The pressurepulse decay is a preferred technique for determining permeability of unconventional gas reservoir rocks. The pressurepulse decay often shows quite different characteristics during the early time and the later time. Most approaches for estimating the permeability proposed in the literature are required to use the latertime pressurepulse decay measurements. However, the latertime data are often selected subjectively, lacking a universal criterion. In this paper, a method of differentiating the earlytime and latetime behavior for pressurepulse decay test is proposed. The analytical results show that the critical time (dimensionless time) of early/latetime decay characteristics mainly depends on the volume ratios, and it increases first and then decreases with the volume ratios. The critical time for cases with same chamber sizes is much less than that for cases with unequal chamber sizes. Applicability of the proposed methods is examined using a numerical simulator, TOUGH+REALGASBRINE. The numerical results show that the pressure gradient along the sample varies nonlinearly at the early time and becomes a constant at the late time. Then, the proposed method is applied to real data for permeability estimation. It is found that the earlytime behavior is negligible as the volume ratio takes on small values. As the volume ratios increase, the deviation becomes significant and considerable permeability errors will be produced if these earlytime data are used.
Permeability is typically considered the critical parameter for commercial gas production [
The governing equation for fluid flow in the core sample using the pressurepulse decay method can be specified as follows [
Brace et al. [
The general analytical solution of equation (
was presented by Dicker and Smits [
The analytical solution of equation (
Jones [
Therefore, the latetime solution for predicting permeability was then given as
Brace’s solution of equation (
Dicker and Smits [
A comparison of the general solution (equation (
Figure
The first six terms of the series in equation (
Figure
The evolution of critical time
The accuracy in permeability measurement mainly depends on the slope of latetime solution. To investigate the influence of contribution ratio
Error coefficient
Volume ratios  90%  95%  99%  99.9%  99.99%  

Slope 

Slope 

Slope 

Slope 

Slope 
 

0.020  0  0.020  0  0.020  0  0.020  0  0.020  0 

0.197  0  0.1967  0  0.1967  0  0.197  0  0.197  0 

1.710  0.185  1.709  0.089  1.707  0.021  1.707  0.001  1.707  0.001 

7.672  10.645  7.361  6.165  7.055  1.745  6.950  0.228  6.937  0.041 
Different chamber sizes between upstream and downstream have been chosen by some scholars to have a speedup of the measurement [
Critical time of cases with unequal chamber sizes.
A numerical simulator, TOUGH+REALGASBRINE (TOUGH+), is employed to simulate pressurepulse decay tests. TOUGH+ is based on a finite volume approach and is a successor to the TOUGH2 family of codes for fluid and heat flow [
Modeling parameters of numerical tests.
Parameters  Case 1 ( 
Case 2 ( 
Case 3 ( 
Case 4 ( 

Chamber volume (cm^{3})  196  19.6  19.6  1.96 
Core length (m)  0.1  0.1  0.1  0.1 
Core diameter (m)  0.05  0.05  0.05  0.05 
Pore pressure (MPa)  1  1  1  1 
Pulse pressure (MPa)  0.1  0.1  0.1  0.1 
Porosity  0.01  0.01  0.1  0.1 
Figure
Pressure responses along the length of the sample at different real times: (a) case 1, (b) case 2, (c) case 3, and (d) case 4.
In this section, we applied the method of differentiating the earlytime and latetime behavior into laboratory permeability tests. Previous studies [
Experimental setup for pulsedecay permeability measurements. 1: gas cylinder; 2: air booster pump; 3: gas reservoir; 4: pressure reducing valve; 5: vacuum pump; 6: upstream chamber; 7: triaxial cell; 8: rock sample; 9: differential pressure transducer; 10: downstream chamber; 11: constant temperature oven; 12: computer; 13: data acquisition system; 14: hydraulic water pump.
Experimental schemes of test 1 and test 2.
Test 1  Test 2  

Rock type  Shale  Rocklike material  
Sample size 

 
Testing condition  Constant effective stress  Varied effective stress  
Volume ratio 

0.059  1.32~1.60 

0.069  1.55~1.88 
TSSR system is composed of five main units: triaxial cell, loading unit, temperature control unit, vacuum unit, and data collection unit. The experimental setup (shown in Figure
Shale and rocklike sample, with significant difference in pore volume, were selected for experimental study. The rocklike sample, with dimensions of 50 mm in diameter and 100 mm in length, was made of a mixture of ordinary Portland cement, sand, and water at a ratio of 1 : 1.2 : 0.65 [
Blocks of shale sample were first extracted from a quarry located in Liutang in Shizhu Country, China. Then, a cylindrical core sample with dimensions of 50 mm in diameter was sampled from the shale block. The length of the shale sample was cut to 33.5 mm for two purposes. For one thing, the shorter the length is, the smaller the pore volume will be. For another, a short length helps to reduce measurement time since shale has ultralow permeability. The density of the shale sample is 2.55 g/cm^{3}, and the porosity is 2.0%. Figures
Rocklike sample: (a) vertical section, (b) cross section, and (c) image from CT scanning.
Shale sample: (a) vertical section, (b) cross section, and (c) image from SEM scanning.
The permeability tests were carried out using the conventional pressurepulse decay proposed by Brace et al. [
When the dimensionless differential pressure is plotted with dimensionless time, the curves of test 1 will be similar and situated close to each other due to the constant pore volume. Examination of Figure
The relationship of dimensionless pressure and time: (a) shale sample and (b) rocklike sample.
Figure
Experimental schemes and permeability results of test 1 and test 2.
Sample  Size (mm)  Confining pressure (MPa)  Pore pressure (MPa)  Permeability (m^{2})  

Test 1  Shale sample 

5  2 

6.5  3.5 


8  5 


9.5  6.5 




Test 2  Rocklike sample 

5  2 

10  2 


15  2 


20  2 

The earlytime pressure decay data cannot be used for permeability estimation in pressurepulse tests though it is useful for sample heterogeneity investigation. A method for differentiating the earlytime and latetime behavior is proposed. It is validated by a TOUGH+ simulator and is applied into the laboratory tests for permeability calculation. Based on the work completed, the conclusions are summarized as follows:
As the volume ratio increases, the critical time appears to have a declining trend after an initial ascent. The critical time for cases with same chamber volumes is much less than that for cases with unequal chamber volumes
The numerical results show that the pressure gradient along the sample varies nonlinearly at the early time and becomes a constant at the late time. The time for pressure gradient to attain constant increases with volume ratios; it could be hours for pressure gradient to attain constant at large volume ratios
The earlytime behavior is negligible when the volume ratio is smaller than 0.1. With the increase of volume ratios, the deviation of pulse decay curve from singleexponential behavior increases, and the latetime data should be used for permeability calculation
All the data and computer codes for this paper can be available by contacting the corresponding author, whose email address is as follows:
The authors declare that they have no conflicts of interest.
This research was supported by the National Natural Science Foundation of China (No. 51374257 and No. 50804060). It was also supported by the China Scholarship Council (CSC) for the second author’s visit at the Lawrence Berkeley National Laboratory. The authors would like to acknowledge Lehua Pan from the Lawrence Berkeley National Laboratory for the continuous encouragement and guidance.