1 INTRODUCTION

There are two types of component-type borehole strainmeters. One is the three-component strainmeter developed abroad (USA, Japan and Australia), in which the 3 measurement components are arranged with a 60° angle between each other. The other is the four-component strainmeter developed in China, whose 4 measurement components have a 45° angle to each other, like the Chinese character “米”. The “Plate Boundary Observation (PBO)”^[1] which was started from 2003 in America planned to establish 175 borehole strain stations along the San Andreas Fault and Southern Alaska at American west coast. Up to now, 85 new Gladwin component strainmeters^[2] have been installed and put into operation. Affected by the American PBO Plan, research and applications of the Chinese four-component strainmeter have developed rapidly during the Tenth Five-Year period. The YRY-4 strainmeter^[3], which is developed by Chi Shun-Liang, has been applied to key earthquake monitoring areas in China (including more than 40 survey points) and obtained observation data of five consecutive years. The RZB-2 and RZB-3 strainmeters^{[4, 5]} which are developed by Ouyang Zu-Xi, have been used in earthquake observations in Xinjiang, Gansu, Sichuan and Chongqing. Now there are 12 sets of the instruments are running, and there will be much more in the Twelfth Five-Year period. Apparently, component strainmeters are gradually becoming the mainstream instrument in crustal strain field observation in the world^[6]. No matter what type of component strainmeter it is, the working principle remains the same. All the gauges utilize the radial displacement sensor to measure relative change of the borehole diameter in rocks. The senso has to be fixed on the underground bedrock (tens to hundreds of meters deep) with expansive cement (or cement). Therefore, the rock, cement and instrument steel tube form a three-layer composite double-bush structure. Significant coupling effects are inevitable due to concentrated wellhole stress and stress transfer in each medium layer. As can be seen from theoretical formulas^{[7, 8]}, all the linear strain measured by each component has two coupling coefficients (A, B). Compared with the traditional body strainmeter (blind direction), the component strainmeter has a prominent progress in functions. It can determine the maximum principal strain, minimum principal strain and direction of the maximum principal strain of the crustal strain field between two moments based on any measured strain values in three directions. But the prerequisite is to know in advance the two coupling coefficients, otherwise this function does not make any sense. So solving these two coupling coefficients is a crucial basic problem. At present, there are two methods to solve this problem in the world: inversion and forward modeling. For inversion, the method proposed by Hart et al.^[9] utilizes the average earth tide measured by baseline laser strainmeters (1~2 km long) to demarcate the three-component strainmeter; Roeloffs^[2] carried out harmonic analysis on strain observation values to obtain the coupling coefficients. In China, Qiu Z H et al.^[7] used the relationship between observed earth tide and theoretical earth tide to invert the coupling coefficients. The forward modeling belongs to traditional methods; Gladwin et al.^[10] established a dual-ring model to utilize complex variable functions to obtain results. Luo M J et al.^[11] derived a two-layer medium model (rock, steel tube) for solving the coupling coefficient formulas. Different from the above-mentioned work, this paper proposes a new idea. Based on the research of Pan L Z^[12], Ouyang Z X^[13] and Chen Y J^[14], we select a double bush model which is close to the actual situation to derive the final theoretical calculation formulas of the two coupling coefficients. This method has a positive meaning for the borehole strain observational study. It can also combine with calculation results of other methods in terms of reliability analysis to play a greater role in monitoring of earthquakes, volcanoes and plate boundaries.

2 THEORETICAL MODEL OF COMPONENT STRAINMETER OBSERVATION

We assume that there is an infinitely large rock plate, at infinity which withstands two uniform horizontal tensile stress, i.e, the maximum and minimum stress σ₁ and σ₂, with the maximum and minimum strain ε₁ and ε₂. In the plate, there is a wellhole with a radius r₃, in which a strainmeter steel tube with inner-radius r₁ and outer radius r₂ is fixed. The steel tube is fixed on the rock with expansive cement (welding) to ensure the continuity of stress and displacement components on the boundary. Therefore, the borehole has a three-layer composite double-bush structure (Fig.1). Let E₁, μ₁, E₂, μ₂, E₃, and μ₃ be the elastic modulus and Poisson’s ratios of steel tube, cement and rock, respectively. We assume that the medium in which the borehole is located is an isotropic elastic medium, and the effect of wellhead and well bottom on the sensor can be ignored according to Hooke’s law. So the axial stress of the borehole is 0. Let θ be the angle between the north direction and any component measuring the relative change of borehole, $\phi $ be the angle between the north direction and the maximum principal stress. We specify that, starting from the north direction, the value of the angle which is rotated clockwise is positive and the value of the angle which is rotated counterclockwise is negative. And the radial strain on the steel tube wall in the θ direction is^[7]

where A and B are constants which are related with elastic parameters and radius of each material layer of the borehole, collectively known as coupling coefficients (also called plane strain and shear strain sensitivity p coefficient in some paper^[7]). And formula (1) is the theoretical model of component borehole strain observations.

For the four-component borehole strainmeter, starting from the north direction to right, if the angle of the first component is θ, then the angle of the second, third and fourth component will increase 45° successively. That is, components one and three are perpendicular to component two and four, respectively. The strain values of the four components can be written as

A and B are very important constants. The maximum principal strain (ε₁), minimum principal strain (ε₂) and the direction of the maximum principal strain ($\phi $) can be calculated by solving equation set according to measured values of any three components (S₁ ~ S₄) after knowing A and B. Similarly, for the three-component strain gauge, the strain values of the three elements can be written as

3 CALCULATIONS OF COEFFICIENTS 3.1 Derivation of Basic Formulas

Under the effect of plane stress, the linear strain of non-porous rock is

then Substituting (9) and (10) into (1) yields Let Then there is

In 1981, Pan L Z^[12] proposed the principle of measuring crustal stress with the borehole lining method and derived related basic formulas. And he gave a particular solution table for solving elastic plane problems. In 1988, Ouyang Z X and Zhang Z R analyzed the boundary conditions and interface conditions of the double-bush borehole, obtained the particular solutions according to Pan’s particular solution, and proposed three linear equations with unknown constants to be determined and the formula of radial displacement on the inner wall of probe steel tube^[13]. In 1990, Chen Y J et al.^[14] further calculated these unknown constants and derived expressions of radial strain on the inner wall of the steel tube in θ direction:

where σ₁ and σ₂ are the max and min principle strain. M₁ and M₂ are only determined by the measurement system (the rock body, expansion cement and steel tube) itself and have nothing to do with external forces. Therefore, they are called characteristic coefficients of the measurement system. Comparing (14) with (17), we can see that Substituting A₁ and B₂ into (15) and (16), we have Obviously we can determine A and B by solving M₁ and M₂.

3.2 Solving Characteristic Coefficients M₁ and M₂

Based on the work of Pan, Ouyang, Zhang and Chen et al. aforementioned, we know that both M₁ and M₂ are constants related to elastic parameters and radius of layers of material^[14], where there is

where x₄ is determined by the following quaternion linear Eqs.(I): The coefficients are and the solution is Substituting (21) into (20), we can obtain M₁. Then, the coupling coefficients can be solved by substituting M₁ into (18):

It should be noted that the use of thick-walled cylinder equation^[15] can also derive exactly the same formulas mentioned above (this will be further explained in another dedicated essay), which indicates that the research work of Pan et al. is reliable.

Besides, there is^[14]

in which y₅, y₆, y₇ and y₈ can be determined by the following ten-element linear Eqs.(II) where α₁, α₂, α₃ and β₁, β₂, β₃ have the same meanings as mentioned previously, and the other coefficients are The unknowns can be solved by using the method of elimination in which the coefficients are where and and and Substituting (24)-(27) into (23), we can obtain M₂. Then, the coupling coefficients can be solved by substituting M₂ into (19):

Then we obtain the expressions of y₁, y₂, y₃, y₄, y₉ and y₁₀ and substitute them into (II). The results show that the derivation of calculation formula of y₅, y₆, y₇ and y₈ is correct.

4 COMPARISON WITH GLADWIN’S RESULTS 4.1 The Focus of Comparison

In 1981, Pan^[12] had considered using functions of complex variables when establishing the double-bush model, and he thought it was a powerful tool for solving elastic plane problems. However, the polar coordinates solution method was finally selected because some practitioners were not familiar with functions of complex variables which not convenient for them (in fact, this research has begun to take shape early in 1977¹⁾ ( 1) Pan L Z. Derivation and discussion of several formulas about measurement of earth stress. Selected Papers of National Professional Meetings of Earth Stress (first volume), National Professional Meetings of Earth Stress of State Seismological Bureau, 1977: 1-41.)). In 1985, Gladwin et al.^[10] established a double-ring observation model, i.e. the instrument steel tube is the first ring and the expansive cement is the second ring. The principle of this method is basically the same as that of the double-bush model, but it utilizes functions of complex variables to solve the two coefficients. Under the effect of plane stress, the radial strain of the probe steel tube is^[10]

where a and b are two constants, G₁ = E₁/[2(1 + μ₁)] indicates the shear modulus of steel tube (i.e. modulus of elasticity), and the radial stress is Gladwin et al.^[10] gave the expressions of a and b where the known items are There are still 6 unknown items (a₁¹, b_-1¹, a_-1¹, a₃¹, b_-3¹ and b₁¹) which do not have clear calculation expressions. Although Gladwin et al.wrote a program to solve this problem, there was no specific description in their paper^[10].

Gladwin et al.^[10] also proposed conceptions of the area strain response factor c and shear strain response factor d

where G₃ represents the shear modulus of the rock, (29) can be further written as^[10] Compare (30) with (1), if the two equations are equal to each other, then c/2 = A and d/2 = B hold, i.e.

4.2 Physical Meanings of the Response Factor

It is not difficult to derive the following equations based on formulas (2), (3), (4) and (5),

in which ε_m is called the observed area strain of the rock (apparent plane strain), S₁ + S₃ and S₂ + S₄ are two basic fixed values in the plane strain field, which reflect that the two area strain recorded by any two mutually perpendicular measuring components are equal to each other. Therefore, (33) is taken as the self-test condition for the four-component strainmeter. (34) and (35) represent the difference between two tested values measured by two mutually perpendicular measuring components. Its physical essence is shear strain^[16], which is called shear strain or differential strain.

For non-porous rock, assuming that there is a cylinder in rock, the radial strain in the θ direction is^[17]

we have where L_m, L_J1 and L_j2 are area strain and shear strain of the non-porous rock, respectively. By (33)÷(37), (34)÷(38), (35)÷(39), we can obtain the borehole coupling coefficients of borehole area strain and shear strain Obviously, the physical essence of response factors c and d is the borehole coupling coefficient of area strain or shear strain observation, respectively. And A and B are 1/2 of c and d, respectively. The real crustal rock strain (the observed strain value/borehole coupling coefficient) can be determined by solving c and d. And the comparability of the observation data of different stations can be achieved, which is another important use of the research work.

For the three-component strainmeter, (40) and (41) also hold. This is because the area strain ε_m recorded by the strainmeter has nothing to do the number of components, and theoretically^[18]

The area strain of the non-porous rock is Combining with (22) we have According to (6), (7) and (8), using similar methods mentioned above can also prove that the three shear strain borehole coupling coefficients are equal to 2B.

4.3 Graphic Correlation of the Two Results 4.3.1 2A and c

Gladwin^[10] provided the relation curve of area strain response factor (c) and shear modulus ratio (Fig.2a). According to his idea, if we set G2/G1 = G21, G3/G1 = G31, substituting E_i = 2(1 + μ_i)G_i(i = 1, 2, 3) into formulas (21) and (22), and taking^[8] G₁ = 8.39×1010 Pa, μ₁ = 0.283, μ₃ = 0.25, r₁=56 mm, r₂=62 mm, r₃=89 mm, then we have

where x₄ = P₁/(P₂ + P₃), and

Mimicing Fig.2a, we make the relation curve (Fig.2b) of 2A and shear modulus ratio (G₂/G₁) and keep them in the same scale. Then we can use drawing software in Windows to make transparency processing and overlap (a) and (b). It can be seen that the two graphical curves are almost completely consistent with each other, fully indicating that 2A = c holds.

4.3.2 2B and d

Gladwin^[10] also provided the relation curve of shear strain response factor (d) and shear modulus ratio (Fig.3a). According to his idea, (28) can be transformed into

The coefficients that need to be transformed in the process of solving y₅, y₆, y₇ and y₈ is

Similarly, mimicing Fig.3a, we make the relation curve (Fig.3b) of 2B and shear modulus ratio (G₂/G₁) and keep them in the same scale. Then we can use drawing software in Windows to make transparency processing and overlap (a) and (b). It can be seen that the two graphical curves are almost completely consistent with each other, fully indicating that 2B = d holds.

The above analysis shows that the present results are identical with Gladwin’s results. The correctness of the two methods is thereby proved by each other, the formulas derived in this paper can obtain the same graphics as that of Gladwins et al.^[10].

5 CONCLUSIONS AND DISCUSSION

(1) With plane stress, the two coupling coefficients of component borehole strain observation can be respectively expressed as

where x4, y₅, y₆, y₇ and y₈ are constants related to elastic modulus, Poisson’s ratio and the radius of layers of material in the measuring system and have very complicated function relations, 2A and 2B are actually c (area strain response coefficient) and d (shear strain response coefficient), respectively provided by Gladwin^[10], of which the physical essences are borehole coupling coefficients of area strain and shear strain observation. The calculation results in this paper are the same as that of Gladwin et al.^[10], but the research method and idea are completely different. The calculation formulas of A and B are common for various types of strain gauges and irrelevant with the number of components of the sensor. If the types of strainmeter and borehole are fixed, related r₁, r₂ and r₃ are known, the elastic modulus (E₁, E₂) and Poisson’s coefficients (μ₁, μ₂) of the probe steel tube and cement can be obtained in the lab tests (generally provided by instrument development companies), what users need to measure are elastic modulus (E₃) and Poisson’s coefficients (μ₃) of the rock at the bottom of the borehole. Therefore, borehole rock core samples need to be collected at stations. A and B can be calculated after knowing the 9 parameters. Because the calculation of x₄, y₅, y₆, y₇ and y₈ is very complex and cumbersome, the authors suggest writing a program to calculate these parameters based on the idea of this paper, which is convenient and less error-prone. Readers can also transform parameters according to their own requirements and do some further theoretical research.

(2) The change characteristic of the coupling coefficient A is: when G₂/G₁ < 0.1, A increases rapidly with the increase of G₂/G₁ and decreases slowly with the decrease of G₂/G₁. The change characteristic of the coupling coefficient B is: when G₂/G₁ < 0.01, B increases rapidly with the minor increase of G₂/G₁ and decreases with the decrease of G₂/G₁, and when 0.01 < G₂/G₁ < 0.2, it decreases rapidly with the decrease of G₂/G₁; but when G₂/G₁ > 0.2, it decreases slowly with the decrease of G₂/G₁. The common characteristics of A and B are: when G₂/G₁ is a fixed value, A and B increase with the increase of G₃/G₁; and when G₂/G₁, G₃/G₁, μ₁ and μ₃ are fixed values, the change of μ₂ has very small impact on both of them.

(3) It should be noted that for a real nonporous rock, A = B = 1/2; for rock with pores, A = 1/(1-μ₃), B = 2/(1+μ₃), if we take μ₃ = 0.25, then A = 1.333, B = 1.600. For borehole strain observation systems, generally A and B are not exactly 1/2 and not simultaneously 1.333 and 1.600. Therefore the radial strain on the inner wall of steel tube is not equal to the linear strain of non-porous rock in the same direction, nor the radial strain on the inner wall of rock with pores in the same direction. And the recorded earth tide and theoretical strain earth tide along the same line are not a paired comparison relationship.

(4) How to measure and calculate the two parameters A and B indoors is the task that needs to be carried out immediately. Elasticity modulus of rock and Poisson’s ratio are indispensable for theoretical calculations. Therefore, we need to carry out sampling tests of rock cores around the sensor, and this sample work is very important foundation work. However, most stations have not done this yet. As time goes some stations will lose some information of cores and their information storage will be incomplete. So from now on we must establish this kind of projects to promote the foundation work.

ACKNOWLEDGMENTS

We are grateful to Su K Z and Qiu Z H for their constructive support and help in the research, and three anonymous reviewers for their pertinent amendment suggestions. This work was supported by the National Science and Technology Support Planning Program (2012BAK19B02-02) and Seismic Industry Special Research (201108009).