KEYNOTE LECTURE: NUMERICAL MODELING OF SEISMICITY: THEORY AND APPLICATIONS

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2 8 th Int. ymp. on Rokbursts and eismiity in ines, A. alovihko & D. alovihko (eds) 013 G RA & I UB RA, Obninsk-Perm, IBN EYNOTE LECTURE: NUERICAL ODELING OF EIICITY: THEORY AND APPLICATION A.. Linkov Rzeszow University of Tehnology, Rzeszow, Poland; Institute for Problems of ehanial Engineering, aint Petersburg, Russia Numerial modelling of seismi events is the basis for quantitative studying and interpretation of seismiity in well-established terms of solid mehanis. In this way we obtain a better basis for making deisions onerning with mining, rok burst mitigation, oil and gas exploration and prodution. The paper summarizes omputational, geomehanial and geophysial rationale for progress in this area. We explain in detail how to employ the theoretial rationale for developing a omputer ode providing both mehanial quantities (stresses, strains, displaement, et.) and seismi data (spatial and temporal distributions of events and seismi harateristis used in mining seismology). It is shown that a onventional ode of the BE, FE, DE or FPC type may be easily adjusted to simulations of seismi and aseismi events by omplimenting it with a number of subroutines, desribed in due ourse. Examples illustrate appliations to problems of mining and hydrauli fraturing. In memory of the founder of syntheti mining seismiity, Professor iklos alamon A Little of History INTRODUCTION Twenty years have passed sine 1993, when Professor iklos alamon formulated the first theoretial rationale and pratial appliation of syntheti mining seismiity. In fat, his key-note address to the 3rd International ymposium Rai (alamon 1993 ) ontained all the germs of generality. He suggested as an urgent task for rok mehanis to evolve a tool whih provides the opportunity of relating seismiity to the hanging mining layout and to geologial environment. And what is espeially signifiant, he suggested the method to furnish the basis of suh a tool (p. 99). Further development has atually followed the line by alamon. The only prinipal improvements onsisted in inluding: oftening of a fratured element whih, as shown by Cook 1965a, 1965b (see also Hudson et al. 197 and Linkov 1994 ), provides a measure of brittleness and instability of rok. Time effets, whih opens an opportunity to aount for aseismi events and to model intervals between seismi events. A natural way to aount for time is to expliitly inlude reep into the onstitutive equations. Napier and alan 1997, alan and pottiswoode 1997 inluded the time expliitly into onstitutive equations for ontat interation when numerially modeling seismiity. The ontat was assumed to be visoplasti. The authors employed the simplest visous element in the programs DIG, INI, and INF, what allowed aounting for aseismi effets. In addition to the mentioned dependenes, the results inluded atalogues of syntheti seismiity and hains of events after a blast. The advantages of this step inluded providing (under ertain assumptions) the main seismi harateristis (loation, energy, et.) and agreement with observations. Nevertheless, in the ited papers, there were limitations as onerned with aounting for softening. The latter was used in a model, whih inludes a softening element in parallel with a visous element. In aordane with the general theory (Linkov 00 ), the latter model exhibits frature aeleration, but atually it eases to generate instability in the form of a jump. It is beause infinite instant stiffness of a visous dash-pot prevents instability. The simplest model apable to simulate both seismi and aseismi events is the Elastiity-oftening-Creep (EC) model (Linkov 1997 ). In this model, in ontrast with that used previously, the softening element is inluded in series (not parallel) with the visous (reeping) element. Its analysis shows (see, e.g. Linkov 1997, 00 ) that the ECmodel aptures the most important effets: deaying aseismi deformation; smooth although aelerating aseismi deformation; and instant instability (seismi event). It appears that near a point of instant instability, there our frature aeleration and that arbitrary short time intervals may be observed near the point of dynami instability. At the level of the earth rust, suh aseismi aelerating movement appears as a so-alled silent earthquake (see, e.g. Linder et al. 1996, Dragert et al. 001 and awasaki 004 ). Revealing the existene of aelerating events fills the gap between slow reep and instant instability (seismi event). Although models used differ in signifiant details, their employment has learly demonstrated that syntheti seismiity mimis observed seismiity in the most essential features (ellers and Napier 001, pottiswoode 001, Linkov 005, 006 ). As noted by Napier 001 : it appears that NUERICAL ODELING OF EIICITY: THEORY AND APPLICATION

3 the expliit failure mode shows some superfiial orrespondene to observed seismi behavior but displays number of shortomings. Note that sometimes modeling in the line of alamon is alled ative, beause it models expeted events without employing the data on observed passed events. In ontrast, passive modeling is based on employing data on events observed in a mine (e.g. Wiles et al. 001 ). Passive modelling uses data on the loation, orientation, radius of raks and displaement disontinuities (DD), obtained from seismi monitoring. They serve as input data for numerial modeling of stress distribution on mining steps. The alulated stresses are used for onlusions on stability and for alibration of the model. The main problem of the passive approah is to transform seismi data into input information on the orientation of elements and values of the DD. The appliations of the passive approah look quite promising. As written in the paper by Wiles et al. 001 : The authors believe that this tehnique provides a unique opportunity for onstrution of mine-wide stability models inorporating major strutural fators. The authors ontinue: ome would argue that sine part of the rokmass behavior ours aseismially, we annot ahieve this goal. We would argue that it is also true that part of rokmass behavior ours seismially till the approah has obvious shortomings: impossibility to use it on the stage of planning, negleting aseismi events and diffiulties with inversion of seismi data into the DD on raks. o far the ative and passive approahes have been used separately. eanwhile, nothing prevents to join them. In fat, this means alibration of the input data, used in diret modeling, by omparing the simulated events with the data of observation. uh joined geomehanial and seismi monitoring, disussed below, presents a hallenge for modern siene and pratie. Rok ehanis and eismiity: Advantages of Integration Why is it important to join rok mehanis and seismiity? Let us reall merits and flaws of the two. Rok mehanis operates with onepts and quantities rigorously established in solid mehanis, suh as displaement, strain, stress, equilibrium, stability, and the like. Consequently, if provided with reliable input data on rok struture, properties, ontat and boundary onditions, it is possible to solve a problem by using modern omputers and numerial tehniques. Unfortunately, the needed input information is limited and unertain. For years, it has raised a doubt in pratial usefulness of numerial simulations. As wrote tarfield and Cundall 1988 : models (mathematial or omputational) were generally thought to be either irrelevant or inadequate. odellers spent a large part of their efforts trying to persuade skeptis that modeling was a useful engineering exerise. Unertainty of input data distinguishes rok mehanis from other engineering fields. If following tarfield and Cundall 1988, we onsider Holling s lassifiation of sienes (Figure 1), we an see that our usual region is the region of good understanding and little information. In ontrast, mining seismology provides vast data on seismi events while there are great diffiulties in their interpretation in well-established terms of solid mehanis needed to make pratial deisions. In Holling s lassifiation, mining seismology is in the area 3 of vast data and poor understanding. Figure 1. Holling s lassifiation of sienes Presently modern numerial tehniques allow us to aount for ombined effet of thousands and even millions of strutural elements and provide understanding of simulated proesses. eanwhile, there are no reliable input data on real rokmass. On the other hand, modern mining seismology proesses millions of events, ontaining data on a rok state, while there is no proper understanding of the state. We see that the advantage of one tool is the shortoming of the other. Clearly, their integration serves to ross-fertilization, whih preserves their benefits and removes the flaws. The sientifi advantage of the integration onsists in joining rih data of seismi monitoring with lear understanding of results of numerial simulation providing well-established mehanial quantities. This moves us into the area 4 on Figure 1 of suffiient data and better understanding. Pratial advantages are also obvious. We obtain a new modern tool for assessment of the state and proesses in roks and for making deisions. The tool allows us to arry out ompletely ontrolled seismologial experiments, play various senarios, alibrate input parameters, validate existing and suggest new methods serving for interpretation of seismi data. And what is also important, it is not expensive: atually, it is the work for a few speialists. Reent Tendeny: Joined ehanial and eismi odelling and onitoring in Real Time An espeial hallenge is to use joined mehanial and seismi modeling and monitoring in real time. Nowadays, Rai8

4 the possibility to implement it is failitated by tremendous progress in omputers, numerial tehniques, information tehnologies and automati measurements. Atually, the passive approah ontains the germs of the joined monitoring. It is visibly reognized by the speialists applying it, who wrote (Wiles et al. 001 ): The integration of deterministi modeling with seismi monitoring offers ruial data regarding loal variability and sensitive features Taken together this learly enhanes our deterministi predition auray of rokmass response to mining. The feasibility of the purpose is evident from the papers by Cai et al. 007 and Dobroskok et al The authors of the first of them used the FLAC and PFC for simulation of seismiity indued by large-sale underground exavations. The seond work employed the ode EI-3D, based on using the general theory (Linkov 005, 006 ) in frames of the hypersingular boundary element method (H-BE). The both papers vividly demonstrate that geomehanial and seismi modeling may be effiiently joined with daily seismi monitoring in mines. It provides important data for making reliable pratial deisions. The trend to meet the hallenge is obvious from merging of ITACA (geomehanis onsulting and software development ompany) with AC (Applied eismology Consultants) in 009. As advertised the site (eptember 6, 01), AC and Itasa have developed advaned, groundbreaking tehniques for orrelating miroseismi field observations with simulated miroseismiity from Itasa's models. The same trend is also seen in transforming the I International by Dr. endeki into Institute of ine eismology (I) in 011. The Institute, among other purposes, aims to ompliment daily geomehanial alulations with numerial modeling of seismiity in real time and to ompare the syntheti seismiity with the data of seismi monitoring. The modeling of stressed state and seismiity joined with diret observations beomes of ruial signifiane for hydrauli fraturing. Hydrauli fraturing is widely used for stimulation of oil, gas and heat reservoirs. Its importane grows with the tendeny to extrat gas from low-permeable shales. The geomehanial data in these problems are even less than in those onerning with mining. In fat, seismi observations are the main soure of information on the hydrauli frature propagation. For this reason, the newest methods for modeling hydrauli fratures employ the observed seismiity to alibrate the input data (see, e.g. Cipolla et al. 011 and resse et al. 011 ). Obviously, the reliability of the methods employed will notably gain when omplimenting them with numerial modeling of seismi events and omparing syntheti seismiity with observations. The papers by Al- Busadi et al. 005, Dobroskok and Linkov 008 show that simulation of seismiity, aompanying hydrauli frature propagation, an be effiiently performed. ope and truture of the Paper The main objetive of the paper is to present the modern theory of joined modeling of mehanial state and seismiity in suh a form, whih provides easy implementation in a omputer ode of a onventional numerial method (BE, FE, FD, DE and the like). The paper also aims to: 1) Distintly outline the fundamental issues, whih stay the same when onsidering a wide variety of appliations. ) Point out flexible elements, whih may vary depending on a partiular problem and/or available omputational failities. The exposition takes into aount the mentioned tendeny to arry out joined modeling in real time. It is aompanied with examples. The struture of the papers is as follows. Firstly, we onsider fundamentals of modeling a single event. This involves onepts of instability in the form of a jump, softening behavior, elasti energy release, elasti rigidity of rok near a flaw, seismi energy, seismi moment and seismi shear. Then we onsider the time effets of two kinds: those governed by proesses external to the soures of events (for instane, mining steps) and those, whih are defined by proesses in the soure of an event. The first group is the same as in onventional alulations of stresses. The seond is speifi and it leads us to the Elastiity-oftening- Creep (EC) model as the simplest model apable to serve for modeling both seismi and aseismi events and intervals between the events. Having the bakground for modeling a single event, we follow the line by alamon 1993 of random seeding raklike flaws in the rok mass and heking their stability under urrent stresses. peial attention is paid to proper presribing the values and statistial distributions of geometrial and mehanial properties of seeded flaws. With these prerequisites, we ome to the general struture of a omputer ode for simulation of seismi and aseismi events. It appears that it is suffiient to ompliment a onventional ode with a set of quite general seismi subroutines, whih may be used in diverse omputational environments. The disussion of subroutines, serving for proessing the output data, is aompanied by examples of various appliations. Brief summary onludes the exposition. NUERICAL ODELING OF EIICITY: THEORY AND APPLICATION

5 ODELING OF A INGLE EIIC EVENT eismiity is a sequene of many individual events. Therefore, to simulate seismiity we need to learly understand and properly model a single event. Having a tool for suh modeling, it beomes possible to randomly generate parameters, defining an event, and to obtain statistial olletion of events alled syntheti seismiity. Instability in a Form of a Jump The essene of a seismi event is a jump from one equilibrium state to another with release of elasti energy. The exess of the elasti energy released in a jump over the energy onsumption manifests itself through elasti waves, registered in (miro)seismi observations. Although wave reeivers are loated at a distane from the soure, seismograms ontain important information on the soure suh as its loation, magnitude, seismi moment, et. (see, e.g. Aki and Rihards 00, Gibowiz and ijko 1994, endeki 1997 and Rie 1980 ). These seismologial data obtained at a distane from the soure may be ompared with that provided by an analysis of the jump in the soure itself. Here we fous on suh an analysis. From the mehanial point of view, a jump is a form of instability with elasti energy exess transformed into kineti energy of waves. Jumps our in rok at different sales. In a loaded speimen, they our on the level of miroraks and appear as aousti emission. It may serve to validate modeling of seismi events under known onditions of loading. In oil and gas prodution, jumps appear as miroseismiity around a borehole or propagating front of a hydrauli frature. In a mine, we have a larger sale: there emerge not only miroseismi events but also large dangerous jumps, rokbursts. Their magnitude in the Righter sale may reah two. oreover, in the earth rust the length and magnitude sales are muh greater; an event may appear as an earthquake. The magnitude may reah eight. Despite the length sale hanges nine orders (from miro meters to kilometres) the essene of a single event is the same: it is the dynami (with the energy exess) instability. Consequently, proper modeling of a single event requires the study of instability and it refers to events on a wide range of sales. Equilibrium, tability and Instability The theoretial anvas of studying instability in mehanis is ommon. Firstly, we need to find a urrent state. To this end, the equations of ontinuum mehanis for a small volume of a medium, assoiated with a mathemati point, are used. They inlude: Dynami or stati equations of motion (equilibrium), whih are atually the Newton s seond law, speified for a point of ontinuum. inematis equations, defining hanges in mutual positions between losely loated points of the medium. Constitutive equations for a volume element, whih onnet dynami quantities of the first group with geometri quantities of the seond. This yields a omplete system of partial differential equations (PDE), defined at points of a onsidered region of the medium. It is solved under presribed initial, ontat and boundary onditions to distinguish the partiular solution of the PDE, orresponding to a partiular problem, from the general solution. In stati problems, whih are of prime interest to our theme, there is no need in initial onditions: we use only ontat and boundary onditions. The ontat onditions inlude onstitutive equations for a ontat element. (In the simplest ase of a perfet ontat, the onstitutive equations express ontinuity of trations and displaements through the ontat). uppose we have found the solution of a problem and know the equilibrium state. Now we want to know if the state is stable or unstable. A rigorous analysis of stability requires strit definitions of what is assumed to be a stable or unstable state. Naturally, the definition differs in different problems. till, the general feature is that instability is always indued by some soure(s) of non-linearity. The problem of stability does not exist for a linear system. Thus we need to inspet the possible soures of nonlinearity. For a massive body of hard rok, in ontrast with slender bodies, we may neglet non-linear terms in equilibrium and kinematis equations. Thus the only andidates to be the soures of instability remain non-linear onstitutive equations for volume or/and ontat elements. These equations are obtained in physial experiments with rok volumes and ontats. Figure presents typial non-linear diagrams for a volume element obtained by using a rigid testing mahine. Figure. Complete diagrams for a volume element Of importane is that the diagrams have desending parts, orresponding to so-alled softening behaviour. As known from the pioneering work by Cook 1965b, under some onditions, softening beomes the soure of Rai8

6 instability with the energy exess transformed into kineti energy of fragments of fratured rok. Note that from the plastiity theory (e.g. ahanov 004 ), we know that suh kind of instability annot our for linear elastiity and hardening that is for deformation on asending parts of diagrams. imilarly Figure 3 presents typial non-linear shear tration-shear displaement disontinuity (DD) diagrams for a ontat (surfae) element. They also have desending (softening) parts. Again they may ause dynami instability (see, e.g. Linkov 1994 ). Figure 3. Complete diagrams for a ontat element Now we need to deide whih of the two types of softening or both of them should be taken into aount when modeling a seismi event? Atually, modern numerial tehniques may aount for the both types. till the surfae type looks dominating as onerns with seismiity in hard rok, at least. As wrote Napier 001 : The fundamental 'building blok' of material failure an then be onsidered to be a 'rak element' rather than a 'partile'. Furthermore, even volumetri softening of a pillar may be modeled by using the dependene between average displaements of its boundaries and average trations on them (see, e.g. Linkov 1994 ). Thus in the problem onsidered, it is reasonable to fous on surfae (ontat) softening. Constitutive Equations for Contat Interation Consider an element of rok surfaes in ontat. Denote n the normal to the element, the tration vetor on that side of the ontat, with respet to whih the normal is outward, u is the vetor of displaement on this side; and u the tration and displaement on the opposite side (Figure 4). Commonly, in problems involving ontat instabilities, the ontat is either not filled or it is filled with a material softer than embedding rok. Then we may neglet bending of a ontat layer. This yields that (i) the ontat interation is haraterized by the DD: u u u, (.1) rather than by the average displaement 1/ ( u u ) and (ii) the tration is ontinuous through the ontat:. (.) Figure 4. Tration and DD vetors on a ontat Therefore, the ontat interation is haraterized by omponents of the vetors and u. In the omponent form, the vetor Equation (.) in 3D presents three salar equations. Three other needed equations express the dependene between the tration and DD vetors: F( u). For irreversible deformations at a ontat, this dependene should be inremental, onneting inrements of the trations with inrements of the DD in a way similar to plastiity theory. Quite general equations of this type may be found elsewhere (e.g. Linkov 1994 ). However, having in mind that the properties of a surfae are unertain and diffiult to find, using a general desription looks impratial. It is reasonable to employ as simple desription as possible to shorten the list of unertain parameters. For our purpose, we shall use the following simplifiations. 1) A flaw (rak), on whih an event may our, is losed if the tration does not reah the tensile strength 0 n or the initial shear strength C. For simpliity, the initial shear strength is presribed by the Coulomb s law: C 0 ( n ) tan, (.3) where is the shear omponent of the tration, 0 is the initial ohesion, is the surfae frition angle. For illustrative purposes we assume that the shear tration vetor is direted along a loal oordinate axis in the plane of shear; thus we assume positive. Then the shear DD is negative, and we shall use u to have a positive value. A ompressive normal tration, like ompressive stresses, is assumed negative. Thus we have: u 0, if 0 and. (.4) n ) When the tration reahes a limit value, there DD our. For the tensile mode ( n 0n ), the DD are defined by the ondition that the tration on an open rak beomes zero: n C 0. (.5) NUERICAL ODELING OF EIICITY: THEORY AND APPLICATION

7 For the shear mode ( C ), the DD are defined in aordane with the piee-wise linear diagrams with desending parts shown in Figure 5: C ( u ) - u u*, (.6) C ( 0 * ) - u u*. (.8) Herein, the shear rigidity is evaluated per unit ontat area; in the onsidered ase ( E / l)sin os. where is the shear softening modulus, u ( * 0 * ) / is the shear DD, orresponding to reahing the residual ohesion. * Figure 6. Example of instability aused by ontat softening Figure 5. implest diagrams for a shear softening ontat The normal DD presribes as un for the shear mode may be u u tan n, (.7) where is the dilation angle. The minus sign in front of u n serves to have a positive value. Reall that we assume u to be negative and that under the aepted definition (.1) of DD, a DD, orresponding to opening, is negative (see Figure 4). Commonly, we may neglet the dilation; then u = 0 for the shear mode. n Equations (.1) (.7) ompletely define the properties of a ontat element. The dependene (.6) aounts for softening what serves for modeling dynami instability. oftening Contat in Rok. Instability We explain dynami instability, aused by ontat softening, by onsidering the simplest system shown in Figure 6a. It presents a speimen with the length l and the elastiity modulus E. The speimen is ut under the angle and it is loaded in a rigid testing mahine. Two elasti parts of the speimen interat along the ontat surfae, for whih shear deformation ours in aordane with the diagram shown by the solid line in Figure 6b. The elasti parts represent an external (with respet to the softening ontat) system, whih aumulates elasti energy. Continuous deformation of the ontat may beome impossible when its shear strength is reahed and softening starts. It is easy to show (Linkov 1994 ) that unique solution of the problem exists only when the softening modulus is less that the shear rigidity of the external system: If the inequality (.8) is not satisfied, then we have either infinite number of solutions, orresponding to any point of the desending part when, or a jump to the residual strength when. The dashed line in Figure 6b shows the orresponding mutual deformation of elasti parts. The hathed area shows the energy exess (per unit area), aompanying the jump from the initial shear strength to its residual value. Thus when, we have dynami instability. The ase is intermediate between stable deformation and dynami jump. In this ase, the elasti energy release equals to the energy onsumption at the softening interfae. Although the energy exess is zero, we shall relate this ase to instability beause the solution losses uniqueness and whatever small exeeding of over results in a dynami event. Thus if the shear strength is reahed, the state is unstable and jump to the residual strength ours when Otherwise, the state is stable.. (.9) From the general theory of dynami instability (Linkov 1987, see also Linkov 1994, 1997 ) it follows that appliability of the simple onditions of stability (.8) and instability (.9) in terms of the elasti rigidity and the softening modulus is quite limited. In general, we need to use onditions in terms of energy exess on virtual displaements. Nevertheless, the simpliity of the onditions (.8), (.9) is very attrative. It appears that it is possible to apply them to modeling seismi events. peifially, they are appliable to rak-like flaws in elasti rok when the shear rigidity of the external system is properly defined. Rigidity of a Crak-like Flaw Consider, for ertainty, a retangular rak-like flaw with the minimal size a and the other side b ( b a) in Rai8

8 elasti roks with the elastiity modulus E and the Poisson s ratio v (Figure 7). We use the loal oordinate system with the axis x normal to the flaw plane, 1 x and x 3 in the plane. The x -axis is direted along the long side of the retangle, the x3 -axis is along the short side. The flaw sizes are small as ompared with the sizes of openings, faults and other strutural elements of a onsidered region. Then the total influene of these disturbing fators and stresses at infinity is inorporated in the total tration t, indued at the loation of the flaw on its surfae. only harateristi of shear rigidity. (The bold line in Figure 8 gives the dependene of k on the ratio b / a ). Then (.11) beomes: 1 E k1, 1 a E k 3. (.1) 1 a Figure 8. Rigidities as funtions of b/a Figure 7. A retangular flaw in elasti rok It is easy to solve numerially a 3D problem for a plane rak of an arbitrary onfiguration in an infinite elasti region under presribed tration at the rak surfae (e.g. Linkov et al ). From the solution we an find the dependene between the average tration vetor and the average DD vetor u : u t, (.10) where is a matrix with known oeffiients; it has the meaning of average (per unit area) rigidity matrix. In partiular, for a retangular rak in isotropi rok, by symmetry, the rigidity matrix is diagonal. Its diagonal oeffiients 1, and 3 may be written as: E k i, (.11) 1 a i where the dimensionless oeffiients k i ( i = 1,, 3) depend on the ratio b / a ; besides the oeffiients k and k depend on the Poisson s ratio. 3 Figure 8 presents the graphs of the oeffiients k i as funtions of b / a for the Poisson s ratio v = 0.5 (Linkov 006 ). Note that in the limiting ase of a plane-strain problem ( b / a ), the oeffiients found analytially are: k 1 /, k /, (1 ) / k 3. From alulations it follows that to the auray of 3% the plain-strain onditions are aeptable when b / a = 4. This implies that the oeffiients k and k 3 differ by the fator 1 at most. Therefore, for simpliity, we may use their mean value k k ) /, whih beomes the k ( 3 In view of (.1), we may write the dependene (.10) between the average DD and average trations in terms of separated normal n, un and shear, u omponents: n 1 un tn, u t. (.13) Equations (.13) notably simplify analysis of the system flaw embedding rok. The unified shear rigidity makes the retangular flaw isotropi in its response to shear, like it is for a irular flaw. By employing this simplifiation, from now on, we shall assume the loal oordinate x direted along the shear tration. Comment. The atual onfiguration of flaws in rok mass is unertain. Thus there is no sense to distinguish between the shear rigidities and 3 and to exatly speify the Poisson s ratio. oreover, as the shear rigidity is ompared with the shear softening modulus, whih is even more unertain, it is enough to use rough estimations. In partiular, sine the shear rigidity hanges only 1.6-fold when the retangle hanges from a square ( b a) to an infinite strip ( b / a ), it is possible to set b a. Then for v = 0.5, the rigidities are: E a, E (.14) a Note that the analytial solution for a irle of the radius R a /, yields the normal rigidity E / a ; the differene with the first of (.14) is only 5%. Thus the onfiguration of a flaw does not influene the rigidity signifiantly. Roughly, the rigidity is defined by the minimal size of a flaw. Using a irular flaw simplifies also input data and estimations of the energy release. For these reasons, alamon 1993 seeded irular flaws. NUERICAL ODELING OF EIICITY: THEORY AND APPLICATION

9 Equilibrium and Instability of a Crak-like Flaw in Rok We are in a position to obtain an approximate solution for a softening rak-like flaw by equating the trations and DD on its surfae to those of elasti rok. Before the tensile or shear strength is reahed ( t n 0n, t C ), Equations (.4) and (.10) yield the obvious result: eismi Charateristis of a ingle Event Consider a flaw with the normal n in rok (Figure 7). uppose that the trations n and are suh that the shear strength is reahed. Then the flaw may experiene softening in aordane with a diagram in Figure 5, orresponding to the urrent normal tration n t n. This diagram is shown by solid line in Figure 9. t. After the tensile strength is reahed ( t n 0n ) Equations (.5) and (.10) give: u t 1,. (.15) At last, if the shear strength is reahed ( t C ), joining the first of (.6) and the seond of (.13) gives the system, defining the shear DD under softening: u ), u t. C Its solution is: u ( t C. (.16) Reall that by the aepted agreement on signs, for a physially signifiant solution, it should be u > 0. ine the numerator in (.16) is non-negative, this implies that a ontinuous solution exists only under the ondition of (.8) form:. If, there is no ontinuous solution: the system jumps to the state orresponding to the residual strength. We see that the ondition (.9) is the ondition of dynami instability for a flaw in elasti rok. For the final state after the jump, the seond line in (.6) and the seond Equation in (.13) yield: u t C ( 0 * ). (.17) Note that the r. h. s. in (.17) is positive. Comment. Reall that the solutions (.16), (.17) are approximate beause we have used trations and DD averaged over a flaw. It is of interest to estimate the auray of suh an approah. It an be done by using the results of the exat solution to the plain-strain problem for a straight shear-softening rak of the length a in elasti rok (Belov and Linkov 1995 ). From inspeting dimensions, it is obvious that the ondition of dynami instability has the form (.9). The value of the rigidity, orresponding to E 1 the exat solution, is Its omparison 1 a E 1 with the approximate value, obtained by 1 a averaging, shows that the error is 9.3%. uh an error looks aeptable when modeling a seismi event. Figure 9. Diagrams of softening element and external system (embedding rok) We are interested in harateristis of the jump ourring when the instability ondition,, is met. Then the deformation of rok surfaes follows the diagram shown by the dashed line in Figure 9. The area between the diagrams represents the energy exess per unit area. Therefore, for the whole flaw area, equal to ab, the total elasti energy exess is: 1 ( 0 * ) W ab 1. (.18) Note that the total elasti energy release is 1 ( 0 * ) Wr ab, (.19) while the total energy onsumption on the softening surfae is W C 1/ ( 0 * ) ab/. Introdue the seismi effiieny fator, defined in seismology as eff ( Wr WC )/ Wr, so that W W eff r. Then Equation (.18) implies that in the onsidered ase the seismi effiieny is: eff 1. (.0) Depending on the softening modulus it may hange from the unity for ideally brittle rok, onsidered by alamon 1993 ( C ), to zero at the threshold of instability. C We obtain a rough estimation of the seismi energy by using the seond of Equations (.14) for a square flaw: ( 0 * ) 3 W 0.56 a eff. (.1) E Rai8

10 Equation (.1) shows that the energy of an event is proportional to the ubed rak size a. The simple estimation (.1) is useful in numerial simulation of seismiity. It serves for the hoie of the average size of flaws, whih define the average level of energy; the latter is available from seismi observations. The seismi shear u, aording Figure 9, is: 0 * u. (.) The tensor of seismi moment in the loal oordinates of the flaw is: seism 0 m Herein, m is the magnitude, defined as usual by the equation m u, where is the shear modulus E (1 ), is the flaw area ( ab). (Reall that we have assumed the loal oordinate x direted along the shear tration, so that the shear DD has the diretion opposite to x what explains the minus sign in front of m ). One nodal plane of an event is normal to the loal axis x 1, another is normal to x. Usual tensor transformation gives the tensor of seismi moment in a global system of oordinates. For brevity, the value m is also alled the seismi moment when there might be no onfusion. Using (.) yields: 0 * m ab. (.3) From (.19) and (.3), it follows that the energy release is proportional to the moment m : W r 0 * m. The energy exess differs only by the seismi effiieny fator: W 0 * e m eff. Therefore, in rough estimations, assuming the seismi effiieny onstant, the energy exess is proportional to the seismi moment. ummary for odeling a ingle eismi Event We have seen that the essene of a seismi event is instability in the form of a jump. It ours when the elasti energy release exeeds the energy onsumption for softening deformation of a flaw. Therefore, when modeling a single event we need to aount for: 1) Elastiity, whih provides the soure of energy. ) oftening, whih defines the energy onsumption. Thus, Elastiity + oftening ingle seismi event This implies that when seeding many flaws, at whih seismi events may our, the minimal input data for a flaw are: 1) Its loation (three global oordinates of the flaw enter). ) Orientation of the flaw plane (two oordinates of the unit normal to the plane; and, for a retangular flaw, orientation of one of the sides in the plane). 3) The size(s) of the flaw (two lengths of a retangle, or one size for a square or irle). 4) ehanial properties of the flaw: its tensile 0 n and initial shear 0 strength, frition angle ρ, residual ohesion *, softening modulus and dilation angle ; in simplified models, one may set 0 n = 0, * = 0, = 0 and/or onsider ideally brittle ontat ( ). 5) Elastiity modulus E and Poisson s ratio ν of rok at the loation of the flaw. The elasti properties of rok and minimal size of a flaw define the shear rigidity of rok with respet to the flaw. It is found from the seond of Equations (.1) and Figure 8 for a retangular flaw, or from the seond of Equations (.14) for a square or irular flaw with the radius R a. The behavior of a flaw in rok depends on the trations t indued on its surfae by external stresses. The stresses are found by solving a boundary value problem for rok masses with openings, faults, inlusions, et. under presribed in-situ stresses. A ommon ode of FE, BE, DE and the like may serve to find the indued stresses. A flaw is losed and it does not influene the rok state when neither tensile, nor shear strength is reahed. A flaw opens when the normal tration beomes equal or exeeds its tensile strength ( t n 0n ) ; then the flaw obtains the DD defined by (.15). A flaw experienes shear deformation when its shear strength is reahed ( t C ). In this ase, there are two options: 1) There is no seismi event if the stability ondition (.8) is met ( ). Then the shear DD is found from (.16); the normal DD is defined by (.7); when setting the dilation angle zero, the normal DD is zero. ) A seismi event (jump) ours if the instability ondition (.9) is met ( ). Then the harateristis of the event (energy release, energy exess, seismi NUERICAL ODELING OF EIICITY: THEORY AND APPLICATION

11 effiieny, seismi shear, seismi moment, et.) are found by using Equations (.18) (.3). ultiple Events: eismiity TIE EFFECT Time effets are obviously present in rok. The first law of rok mehanis is that holes tend to lose up (saying of wise old engineer quoted by van der erwe 1995 ). As onerns with seismi events, they are distributed in time; obviously time effets ontrol intervals between individual events. They lead also to aseismi deformations. Thus to onsider many seismi events and to inlude aseismi events, we need to ompliment the two disussed fundamental fators (elastiity and softening) with time. A multipliity of seismi events is alled seismiity. Therefore, Elastiity + oftening + Time eismiity For events of small magnitude, the term miroseismiity is ommonly used; dynami events in a loaded speimen are alled aousti emission. We assume these terms equivalent. External Time It looks reasonable to distinguish two types of time proesses: 1) Those whih run out of the soure of an event and its lose viinity. ) Those whih our in the soure and its lose neighbourhood. The first group of proesses is external, while the seond is internal with respet to an event. Figure 10 presents examples of proesses ontrolled by external time. They inlude a speimen loaded with stresses, whih hange in time in aordane with a presribed loading path (Figure 10a); mining with hanges of geometry planned by an engineer (Figure 10b). In these two ases the time is just a parameter defining hanges in boundary onditions. The other two examples refer to oil (gas, heat) prodution (Figure 10), where hanges in the stressed state are defined by transient proesses of fluid flow, and propagation of a hydrauli frature (Figure 10d), whih hanges the stresses around it. Various onventional odes of FE, BE, DE, et. may serve to numerially trae the hange of stresses in the external time. We assume that a numerial ode to simulate these hanges is available. From now on we fous on the internal time. Internal Time. Elastiity-oftening-Creep (EC) odel There is obvious evidene of internal time present in seismiity: after a strong exitation of rok, suh as an earthquake, or rokburst, or blast, a asade of seismi events ours during some interval of time. The events ease in time approximately exponentially, what in seismology is alled the Omori s law. The questions are: how to aount for the internal time in the simplest way and how to model aseismi events? Figure 10. Examples of proesses running in external time We may follow the ommon path of solid mehanis to get an answer. As known, the simplest desription, expliitly aounting for the time, is the Newton s visosity law, whih linearly onnets a fore (stress) with a kinemati quantity (veloity, strain rate, DD). In shemes, presenting model behavior, it is shown by a dash-pot (Figure 11a). Figure 11. Using a dash-pot (a) to simulate visous deformation of a volume (b) or surfae () element In onstitutive equations for a volume element (Figure 11b), the dependene between a time-dependent part of the stress ij and the strain rate ij is ij ij with the oeffiient having the dimension of dynami visosity [ ] = stress/strain rate = stress time. For a ontat element (Figure 11), time-depending shear tration linearly depends on the veloity u of shear DD: u with the oeffiient having the dimension [ ] = stress/veloity = stress time/length. The dot over a symbol denotes the derivative with respet to time. There are numerous option for inluding this model into onstitutive equations for embedding rok (volumes) or/and interfaes (surfaes). When hoosing between them, it is reasonable to have in mind that a dash-pot responds absolutely rigidly to instant hanges. Atually its instant Rai8

12 rigidity is infinite. peifially, its reation is absolutely rigid in a jump, whih orresponds to dynami instability assoiated with a seismi event. Consequently, there is no sense to inlude a visous element in parallel with an elasti element when modeling external system, represented by external rok (Figure 1a). uh a system will not provide elasti energy instantly. It is also senseless to inlude a visous element in parallel with a softening element modeling the behavior of a rak-like flaw (Figure 1b). uh a system is unable to onsume energy in a jump. model, beause it inludes the elasti element with the rigidity E l, softening element with the softening modulus and visous (reeping) element with the visosity. The harateristi time of the model is the retardation time: t r. (3.1) E l Figure 14. Elastiity-oftening-Creep model Figure 1. hemes with parallel inlusion of a visous element We onlude that the reasonable way of aounting for internal time is to inlude a visous element in series with an elasti element representing an external system (Figure 13a), or/and with a softening element shown in Figure 13b. Figure 13. hemes with inlusion of a visous element in series In the simplest softening element shown in Figure 5, we have negleted the elasti DD on the flaw surfae. Consequently, to have a time sale, it is reasonable to use the visous element in frames of the elvin-voight model. Then we obtain the simplest model (Figure 14), whih exhibits instant softening and has a harateristi time t r. We all this model the Elastiity-oftening-Creep (EC) The instant reation of the model is that of the softening element with the instant softening modulus. The long term reation is that of a softening element with the long-term softening modulus, defined by equation: (3.) E l The EC-model is the simplest extension of the standard linear body. It redues to the standard body when exluding softening of the upper element in Figure 14, so that it beomes elasti. Below we shall use the opposite option and neglet elasti deformation of the upper element. urely, non-newtonian visous element may be used when appropriate. EC-model, being not muh more ompliated than models used to the date, it provides signifiant advantages. It allows us to aount for rok brittleness, to distinguish between stable and unstable states, to evaluate energy onsumption and to follow damping or aelerating aseismi deformations. Constitutive Equations for EC-odel We shall use the simplified diagrams of Figure 5 and their analytial form (.6). The orresponding EC-model is shown in Figure 15. The properties of the model are as follows. Until the tensile or shear strength is reahed, we assume that a flaw is losed and DD at its surfae are zero. If the tensile strength is reahed, the reation is instant, thus the flaw opens and the tration turns to zero (Equation.5). If the shear strength is reahed, then the upper element of the NUERICAL ODELING OF EIICITY: THEORY AND APPLICATION

13 EC-model behaves in aordane with Equations (.6). We rewrite them with the subsript u at the shear DD to mark that the DD refer to the upper element in Figure 15: C ( uu ) - uu u*. (3.3) C ( 0 * ) - uu u* signs of shear DD and shear tration are those aepted above. The ODE (3.6) is solved under the initial ondition u ( t 0 ) = 0, where t 0 is the moment when the shear strength is exeeded. We shall not write down the obvious solution. Rather we disuss the general solution of a homogeneous ODE, orresponding to (3.6), beause it defines general features of the mehanial system. The general solution, inluding an arbitrary onstant C, is: t u g ( t) C exp. (3.8) t r Figure 15. EC-model (no elasti DD prior softening) For the lower elvin-voight element, we have: E u ) ( u ), (3.4) l ( l l where the subsript l marks that the DD refers to the lower element. The tration is the same in the upper and lower element beause they are joined in series. For the same reason, the total DD is the sum: E u ) ( u ). (3.5) l ( l l The total normal DD may orrespond to dilation in aordane with (.7). Joined ystem of EC-urfae Element and Rok We join Equations (.13) of the external system with Equations (3.3) (3.5) for a flaw desribed by the EC-model. If neither tensile nor shear strength is reahed, the flaw is losed; its DD are zero. If the tensile strength is reahed, the DD are defined by Equation (.15). If the shear strength is reahed ( t C ), the joined system yields the ordinary differential equation (ODE): d( u dt ) ( u t r ) ( t C ), (3.6) t where t r is the retardation time defined by (3.1), r / 1 1/,, (3.7) / 1 / 1 is the long-term softening modulus defined by (3.); note that (3.) implies inequality. When deriving the ODE (3.6) we negleted the visous deformation, whih ourred long before the urrent time, and the time derivative of the indued tration. The agreements on the From (3.8) it is lear that a solution exponentially grows (deays) when < 0 ( > 0). The sign of depends on the signs of numerator and denominator in the first of (3.7). This implies that there are three types of shear motion depending on partiular values of, and : 1) Instant instability (jump) ours when. Then we have a seismi event with harateristis disussed in the previous subsetion. Otherwise, the state is stable ( ). In this ase, the deformation ours in aseismi (without energy exess) form. Although it is ontinuous in time, there appear two additional options (we do not onsider the exlusive ase, whih is not of pratial interest): ) Deaying aseismi motion ours if. 3) Aelerating aseismi motion ours if. We see that the EC-model desribes both seismi and aseismi events. For the latter, the motion may be very fast when the instant softening modulus is lose to the rigidity :, when. This explains why intervals between observed seismi events are different; the differene may be of several orders. The fast aseismi motion explains also so-alled silent earthquakes (see, e.g. Linder et al. 1996, Dragert et al. 001 and awasaki 004 ). From the analysis it appears that, atually, we need to presribe only three parameters t r, / and / to model the deformation of a flaw after its initiation by external shear tration. Aseismi Behavior and Charateristis of a ingle Aseismi Event We have seen that aseismi motion at flaw surfaes ours when the stability ondition is met. The motion runs with aeleration if the long-term modulus exeeds the rok rigidity ( ). Otherwise, it deays in time. In the both ases, if the residual shear strength is reahed in the ourse of motion, we have for the final shear DD: Rai8

14 u The time t f t f f t C 0 *. (3.9) of reahing the residual shear strength is: t r 1 ln 1 / / u* 1 u, (3.10) where u ( t C )/( ), u u / * *, u ( * 0 * ) /. Equations (3.9) and (3.10) refer to both aelerating and deaying motion. ODELING OF EIICITY. INPUT DATA General Considerations With the EC-model, we an simulate a single seismi or aseismi event arising at a flaw with presribed position of its enter and orientation of its plane. It is assumed that there is a omputer ode, whih provides stresses at the flaw loation as a funtion of external time. Then if at some instant the tensile or shear strength is reahed, we have an event (seismi or aseismi) with known time, loation, orientation, seismi and aseismi DD and other harateristis disussed above. Obviously, when having many flaws, we simulate events distributed in spae and time. Therefore, to model seismiity, it is suffiient to seed a set of flaws in an area of interest and to hek the state of eah of them on steps of external time. There are two omments to this general sheme. Firstly, imagine that the flaws are seeded in a natural state of rok, whih is not disturbed by mining or oil (gas) prodution. We may expet that in situ trations at some of the flaws exeed the shear strength. Then, in aordane with the disussion, there should be DD at suh flaws. However, these DD may be referred to times long before the urrent time. Clearly these DD should be exluded from the totality of seeded events. In a omputer ode, it has to be done by a speial subroutine heking seeded flaws under in situ stresses and exluding those initiated by these stresses. We shall all this subroutine ExlusInitu. eondly, an event arising at a flaw on a time step leads to DD on the flaw. The DD notably hange the stresses around the flaw. The hange of stresses may be strong enough to initiate some of the neighbouring flaws. Eah of the initiated flaws, in its turn, experienes DD, influenes its neighbours and may initiate new flaws. And so on, till new flaws are initiated. In this way, there may arise hains of seismi and aseismi events. Obviously, if the density of flaws is small, they pratially do not interat. Then there are no hains of the type. On the other hand, if the density is too high, there arises a hain reation, involving pratially all the flaws. The ase intermediate between the two is of major interest for simulation of seismiity. In this ase, we may expet that it will be possible to model the dependene of Omori type. Therefore, it is of value to find the range of flaw density, in whih a hain reation does not arise while the interation of flaws is suffiient to produe suessive asades of events. Below we shall define the range. Deterministi Input Data We have assumed that a omputer ode is available for alulating stresses at eah point of a region as funtions of the external time. The input data of the ode ommonly ontain: 1) Initial geometry of a problem; speifially, for a mining problem, we presribe the depth, dip angle, strike angle, ontours of pillars and openings. ) Physial properties of a medium. 3) In situ stresses. 4) ome additional information for a partiular problem, suh as pumping rate and fluid visosity for hydrauli fraturing, pore pressure and permeability of rok for oil (gas, heat) prodution, et. All these data are presribed in the same way as in ases when a ode is used without simulation of seismiity. Normally these data are deterministi. The influene of the external time is also deterministi. peifially, in mining problems, we plan the hange of mining geometry in time; in hydrauli fraturing, we follow the frature propagation and the hange of the net-pressure in time steps of alulations. Briefly, we assume deterministi all what onerns with the performane of a onventional ode when there is no simulation of seismiity. The following disussion entirely refers to parameters speifi for modeling seismi and aseismi events. They inlude geometri input data on positions and orientations of randomly seeded flaws, data on flaw sizes and density, data on mehanial properties of flaws. In the next subsetion we speify presribing these data. In a omputer ode, a speial subroutine produes these data. We all it FlawInput. Input Data on Position and Orientation of eeded Flaws We seed flaws in a parallelepiped with the sides parallel to the global oordinates, with the sizes X 1, X X 3 and volume V X1X X 3. The sizes notably (three- to five-fold) exeed those of the region of interest, where we want to model seismi and aseismi events. The uniform random distribution is used for: 1) Three oordinates of the flaw enter, hanging in the intervals [ X 1 /, X1 / ], [ X /, X / ] and [ X 3 /, X 3 / ] for the global oordinates x 1, x and x 3, respetively. ) Dip angle of a flaw hanging from 0 to π/. NUERICAL ODELING OF EIICITY: THEORY AND APPLICATION

15 3) trike angle hanging from 0 to π. 4) Rotation angle in the dip plane, hanging from 0 to π (for a retangular flaw). 5) Ratio b / a, hanging from 1 to 3 (for a retangular flaw). In general the total number of geometri parameters is seven. For square flaws ( a b), the item (v) is not used; the number of parameters is six. For irular flaws ( a R), the item (iv) is not used, as well; the number of parameters is five. For the size a of a flaw, we use the exponential distribution of the probability density: 1 a P( a) exp, (4.1) l l where l is the average length of the smallest side. The average length l and the total number N of the flaws are presribed as explained below. Input Data on ehanial Properties of eeded Flaws hear rigidity. Having the input data on the rok elastiity modulus and the sizes of flaws, we find the shear rigidity from the seond Equation (.1) or (.14). 0 Tensile strength n. Commonly we may assume that the tensile strength is zero: 0 = 0. n hear strength 0, *,. The shear strength is haraterized by the initial ohesion 0, residual ohesion * and the ontat frition angle. As follows from Equations (.18), (.19), (.) and (.3), atually we need the differene 0 *, rather than 0 and * separately. Therefore, we may set * = 0. To redue the number of quite unertain parameters, like 0 and, we may presribe the latter two quantities by their average values. In partiular, for hard roks, aording to alamon 1993, the initial ohesion hanges from 0.0 to 4.3 Pa with the statistially mean value 0 =.5; the frition oeffiient tanρ hanges from 0.5 to 1 with the mean frition angle = When high onfining pressure prevents shear, the number of initiated events may be inreased by dereasing the frition angle to 0 or even to 10. hear softening modulus. For the presribed shear rigidity, presribing the softening modulus is equivalent to that of the ratio /. The latter defines the instability ondition and seismi effiieny. As both and derease with growing rak size, they are partly orrelated. till their ratio is a random quantity and quite unertain. We may presribe the ratio / as a sum of the determined (mean) part a and the random part b f, uniformly distributed on the interval [ b, b ]: a b f, (4.) where a b, f is a random value uniformly distributed on the interval [-1, 1]. The values of a and b in (4.) may be presribed by using the following onsiderations. If a b 1, then / 1; hene in aordane with the instability ondition (.9), all the modeled events will be seismi. In modeling, this hoie serves to neglet aseismi events. If a b < 1, then / < 1; hene all the modeled events will be aseismi. In modeling, this hoie serves to model only aseismi events. If a 1 b, the ratio of numbers of modeled seismi events N to that of aseismi events N is: N N A b b ( a ( a 1). (4.3) 1) Thus, by an appropriate hoie of a and b, one may adjust the ratio of seismi to aseismi deformation to data of observations. Note that when a = 1, Equation (4.3) implies that the number of seismi events equals to the number of aseismi events for any value of b. The mean seismi effiieny of seismi events is: a b 1 eff. (4.4) a b 1 Therefore, one may hoose the parameters to presribe the mean value of. eff A a and b Internal time sale t r. In the EC-model, the time sale is defined by the only ombination with the time dimension, whih is the retardation time t r / El of the elvin-voight element. Both the visosity and rigidity E l of this element are perhaps the most unertain for rok. eanwhile, the internal time is no more than a parameter ordering events in a sequene. Note also that aording to (3.10), the mean time of aseismi events is haraterized by t r / rather than t r. till, as in many ases the mean value of is of order of unity, the retardation time is typial for the majority of aseismi events. The ontat visosity being unertain, the time sale is onditional. Clearly, it should be less than a typial time step of the external time. It may be roughly estimated if a dependene of Omori type is available from observations. uppose we have dependene of the number of seismi events, ourred in equal time intervals after a strong exitation of rok (Figure 16a). The dependene is approximated by deaying exponent (Figure 16b) as N N exp( t / t ). (4.5) 0 O Then we may assoiate the retardation time with the harateristi time t O of the Omori dependene (4.5) Rai8

16 to a hain reation. For 3D problems, the range was established by numerial experiments (Linkov 006 ). It is: (4.10) Figure 16. Dependene of Omori type for the number of seismi events after exitation of rok Rigidity of elvin-voight element E l. It an be shown (Linkov 006 ) that for the distribution (3.1), the mean portion of aelerating events in the total number of all aseismi events is 1/ ( / El ) / b if either (i) 1 a b and a / El 1, or (ii) a > 1 and 1 a / El b. In these ases we may presribe the rigidity E l by hoosing its value, whih provides a needed portion of aelerating events. ean length of flaws l. The mean length l of seeded flaws may be defined via the expeted level of energy W l of events. Then, by using a l in (.1), we obtain: 1.13 Wl l. (4.6) / E 3 ( eff 0 * ) Note that the expression under the ubi root in (4.6) has the order of elasti energy released from a unit volume of rok under the stress drop. 0 * Flaw density and the number N of seeded flaws. We have mentioned that the dominant fator ausing hains of events is flaw interation. The interation is signifiant at distanes less or omparable with the flaw minimal size a. Therefore, in problems onerning with modeling seismiity it is reasonable to introdue the flaw density as the ratio of the mean length l of flaws to the mean distane L between them: l. (4.7) L In a 3D volume V with uniform distribution of N flaws, the mean distane between their enters is: 3 N V L. (4.8) From (4.7) and (4.8) we obtain the total number of flaws to be seeded in a volume V to have the density : 3 N V L. (4.9) When using (4.9) we need to speify the range of the density, in whih interation of flaws is strong enough to produe hains of events while it is not too strong to lead When the density is below the lower threshold, there is atually no flaw interation. Consequently, there are no hains of events. When the density is above the upper threshold, the interation beomes so strong that there arise hain reation involving almost all the seeded flaws into deformation. A partiular value of the density in the range (4.10) may be hosen from additional onsiderations. Consider rare strong events with the energy W str notably, say 00-fold, exeeding the mean energy W l. The number N str of suh strong events is first units, say N str = 1. Then assuming the number of seismi events proportional to the number of flaws apable to produe events of presribed energy, we have from (.1) and (4.1) the expeted number N of all seismi events: W str N 3 N str exp. (4.11) Wl Naturally, the total number of seeded flaws should be at least an order greater than the expeted number of seismi events. Denote k the fration of flaws, whih produe seismi events, so that (4.11), we obtain the estimation: N 1 N k N N. When using 1 W str N N exp 3 str. (4.1) k N Wl 3 When having N, Equation (3.19) defines l N /V. The found should be within the range (4.10). For a partiular problem, the number k N is not known in advane. Consequently, there might be need to perform a series of alibrating runs with a omputer ode. They start from a guess regarding k N. Then for a presribed ratio W str / W l, the value of N beomes available from (4.1) and (4.9) serves to find the density. (The latter should be within the range (4.10)). The first run provides the portion k N of flaws, whih produe seismi events. The found k N serves for the next run and so on. If at some step the density is outside the range (4.10), we need to repeat alibration by taking another value of N str, say 0.1, or, or 5. Another, although limited option, onsists in hanging the volume V of seeded flaws. Finally we obtain the needed values of N and. Comment. Calibration is notably simplified when having the dependene frequeny-magnitude of Gutenberg- Righter type obtained from field observations. We shall disuss this dependene below in etion Output Data. NUERICAL ODELING OF EIICITY: THEORY AND APPLICATION

17 Initialization INITIALIZATION AND TIE TEP The generated input data on flaws are firstly analysed on the initialisation step of alulations. As mentioned, it onsists in onsidering in situ state of rok and exlusion those of seeded flaws, for whih in situ trations exeed the tensile or shear strength of a flaw. It is done by a subroutine, whih we have alled ExlusInitu. It presents a yle over N seeded flaws with heking for eah of them if its strength is exeeded. These flaws are exluded from further alulation and the number N is respetively diminished. The remaining flaws are assumed ative in the sense that they are losed and able to experiene seismi or aseismi DD under indued trations. teps in External Time After the initialisation, the ode starts alulations in a yle of external time steps. At the beginning of a step, a onventional (basi) ode alulates trations at the loation of eah ative flaw. For brevity, we all them onventional trations. Besides those, there arise additional trations aused by the DD on flaws, whih have been ativated on previous steps. We all their sum at eah of the urrent ative flaws flaw-trations. At the beginning of the first step, the flaw trations are zero. We neglet the bak influene of the flaw-stresses on the onventional stresses. This serves us to use the basi ode unhanged. Normally, it does not lead to notable loss of auray. till, if there is need, the bak influene may be aounted for by an iteration inluded into the basis ode. At eah of the time steps, we perform alulations in a similar way. It is done by a subroutine, whih we all eismtimetep. Consider a typial time step. For it, we have onventional and flaw-trations at flaws, whih are ative to the beginning of the step. Calulations always start from the zero-stage and may inlude a number of stages depending if there arise aseismi events. peifially, if there appear aseismi events on the zero stage, it is followed with the fist stage. If on the first stage there appear new aseismi events, it is followed with the seond stage, and so on until aseismi events ease to appear. The alulations are performed as follows. Zero-stage. At eah ative flaw, we add the flawtrations to the onventional trations and hek if the summary trations exeed the tensile or shear strength. If the strength is exeeded, we alulate the DD arising at the flaw by using Equation (.15) for the tensile mode, and (.17) for the shear mode. Besides we hek if the orresponding event is seismi or aseismi. Then ommon quadratures of the H-BE provide stresses aused by these DD at any point of a medium (see, e.g., Linkov et al ). The alulation is performed by a subroutine, whih we all TraFlaw3D. As a result, we obtain additional trations aused by the DD arisen at an ativated flaw at loations of other ative flaws. The additional trations are summed and stored separately for seismi and aseismi events, whih have appeared on the zero-stage. After heking all the ative flaws, those of them, whih have produed seismi events, are exluded from ative and delared passive flaws. The sum of their additional trations is immediately added to flaw-trations. Then we start the next yle, repeating the same alulations for the new set of ative flaws and with updated flaw-trations. The yles are repeated until new seismi events ease to appear. Then we turn to the flaws, whih produed aseismi events. Now we exlude them from ative flaws, delare them passive and add the sum of trations indued by them to flaw-trations. The stage is over. First and following stages. The new sets of ative flaws and flaw-trations are used in the same way as that desribed for the zero-stage. If there arise new seismi events, we onsider them appeared after those, whih have been simulated on the previous stage. The time lag is assumed to be equal to the harateristi time of aseismi events. In this ase, the Omori-type dependene may be modelled. The stages are repeated until new aseismi events ease to appear. Then the urrent time step is over, and we may start the next step of external time. aved Information In time steps, a omputer ode saves data on eah event, seismi or aseismi, simulated. The information on an event inludes: 1) Its number in the numeration of events in the sequene of their arising; the number of the flaw (in initial numeration of the flaws), on whih it appeared; the input data for the flaw gives loation of the event, orientation of its plane and presribed mehanial properties. ) The time step, at whih the event is simulated. 3) The type of the event, seismi or aseismi. For a seismi event, its mode, tensile or shear. For an aseismi event, its harater, aelerating or damping. 4) The stage within the time step, on whih the event ourred. 5) The yle (for a seismi event) within the stage, on whih the event ourred. 6) Charateristis of a seismi or aseismi event, desribed in previous setions. These data are sorted and analysed by output subroutines Rai8

18 General Considerations OUTPUT DATA imulation of events notably extends data on a rok mass state. In addition to ommon mehanial data on stresses, strains and displaements, provided by a onventional ode, we obtain data on seismi and aseismi events aused by hanges of stresses. These data are similar to those observed in mines or around a hydrauli frature, or oil (gas, heat) prodution well. peifially, we obtain data on temporal and spatial distributions of events. For eah simulated seismi event, we have its mode (tensile or shear), oordinates of its soure, orientation of the plane, nodal planes, DD, energy, seismi moment, seismi effiieny, stress drop and seismi shear. For eah aseismi event, we have its type (aelerating or damping), time and stage of its ourrene, harateristi duration, stress drop and final shear DD. What is of speial signifiane, in ontrast with observed seismiity, for any event, we exatly know the mehanial state, at whih the event ourred. These data on events, plaed in output files, undergo statistial proessing. The latter may inlude building the tallies of the listed output harateristis; graphial plotting of Gutenberg-Righter and Omori dependenes; delineation of regions of seismiity onentration suh as pillars, edges of seams, moving front of a propagating hydrauli frature, et. This is done by postproessor subroutines, whih we assume joined in a single subroutine, alled FlawOutput. Consider some of post proessing ations applied to mining and hydrauli fraturing. Frequeny-agnitude Dependene The frequeny-magnitude dependene presents logarithm of the number N W of seismi events with the magnitude exeeding m W. Aording to alamon 1993, the magnitude of a seismi event in Gutenberg-Righter sale is onneted with the energy W measured in J, as: m W / 3(logW 1.). (6.1) A typial graph of the frequeny-magnitude dependene is shown in Figure 17 by the solid line. It is obtained when simulating seismiity in a deep gold mine of outh Afria (Linkov 006 ). The simulation is performed for mining onditions similar to those, for whih summarised data of seismi observations are available (data by Anderson reprodues by Napier 001 ). The dashed line in Figure 17 shows the urve for observed events. The agreement looks impressive. Note that in mining pratie there are atually no events with the magnitude exeeding. This explains the drop of the urve near m W = 1.5. On the other hand, there are very many events with small magnitude ( mw < 0), whih are not aounted for in earth seismology. This explains lose to horizontal part of the urves at small magnitudes. The average slope of the urve is lose to It is in ontrast with the earthquake observations, for whih the Gutenberg-Righter dependene is lose to a straight line with the slope -1. Figure 17. Dependenes of Gutenberg-Righter type For simulated seismiity, aompanying hydrauli fratures, the frequeny-magnitude dependene is similar to that shown in Figure 17, but the magnitude of the strongest events is four orders less (Dobroskok and Linkov 008 ). It is beause the net-pressure, whih auses the frature propagation and defines the hange of stresses, is two-three orders less than the stress hange indued by mining in deep gold mines. Omori Dependene Numerial simulation of exponentially dereasing number of seismi events after a strong exitation, orresponding to the stope advane in time, has led to an unexpeted result (Linkov 005, 006 ). It appeared that to simulate the Omori dependene (Figure 16), it is neessary to seed at least two sets of flaws with different linear sales. One of them ontains flaws having mean length of order of ten meters; the other is of the mining step (first meters). The Omori dependene arises as the influene of rare aseismi events, generated by large flaws of the first group, on small flaws of the seond group, whih generate the majority of seismi events. Figure 18 presents typial results obtained in this way for long-wall advane in three time steps. It shows the numbers of events at stages within eah of the time steps. Joined Geomehanial and eismi odeling and onitoring In the Introdution we have mentioned about the need and easy implementation of joined geomehanial and seismi modeling and monitoring. Figure 19 shows the distribution of projetions of simulated seismi events to the horizontal mining plane when the long-wall mining advanes 10 meters downward (along the x3 -diretion). The solid line shows the ontour of the mined area at the end of a mining step. Evidently, the events follow the hange of the geometry. NUERICAL ODELING OF EIICITY: THEORY AND APPLICATION

19 Figure 18. Numbers of seismi events on stages within three time steps Figure 19. Projetions of simulated seismi events to the mining plane The distribution of the events in the vertial rosssetion orthogonal to the mining line is shown in Figure 0. We may see that the simulated events group near the mining front. They are distributed in the roof and floor mostly beyond the wedge-like zone. Figure 0. Projetions of simulated seismi events to the vertial plane orthogonal to the mining line Remarkably, almost no events our in the seam just ahead of the mining front. The reason is that the shear stresses are small along the middle plane of the seam what prevents shear events. eantime, alulation of mehanial quantities shows that the onentration of ompressive normal stresses in this zone is very high. It implies that this zone is extremely dangerous as onerns with unfavorable manifestations of rok pressure. The example learly demonstrates that there are ases when seismi monitoring alone is not suffiient for obtaining reliable pratial onlusions. The need to ompare the observed seismiity with simulated events and to ompliment it with geomehanial modeling is obvious. Validation and Calibration of Input Data We have already mentioned about extreme value of diret observations of seismiity for proper presribing the input parameters on the seeded flaws. They are also important for improving a geomehanial model. For instane, the observed seismiity may serve to dislose important geologial features like faults, dykes, zones of signifiant hanges of mehanial properties and input data on in-situ raks. An impressive example of the kind is appliation of the unonventional frature model (UF) to numerial modeling of hydrauli fratures in low-permeable shales (Cipolla et al. 007 ). The input data of the numerial method, based on the UF, inlude parameters on the supposed insitu rak net in the area of simulation. These parameters are alibrated after obtaining data on spatial distribution of seismiity observed during hydrauli fraturing. Presently the method does not employ syntheti seismiity. urely its reliability will inrease when omplimenting alulations of onventional mehanial quantities with numerial modeling of seismiity. Inversion of imulated Data. PCA and R ethods to Delineate Geometrial Features For events simulated as desribed in the previous setion, we know exat geometry, alulated stresses, strains, DD and displaements on time steps. Thus there is expliit orrespondene between seismiity and mehanial quantities. Forget, for a while, the knowledge on the geometry and mehanial quantities. Now we have only seismi data similar to that observed in rok. Consequently we may apply methods, used in mining pratie for interpretation of observed seismiity, to the syntheti data. If the onlusions, obtained by a method of inversion, agree with the underlying mehanial state, the method is aeptable. Otherwise, it needs improvements. Thus simulation of seismiity opens an opportunity for validation of existing methods of inversion. It may also serve for suggestion of new methods, adjusted to peuliarities of a partiular problem (e.g. Dobroskok and Linkov 008, 009, 011, Dobroskok et al. 010 ). Consider, as an example, traing of hydrauli frature by employing seismi data on loation and time of miroseismi events Rai8

20 Figure 1 presents projetions of simulated events to the frature plane. They were obtained in alulations for a frature of the height 100 m propagating in 15 time steps, 10 m eah. Thus the final frature is a retangle with the height 100 m and the length 150 m: 0 x 1 150, 50 x 50. We may see that on whole the simulated events group within and near this retangle. ourring per time needed for frature propagation of about 10 meters. At least half of events are onentrated near the front line. The onventional method of analyzing geometrial features is the priniple omponent analysis, PCA (see, e.g. Jolliffe 1986 ), urrently employed in the oil (gas) prodution industry. It gives the orientation and three prinipal axes of an ellipsoid ontaining the majority of events. It does not provide tallies. The omparison of the PA and R shows that when applied to the totality of the events, the both methods are apable to ath the mining or frature plane with reasonable auray. However, the totality of events does not provide unambiguously estimation of the sizes of ative zone (see, for example, Figure 1, where sattering of the events is notably out of the hydrauli frature oupying the area 0 x 1 150, 50 x 50 ). Figure 1. Projetions of simulated seismi events to the plane of hydrauli frature Now we forget about the atual frature plane and try to guess its position from the syntheti seismiity. To this end, we apply the method of strip ray sanning (R), speially suggested for traing geometrial hanges (Dobroskok and Linkov 008 ). The method employs ounting simulated or observed events within a semi-infinite strip of a fixed width w and arbitrary orientation of the normal to the strip plane. The normal sans the hemisphere, and the orientation, orresponding to the maximal number of events within the strip, is assoiated with the normal to the plane ontaining the soure, whih indues the events (in the onsidered ase, the frature plane). Projetions of the events to this plane are used for further analysis. We find tallies of events along various azimuths (rays) in the plane. The tallies ontain information on geomehanial features within the plane. For instane, tallies obtained for a time step may serve to onlude on the position of the frature front at this moment. peifially, we may assoiate the tally having the most distint peak with the diretion of the front propagation. The most smeared tally, whih normally orresponds to the orthogonal diretion, is assoiated with the front line. As an example, Figure shows the tally of the events ourred at the sixth time step of simulated hydrauli frature propagation. The atual position of the front is at the distane 60 m from the origin. We may see that the maximal number of events remarkably agrees with the front loation. The number of events ahead of the front does not differ signifiantly from their number behind the front. This means that in pratie, the front line may be assoiated with the middle of events Figure. Tally of events simulated at the sixth time step of hydrauli frature propagation Further analysis shows that for traing hanges in geometry, it is muh more informative to use statistially signifiant groups of events on time steps. Appliation of the PA and R to these groups allows us to delineate the mining or frature front and to find the diretion of the propagation in time. The methods give lose onlusions, when applied to the seismiity indued by long-wall mining. In ontrast, the results for hydrauli fraturing indiate that the R is superior over the PCA. On the other hand, the PCA gives estimation of the strip width w, presribed in the R. Thus taken together, the methods omplement eah other. Using the syntheti seismiity has allowed us to establish the auray, thresholds of appliability and relation between the methods. Comment. Presently, in pratie, only the totality of the observed miroseismi events is used for making onlusions on a hydrauli frature (e.g. Cipolla et al. 011 and Wright 007 ). eanwhile, the analysis based on numerial simulation of seismiity learly shows the advantages of employing groups of events on time steps providing statistially signifiant number of events. It looks reasonable to use this onlusion in pratie. NUERICAL ODELING OF EIICITY: THEORY AND APPLICATION