven open by a fluid overpressure that varies linearly from 10 MPa at the fracture centre to 0 MPa at the fracture tip. The host rock has a uniform Young's modulus of 10 GPa, a Poisson's ratio of 0.25, and is dissected by vertical joints and horizontal contacts, each of which is modeled as an internal spring of stiffness 6 MPa m−1. The number of joints and contacts, and their location with respect to the hydrofracture tip are varied in different model runs. The results of the analyses indicate that the tensile stresses generated by overpressured hydrofractures open up joints and contacts out to considerable distances from the fracture tip, so that they tend to link up to form a hydraulic pathway. Using the same Young's modulus, Poisson's ratio, and internal spring constant for joints as in the hydrofracture models, boundary-element models were made to study the interaction stresses that cause neighbouring joints to become interconnected through the growth of linking transverse fractures that, ultimately, may evolve into shear fractures. The models were subjected to tensile stress of 6 MPa acting normal to the joint planes as the only loading. The offset (horizontal distance) and underlap (vertical distance) between the adjacent tips of the joints were varied between model runs. The results show a concentration of tensile and shear stresses in the regions between the neighbouring tips of the joints, but these regions become smaller when the underlap of the joints decreases and changes to overlap. These stress-concentration regions favour the development of transverse (mostly shear) fractures that link up the nearby tips of the joints, so as to form a segmented shear or extension fracture. Analytical results on aperture variation of a hydrofracture in a homogeneous, isotropic rock are compared with boundary-element results for a hydrofracture dissecting layered rocks. The aperture is larger where the hydrofracture dissects soft (low Young's modulus) layers than where it dissects stiff layers. Aperture variation may encourage subsequent groundwater-flow channeling along a pathway generated by a hydrofracture in layered rocks.
KeywordsHydrofracturesRock discontinuitiesFracture growthFluid transportHydrogeology
Nous proposons que le développement des fractures est le processus de base qui génère et maintient la perméabilité des roches indurées de socle. De nombreuses fractures en extension se forment en tant qu'hydrofractures, tandis que de nombreuses fractures de cisaillement (et en extension) se développent par la formation de fractures transverses qui mettent en connexion les parois adjacentes de fractures existantes. Dans une analyse des éléments aux limites, les hydrofractures sont modélisées comme si elles étaient maintenues ouvertes par une surpression de fluide qui varie linéairement de 10 MPa dans la fracture à 0 MPa sur sa paroi. La roche magasin possède un module de Young uniforme de 10 GPa, un rapport de Poisson de 0.25, et est recoupée par des fractures verticales et des joints horizontaux, chacun étant modélisé comme une source interne avec une rigidité de 6 MPa m−1. Le nombre de fractures et de joints et leur localisation par rapport aux parois de l'hydrofracture varient dans les différents traitements de modélisation. Les résultats des analyses indiquent que les contraintes de tension générées par les hydrofractures en surpression ouvrent les fractures et les joints sur des distances considérables à partir des parois de la fracture, de telle sorte qu'elles tendent à se connecter pour former un cheminement hydraulique. En utilisant le même module de Young, le même rapport de Poisson et la même constante de source interne pour les fractures que dans les modèles d'hydrofracture, des modèles d'éléments aux limites ont été élaborés pour étudier les contraintes d'interaction qui provoquent l'interconnexion de fractures voisines grâce à l'extension de fractures transverses qui, finalement, peuvent évoluer en fractures de cisaillement. Les modèles ont été soumis à un effort de traction de 6 MPa appliqué normalement aux plans de fracture comme unique charge. Le déplacement (en distance horizontale) et l'écartement (en distance verticale) entre les parois des fractures ont varié selon les différents traitements. Les résultats montrent une concentration de contraintes de tension et de cisaillement dans les secteurs entre les parois des fractures, mais ces secteurs se réduisent lorsque l'écartement des fractures diminue et devient un recouvrement. Ces zones de concentration des contraintes favorisent le développement de fractures transverses (principalement de cisaillement) qui mettent en relation les parois voisines des fractures de manière à former une fracture segmentée de cisaillement ou d'extension. Les résultats analytiques sur la variation de l'ouverture d'une hydrofracture dans une roche homogène et isotrope sont comparés aux résultats des éléments aux limites pour une hydrofracture recoupant des roches litées. L'ouverture est plus large lorsque l'hydrofracture recoupe des couches tendres (module de Young faible) que lorsqu'elle recoupe des couches rigides. La variation de l'ouverture peut favoriser la chenalisation pour un écoulement souterrain subséquent le long d'un cheminement généré par une hydrofracture dans des roches litées.
Se propone que el crecimiento de fracturas es el proceso básico de generación y mantenimiento de la permeabilidad en rocas sólidas (roca madre). Muchas fracturas extensivas crecen por fracturación hidráulica, mientras que muchas fracturas de cizalla (y extensivas) lo hacen mediante la formación de fracturas transversales que conectan los extremos adyacentes de fracturas existentes. Por medio de un análisis de elementos de contorno, se ha modelado el crecimiento de las fracturas hidráulicas por un exceso de presión, la cual varía linealmente entre 10 Mpa en el centro de la fractura y 0 Mpa en el extremo. La roca madre tiene un módulo de Young uniforme de 10 Gpa, un coeficiente de Poisson de 0.25, y está diseccionada por diaclasas verticales y contactos horizontales que son modelados, cada uno de ellos, como fuente interna de rigidez igual a 6 MPa m−1. El número de diaclasas y de contactos, así como su situación respecto al extremo de las fracturas hidráulicas, han sido modificados en diferentes pasadas del modelo. Los resultados de los análisis indican que las tensiones generadas por fracturas hidráulicas sobrepresionadas abren las diaclasas y contactos a distancias considerables del extremo de la fractura, de manera que tienden a unirse y a formar caminos o vías de flujo. Utilizando para las diaclasas el mismo módulo de Young, coeficiente de Poisson y manantial interno constante que se empleó en los modelos de fracturas hidráulicas, se ha elaborado modelos de elementos de contorno para estudiar las tensiones de interacción que causan la interconexión de diaclasas vecinas por medio del crecimiento de fracturas transversales de enlace, y que pueden llegar a convertirse en fracturas de cizalla. Los modelos fueron sometidos a tensiones de 6 Mpa normales a los planos de diaclasas como única carga. Se ha modificado el acomodo o distancia horizontal ("offset") y la distancia vertical ("underlap") entre extremos adyacentes de las diaclasas en las diversas pasadas del modelo. Los resultados muestran una concentración de tensiones y cizallas en las regiones situadas entre extremos vecinos de las diaclasas, pero estas regiones son menores conforme la distancia vertical de las diaclasas decrece y cambia a solapamiento de techo ("overlap"). Estas regiones de concentración de esfuerzos favorecen el desarrollo de fracturas transversales (mayoritariamente de cizalla) que enlazan los extremos vecinos de las diaclasas para formar una fractura segmentada de cizalla o extensiva. Se compara los resultados analíticos de la variación de la apertura de una fractura hidráulica en una roca homogénea e isótropa con los resultados del modelo de elementos de contorno en fracturas hidráulicas que diseccionan rocas estratificadas. La apertura es mayor si la fractura hidráulica disecciona capas deleznables (módulo de Young pequeño) que cuando lo hace en capas rígidas. La variación de la apertura puede favorecer el acanalamiento del flujo de aguas subterráneas a lo largo de caminos generados por una fractura hidráulica en rocas estratificadas.
Flow of fluids in solid rocks is commonly very different from flow in sediments. Most solid rocks contain systems of fractures due to tectonic forces, occasionally the result of excavations, which bound blocks of intact rock. In crystalline rocks, the intact rock has normally such a low permeability that groundwater flow occurs mainly along joints, faults, contacts, and other discontinuities. This means that the flow of water in the rock is largely determined by the distribution of interconnected fractures and the way in which their apertures respond to the associated stress field.
All tectonic fractures are either primarily extension fractures or shear fractures. These can be distinguished based on the relative displacement across the fracture plane. In an extension fracture the displacement is perpendicular to, and away from the fracture plane; in a shear fracture the displacement is parallel with the fracture plane. Extension fractures grow in a direction perpendicular to the minimum principal compressive stress (considered positive). They include two main types: tension fractures and hydrofractures. Tension fractures form when the minimum principal compressive stress is negative. They are mostly limited to shallow depths in areas undergoing active extension, such as rift zones (Gudmundsson 1992). By contrast, hydrofractures can form at any depth, provided the total fluid pressure is equal to the sum of the minimum principal stress and the tensile strength of the rock. They comprise fractures driven open by any kind of fluid, such as magma (dykes, sheets, and sills), geothermal water (mineral veins), oil, gas, and groundwater (some joints).
Shear fractures with significant displacements are referred to as faults. They are commonly major water conduits (Bruhn et al. 1994; Caine et al. 1996; Evans et al. 1997; Haneberg et al. 1999; Faybishenko et al. 2000). Many faults develop from smaller fractures, commonly sets of joints and extension fractures (Gudmundsson 1992; Cartwright et al. 1995; Crider and Pollard 1998; Acocella et al. 2000). The permeabilities of fracture sets and faults change much during their development. The state of stress in a particular area controls the activity of existing faults and fracture sets and may also initiate new, water-conducting fractures. Many old fracture sets and faults are reactivated when the controlling stress field changes, in which case they may increase the temporary average permeability of a site by several orders of a magnitude (Gudmundsson 2000a). The overburden pressure may also affect the apertures of fractures. Although considerable progress has been made in recent years concerning the general effect of normal and shear stresses on the permeability of a rock mass, these effects appear complex (Gentier et al. 2000) and the details are still not well understood.
This paper has three principal aims. First, to summarise general field data on extension fractures and shear fractures (faults) as a basis for modeling. These data are taken from several areas studied by the authors, but primarily from Iceland, England, and Norway. The second aim is to develop analytical and numerical models of extension fractures and shear fractures. Here the focus is on their growth. In particular, new numerical models are presented of the growth of fractures through linking up of existing discontinuities such as contacts and joints in the host rock. Also provided are analytical and numerical models of aperture variations of extension fractures and how these may reflect the mechanical properties of a layered host rock. The third aim is to present some general analytical models of fluid flow in fractures, based on the field data and the results of the modeling.
Field Examples of Extension Fractures
In many hydrofractures, such as those generated by gas, oil, and groundwater pressure, the fluid may disappear when the fracture has formed. These fluids are presumably responsible for the formation of many joints, as initially suggested by Secor (1965) and discussed in standard textbooks (Twiss and Moores 1992; van der Pluijm and Marshak 1997). Other hydrofractures, however, are generated by fluids that freeze or otherwise solidify in the fracture subsequent to its formation. These hydrofractures include magma-driven fractures such as dykes, sills, and inclined sheets, as well as mineral veins. Of these, perhaps the most relevant, and easiest to study, are mineral veins.
Field Examples of Shear Fractures
In vertical sections, the growth of small faults across mechanical layers is also indicated by examples from west Norway (Fig. 6). Normally, fractures have difficulty in crossing open, subhorizontal discontinuities, as are exemplified by exfoliation (sheets, sheeting) fractures in the gneiss in west Norway (Fig. 6). Exfoliation fractures form as a result of high compressive stresses. In west Norway, such stresses may have been generated during rapid erosion by glaciers, or alternatively as a result of deglaciation and associated postglacial uplift and bending (Gudmundsson 1999).
The process of linking up of segments and growth of shear fractures at various scales, including large faults, is reviewed by Pollard and Segall (1987), Crider and Pollard (1998), Acocella et al. (2000) and Mansfield and Cartwright (2001). The general process is that fractures link up at different scales into larger and larger segments of the main fault. For strike-slip faults, the linking up of fractures into larger segments is commonly a complex process (Cox and Scholz 1988; Bergerat and Angelier 2001). By contrast, for most dip-slip faults, particularly normal faults, the process of linking up is relatively simple (Figs. 7, 8).
Modeling the Growth of Extension Fractures
Here the formation of a groundwater pathway through the linking up of discontinuities in the host rock is considered. The mechanism of pathway formation considered here is the propagation of a hydrofracture under internal fluid overpressure. This is the only loading applied. Numerical models were made of the pathway formation using the boundary-element program BEASY (1991). Well-known theoretical considerations suggest that hydrofractures are normally extension fractures (Anderson 1951; Hubbert and Willis 1957). For natural hydrofractures such as those resulting from the intrusion of dykes and mineral veins, this suggestion is generally supported by field observations (Pollard and Segall 1987; Gudmundsson 1995; Gudmundsson et al. 2002). Consequently, hydrofractures are here modeled as mode I cracks, that is, as pure extension (opening) fractures (Twiss and Moores 1992).
A typical Poisson's ratio of 0.25 is used in all the models (Johnson 1970; Jumikis 1979). A uniform Young's modulus of 10 GPa is used, a value that corresponds to the lower range of laboratory values of Young's modulus for gneiss, 3–70 GPa (Johnson 1970; Jumikis 1979; Bell 2000). This low value is used because the in-situ values of Young's modulus are commonly much lower, particularly for near-surface rocks, than values for intact, small rock samples measured in the laboratory (Goodman 1989; Priest 1993).
The models differ largely in the location and size of the host-rock discontinuities, and the location of the tip of the hydrofracture with respect to the discontinuities. The term "discontinuity" is traditionally used for any type of mechanical break in a rock. In the models, however, most discontinuities represent either contacts or joints. Thus, for brevity, the vertical discontinuities are frequently referred to as joints, and the horizontal discontinuities as contacts. Each discontinuity is modeled as an internal spring with a stiffness ("strength") of 6 MPa m-1 (Fig. 9). Such a low value is assumed to be appropriate for a joint or extension fracture with a consolidated, elastic but soft infill, or a contact with weak sedimentary or pyroclastic material. These stiffness values are thought to be primarily appropriate for the upper part of a rift zone, such as in Iceland (Figs. 7, 8). For comparison, weak tuff layers, as are common in volcanic successions, have small-sample laboratory Young's moduli as low as 50 MPa, and weak mudrocks (e.g. marl) have Young's moduli as low as 3 MPa (Bell 2000). Numerical models have also been made with zero-discontinuity stiffness, corresponding to an open joint or contact, with results that are very similar to those for the stiffness of 6 MPa/m-1 used here. In all the models, the lengths are given as fractions of the height which is a unit.
Modeling the Linking up of Offset Joints
In this section examples are presented of the linking up of offset joints into interconnected fracture sets that may eventually develop into shear fractures and, then, major faults. All these models take as starting points field examples of evolving fracture systems and faults in west Norway, in particular on the islands of Sotra and Øygarden, west of the city of Bergen. All the models were run using the boundary-element program BEASY (1991).
With reference to Fig. 5, a general model of an en-echelon system is first presented subject to a tensile stress of 6 MPa. The model consists of eight extension fractures, each modeled as an internal spring. The model height is one unit. Each fracture has a length of 0.1 unit, and the upper and lower tips of each adjoining pair of fractures are at the same horizontal level. The underlap or vertical distance between the adjacent tips is thus zero, whereas the offset or horizontal distance between the adjacent tips is the same for all the fractures and equal to 0.1 unit.
The results (Fig. 15) show that there are stress-concentration regions between the adjacent tips of each pair of fractures. In these regions, there is concentration of shear stress and also tensile stress (not shown). This indicates that transverse fractures, many of which are shear fractures whereas others could be hybrid fractures, tend to develop between the adjacent ends of the extension fractures and link them up into a single, segmented fracture. When the initial en-echelon extension fractures become interconnected through transverse fractures, the percolation threshold of the fracture set is reached so that the fracture set acts as a single, segmented and interconnected cluster of fractures that can conduct groundwater from one end to the other. Such a development has already occurred in Fig. 5.
Photoelastic studies (Gudmundsson et al. 1993) support this conclusion; they show that once the underlap changes to an overlap, that is, when c becomes negative, the stress-concentration region between the nearby tips decreases in size. These results indicate that the most favourable configuration of offset extension fractures for the development of interconnecting shear fractures between the nearby ends is that of a significant underlapping. This conclusion is supported by the geometry of the en-echelon fracture set in Fig. 5. There the transverse fractures have developed while there was a significant underlapping of the original extension fractures.
As the offset b decreases, that is, when the original extension fractures become more nearly collinear, the shear-stress intensity decreases (Fig. 18), whereas the tensile-stress intensity increases (Fig. 19). Thus, collinear extension fractures have the greatest probability of in-plane propagation, which would lead to linking up (through extension fractures) of the nearby ends. Thus, the original configuration of the fractures in Fig. 5 favours shear-stress concentration and transverse-shear fractures, whereas collinear-extension fractures favour tensile stresses and the growth of an extension fracture into a single, segmented fracture. In both cases, however, the tensile loading leads to stress concentrations that favour the linking up of the extension fractures—either into a fracture with en-echelon segments linked by transverse-shear fractures, or into a single, segmented (but collinear) extension fracture, depending on the configuration of the original extension-fracture set.
Groundwater Transport in Fractures
Volumetric Flow Rate
Aperture Size in a Homogeneous Rock
Equations (4) and (5) show that volumetric fluid transport of a fracture depends on its aperture size, b, which in turn is a function of the fluid overpressure. The fluid overpressure itself, \( P_0 \), can vary along the plane of the hydrofracture. The overpressure variation depends on initial excess pressure, \( p_e \), in the source aquifer (Eq. 3), the buoyancy term (Eqs. 4, 5), the local dip of the hydrofracture (Eq. 5), and the stress changes in the host rock. A general analytical solution for the aperture-size variation along a hydrofracture is obtained by considering a mathematical crack, a line crack, located along the vertical y-axis. In this solution, the effects of buoyancy, fracture dip, and host-rock stress are all included in the fluid-overpressure function. The crack is then defined by x=0, −a≤y≤a and is supposed to be subject to an internal fluid overpressure given by the even function p(y)=p(−y), so that the pressure is the same inside the fracture, above and below the x-axis.
This is the plane-strain formula for a two-dimensional, elliptic through crack, a "tunnel crack" (Sneddon and Lowengrub 1969; cf. Gudmundsson 2000b), showing that a constant fluid overpressure opens a hydrofracture in a homogeneous, isotropic rock into a flat ellipse.
Aperture size in a Layered Rock
When the host rock is layered, many hydrofractures become arrested at the contacts between the layers (Figs. 4, 11; Gudmundsson and Brenner 2001) but some are still able to propagate through many layers. If the layers have different mechanical properties (Figs. 4, 6), the variation in aperture size of a hydrofracture can be very different from that presented for the homogeneous and isotropic rock above. In particular, the Young's moduli of the layers through which the hydrofracture propagates have large effects on the resulting aperture size. Layers with a low Young's modulus are referred to as soft and those with a high Young's modulus as stiff, as is traditional in engineering rock mechanics (Hudson and Harrison 1997).
A boundary-element model was made of the variation in aperture size in a fracture subject to constant fluid overpressure of 6 MPa but dissecting 10 layers with different Young's moduli. To bring these effects into focus, the Young's moduli used cover the normal static range for common rock types, that is, from 100 GPa for the stiff layers, through 10 GPa for the moderately stiff layers, to 1 GPa for the soft layers (Johnson 1970; Jumikis 1979; Afrouz 1992; Bell 2000).
Groundwater transport in solid rocks is commonly largely, and sometimes entirely, controlled by the permeability of the associated fracture networks. For example, in a fractured but otherwise impermeable rock, all the groundwater transport would be along the fractures, irrespective of the hydraulic gradient. For groundwater flow to take place along a fracture network, the fractures must be interconnected in such a way that the percolation threshold is reached. In this paper, two mechanisms have been considered by which fracture networks, such as systems of joints, and contacts become interconnected.
One mechanism is the linking up of fractures in a rock subjected to tensile (or any other external) loading. If the original fractures are offset, they link up primarily through transverse-shear fractures but, if the original fractures are collinear, they link up through extension fractures. When such fracture sets reach certain crustal depths (Eq. 1), or otherwise become subject to shear stress, they may develop into shear fractures and, eventually, faults. The other mechanism is the growth of extension fractures as overpressured hydrofractures. The pathway of the hydrofracture is then primarily generated by opening up and linking contacts, joints, and other discontinuities in the host rock.
Both mechanisms assume the existence of outcrop-scale discontinuities, such as joints and contacts. This assumption is normally justified because most rocks contain joints, and many of these are generated during the formation of the rock. This applies in particular to igneous and sedimentary rocks. For example, columnar joints are very common in lava flows (Figs. 1, 2, and 13), from which larger fractures may develop by the mechanisms discussed above. Rock-mechanics experiments as well as field observations indicate that, in rocks under compressive stresses, shear fractures may form by a mechanism that does not involve the linking up of outcrop-scale discontinuities (Paterson 1978; Twiss and Moores 1992). A discussion of this mechanism, however, is beyond the scope of the present paper.
In this paper, the numerical models of hydrofracture propagation focus on pathway formation where the entire fracture-forming fluid disappears subsequent to the fracture formation. This is presumably a common situation where the fracture-forming fluid is groundwater but, as a rule, it is not the case when the fluid is hydrothermal water or magma. Hydrothermal veins are very common in active fault zones (Gudmundsson et al. 2002). Although some veins may reach lengths of hundreds of metres (Bons 2001), presumably most have lengths of the order of metres or less (Gudmundsson et al. 2002). Thus, although mineral veins are commonly barriers to transverse groundwater flow, individually they are normally too small to have significant effects on regional groundwater flow.
Regional dykes are also emplaced as hydrofractures where the fluid is magma that becomes solidified in the fracture subsequent to its formation. Some dykes, particularly thick and massive ones, are barriers to transverse flow of groundwater. This follows because the dyke rock is commonly dense basalt of low matrix permeability. Other thick dykes, however, form in many magma injections, each of which may develop transverse columnar joints on solidification (Gudmundsson 1995). These dykes, some of which reach tens of metres in thickness, may be relatively permeable because of the columnar joints, and the same applies to many thin dykes. Thick, fractured dykes may act as sources of groundwater (Singhal and Gupta 1999). In some dykes, however, the columnar joints are partly, or entirely, filled with secondary minerals, in which case even thin dykes may be barriers to transverse groundwater flow.
Because the dyke rock has commonly very different mechanical properties from that of the host rock (granite, gneiss, sedimentary, and pyroclastic rocks), stresses tend to concentrate at the contacts between dykes and host rocks and generate groundwater conduits. Thus, dykes, even those that are barriers to transverse flow, are commonly good conduits of dyke-parallel flow of groundwater. Regional dykes that extend for many kilometres, or tens of kilometres, on strike and down the dip, can have very significant effects on the regional groundwater transport. In many areas, for example, in Iceland and Tenerife (Canary Islands), dykes collect groundwater and may transport it over long distances. Because the dykes are hydrofractures, they form groundwater pathways by linking up of discontinuities. Thus, groundwater flow along dykes follows pathways similar to those generated in the numerical models discussed above (Figs. 9, 10, 11, 12, and 14). It follows that the models presented here indicate not only the pathways of groundwater flow inside open hydrofractures, but also groundwater flow along the margin of dykes and other solidified hydrofractures.
We thank the Hydrogeology Journal referees for helpful comments. This work was supported by a grant from the European Commission (contract EVR1-CT-1999-40002), several grants from the Norway Research Council, and a PhD grant from Statoil (to Agust Gudmundsson) for Sonja L. Brenner.