Mathematical Model to Locate Interference of Blast Waves from Multi-Hole Blasting Rounds

Maximum charge per delay in a blasting round is universally accepted as the influencing parameter to quantify magnitude of vibration for any distance of concern. However, for any blasting round experimental data reveals that for same charge per delay magnitude of vibration varies with total charge. Considering linear transmission of blast waves, the paper firstly investigates into the influence of explosive weight, blast design parameters and geology of strata on magnitude and characteristics of vibration parameters and thereafter communicates that possibly interference of blast waves generated from same and different holes of a blasting round result into variation in vibration magnitude. The paper lastly developed a mathematical model to evaluate points of interference of blast waves generated from singleand multi-hole blasting round.


Introduction
Energy used for ground vibration is about 7% of the quantity generated in the form heat of explosion [1][2][3][4].Detonation of explosive generates both acceptable and non-acceptable consequences.The non-acceptable consequences of blasting viz., magnitude of vibration, audible and in-audible air concussion, excessive throw and dust and fumes causes uncomfortable and unhygienic condition around the mining area.The increased environmental awareness and regular public complaints in the recent years have evolved into legislation to limit maximum magnitude of vibration for safety of structures around the mining sites [5][6][7][8][9].Empirical, analytical and numerical predictor techniques have been developed to determine vibration magnitude for any distance of concern [10][11][12][13].All the predictor Equations have considered maximum charge per delay as the main parameter for derivation of empirical Equation.The work of recent researchers' states that charge per delay cannot be the only single parameter to quantify magnitude of vibration [14][15][16][17][18]. Charge weight scaling model introduced sliding time window over which the scaled charge weight should be added for determination of maximum charge weight per delay in a blasting round.Authors also used a Dynamic Finite Element Model for single blast hole to evaluate vibration as a function of VOD of explosive [19].The author used the Seed Waveform Model to define the wave propagation characteristics.Various au-thors have also worked upon blast wave characteristics for prediction of vibration magnitude for any distance of concern [20,21].
Review of vibration data monitored at various sites for different distances of concern illustrates that VOD of explosive, time interval between detonation of different delays in same or different holes of a blasting round and propagating velocity in transmitting medium quantifies magnitude of vibration at a distance of concern.Since, duration of vibration for elemental charge depends upon charge quantity detonated in unit time, borehole pressure and linear charge concentration viz., density and diameter of explosive, the paper have attempted for segmental analysis of explosive length to illustrate the vibration propagation.Since, most of the seismographs record vibration at every 0.001 seconds time interval, the theoretical model evaluates propagation characteristics for infinitesimal explosive length detonated in every 1 milliseconds i.e., segmental analyses of explosive column for every blast hole and all holes in a blasting round.The paper uses linear super-imposition concept to evaluate theoretical model for determination of points of interferences during its transmission.

Wave Behaviour
Propagation characteristics of blast waves vary according to the characteristic properties of rock strata and angle of incident of stress wave on joint plane.The quantum of energy absorbed, reflected or refracted depends upon thickness of each stratum, smoothness of joint plane and filling material within joints.At the point of incidence, the angle of incident (i) will be equal to the angle of reflection (r) and ratio between the angle of incident and refracted wave (refr) will depend upon ratio of densities and P-wave velocities of two mediums.Transmission of blast induced stress wave being a function of Poisson's ratio, , friction angle, , and orientation of structural plane, , with respect to incidence of stress wave, the magnitudes of transmitted and reflected wave can be determined with the help of transmission coefficient (A s ) and the reflection coefficient (A z ) of P-wave, Equation 1 and 2 respectively.The pressure reduction of transmitted wave during its transmission can be determined with the help of Equation 3 [22].The velocity of P-wave and Swave viz., C p and C s in rock medium can be determined with the help of Equations ( 4) and ( 5) respectively [23].

Experimental Evaluation
According to conservation of energy, explosive energy on detonation results into equal and opposite reaction.
The effective reaction for any blasting are heat, temperature, light, sound, fragmentation, throw and vibration.
The undesirable or non-acceptable forms are noise/air overpressure, vibration and excessive throw, if any, causing structural and human damage.For same charge per delay and varied total charge variation in vibration magnitude illustrates the characteristics of cooperation of explosive weight detonated in same and different delays of a blasting round (Tables 1 and 2).In Table 1   with an increase in distance of measurement indicates cooperation of blast wave (Sl.Nos. 2 & 3).However, magnitude of displacement decreases possibly due to reduction in its energy component.Similarly, comparing the vibration parameters for multi-hole firing it is observed that acceleration decreases with an increase in distance of measurement (Sl.Nos.4-7, Table 2).But, the magnitude of displacement computed by the seismographs indicates no certain characteristics, indicating the influence of interference of blast waves generated from different blast holes.Comparing magnitude of acceleration between single-and multi-hole blasts indicates that magnitude of acceleration is always higher with an increase in total charge.In comparison to single-hole blasts, magnitude of displacement for multi-hole blasts, except for Sl.No. 6, is observed to have less magnitude, indicating loss of energy during its transmission.Comparing the arrival time of blast waves between single-and multihole firing it is observed that the arrival time is always higher for multi-hole shots, indicating interference of blast waves during their transmission.Therefore, attenuation of vibration magnitude i.e., vibration propagation varies with linear charge concentration, charge length, diameter of explosive and VOD of explosive.Similarly, for same charge per delay, magnitude of vibration measured at same place may also vary with distribution characteristics of explosive weight in blast holes.

Theoretical Evaluation
Propagation velocity and intensity of stress generated due to detonation of explosive varies with energy contained in it during the time of detonation.Velocity and intensity of wave varies with rate of energy loss due to absorption in the preceding medium/layer.Energy loss being proportional to energy contained, loss of energy will be more at initial period (closer distance) than that observed at far off distances i.e., attenuation of vibration magnitude will be faster at closer distance than that observed at far off distances.At closer distance, interference of blast waves is influenced by enhanced charge length/concentration, ratio between total charge and charge per delay and delay timing between two initiations made in same or different holes of a blasting round.Interference of blast waves from different holes result into constructive or destructtive interference i.e., magnification or reduction in resultant magnitude.Constructive interference causes amplification in magnitude and sustains for longer duration.
Destructive interference, on the other hand, results into low magnitude and sustains for lesser time duration.For far off distances, magnitude of vibration measured is the resultant impact of interference of blast waves generated from different delays of a blasting round.Attenuation characteristics in this zone are also very slow.At such distances, wave transmitted from different holes of a blasting round adds to the less energy contained waves during the path of transmission to quantify duration and magnitude of vibration.The waves at such distances take longer time to pass through any element of construction (civil or rock) or particle of medium coming in its path and generates poor stress and strain rate.Based on variation in measured vibration data, the paper attempted to evolve a theoretical model to understand the philosophy of interference of blast waves from single blast hole for both monolithic and bedded strata (Figures 1 and 2).Considering the concept of propagation of waves through bedded strata, the paper evolved a mathematical model for an array of holes in a blasting round.For computer simulation, arbitrary point, (X, Y, Z), in the transmitting medium has been considered as the point of interference.For ease in calculation, the paper assumed  the coordinate of bottom of first hole initiated in a blasting round as (0, 0, 0), Figure 3.

Let
Velocity of detonation of explosive = V e ms -1 ; Sonic velocity of transmitting medium = V m ms -1 ; Depth of hole = h, m; Distance of sensor = h 1 , m; At 0.001 seconds length of explosive column detonated = V e × 10 -3 m; Center of gravity of explosive detonated in 0.001 seconds = (V e × 10 -3 )/2; Similarly, for next sector, center of gravity and corresponding radial distance to the sensor can be determined from Equations (6b) and (6c) respectively.
Therefore, the general form of Equation to determine center of gravity from bottom of blast hole for each sector and the corresponding radial distance to the sensor can be determined with the help of Equations (6d) and (6e) respectively.

  th 3 e
Center of gravity for n section 2n 1 V 10 2 Radial distance for n section If  (n) is the angle between the horizontal plane and line of propagation of blast wave generated from sector "n", then where n is the term or section of explosive detonated from bottom of the hole.Interference of propagating blast waves will depend upon cumulative impact of explosive column detonated in each sector i.e., detonation velocity of explosive column, sonic velocity of transmitting medium, delay in detonation timings for each sector and phase difference between them.The resultant impact due to collision of waves detonated from two sectors will be of the form as shown in Figure 4 Magnitude and direction of resultant can be evaluated with the help of Equations (7a) and (7b) respectively.
Considering magnitude of vibration generated by two sectors as V 1 and V 2 with phase angle, , the resultant impact of first two sectors i.e., R and  will be the input parameter to determine the second resultant when the sensor is struck by blast wave detonated from third sector.The process will continue till all sectors along the length of hole are detonated.The waves generated from each sector will also suffer reflection and refraction when interfered by any fracture or joint plane and therefore the distance between the source and measuring location (sensor) will vary for each sector.The cumulative distance impact for total charge length can be determined with the help of Equation (7c).Considering USBM empirical Equation as best-fit predictor Equation, the magnitude of vibration measured at any place due to the impact of total charge length can be determined with the help of Equation (7d).Here the resultant impact from first two sectors will be input parameter when the sensor is struck by wave generated from third sector and the process will continue till total charge length under consideration is evaluated.Here, instead of charge per delay, Q, (kg), the paper considered "Q" as the quantity of explosive (kg) detonated in 1 millisecond.For monolithic rock mass and close field monitoring, probability of interference will be minimized.However, for longer distance of measurement, constructive or destructive superimposition of waves may take place during the path of travel.
where, πr 2 V e × 0.001; and /m 3 ).T interference of blast w Q =  r = radius of blasthole (m);  = density of explosive (kg he mathematical model to locate aves generated from two sectors of same blast hole is given below: Suppose, Distance traveled up to the point of interfe de rence on tonation of sector S 2 will be Equating above two Equations, we have Therefore, knowing the sonic velocity of rock me (V

Single-Hole Blast in Bedded Strata
ss character- D  dium m ) and time of detonation of both the sectors viz., T and T 1 , location of interference from two sectors, if any, can be identified/determined by the method of iteration.Since, a number of interference points can be identified, the same can be evaluated by computer simulation method.
For bedded strata, depending upon rock ma istics, blast hole may be loaded in single-or multi-deck systems.Figure 4 shows schematic diagram of a blast hole loaded in multi-decks.Initiation of holes may be carried out by either detonating cord or NONEL system.Similarly, Figure 3 shows two-deck charging with NONEL system of initiation.With detonating cord system of initiation i.e., top initiation, the primer or booster is first detonated to trigger or detonate the column explosive in each deck.Therefore, it can be well understood that due to more number of strata for lower decked explosive column and delayed time of initiation for lower decks there will be minimum possibility of interference of blast waves generated from two different decks.However, when soft stratum is overlain by hard strata the probability of interference of blast waves generated from different decks cannot be ignored.For NONEL system of initiation i.e., bottom initiation, probability of interference of blast waves generated from different decks may take place.Probability of interference of blast waves generated from different decks of same blast hole is minimum when compact hard strata lies above loose strata.However, for vice-versa strata condition i.e., soft layer overlying the hard strata, probability of interference of waves at any distance of concern cannot be ignored.
In Figure 4 the sensor is located at a horizontal dista nation time for bottom deck = t; t 1 ; ck = l; s = d; ration wave generated fr f vibration wave generated fr n detonated in 0.001 secon s of explosive detonated in bottom de of explosive detonated in upper de of hole = (0, 0, 0); aves = (X, Y, Z) r three dimension analysis X-and Y-axis are consi nce h 1 from center of blast hole of depth h.The path of travel by blast waves from individual sector and each deck are shown by dotted lines viz., AS 1 , BS 1 and CS 1 .The path of travel through bedded strata will always suffer refraction when transmitted through different mediums and actual distance of travel will always be more than the actual linear length.The angle of refraction will depend upon characteristics of two consecutive transmitting mediums.Energy absorbed in each layer will also be different and directly proportional to magnitude of energy contained in it.
Let Deto Detonation time for next above deck = Length of explosive column for bottom de Length of explosive column for upper deck = l 1 ; Length of non-explosive material between two deck VOD of explosive = V e m/s; Average P-wave velocity of vib om bottom deck = V m m/s; Average P-wave velocity o om upper deck = V m1 m/s; Length of explosive colum ds = V e × 10 -3 m; Number of sector ck = l/(V e × 0.001); Number of sectors ck = l 1 /(V e × 0.001); Coordinate of bottom Coordinate for interference of two blast w .Fo dered along the horizontal plane i.e., along length and width of blasting patch and Z-axis in vertical plane i.e., along depth of blasthole (Figure 3).The coordinate of center of gravity of bottom most and subsequent sector of explosive column in bottom deck will be (0, 0, 0.001V e /2) and {0, 0, 0.001(V e + V e /2)} respectively.Similarly, for next deck of explosive column, the center of gravity for bottom most and subsequent sector will be (0, 0, (l + d + 0.001V e /2)) and [0, 0, {l + d + 0.001(V e + V e /2)}] respectively.The generalized term for center of gravity for each sector for bottom and upper deck of explosive column from the base of blast hole for two deck system of charging will be of the form given in Equations (9a) and (9b) respectively.
where, ector number from the bottom of lower deck (p er from the bottom of upper deck (p k = l + d Interference of waves e from sectors detona n = s oint of initiation) n 1 = sector numb oint of initiation) take plac ted either same or different decks.Firstly, considering interference e of waves from sectors detonated in same deck.If the distance travelled by wave from different sectors of same deck to the point of interference be D and D 1 respectively, the mathematical Equations for determining the magnitudes for each will be of the form as given in Equations (10a) and (10b) respectively.If time of detonation of two waves under interference are t and t 1 and the time of interference is t 2 , then distance travelled by each wave can be determined with the help of Equations 10(c) and (10d) respectively.Equating Equations (10a)-(10d), location of interference, by the method of iteration, can be determiend from Equation (10e).
Equating Equations (10c) and (10d), we have Similarly, considering interference of waves due to detonation of explosive from different decks, the linear distance travelled by each can be determined by using Equation (11).Interfenece of blast waves will depend upon characteristics of transmitting medium and initiation timing of two sectors.
Let, tion time of sector from lower deck = t 1 ; Initia Initiation time of sector from upper deck = t 2 ; Time of interference of waves from two sectors = t 3 ; el to deck to travel to th on wave generated by th tion wave generated by th ctor to the point of inte Time available for the sector from lower deck to trav the point of interference = (t 3 -t 1 ); Time available by a sector from upper e point of interference = (t 3 -t 2 ); Average sonic velocity of vibrati e sector from lower deck = V m ; Average sonic velocity of vibra e sector from upper deck = V m1 .Distance traveled by the first se rference, Distance traveled by the secon in d sector to the point of terference,   If θ (n) and β (n1) be the angles of re where, an -1 (h -0.001 × (2n -1)V e )/h 1 ; β )/h 1 ; nd location of rizontal distance traveled by both the se transmitted waves with spect to horizontal plane, then the horizontal distance for each can be detrmined by using Equations (12c) and (12d) respectively.h horizontal distance between blasthole a interference.Since, the ho ctors are same, Equations (12c) and (12d) reduces to as given in Equation (12e).
Knowing sonic velocity of rock medium, ti de h constant sonic velocity an

For Array of Blastholes
blasting round, the coor-

Conclusion
rom different sectors of same or dif-me of tonation of each sector in a blast hole and time of triggereing of sensor, time taken by the blast wave to reach the location of interference can be determined by the method of iteration.Evaluation of the above Equation being cumbersome and time consuming, computer simulation technique should be adopted to evaluate and locate the point of interferences.
For monolithic rock mass wit d minimum delay time between detonation of two decks i.e., 25 ms, there will be minimum possibility of interference of blast waves generated from same blasthole.However, probability of interference of blast waves cannot be ignored when length of explosive column is high for single hole blast or when more number of holes are detonated in a blasting round.Probability of interfereence is maximum for down-the-hole initiation and when soft strata is overlaying hard strata.For bedded strata, blast waves undergo refraction when transmitted from one medium to another and sonic velocity varies with strata characteristics.
For an array of blast holes in a dinates of bottom or base of each hole in three-dimensional form can be represented by matrix, "A".Here, the first element of the matrix (on top left corner) indicates the first hole initiated in the blasting round.The row and column of the matrix represents number of holes in a row and number of rows in blasting round respectively.Therefore, for "m" number of holes in a row and "n" number of rows, the matrix "A" will be of order n x m.For each row, the coordinate of bottom of each blasthole is formed by an increment of magnitude, s, where "s" indicates the spacing of blastholes in a row.Similarly, number of elements in each column of matrix represents number of rows in a blasting round, the matrix is obtained by an increment of magnitude "b", where "b" represents burden between rows.For each blast hole the coordinates of each sector of explosive column i.e., the coordinates of "z" can be obtained in the form of a sub-matrix for each blasthole (element) within the matrix "A".The coordinates of each sector of explosive column within the blasthole can be obtained by adding 0.001 V e to z-axis coordinate for each element in the matrix "A".The dimension of this matrix will depend upon depth of blasthole, charge length and VOD of explosive column.The matrix will have its elements in single column and the order of the matrix will be of order w × 1, where, "w" represents number of sectors denoted in a blast hole.Matrix "B" represents sonic velocity of transmitting medium.The dimension of matrix "B" will depend upon number of layers encountered by blast wave.Order of this matrix will be 1 × o, where "o" represents the number of layers of rock medium.Considering thickness of each strata and direction of blast wave, actual length of travel in a particular medium can be deterimed.When wave is transmitted from one medium to another, rate of energy absorbed and angle of refraction can also be determined with the help of Equations ( 1)-( 5).Knowing the actual blasting time and triggering time of the instrument at a given location, coordinates of point of interference of waves, if any, can be detrmined by the method of iteration.An arbitary, three-dimensional coordinate, is required as an input to simulate the method of iteration technique and evaluate the locations of interference for each wave.
Wave generated f ferent holes detonated in same or different delays of a blasting round causes constructive or destructive interference of blast waves.The characteristics of interference of blast waves i.e., phase of interaction, quantifies magnitude of vibration and its corresponding frequency at any distance of conceren.At longer distance, time duration between detonation of explosive and triggering of sensor increases and multiple interferences of blast waves take place during the path of travel.At such distances, seismographs receives the resultant impact of interference of blast waves generated from different sectors of same or different holes of a blasting round.At such distances attenuation of vibration magnitude is slow and is less.At close distance, probability of interference will be minimized due to high propagation velocity of blast waves.At such distances probability of interference is from holes detonated in separate delays and attenuation is also fast.At intermediate distance, magnitude of vibration depends upon P-wave velocity of blast waves in different transmitting rock layers, time of detonation of each delay, VOD of explosive, charge per hole and number of holes in a blasting round.Therefore, for any blasting round reduction or magnification in magnitude of vibration will, depend upon phase of interaction of each blast wave generated from different sectors.

Figure 1 .
Figure 1.Schematic diagram of blast hole in monolithic rock mass.

Figure 3
Figure 3 Schematic diagram of blasthole loaded in two decks.