ISRM Suggested Method for Uniaxial-Strain Compressibility Testing for Reservoir Geomechanics

Appendix

This appendix provides one suggested method for determining the pressure and differential load effects on the deformation and load measurements of the apparatus. This (or any other) calibration procedure should always be followed by a verification procedure that measures the deformation of a standard specimen of known properties and verifying that the measured and corrected deformation matches that predicted from the known standard elastic constants.

One technique to calculate the confining pressure correction is to mount all in-vessel instrumentation to a specimen with known elastic properties (such as aluminum 6061-T6) and to vary hydrostatic stress (see Fig. 1a) to characterize the excess strain response of the system. In order to calculate excess system strain (as in the specimen platens), a differential (i.e., axial minus confining) stress correction can be calculated using a similar specimen assembly and varying the differential stress (see Fig. 1b).

Once all corrections are characterized, they can be integrated into the signal conditioning or control and data acquisition software to provide real-time correction for all sensors, which is necessary to provide accurate control, especially to maintain uniaxial-strain conditions. A suggested transducer-corrected output “y” can be written as follows:

$$y = mx + b + \beta P_{\text{c}} + \gamma \sigma_{\text{d}} ,$$

(61)

where mx + b is the transducer output, calibrated at ambient pressure and temperature,

• $+ \beta P_{\text{c}}$ is the confining-pressure correction coefficient β and P _c the confining pressure (one possible method for calculating the coefficient for deformation sensors is described in "Calculation of Pressure-Correction Coefficient for Deformation" section and for internal load sensors in "Calculation of Pressure-Correction Coefficient for an Internal Load Sensor" section), and

• $+ \gamma \sigma_{\text{d}}$ is the differential stress correction coefficient γ and σ _d the differential stress (one possible method for calculating the coefficient is described in "Calculation of Differential Stress-Correction Coefficient for Platen Deformation" section).

• y is the corrected net deformation (=change in distance between two reference points) or load.

To verify accuracy of all instrumentation calibration and characterized corrections, a specimen of known material properties (aluminum 6061-T6 or similar) can be loaded using non-simultaneous hydrostatic- and differential-loading cycles, and the results compared to the known material properties (see Figs. 10, 11, for example, results from an aluminum 6061-T6 billet). An acceptable tolerance for such a comparison would normally be ±5 % on Young’s and Bulk moduli and ±10 % on Poisson’s ratio. An additional verification can be to run a uniaxial loading path on an aluminum specimen in order to check radial control.

Fig. 10

Measured Young’s modulus (left) and Poisson’s ratio (right) of an aluminum 6061-T6 billet during a differential loading cycle. (The results are within about 2 % of expected values after incorporation of sensor differential load corrections)

Fig. 11

Measured bulk modulus of an aluminum 6061-T6 billet during a hydrostatic loading cycle. (The results are within about 2 % of expected value after incorporation of sensor pressure correction)

1.1 Calculation of Pressure-Correction Coefficient for Deformation

To calculate the pressure effects, a hydrostatic test is run on a reference billet at stress levels similar to the rock specimens to be tested. The strains (axial, radial #1, radial #2 in this example) are plotted on the y-axis against the confining pressure on the x-axis. The slopes of these lines will be calculated and corrected so that they are equal to the theoretical slope of the reference billet material of known Young’s modulus E and Poisson’s ratio ν. Those corrections are then converted back to the calibrated deformation units. The theoretical slope of the material can be calculated as follows:

$$\epsilon_{\text{vol}} = \epsilon_{\text{a}} + \epsilon_{r1} + \epsilon_{r2}$$

(62)

$$K = \frac{{\mathop \sum \nolimits_{1}^{3} \sigma_{ii}}}{{3\epsilon_{\text{vol}}}}$$

(63)

$$C = \frac{{d\epsilon_{\text{vol}}}}{{dP_{\text{c}}}} = \frac{1}{K}$$

(64)

$$\text{Theoretical slope} = \frac{{\epsilon_{\text{vol}}}}{{\mathop \sum \nolimits_{1}^{3} \sigma_{ii}}} = \frac{1}{3}C$$

(65)

where K is the bulk modulus, σ _ii is the principal stress, ϵ _vol is the volumetric strain, ϵ _a is the axial strain, ϵ _r1 is the radial #1 strain, ϵ _r2 is the radial #2 strain, P _c is the confining pressure, and C is the bulk compressibility.

The theoretical slope as defined above is the slope of an individual strain. Since C is obtained from the volumetric strain, which is the axial strain (ϵ _a) + the two radial strains (ϵ _r1, ϵ _r2), C must be divided by 3 to obtain the slope for an individual strain (assuming isotropic material). It is necessary to obtain a second-order, least-squares polynomial-curve fit for each strain versus the confining pressure. The curve fits for the measured and theoretical lines are given by the following:

$${\text{Measured strain}} = b_{1} P_{\text{c}} + b_{2} P_{\text{c}}^{2}$$

(66)

$${\text{Theoretical strain}} = c_{1} P_{\text{c}} + c_{2} P_{\text{c}}^{2}$$

(67)

If the reference billet is a linear elastic material (like aluminum 6061-T6), the theoretical curve is linear (c ₂ = 0). The correction coefficients, $a_{1}$ and a ₂, are obtained as follows:

$$a_{1} = c_{1} - b_{1}$$

(68)

$$a_{2} = c_{2} - b_{2}$$

(69)

There should now be two correction coefficients for each of the strain channels. These correction coefficients can now be applied to the test data. They are applied as follows:

$${\text{Corrected strain = uncorrected}}\;{\text{strain + pressure effects}}$$

(70)

$${\text{Corrected}}\;{\text{strain }} = Y + a_{2} P_{\text{c}}^{2} + a_{1} P_{\text{c}}$$

(71)

where a ₂ is the second-order correction coefficient, a ₁ is the first-order correction coefficient, and Y is the uncorrected measured strain.

The corrected strain is then converted back to the deformation using the appropriate reference billet dimension. For example, the pressure corrected net axial deformation would be

$$y = mx + b + \beta P_{\text{c}} = YL_{0} + \left( {a_{1} P_{\text{c}} + a_{2} P_{\text{c}}^{2} } \right)L_{0} ,$$

(72)

where L ₀ is the reference billet length, βP _c is (in this example) a second-order pressure correction: βP _c = (a ₁ P _c + a ₂ P ²_c )L ₀.

The radial-deformation correction is done in an analogous way, using the diameter instead of the length.

Changes to the jacketing material, mounting platens, or mounting positions of individual sensors may alter the correction coefficients. It is necessary to keep the specimen-assembly geometry consistent with the geometry of the reference billet test. Temperature may also alter the correction coefficients, so the billet test needs to be done at the approximate (±5 °C) test temperature.

1.2 Calculation of Pressure-Correction Coefficient for an Internal Load Sensor

Internal load sensors are preferred to eliminate issues with correcting external load measurements for friction between the loading ram and its associated seals in the pressure vessel. Such friction corrections are hysteretic with ram movement direction and normally more difficult to quantify and reliably correct for. Typical internal differential load sensors, however, normally have a small, linear pressure effect that needs to be corrected for. This is easily accomplished by cycling the confining pressure in the vessel with no axial load applied (i.e., with a gap in the loading stack) and could be done simultaneously with the deformation sensor pressure-correction coefficient determination. The negative of the slope of the measured load versus pressure provides the added load sensor pressure-correction coefficient:

$$Ld_{\text{corr}} = \, Ld_{\text{meas}} + \, a_{\text{Ld}} P_{\text{c,}}$$

(73)

where Ld _corr is the corrected loading force, Ld _meas is the measured loading force, and a _Ld P _c is a first-order pressure correction.

Figure 12 provides an example internal load cell pressure-correction determination.

Fig. 12

Example internal differential load sensor pressure-correction determination of a _Ld = 0.025 kN/MPa (0.039 lbf/psi)

1.3 Calculation of Differential Stress-Correction Coefficient for Platen Deformation

A typical specimen stack may include both the specimen and mounting platens between the measuring devices for axial deformation. Therefore, the axial deformation measured by the sensors is measuring the deformation of the specimen as well as the deformation of the mounting platens (and any drainage or flow distribution disks, if present). To obtain the true axial deformation of the specimen, the deformation of the mounting platens (and drainage disks) must be subtracted out of the sensor measurement. Here we describe a method for determining this deformation correction.

During testing, there are three unknowns: the deformation from the strain of the specimen, the “closure” of the specimen-platen interfaces, and the deformation from the strain of the mounting platens (we assume no drainage disks here). The deformation of the specimen will vary from test to test depending on the elastic properties of the specimen, and the interface closure should be the same for similar specimen-end treatments and should be negligible above a certain stress level. The effective modulus of the mounting platens is the same from test to test, provided the material and geometry of the mounting platens remain the same. Therefore, to determine the effective modulus of the mounting platens and therefore the platen strain, an unconfined compression test is run on a reference billet with a known Young’s Modulus (Aluminum 6061-T6 recommended) at stress levels similar to the specimens to be tested. The true reduction in the specimen length is given by:

$$\Delta L_{\text{s}} =\Delta L_{\text{m}} {-} \Delta L_{\text{mp}} ,$$

(74)

where $\Delta L_{\text{s}}$ is the specimen-length reduction, $\Delta L_{\text{m}}$ is the measured-length reduction, and $\Delta L_{\text{mp}}$ is the mounting-platens length reduction.

Hooke’s law states that

$$E = \frac{\sigma}{\epsilon}$$

(75)

Therefore, the length reduction in a specimen is as follows:

$$\Delta L_{\text{s}} = \frac{\sigma }{E}*\left( {\text{length of the sample}} \right)$$

(76)

The length reduction in the reference billet is given as:

$$\frac{\sigma }{E}*L_{0} = \Delta L_{\text{m}} - \frac{\sigma }{{E_{\text{mp}} }}*l_{\text{mp}} ,$$

(77)

where σ is the axial stress difference, E is the Young’s Modulus of the reference billet (e.g., Al 6061-T6), L ₀ is the length of the reference billet, E _mp is the effective mounting-platen modulus, and l _mp is the length of the mounting platens within the measurement device.

In the above equation, all variables are known except E _mp. The axial deformation and σ are recorded during testing, the different lengths are measured, and E for aluminum 6061-T6 or other reference billet can be found in most metals handbooks (e.g., Davis 1998).

To obtain E _mp, it is necessary to find the slope of the line of the stress–strain curve of the mounting platens. This is given by

$$\epsilon_{\text{mp}} = \frac{{\Delta L_{\text{m}} - \left( {\frac{\sigma }{E}*L_{0} } \right)}}{{l_{\text{mp}} }}$$

(78)

where ϵ _mp is the strain of the mounting platens. This equation shows how to determine the strain in the mounting platens. To obtain the effective mounting-platens modulus, a linear regression must be carried out with the differential stress on the y-axis and the mounting-platens strain on the x-axis. The slope should be determined for all the compression cycles. The effective mounting-platen modulus is the average of these slopes. It is important to remember that the value will vary depending on the material and geometry of the mounting platens.

Once the effective mounting platen’s modulus is known, we can return to the equation for true specimen deformation:

$$\Delta L_{\text{s}} = \Delta L_{\text{m}} - \Delta L_{\text{mp}}$$

(79)

$$\Delta L_{\text{s}} = \Delta L_{\text{m}} - \frac{{\sigma *l_{\text{mp}} }}{{E_{\text{mp}} }}$$

(80)

Therefore, the fully pressure- and load-corrected net deformation measurement would be as follows:

$$y = mx + b + \left( {a_{1} P_{\text{c}} + a_{2} P_{\text{c}}^{2} } \right)L_{0} - \frac{{\sigma *l_{\text{mp}} }}{{E_{\text{mp}} }}$$

(81)

Again, changes to the jacketing material, mounting platens, or mounting positions of individual sensors may alter the effective mounting-platen modulus. It is necessary to keep the specimen-assembly geometry consistent with the geometry of the reference billet test.

This correction for axial deformation is linear, and there can sometimes be some remaining nonlinear deformation at low stress (e.g., see Fig. 10). If this amount of nonlinear excess deformation is acceptable, then no further correction is required. If, for the specimen being tested, it is not acceptable, then an additional correction, nonlinear with differential stress, could be considered. However, this additional correction may detrimentally affect the specimen data, depending on actual rock response. Alternatively, the operator should ensure that the deformation (and stress) ranges over which the specimens will be tested are tightly bound by the load- (and pressure-) effect correction. In addition, the load cell should be chosen such that its sensitivity is appropriate to the anticipated specimen response.