 Good morning, everyone. So today will be an entirely different day from the previous ones, because we will go to seismology. So seismology is among you. For some of you, some parts may be easy, but they are meant, of course, for the other ones. While you might get more food along the way from the expert seismologists that we call to lecture today. So we will also have a lot of exercises. And the exercises contain a software that we hope you might be able to use also in your future. So I'll give the word to Tarsten. OK, so good morning again. And as Eleonora already announced, we slightly slowly moved to seismology. But actually, my first presentation here is still on a subject that was heavily discussed yesterday on the fluid field fracture growth, or how do dikes move and grow? So these are general questions. How do fluid field cracks grow in general? What can we learn from the shape and the growth path, especially, and how this is linked to induced seismicity, to seismicity in general? So I will concentrate a bit more on the basics in terms of the theoretical concepts in this presentation directly after my presentation. Cindy will show examples on dikes and induced seismicity, much more than I do. So the first section, how do fluid field fractures form and grow? And we need to have a look into theoretical concepts for this. But before, I have at least also one nice figure of a picture photo of a dike. This is a dike in Germany in the Eifel. And I mean, you all experts here, you have seen a lot of dikes on the poster outside. You see very nice example. Here's another one. You see the dimension of magnetic dikes. The opening can be up to several meters, length overall to up to several kilometers. I like this figure because if you'll carefully look to the out drop here, you can also see indications of faulting. And we don't know for sure, but it's likely that this faulting occurs during the dike intrusion. And this is exactly the subject of today. How is this emplacement formed? And how do dikes grow? But also how are the earthquakes linked to this process? And yesterday, we have also already discussed the problem of magnetic reservoirs. And in the second part of my presentation, I will concentrate also on geometry that is the war presented by this sketch here, where you try to couple also to the reservoir because large dikes, kilometers long as in this sketch, for instance, for lateral intrusions or rifting episodes, they have to be fed from large magnetic reservoirs as seen here. If you look to the, well, you can find examples of exposed solidified magnetic dikes at the surface. This is a figure taken from Dilani and Pollard, where a complete segment of a dike has been exposed and the opening shape has been measured. And this gives a good impression. It's in scale how this possibly looks. And you see it's not a planar crack. You see there's a small curvature inside, but it's not very large actually. So the first question is how, what is the expected theoretical shape of opening? And maybe many of you know this. And I will not go into the equations, only give a reference in principle. This is a theoretical concept, for instance, that has been derived for two-dimensional parallel infinite cracks as seen here, so-called Griffith crack. And the geometry is nicely sketched in this figure, which is taken from Pollard and Siegel in a review paper, 87. And the main concept of this type of, simply, of 2D crack is that the length in one dimension is infinite. This is the crack tip here. This is the, here it's called width, or we often say length of the crack. Half length is A. And then you can derive equations, how the displacement and the stress field changes if this crack is in place in a homogeneous rock. Important is to recognize that here we go to crack problems. These are boundary value problems. So we cannot start with equations of elastic continuum. In principle, we need to solve the combined problem of sources of these cracks in an elastic medium. And the boundary conditions are very important. And they are given in a sense that the stress is continuous at the surface over the crack plane. And this figure also, if you look carefully, distinguished between the remote stress or the regional stress that is given far away from the crack and the stress, which is, for instance, the pressure inside this crack if it's fluid field that is acting on the surface. And responsible for the opening shape is the difference between the confining or stress here and the remote stress, which is, for instance, described in the term driving stress. So in my notation here, this is given by this delta sigma, the driving stress. And this is actually the boundary condition. So if you know the driving stress, you can estimate the opening shape or the shearing shape of such a crack. Maybe I should also give a note or indicate that you have three options of driving stresses. One is normal stress. And the other two are shear stresses, either in this or the other directions. And these are associated with three types of opening mode 1, 2, mode 3, as many of you probably also know. So the equations of opening can be derived in for such a simple crack if the driving stress is constant. For instance, a constant internal pressure in a medium without any remote stress. Then the opening shape is plotted here, is elliptical. And the equation for the opening is seen here. So here you see the half length of the crack. These are elastic modules. And these are the driving stresses. If it's a mode 1, which is an opening crack, it's sigma 1. And these are the ones associated with the in-plane shear and tiering mode, so anti-plane shear crack. And this is the opening. So it's plotted here. You see three curves, a comparison between analytical and numerical methods. And this is the crack tip. And the opening is also associated with a small rotation if you look here. So this is more or less what we have seen in this figure. And this is a geological outdrop. But more of interest for our purpose, for the question of how do cracks grow, is the stress. So from this equation, you can also derive stress fields. For instance, a very much of interest is the stress field at the crack tip. And there you find for this grip of cracks there's a stress singularity. This is the crack tip. And again, we have analytical solution and numerical solutions. But important is that here the stress goes to infinity in theory. In this linear elastic fracture theory, it's really infinity. So there's a stress singularity. And the stress singularity can be approximated at the crack tip by this formula. And interesting is that this is more or less for all types of cracks, always the same shape in this approximation. And therefore, parameters have received special attention. You see that the stress, the stress singularity, very close to the tip, is expressed by a strength. This is a factor in front of the singularity. Then there's always this 1 over r1 singularity. And r1 is measured from the crack tip in this case. So the distance to the crack tip. So it shows how quickly the singularity attenuates. And then there's a shape factor. This can vary for different modes of cracks. But important is that the strength of the singularity can be measured by a parameter which is defined as stress intensity factor. And this stress singularity or the strength, the stress intensity factor in principle defines whether a crack can be stable or not. If this is too much larger, the stress singularity than the tendernal strength of the rock, then the crack would grow and would be instable. And I mean, the stress intensity factor can vary a bit for this simple crack with a constant driving stress it is given here. But important, and it always depends on the driving stress in a linear manner. But it also depends on the length of the crack. And this is an interesting observation because it means the longer the crack, if you keep the driving stress constant, the internal pressure, for instance, the larger the stress singularity becomes. And the more instable this crack would be. So this is something that is possibly unexpected. OK, we have these 2D Griffith cracks and know all the theory for those. But in reality, it's clear that fluid fractures are typically 3D fractures. And for instance, a penny-shaped crack or a circular crack or elliptical crack is much more realistic. Here, you see also a comparison to an observation. This was a hydrofracking experiment in central Australia in the Cooper Basin. You see seismicity, and this confirms very much. This is a cross-section. But if you look overall to the seismicity, it confirms very much that this fracture was growing in an elliptical or a penny-shaped mode. It is fortunate or simply a fact that the equations of for the stress intensity factor for the opening do not change a lot if you go from a Griffith crack to a penny-shaped crack. There's a factor of P over 4 that this means that often the theory of the 2D crack can be used as a first-order approximation also partly for these cracks. If you only look in one dimension, for instance, and study crack growth, for instance, from bottom to top upward. So here in this experiment, you also see some indication of the stress field. And everybody would expect that this penny-shaped crack is oriented, that it opens in the direction of the minimum compressive stress, the least compressive stress. But is it always the case to all fluid field cracks open in this direction? And more important, even do they grow in the direction of the maximum compressive stress or in the plane of the maximum and intermediate stress? So what is your guess? What is your opinion? Both cases are true. So I will try to show that the first is true, but the second one is not necessarily true. And this is interesting and important because it has some implications. So if you want to answer the question in which direction do cracks grow, you need to study the crack tip and the stress intensity factor, but you need fracture criterion. And the typical or the valid criterion or established one is the Griffith criterion. Can be given in this term as seen here, which we use, actually, if we do numerical simulations, stating that if we define the strain energy that is released with increasing this crack a small amount by this term, so energy change with length, then the fracture criterion would state that this has to be larger than the specific threshold, which is the surface energy that is needed to create a small extension of this existing crack. And additionally, you can require or you can ask that this should also be maximal if you want to define the direction of growth. So an alternative, what is more used in engineering community is simply criterion based on the stress intensity factor stating that the crack becomes unstable if the stress intensity factor becomes larger than material property that is the fracture toughness. The fracture toughness is related then to the surface, to the specific surface energy. But the formulation in terms of energy release is more interesting if you really want to understand how cracks grow and maybe if they curve, or do not curve, because the second one can only show you whether crack is stable or not, but do not tell you exactly in which direction the crack will grow. So if you want to, then we can estimate this strain energy. For instance, if we know the opening shape of the crack and we know the driving stress, we have a normal and a shear component, so it's a sum of both. And we can integrate over the whole crack surface. This gives the strain energy and the strain energy change with length. It's then simply this first order approximation that it's the difference between the energy in two stages divided by the length increase. And this is related to the stress intensity factor as you see here. So numerically, you can, for instance, solve this. If you have a method that can estimate for a given driving stress the opening shape of the crack, you can simply test. You increase length by a unit step and you can estimate, calculate the opening and then calculate the strain energy from the difference by this formula and then try to understand when is it maximum in which direction and then whether your fracture criteria is fulfilled and then you can simulate crack growth. And I've done this long time ago and take some examples from this paper and this time, which are, I think, quite instructive. Simply simulations, how do cracks go? And we start with empty cracks. So you see, this should be an airfield or empty crack and if you put it in a crust at large depth and you have confining pressure, compressive stress from all sides and there's no deviatoric stress then this crack would do nothing. It would still be addressed and would not start to grow because the fracture criteria is not fulfilled. If you put it in an extensional regime, poorly extensional, for instance, close to the surface, it might happen and a volcano slope, for instance, that you have really extensional stresses, then this empty crack would start to grow and it would grow in its own plane. There's no deviatoric stress here in this medium and it would not stop to grow. It would grow until infinity if you don't change the stress field. If you have a mixed load, so there's shear stress present, then you can observe that the growing is in direction of the maximum compressive stress, as you see here by this arrow. So the crack turns and grows in this direction but then grows until infinity. If you fill this crack with fluids and you assume that this is a finite volume crack, then the figure changes a bit for these simulations in the overall compressive regime. This would also be stable but the opening is not zero. It is a finite value because there's fluid inside that is compressed until the boundary conditions is fulfilled and you have equivalence of the internal and the outer normal stress. If you have it in an extensional regime, this crack would also grow in its own plane but after a while it stops because you have a decompression with volume increase. So this would not grow to infinity and if it's a mixed load, it would also turn as you see here and then after a while it would stop. Interesting is also that this turning here is different. If it's an empty crack, it's a sharp like it's nearly a kink but it's still a smooth curve but this curvature here has a much larger curvature radius if it's a fluid field and there's also already tells something on the growth path. There is a difference and this difference actually comes from the what I call self-stress of the crack. So what is this with the fluid field crack? What is different in the fluid field crack? You assume that you have a finite volume crack so this is a closed system and the fluid cannot disappear. If you make this assumption then the internal pressure can vary, can change. It is always trying to be in to fulfill the boundary conditions with elastic media and you can in principle for instance if the crack grows here are two examples of how such growth can look like unilateral or bilateral from a highly over-pressurized crack to a normal over-pressurized crack then the volume would possibly change. What is constant is the fluid mass but the fluid volume can slightly change and can change according to the compressibility of the fluid, the internal pressure. So this is the ambient pressure in the crack can change, can adapt to the environmental ambient stress and this is an important concept for fluid field cracks if they are of finite volume. So you can continue with these simulations and for instance study the effect of stress gradients. Stress gradients means that as seen in this figure for a poorly compressional stress regime but you have no deviatoric stress, no shear stress involved but overall the stress is larger here and smaller here. For instance there's a volcanic load and if you go away from the volcanic load you have an inhomogeneous stress field maybe you have stress gradients. If this crack would grow here and we make simulations we see that it is not growing in its own plane as we have seen before but it is curving in the direction of the maximum gradient of the stress. So the gradient has an influence of the strain energy release and therefore this is turning. But it's in, this turning depends on the volume enclosed in this crack and it's very smooth if this is a large volume crack which finally would mean this is a dike that is very long and has a large volume and it's the opposite if it's a small volume crack. More interesting is if you have a deviatoric stress and a gradient as you see here. So the axes in this direction and in this direction of the remote stress are not the same. So maximum compressive stress is in this direction, up down in this figure. The gradient is in horizontal direction so the gradient only would try to turn the crack in this direction, the deviatoric stress would try to turn the crack in this direction. Overall, it takes some path in between. It means, it answers already this question, it's not always growing in direction of maximum compressive, there are reasons for instance, stress gradients that can lead to a deviation of this direction and it also means that this is a crack propagating or growing under mixed mode loading. There's also sheer motion involved. It's not only opening and sometimes we observe this in, for instance, the rifting episodes and maybe gradients can also be a reason for this. If the gradient and the maximum compressive would point out of being the same direction the crack immediately turns in this direction and moves in this way. We also see another feature here that is the crack itself has moved as a whole and so wholesale movement or crack propagation as we sometimes call it. And this is maybe better explained in the next figure when we come to buoyancy controlled cracks. But I also like this figure because it's a very extreme simulation but here I changed not only the gradient as before and to have some sheer stress but the sheer stress is also increasing in this direction and then you see that in comparison to this figure where it simply turns and grows in this direction there is according to this numerical stream also a mode of crack growth that is poorly sheer in principle. Also it's a mixed mode. So it's still fluid filled opening but growing in a sheer direction because sheer stress is so large. So it is maybe not so... This is a theoretical result and I have no example where this has been observed and it's so clear so far. But coming now to another question this is this wholesale movement. It can happen if you have lateral gradients of stress as we have already seen in the simulation but it's much easier to explain if you have ascent of dykes and Claude Chopin yesterday already showed this example and explained very well the theory behind the gradient of the driving stress in this case comes from the density difference and from the apparent buoyancy. It's not a true buoyancy force but it's an apparent buoyancy because responsible for the growth is always the strain energy release. So this figure has been discussed by Claude Chopin. You see the, let's say the driving stress as the difference between the ambient stress in the rock changes the depth and the internal stress gradient from the density column above and if the density is smaller than the density of rock the density of the fluid is smaller than the rock there is an over pressure at the top and at the bottom, well at some point the pressure is zero or the driving stress is zero and if you go deeper it will even be under pressure lower pressure, internal than outside and then you can find an opening shape that is plotted here which is this tear drop shape and this is a well-known solution that so-called Werthmann solution and in this mode this is very interesting because here at the top we will see or you can show that the stress intensity factor is exactly zero while here it can be very large so the rock can break on top and on the bottom end, on the lower tip it would, the crack would have tried to close itself even if there is a channel that is left behind him and this is thought to be the mode of dyke growth dyke ascent from bottom to top. This was all for a 2D solution for this type of Griffith crack but you can also simulate how it would change if you go to penny-shaped cracks and in principle you have the same shape there is only a small change in the internal over pressure before we had a factor of 0.5 and if you go to this 3D crack it's 0.7 about half length of the crack times this gradient maybe you see here again that this function is defined as an ambient stress this is the one that adapts and the gradient that is depending on the density difference so we can also understand the growth of penny-shaped crack or the opening of penny-shaped cracks and if they are in this mode they would break the rock on top, leave a channel behind and you can also simulate this and it was often, for instance, observed very well by Eleonora and described in some papers for gelatin cracks you can observe this mode of how it's an upward moving of crack propagation for these cracks in gelatin and you can simulate this here with the boundary element method you can simulate the opening shape and you see then the channel behind that is the path of the crack and this was the dashed line we have seen in the figures before this is let's say the theoretical solution of such a growing crack and it's clear that this crack will also change the stress field in this room and this has also been simulated with this boundary element method you see three cross sections one is a horizontal plane and two vertical plane so this is the plane we have seen before this is orthogonal to this and you see that there are large coulomb stress changes mainly at the crack tip and if you would think on which earthquakes would they induce then you can recognize that the mechanism of the earthquakes can change a lot the stress field is very inhomogeneous if such a crack propagates upward so you can explain normal faulting thrust faulting and also partly shear faulting events in such an environment so the expectation for earthquakes in the surrounding rock is that because of the heterogeneous stress field that the mechanism can vary a lot another question we have only looked to stability but it's clear a big question the key question is also how fast do these cracks grow and move wholesalely and this is more difficult to understand and to find theoretical solutions but in principle the idea is that the fluid has to move during the growth of the fracture so maybe it moved upwards here and downwards at the side if in this presentation and the flow causes some viscous pressure to drop and this would cause during flow that the overall pressure in the crack will be reduced but if you try to simulate this it's an approximation but anyway the result mainly shows that overall this gradient is reserved so the opening shape should be the same it's only the ambient pressure that is on most part of this crack reduced except of the crack tip which is a singularity for the flow and where you have to introduce some approximations and where this simple move of the internal stress field will change so I will not go more in the theory of this flow and velocity of growth but we have for instance lab experiments to compare theories with this for instance these gelatin cracks and you can see that the flow velocity the growth velocity or ascent velocity is larger if the length of the crack is larger meaning also the stress intensity factor at the top is larger when the growth velocity is larger here it's the growth velocity of such a crack in the gelatin experiment is plotted as a function of time and it's smaller if the volume of fluid embedded in this crack is smaller so in terms of dikes it means a very long diameter ascent faster than a very small one but it's also interesting and therefore I include this figure also it's a bit complex that all this theory would also predict that you need a critical length of the crack until it starts to growth this is coming from the fracture criterion and we have seen that the stress intensity factor depends on the length meaning the stress singularity depends also on the length of the crack and only if the length is so large that you meet this criterion the crack would grow and you can try to estimate how large could this be for magma dikes, for water filled dikes and it differs a lot for magma filled dikes it can be in the range of kilometers actually depending on the level in the crust where you simulate this so this was a simulation for dikes in a mantle and then the velocity at the point of the critical length of the crack is very small in the beginning but it accelerates quickly if the crack becomes longer if there is a mechanism to fill in more fluids then the critical amount of fluid it can grow fast but there is also the option that they grow very slow you cannot really know this and overall the growth velocity the ascent velocity for dikes is in the range of meters, centimeters to 10 meters per second so this can be typical velocities grow velocity of dikes so we have then used this and simulated several examples but I possibly will go very quick over this because I think also Eleanor Nora will also show some more examples possibly in her presentation but only so you see some figures to show the principal effects of these simulations of fracture, whole fracture movement you can try to simulate how crack ascent in a crust let's say this is the crust this is the mantle for coming from the deep mantle a dike with a density of given here 2,950 in a mantle with 3,300 kilogram per cubic meter and in the crust the density is smaller so you have the so-called level of neutral buoyancy here and let's say the effective driving stress in principle or this gradient of the stress profile in the rock is seen here and the one in the so the presentation is always that the one in the dike would be simply vertical because it's a reduced scale so you see this crack would grow and it would stop here this is then the stopping point because of the level of neutral buoyancy this is in a stress field that has no deviatoric stress but this can change if you go to a extensional regime or compressional regime in the crust if you go to an extensional regime you in principle observe the same except that maybe the crack intrudes a bit shallower and becomes a bit longer but this actually is the is the situation that has been described yesterday by Glotropar in this situation this crack in this stress field would have a tendency to grows continue to grow laterally but still be a vertical dike if you have a compressional stress field you can turn the crack into the dike into sills but whether and at which level they turn depends again on this volume that is trapped in this crack so if they are long dikes or short dikes have a large volume small volume you can break through the zone of compressional stress if it's a very large volume dike even if you are in such a stress regime so this can explain for instance magmatic underblading in the crust in a compressional stress environment you can study the loading effect of a mountain of a volcano and you see that the path is blocked here have a tendency to attract dikes which can explain the formation of volcanic centers or in a compressional regime you would see even with this loading from a mountain these dikes would not reach the very shallow surface but would accumulate in a deeper level and possibly can explain the formation of dikes so this is a work by Francesco and Eleonora recently for instance studying the effect of gravens which has the opposite effect as one from mountains in principle but there is much more you can learn than Eleonora will show this possibly in her talk a bit better you can also study and try to investigate what would happen in the mantle if you have a mantle flow the mantle flow would change the stress field and you would and you can explain for instance in a subduction zone where this is the subducting slab this is the mantle wedge above the subducting slab if you use analytical solutions of the mantle flow in the stress field that is involved in this mantle flow you find that the fluid field drags would have a tendency to move away from the subduction zone if these drags are generated here and to reach the surface or shallow levels in some distance to the dredge also something that is observed that volcanic chain usually is offset in comparison to the subducting trench and this offset has a characteristic length always where the subducting slab is in a range about 100 kilometer deep and this mechanism possibly can explain that fluid field drags have a tendency to move in this direction this includes also water field drags but also magmatic dikes also at mid ocean ridges you can simulate the drag dike growth in the mantle and you find that there's again an attraction point not from the load but also from the only from the mantle flow at mid ocean ridges and it's interesting that this solution was presented from the assumption that you have dike ascent mechanism as we have explained just before is very similar to the mechanism that comes out from another model that is porous flow of magmas so they also have the same tendency so in principle from the observation you would not be able to distinguish these two models so these are the basics of isolated single cracks but it's clear I think to everyone that there's also a lot of crack of interaction possibly that can take place and this will also change partly the growth mode of these or the growth path of these cracks and a very well known example is this of a crack-crack interaction because there are many expressions in geology this is a photo of a mud layer that has been dried in the sun and you see this features and this is an interaction of two opening cracks overlapping and actually I just looked here you see a very nice crack in this ball and you can see the same features here overlapping cracks growing in this ball and you can also observe this for dike segments I've seen here in this figure on a small scale and we have seen the crack scale but also for mid-ocean ridges you see these overlapping spreading centers so this is an effect of the interaction of the two cracks and these numerical simulations can reproduce exactly this shape What would that look like in the spring? What would it look like in the spring? Well, actually it's not so different I mean if you have a lateral dike like these mid-ocean ridges you can assume they have a length cutting the whole plate of crust and then it's basically a 2D problem If you have really 3D ascending dikes they can also interact and then it simply depends on they have a finite length in the third dimension it will depend on whether they overlap or not And if three dikes in the photo what is new in you when you get them joined together? Well, actually, I mean this is the example I had to show this in a dike out drop and this is a kind of a nostrilol dikes the dike below is a larger feature but also these segments these inertial segments growing slowly upwards but growing at the same time horizontally they show these overlapping features You can also model this interaction by assuming what happens if you have a dike that solidifies or in some level in the crust and then you have a second dike and a third dike and they all feel the stress field that has been created by the first dike the second one, the third one and it's interesting then that this overall creates a stress field that finally leads to this is exaggerated, this is a true scale leads to an accumulation of dikes and it's very similar to what you would observe for a magmatic, a spherical magmatic body so the stress field that comes out so it can explain that multiple dike intrusions over a long time can lead to a stress field and finally to a formation of magma chambers Here this is not yet the figure of the stress field but it shows the surface effect so it can explain the formation of volcanic centers and even some periodicity in volcanic centers for instance along the ocean ridge and here this is a plot of the stress field so you see the maximum compressive stress is more or less similar to what you would expect from an over-pressurized spherical source so this was mainly now so far on the I have to check the time so this was mainly on the growth of on the assent of single dikes or partly interacting dikes that we have never considered the magma source and this is another problem where the growth is controlled by the injection of fluids and there can be different examples as I said there can be the lateral intrusions fed by magma reservoirs, rifting episodes we think also that we see possibly these effects for mid-gross earthquake swarms for instance in northwest Bohemia or in other regions or very classical hydro-hydraulic fracturing experiments is an experiment of an injection and fluid field growth and we start with the hydraulic fracturing experiment because the model that I show here has been derived exactly for this case to understand hydro-fracturing and we worked a bit on this there's no magma chamber there's a drill hole usually it's indicated here and this is packed and then you have a high over-pressure to to create a hydro-fract that grows in two directions typically and we were especially interested in the observation that partly this growth is asymmetric so in one direction it grows faster than in the other direction so the two wings of the hydro-fract are possibly not symmetric it's asymmetric in terms of length and the possible reason for this is that there are also stress gradients present as you see in the sketch here driving stress gradients this can be buoyancy related but they can also be related to the background stress or in a gas field as we will see in the example it can be possibly related to the pore pressure change in the gas field in the porous matrix and then it's interesting that after you shut off you close your valve and you stop the injection there is an after growth of these cracks first be directional but then it can also happen if you have gradients that this is unidirectional after growth and this is a very simple model that has been derived but it explains already very well several features you can observe as you will see here and the principle assumption is maybe best explained with these sketches where you see different stages of this injection process the injection phase and if for a hydrofract you can do it very simple that you say the injection pressure is always the same during the injection phase it's controlled by humans and then the crack growth and this is the upper the eight part of this figure the length of the crack is seen here the injection point is at zero and the pressure profiles are plotted here and this is the limiting let's say pressure that is needed to further grow as soon as this black profile which is the effective pressure or what I call driving pressure before reaches this threshold this defines the tip of the crack and the gradient so the no flow or background driving stress is without flow is seen here this is what Klosho part mentioned with the static solution hydro static solution and principle so introducing a flow you have to consider that there's a viscous pressure viscous pressure drop related to the viscous flow and the pressure crack there and if you approximate this to a first order by a linear law then you have this linear lines as you see here this is leading to the effective pressure over pressure in the crack during the process of growth so we use this we use this laminar flow approximation for this linear viscous pressure drag and try to estimate the velocity and the length of these cracks and it's possible to find analytical solutions that have been described in detail in this paper I will not go in the details of the derivation it would take too much time but it's all published in 2010 but interesting more interesting is to look now what is the concept after you stop the injection then you relax the boundary condition that the injection pressure is at the injection point always P-naught so the crack continues to grow because you still have the possibility of flow and fulfilling the fracture criteria to both sides but the ambient pressure would quickly decrease because of the expansion of the crack the internal pressure will decrease quickly so this is a interim phase which is not lasting very long but if there are these stress gradients present after a while after this point the crack would not if the red line is at this level the crack would not grow further in the short wing direction so growth would stop here but in the other side the crack can still grow and then you have this situation this is the post injection phase oh this is wrong with the uni-directional growth and only in one direction the ambient pressure still decreases until you finally will have the red point here and then you are in the situation as we form more or less with the Batman crack and maybe the crack stops or it slows or it grows very slowly holds a movement mode so in the first phase trying to with this analytical model it's interesting that you can find solutions for the length of the two wings as a function of time they also depend on the exact shape of this long wing and short wing growth depends on viscosity also and some other assumptions but the ratio between the long and short wing is independent of viscosity and only depends on the injection pressure and the pressure gradient so if you can measure during a hydro-frag experiment the two wings as a function of time in such a curve and you can form this ratio and you can estimate the gradient and the internal pressure in the injection phase during the injection phase and it's interesting to ask whether this can possibly be adopted or used also for magmatic intrusions if you can measure by seismicity the length of these two wings in the post injection phase where you have this redistribution of pressure and only growth in one direction there's also a self-similar solution with this analytical model and this tells that the extension at the point when the crack stops to grow in one direction and only grows in the second direction of the long wing and then it extends further but the overall extension is always 1.5 of the length that has been reached just in the moment when it starts to grow unilateral instead of bidirectional the shape, how quick it reaches this value here depends on other parameters but the overall extension is always 1.45 independent of any parameter so it's a good test of the model whether this model is valid but it also gives an option for instance for a magmatic intrusion from a reservoir if you can measure the overall length that you can try to estimate the stop of the injection period in terms of length but also in terms of time so I think I should summarize here the findings for fluid field crack growth and I'm a bit afraid that I have not much time to show examples of seismicity in these last 10 minutes but first summarize what I explained here so fluid field crack growth is controlled by three factors overall one is the orientation of the least compressive stress this is a strong factor this is also what is classically assumed to be the main factor or the only factor but there is also an effective driving stress gradients this can be buoyancy or can be gradients in the background stress and the third one you've seen in the simulations is the self stress that is generated by the cracks if a crack is long if it has a large opening the stress in homogeneity created by the crack itself is very large and is always at the tip and it influences the overall stress field and this leads to the smooth curvature and it's changing if you have a large volume crack or a small volume crack and this also controls the path for instance we have seen yesterday Glotropascio this nice example of the shiprock ex intrusions which we are exposed and you see the path of many dykes and if you look to the original figure of this geological study mapping all these you see some dyke path is crossing the others which if all would happen let's say at the same time you would not expect this but it possibly can also be an influence of the overall pressure in the intrusion because the path if it turns is different if the volume is not the same so something that has to be kept in mind so the growth is also influenced by the crack interaction and this for instance can be used to understand or to study localised volcanic centres another point I have not mentioned is that crack-crack interaction can also lead to stopping of cracks if you have a sill and the dyke reaching the sill it will immediately stop it's in principle a crack-crack interaction problem and it's also a second mechanism to form reservoirs and then we have seen that the penny-shaped crack is also a basic solution I've not really shown but only indicated in the figure that it's not only always circular it can also be elliptical and this depends a bit on the intermediate stress and the change of the intermediate stress to the maximum compressive stress but the penny-shaped crack is very much related also to this Wehrmann crack where you start from a penny-shaped crack and this is growing in one direction and leaving a channel behind so the whole movement of an overcritical crack is an important solution and you can explain these vertical dykes and horizontal sills that are moving as a whole if you have effective stress gradients and then also the injection-related figure in the last examples I've given have shown or I tried to show that asymmetric crack growth is possible b-lateral growth but not at the same speed in two directions and the time function depends on the fracture toughness so this rock parameter but also on the fluid viscosity if you can measure the time function very precisely you have a chance to estimate these parameters from these experiments but the ratio of the long and the short wing depends only on the stress grad and it gives a good option to try to estimate these parameters this parameter and after the injection you have this self-similar solution for first b-lateral and then unilateral expansion with this interesting result that the overall expansion is always 1.5 the length at the end of the injection phase which is also an interesting feature and the last point I mentioned here I had to read it myself crack opening and stress buildup in rock explains b-lateral front of seismicity during injection and the front of back front of seismicity this is something I've not yet shown here actually I wanted to make this in gray because this is something maybe if I have three minutes more five minutes which I can directly show in one example which is coming next I have in the presentation I will give you some basic things on seismicity how it is created but since Cindy will show examples on this anyway I have many slides that are shadowed here are these examples how does it... so we can see one example or two examples maybe quickly for this injection control fracturing so this shows the basic principle and the experiment I will show has a similar geometry you have a bore hole you have a hydrofract there and this is observed from other bore holes for instance with seismometers so that you can measure the seismicity then there is a penny-shaped crack growing and if you plot the distance to the injection point versus the occurrence time you see the growth of the crack tip in principle because there the stressing rate is larger most of the seismicity is occurring there but you already see if you refract that there are no earthquakes occurring here this is a typical stress shadow effect and in the next slide we use the same experiment but on a larger scale in the gas field where you here see the bore hole and there have been different stages of hydrofract experiments this is the... in colors are different stages so experiments at different times different factors and we concentrate only in blue one here we plot the direction of the distance of seismicity in this plane to the injection point we'll see in this distance time plot this asymmetric growth but you also see it here most of the seismicity is in one direction and not in the other direction and it was confirmed that this is not a location problem it's really a fact that the growth was asymmetric so if you plot for such a stage the distance over time you'll see here the seismicity cloud and then you see and we know the injection phase was this long and you see the p-lateral growth and after stopping the injection phase you see this unilateral growth only in one direction not in the other direction stopping here in terms of seismicity and there's much fewer events occurring here and this is the so-called back front the forefront from this fracture model is indicated here in red and blue lines and this means this is the tip of the crack and we've seen there's a large stress singularity and large stress rates so you can explain that this is the front of seismicity when it grows there are many shear cracks occurring in the neighborhood of the crack tip but then you also have this opening shape and this changes with time and at the point where the opening shape is largest and if it moves in one direction then it closes again behind this opening and then this can define a point where you have a stress shadow because it was large opening then it comes smaller and this you can calculate isn't it? It is estimated here in this blue line and it fits roughly already the observation and the prediction of when it should stop is exactly at this point and also this fits very well this observation of the extension from this point to this point here so we had this we confirmed this with a second experiment it's more or less similar but there's one problem if you try to interpret seismicity you see very well you can make these plots you can try to figure out what is the front and the back front but the seismic events are cannot be described in a deterministic way and you have always some events here you have some here that in a deterministic world should not occur here because the earthquake generation problem is more difficult and you need a probabilistic scheme so you should have a way to translate this frag model into seismicity theoretical seismicity occurrence seismic rates, seismicity rates and therefore you need a seismicity model that is based on physics in terms of stress changes for instance the rate and state model can do this and we have tried to do this here from the known solution of our analytical field we know the internal pressure field and the injection phase and the post injection phase and we can use this as input in a rate and state model and we can make predictions on the rate of seismicity how we expect it and then compare directly to the rate of events and this is a much better way to compare seismicity to fracture models and this is here shown in principle for a hydrofrack but the same approach is I think very important to do for dyke intrusions if you have seismicity if you want to really learn something on the dyke itself and the properties of the dyke you should couple all these fracture and dyke models with the seismicity model and try to compare rates of seismicity with theoretical rates observe with theoretical rates okay I think I stop here because my time is over for now this is a very hot topic and there is also Paul Siegel has worked on this the magnitude if I talk about rates of events it's another subject as the magnitudes and the most simple assumption is always that you assume a Gutenberg Richter a magnitude distribution a normal frequency magnitude distribution you can apply this to your model and then you can already compare it but it's debated at the moment if this is maybe too simple and more sophisticated models for instance try to distinguish between poorly triggered models and induced models so there's always a kind of background seismicity that you can expect or faults that are pre-existing that have already shear stress and that can be triggered so rupture can be triggered on these faults and these earthquakes can grow very large and you have the stress field that you only produce because of the dyke or the hydrofrag itself and earthquakes generated by this process they would stop after they reach the end of the stress field that is perturbed so they would not grow very large meaning they would produce only small earthquakes and the other concept would allow also for larger earthquakes so a modern view is to try to combine these two aspects but it's still a matter of debate as far as I know and there's no established model to really explain the magnitude distribution completely what were the highest? these were very small events they were measured from boreholes and not from the surface and typically I don't know exactly here now I don't remember but they are in the range of below zero, one or below zero in this range, so very small events are there three things that you would get put so heterogeneity? what about heterogeneity? yeah maybe I should have mentioned this also it's clear that this can also for instance stop a dyke and influence the growth the heterogeneities in the rock I think it's important but especially in a region when the over pressure is not very large and the dyke is growing very slowly then you will also see influences from the rock and the heterogeneities if you think on a, it's a bit similar to hear these hydrofrac experiments if you think to the ejection phase also for a rifting episode the over pressure is really large I think the dyke will not care too much about the heterogeneities in the rock because the self stress for instance is so large that it simply goes through many things I think it also holds for the reactivation in principle where you would assume that the fracture toughness is more or less zero yes the dyke will be larger it's easier to reactivate it so let's say you need a smaller internal pressure to reactivate it in principle you can compare this maybe with a hydrofrac experiment where you have the first frac and then you have a refrac where you refrac in the pre-existing crack and you can measure the pressure and you can find that the re-injection pressure is smaller it's not the same as the first one to fracture the origin rock and the same would also be for natural processes I think the answer is yes and no both can happen it's a bit the same as we have discussed before it depends on the material injected and on the internal over pressure if this effect is much larger than the effect from structural heterogeneities then possibly it will not follow this but it definitely will follow a pre-existing crack if you are in a low pressure internal low pressure regime let's say maybe it's better or it's interesting to compare magmatic intrusions with cold fluid intrusions so the difference is if you the critical length is much smaller if you think on hydrofrax and you think on the process that you have wholesale movement of water filled cracks or gas filled the cracks are much smaller in length only 100 meters or even less maybe and the opening is very small it's in micrometer range so you cannot easily see and this also means the stress the self generated stress is not very large and these fluid filled cracks very likely follow pre-existing faults very much the intrusion with an extreme case several meters of opening the stress generated by this intrusion is so large that possibly it will simply cut a fault and do not go through