CN117189092B - Soft rock ground stress testing method based on drilling cuttings particle size distribution - Google Patents
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- 238000005553 drilling Methods 0.000 title claims abstract description 95
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Abstract
The invention relates to a soft rock ground stress test method based on drilling cuttings particle size distribution, which combines the output energy of a drilling machine, the drilling cuttings quantity and the drilling cuttings particle size distribution in the drilling process to calculate the in-situ ground stress of the soft rock, wherein the test is divided into 3 parts: drilling machine data monitoring and energy calculation, drilling cuttings data monitoring and crushing energy calculation and ground stress back calculation. The invention is very suitable for coal mine production, can effectively utilize gas extraction drilling, realizes one-hole multi-purpose, promotes drilling informatization, and can be used for further analysis of fine gas geological structures of mines in the future.
Description
Technical Field
The invention relates to the technical field of rock-soil exploration and measurement, in particular to a soft rock ground stress testing method based on drill cuttings particle size distribution.
Background
The ground stress is a common geological parameter in mine excavation production activities, and the measurement and distribution inversion of the ground stress have very important significance for mining, underground space construction and geological disaster prediction. The conventional ground stress measurement method includes: the core stress relief method, the hydraulic fracturing method, the strain recovery method, the borehole breakout method, the acoustic emission method, and the test method (such as the optical fiber stress method) of a few embedded sensors are very dependent on the hole sealing technology, so that the repeatability of the test result is poor.
Most of the ground stress measuring methods are only suitable for being applied to hard rock formations, the coal bed is a weak layer in the rock formations, the intensity of the ground stress measuring methods is lower than that of most of the rock formations, and the properties of the coal bed with the lowest intensity of the ground stress measuring methods are similar to those of soil, so that the ground stress data of the coal bed adopted by coal mining at present are mostly derived from the measuring results of the roof-floor rock formations, and the ground stress test characterization method suitable for soft rock is not yet available. Therefore, there is a need for a method of testing the ground stress for soft rock.
Disclosure of Invention
The invention aims to overcome the defects, and provides a soft rock ground stress testing method based on drilling cuttings particle size distribution, which combines the drilling machine output energy, drilling cuttings amount and drilling cuttings particle size distribution in the drilling process to calculate the in-situ ground stress of soft rock, and the result can be used for guiding coal mine safety production and for geological information characterization and disaster prediction, in particular to underground coal rock dynamic disaster risk prediction.
The purpose of the invention is realized in the following way:
a soft rock ground stress testing method based on drill cuttings particle size distribution comprises the following steps:
firstly, constructing an upward through-layer drilling hole with the diameter phi (unit m) in a roadway, and collecting drilling cuttings when the drilling hole enters a coal stratum to be tested;
step two, collecting torque T (unit N ∙ m) and rotating speed N of the drill rod in real time by utilizing a sensor r (r/min), the effective power of the drilling machine in the drilling process is P=TN r 9.549 (unit W);
step three, calculating the input energy W of the drilling machine in the rock stratum to be tested 1 (unit J),wherein t is the drilling time of the drilling machine in the rock stratum to be tested;
obtaining the particle size distribution of the rock drilling cuttings to be measured by adopting a screening weighing device, placing the rock drilling cuttings to be measured into a multi-stage screening device, after screening is completed, sequentially obtaining the mass of the drilling cuttings on each screen by using an electronic balance, and recording the diameter of the screen hole of the nth screen as d n The mass of the drilling cuttings on the sieve is M n (kg unit), the outer surface area of the drill cuttings on the nth screen is S n = 12M n /(ρd n +ρd n-1 ) Is the apparent density (unit kg/m of drill cuttings 3 ) When n is 1, d n-1 =2d 1 ；
Step five: calculating the external surface area of the drill cuttings on all the screens, and accumulating to obtain the total external surface area S=S of the drill cuttings 1 +S 2 +∙∙∙S n ；
Step six: taking all drill cuttings on a first screen, and putting the drill cuttings into a drop hammer crushing device;
step seven: checking the gas tightness of the device, starting a vacuum pump, vacuumizing for 24 hours, and then filling high-pressure gas into the crushing device, wherein the gas pressure is equal to the in-situ gas pressure of the coal seam, so that the gas is adsorbed in a constant temperature chamber for 24 hours, and timely supplementing after the pressure is reduced during the period, so that the gas pressure is ensured to be slightly higher than the in-situ gas pressure of the coal seam all the time;
step eight: the method comprises the steps of (1) impacting drill cuttings containing gas by using a drop hammer breaking device, wherein in the process of measuring a common firmness coefficient, a drop hammer with a weight m freely drops from a height h to impact the drill cuttings in a base, repeating the drop hammer for 3-5 times, and recording the drop hammer times as a;
step nine: calculating total energy E=mgha (unit J) input by the drop hammer, putting drill cuttings fragments in the base into a screening device, and repeating the fourth and fifth steps to obtain the external surface area S' of the drill cuttings after the drop hammer is broken;
step ten: calculating to obtain the breaking surface energy consumption gamma=E/S' (unit J/m) of the tested soft rock stratum under the drop hammer condition 2 ) The energy consumption of the crushing surface under the drilling condition and the surface energy consumption under the drop hammer condition are in a linear relation, and the ratio is zeta, so that the energy consumption of the crushing surface under the drilling condition is gamma=zeta E/S';
step eleven: calculating the total energy W consumed by rock breaking during the drilling process 2 =γS；
Step twelve: calculating the rock breaking energy W provided by ground stress in the drilling process 3 =W 2 -W 1 ；
Step thirteen: dividing the crushing energy provided by the ground stress into two parts, namely the deformation energy W of the drilled rock fragments under the in-situ condition 3，2 And the work W performed by the stratum on the drilled rock debris 3，1 Assuming the borehole as a plane strain condition, wherein the work done by the infinite hydrostatic pressure environment stratum on the borehole is W 3，1 =2π(1+ν)p 2 Φ 2 L/(3E Y ) The in-situ deformation energy of the rock mass drilled in the drill hole is W 3，2 =3π(1-2ν)p 2 Φ 2 L/(8E Y ) Wherein E is Y V is Young's modulus (unit Pa) and Poisson's ratio of the rock stratum to be measured, p is in-situ stress (unit Pa) of the rock stratum to be measured, and L is length (unit m) of the drill hole in the rock stratum to be measured;
step fourteen: according to W 3，1 +W 3，2 =W 2 -W 1 And (3) combining the formulas in the steps to obtain a ground stress calculation method of the rock stratum to be tested:
。
further, in the first step, a drilling machine is arranged in the roadway, a drilling machine torque and rotation speed monitoring device is arranged on the drilling machine, and a drilling rod of the drilling machine penetrates upwards into a coal rock stratum to be tested.
Further, in the third step, the screening and weighing device comprises a screening system and a weighing system, the screening system can adopt a multi-stage screening device, and the weighing system can adopt an electronic balance.
In the sixth step, the drop hammer breaking device comprises a gas sealing device, a drop hammer and a drop hammer protection cylinder, and the particle size is larger than d 1 The drill cuttings are placed at the bottom of the drop hammer protection cylinder, a lifting drop hammer is arranged in the drop hammer protection cylinder, a gas sealing device is arranged on the top surface of the drop hammer protection cylinder, and the top of the drop hammer protection cylinder is connected with a gas source.
In the sixth step, the drop hammer protection cylinder is further connected with a gas pressure gauge and a vacuum pump respectively.
In the eighth step, a dropping hammer with the weight of 2.4kg freely drops from the height of 0.6m, the drill cuttings in the base are impacted, the dropping hammer is repeated for 3-5 times, and the dropping hammer frequency is recorded as a.
Further, in step ten, in order to improve the accuracy of the test result, the experimental test is independently carried out on the xi of different rock layers to be tested.
Compared with the prior art, the invention has the beneficial effects that:
the ground stress test method provided by the invention has more relaxed requirements on rock hardness, and is suitable for soft rock, especially coal seams.
The ground stress testing method provided by the invention is very suitable for coal mine production, can effectively utilize gas extraction drilling holes, realizes one-hole multi-purpose, promotes drilling informatization, and can be used for further analysis of fine gas geological structures of mines in the future. Meanwhile, the method for measuring the ground stress is relatively simple, can be divided into three modules, namely drilling energy calculation, crushing energy calculation and ground stress energy calculation, and is hopeful to develop a full-automatic underground ground stress measuring device while drilling based on the method and promote development of intelligent disaster early warning analysis of a mine.
The testing method is simple, and particularly aims at coal mine engineering, new equipment and new working procedures are not needed, the testing cost is reduced, and the original underground drilling holes of the coal mine are fully utilized. The invention utilizes the dense gas extraction drilling holes, and can obtain detailed ground stress distribution data.
Drawings
Fig. 1 is a schematic diagram of the scheme of the invention for the construction of through-layer drilling and the collection of characteristic parameters of a drilling machine.
Fig. 2 is a schematic diagram of a cuttings quantity and cuttings size distribution test protocol of the present invention.
Fig. 3 is a schematic diagram of a gas-containing drill cuttings breaking surface energy consumption testing scheme of the present invention.
Fig. 4 is a schematic diagram of the calculation principle of the ground stress based on the hydrostatic pressure environment and the elasticity theory according to the present invention.
Wherein:
the device comprises a coal stratum 1 to be tested, a drill rod 2, a roadway 3, a drilling machine torque and rotation speed monitoring device 4, a drilling machine 5, drill cuttings 6, a screening system 7, a weighing system 8, a gas sealing device 9, a drop hammer 10, drill cuttings 11 with the particle size larger than d1, a drop hammer protection cylinder 12, a gas pressure gauge 13, a vacuum pump 14 and a gas source 15.
Detailed Description
In order to better understand the technical solution of the present invention, the following detailed description will be made with reference to the accompanying drawings. It should be understood that the following embodiments are not intended to limit the embodiments of the present invention, but are merely examples of embodiments that may be employed by the present invention. It should be noted that, the description herein of the positional relationship of the components, such as the component a being located above the component B, is based on the description of the relative positions of the components in the drawings, and is not intended to limit the actual positional relationship of the components.
Examples
Referring to fig. 1-4, fig. 1 depicts a schematic diagram of the embodiment of the through-the-layer drilling construction and the characteristic parameter collection scheme of the drilling machine. As shown in the figure, in the soft rock ground stress test method based on drill cuttings granularity distribution, in underground engineering, in order to avoid the influence of a stress change area near a rock wall on a measurement result, a ground stress test drill hole is a through-layer drill hole, preferably an upward hole, and the test is divided into 3 parts: the drilling machine data monitoring and energy calculation, drill cuttings data monitoring and crushing energy calculation and ground stress back calculation specifically comprise the following steps:
step one, as shown in fig. 1, a drilling machine 5 is arranged in a roadway 3, a drilling machine torque and rotation speed monitoring device 4 is arranged on the drilling machine 5, a drill rod 2 of the drilling machine 5 penetrates into a tested coal rock layer 1 upwards, an upward penetrating drill hole with the diameter phi (unit m) is constructed in the roadway, and when the drill hole enters the tested rock layer, drill cuttings 6 begin to be collected;
step two, collecting torque T (unit N ∙ m) and rotating speed N of the drill rod in real time by utilizing a sensor r (r/min), the effective power of the drilling machine in the drilling process is P=TN r 9.549 (unit W);
step three, calculating the input energy W of the drilling machine in the rock stratum to be tested 1 (unit J),wherein t is the drilling time of the drilling machine in the rock stratum to be tested;
referring to fig. 2, a screening weighing device is used to obtain the particle size distribution of the rock stratum drilling cuttings to be tested, the screening weighing device comprises a screening system 7 and a weighing system 8, the screening system 7 can be a multi-stage screening device, and the weighing system 8 can be an electronic balance;
placing rock stratum drilling cuttings to be tested into a multi-stage screening device, after screening is completed, sequentially obtaining the mass of drilling cuttings on each screen by using an electronic balance, and recording the diameter of the screen hole of the nth screen as d n The mass of the drilling cuttings on the sieve is M n (kg unit), the outer surface area of the drill cuttings on the nth screen is S n = 12M n /(ρd n +ρd n-1 ) Is the apparent density (unit kg/m of drill cuttings 3 ) When n is 1, d n-1 =2d 1 ；
Step five: calculating the external surface area of the drill cuttings on all the screens, and accumulating to obtain the total external surface area S=S of the drill cuttings 1 +S 2 +∙∙∙S n ；
Step six: taking all drill cuttings on a first screen and putting the drill cuttings into a crushing device; referring to fig. 3, the crushing device comprises a gas sealing device 9, a drop hammer 10 and a drop hammer protection cylinder 12, and the particle size is larger than d 1 The drill cuttings 11 of (1) are placed at the bottom of a drop hammer protection barrel 12, a liftable drop hammer 10 is arranged in the drop hammer protection barrel 12, a gas sealing device 9 is arranged on the top surface of the drop hammer protection barrel 12, the top of the drop hammer protection barrel 12 is connected with a gas source 15, and the drop hammer protection barrel 12 is also respectively connected with a gas pressure gauge 13 and a vacuum pump 14.
Step seven: checking the gas tightness of the device, starting a vacuum pump, vacuumizing for 24 hours, and then filling high-pressure gas into the crushing device, wherein the gas pressure is equal to the in-situ gas pressure of the coal seam, so that the gas is adsorbed in a constant temperature chamber for 24 hours, and timely supplementing after the pressure is reduced during the period, so that the gas pressure is ensured to be slightly higher than the in-situ gas pressure of the coal seam all the time;
step eight: impact is carried out on drill cuttings containing gas by using a drop hammer breaking device, and the process refers to a common firmness coefficient measuring process, namely: a drop hammer with the weight of 2.4kg freely drops from the height of 0.6m to strike drill cuttings in the base, the drop hammer is repeated for 3-5 times, and the drop hammer frequency is recorded as a;
step nine: calculating total energy E=14.1a (unit J) input by the drop hammer, putting the drill cuttings fragments in the base into the screening device in fig. 2, and repeating the fourth and fifth steps to obtain the external surface area S' of the drill cuttings after the drop hammer is broken;
step ten: calculating to obtain the breaking surface energy consumption gamma=E/S' (unit J/m) of the tested soft rock stratum under the drop hammer condition 2 ) The crushing surface energy consumption under the drilling condition and the surface energy consumption under the drop hammer condition are in a linear relation, and when the ratio is recorded as xi, the crushing surface energy consumption under the drilling condition is gamma=xi E/S', and in order to improve the accuracy of a test result, experimental test verification is preferably carried out on different rock formations to be tested;
step eleven: during calculation of the drilling process, rock is brokenTotal energy consumed W 2 =γS；
Step twelve: calculating the rock breaking energy W provided by ground stress in the drilling process 3 =W 2 -W 1 ；
Step thirteen: dividing the crushing energy provided by the ground stress into two parts, namely the deformation energy W of the drilled rock fragments under the in-situ condition 3，2 And the work W performed by the stratum on the drilled rock debris 3，1 Assuming the borehole as a plane strain condition, as shown in fig. 4, there are: the work done by the infinite hydrostatic pressure environment stratum on the drilled hole is W 3，1 =2π(1+ν)p 2 Φ 2 L/(3E Y ) The in-situ deformation energy of the rock mass drilled in the drill hole is W 3，2 =3π(1-2ν)p 2 Φ 2 L/(8E Y ) Wherein E is Y V is Young's modulus (unit Pa) and Poisson's ratio of the rock stratum to be measured, p is in-situ stress (unit Pa) of the rock stratum to be measured, and L is length (unit m) of the drill hole in the rock stratum to be measured;
step fourteen: according to W 3，1 +W 3，2 =W 2 -W 1 And (3) combining the formulas in the steps to obtain a ground stress calculation method of the rock stratum to be tested:
。
in this embodiment, there is a precondition for application: w (W) 1 Cannot be far greater than W 3 I.e. the drill energy should be of the same order as the ground stress energy, or the drill energy should be less than the ground stress energy, to avoid W 2 ≈W 1 (this can lead to a large range of ground stress energy errors or even negative values), so that the drilling rig power should be reduced as much as possible while ensuring drillability when measuring ground stress using the above-described method, or the method should be applied in deep formations.
In this embodiment, when a plurality of boreholes are constructed on the same tested rock stratum to measure the ground stress, the steps six to ten are not required to be repeated for a plurality of times, and when the physical property of the rock stratum is ensured to be not greatly changed (the tested rock stratum in the same geological unit), the surface energy consumption measurement is only required to be carried out once.
In this embodiment, when a plurality of boreholes are constructed to measure the ground stress, the distance between the boreholes should be ensured to be sufficiently large (more than 2 times the width of the borehole pressure relief area can be obtained based on numerical simulation) so as to avoid the influence of the borehole construction on the ground stress measurement result of the adjacent boreholes.
In the embodiment, the gas has important influence on the mechanical behavior and crushing property of the coal body, see 'coal mechanics', and in order to further exclude the influence of the gas on the calculation result of the ground stress, the ground stress energy W is calculated 3 The Young's modulus and Poisson's ratio of the gas-containing coal rock mass should be used.
Working principle:
soft rock has strong viscoelasticity, the structure of the soft rock continuously flows in a long geological process until the soft rock is in an equilibrium state, the in-situ stress approaches to the hydrostatic pressure state, and many students default the in-situ stress of a coal bed to be the hydrostatic pressure, which is described in the outburst of coal and gas. Therefore, in-situ stress measurement of soft rock does not need to accurately represent three-dimensional stress, and only reasonable hydrostatic pressure needs to be estimated. In the rock crushing engineering, the particle size distribution generated after rock crushing is related to input energy, and the crushing energy consumption of the rock is in direct proportion to the newly increased surface area of the fragments. According to the existing empirical model, the energy consumed in the crushing process can be obtained through calculation according to the particle size distribution after rock crushing.
During the drilling process, the breaking energy of the rock is derived from the sum of the in-situ deformation energy provided by the ground stress and the input energy of the drilling machine, namely: rock breaking energy consumption = ground stress energy + drill energy. Obviously, for the same rig (assuming an approximate fixed amount of rig energy), the larger the cuttings amount, and the more energy is expended in rock breaking, from which it can be inferred that the ground stress is greater there.
More precisely, if the torque and rotational speed of the drilling machine can be monitored in real time, the drill energy can be further obtained, and the rock breaking energy consumption can be obtained from the drill cuttings amount and the drill cuttings particle size distribution, the ground stress energy will be accurately calculated from the aforementioned analysis. Under hydrostatic pressure, the relation between the ground stress energy and the ground stress can be analyzed by adopting elastic mechanics, and the ground stress value of the soft rock can be obtained by combining the characteristic data of the drilling machine and the drilling cuttings with a theoretical calculation model.
The underground coal mine is provided with a large number of layer-transmitting drilling holes which are only used for gas pressure measurement, extraction and verification, and if the data of the drilling holes are utilized, the ground stress distribution of the whole production space is obtained, so that the method has great significance for the safe production of the coal mine.
The foregoing is merely a specific application example of the present invention, and the protection scope of the present invention is not limited in any way. All technical schemes formed by equivalent transformation or equivalent substitution fall within the protection scope of the invention.
Claims (7)
1. The soft rock ground stress testing method based on the drill cuttings particle size distribution is characterized by comprising the following steps of:
firstly, constructing an upward through-layer drilling hole with the diameter phi in a roadway, wherein the unit of phi is m, and when the drilling hole enters a coal stratum to be tested, collecting drill cuttings;
step two, collecting torque T and rotation speed N of the drill rod in real time by utilizing a sensor r Units of T N ∙ m, N r The effective power in the drilling process of the drilling machine is P=TN r 9.549, P being in W;
step three, calculating the input energy W of the drilling machine in the rock stratum to be tested 1 ，W 1 Is represented by the unit J of (2),wherein t is the drilling time of the drilling machine in the rock stratum to be tested;
step four, obtaining the particle size distribution of the rock stratum drilling cuttings to be tested by adopting a screening weighing device, putting the rock stratum drilling cuttings to be tested into a multi-stage screening device, after screening is finished, sequentially obtaining the quality of the drilling cuttings on each screen by using an electronic balance, and recording the diameter of the screen hole of the nth screen as d n The mass of the drilling cuttings on the sieve is M n ，M n In kg, the outer surface area of the drill cuttings on the nth screen is S n = 12M n /(ρd n +ρd n-1 ) The apparent density of drill cuttings is expressed in kg/m 3 When n is 1, d n-1 =2d 1 ；
Step five: calculating the external surface area of the drill cuttings on all the screens, and accumulating to obtain the total external surface area S=S of the drill cuttings 1 +S 2 +∙∙∙S n ；
Step six: taking all drill cuttings on a first screen, and putting the drill cuttings into a drop hammer crushing device;
step seven: checking the gas tightness of the device, starting a vacuum pump, vacuumizing for 24 hours, and then filling high-pressure gas into the crushing device, wherein the gas pressure is equal to the in-situ gas pressure of the coal seam, so that the gas is adsorbed in a constant temperature chamber for 24 hours, and timely supplementing after the pressure is reduced during the period, so that the gas pressure is ensured to be slightly higher than the in-situ gas pressure of the coal seam all the time;
step eight: the method comprises the steps of (1) impacting drill cuttings containing gas by using a drop hammer breaking device, wherein in the process of measuring a common firmness coefficient, a drop hammer with a weight m freely drops from a height h to impact the drill cuttings in a base, repeating the drop hammer for 3-5 times, and recording the drop hammer times as a;
step nine: calculating the total energy E=mgha, unit J of E, putting the drill cuttings fragments in the base into a screening device, and repeating the fourth and fifth steps to obtain the external surface area S' of the drill cuttings after breaking by the drop hammer;
step ten: calculating to obtain the breaking surface energy consumption gamma=E/S' gamma of the tested soft rock stratum under the drop hammer condition 2 The energy consumption of the crushing surface under the drilling condition and the surface energy consumption under the drop hammer condition are in a linear relation, and the ratio is zeta, so that the energy consumption of the crushing surface under the drilling condition is gamma=zeta E/S';
step eleven: calculating the total energy W consumed by rock breaking during the drilling process 2 =γS；
Step twelve: calculating the rock breaking energy W provided by ground stress in the drilling process 3 =W 2 -W 1 ；
Step thirteen: dividing the crushing energy provided by the ground stress into two parts, namely the deformation energy W of the drilled rock fragments under the in-situ condition 3，2 And the work W performed by the stratum on the drilled rock debris 3，1 Assuming the borehole as a plane strain condition, wherein the work done by the infinite hydrostatic pressure environment stratum on the borehole is W 3，1 =2π(1+ν)p 2 Φ 2 L/(3E Y ) The in-situ deformation energy of the rock mass drilled in the drill hole is W 3，2 =3π(1-2ν)p 2 Φ 2 L/(8E Y ) Wherein E is Y And v is Young's modulus and Poisson's ratio of the rock stratum to be tested, p is in-situ stress of the rock stratum to be tested, E Y The units of v, p are Pa, L is the length of a borehole in the rock stratum to be tested, and the unit of L is m;
step fourteen: according to W 3，1 +W 3，2 =W 2 -W 1 And (3) combining the formulas in the steps to obtain a ground stress calculation method of the rock stratum to be tested:
。
2. the soft rock crustal stress test method based on drill cuttings particle size distribution according to claim 1, wherein: in the first step, a drilling machine is arranged in the roadway, a drilling machine torque and rotation speed monitoring device is arranged on the drilling machine, and a drilling rod of the drilling machine penetrates upwards into a coal rock stratum to be tested.
3. The soft rock crustal stress test method based on drill cuttings particle size distribution according to claim 1, wherein: and step three, the screening and weighing device comprises a screening system and a weighing system, wherein the screening system adopts a multi-stage screening device, and the weighing system adopts an electronic balance.
4. The soft rock crustal stress test method based on drill cuttings particle size distribution according to claim 1, wherein: in the sixth step, the drop hammer breaking device comprises a gas sealing device, a drop hammer and a drop hammer protection cylinder, and the particle size is larger than d 1 The drill cuttings of the drill are placed at the bottom of a drop hammer protection cylinder, a lifting drop hammer is arranged in the drop hammer protection cylinder, and the top surface of the drop hammer protection cylinderThe top of the drop hammer protection cylinder is connected with a gas source.
5. The soft rock crustal stress testing method based on the particle size distribution of drill cuttings according to claim 4, wherein: in the sixth step, the drop hammer protection cylinder is also respectively connected with a gas pressure gauge and a vacuum pump.
6. The soft rock crustal stress test method based on drill cuttings particle size distribution according to claim 1, wherein: and step eight, freely falling a drop hammer with the weight of 2.4kg from the height of 0.6m, impacting drill cuttings in the base, repeating the drop hammer for 3-5 times, and recording the drop hammer times as a.
7. The soft rock crustal stress test method based on drill cuttings particle size distribution according to claim 1, wherein: in step ten, in order to improve the precision of the test result, experimental tests and verification are carried out on different rock strata, wherein xi is independent.
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|---|---|---|---|---|
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