Experimental and Theoretical Study on Comparisons of Some Gas Permeability Test Methods for Tight Rocks

Abstract

Precise gas permeability measurements of tight rocks are important for the exploration of unconventional reservoirs. Steady-state and varied unsteady-state gas permeability tests were conducted on a heat-treated siltstone. Brace’s approach ignores the sample pore volume resulting in the underestimation of sample permeability, and its discrepancy with the sample pore volume corrected permeability depends on the ratio between the sample pore volume and reservoir volumes. Permeability in the pulse-decay test ($k_{{\text{T}}}$) is higher than that based on Brace’s approach ($k_{{\text{B}}}$) but lower than the sample pore volume corrected permeability ($k_{{\text{C}}}$) in the downstream pressure build-up test. It is theoretically and experimentally proved that the transient flow approach is equal to the Brace’s approach as both assume the validity of Darcy’s law and ignore the sample pore volume. Variations of the gas permeability are dominated by gas slippage and the effective stress at low and high pore pressures respectively. The apparent steady-state permeability is higher than the sample pore volume corrected permeability at small upstream pressures, while its intrinsic permeability is always lower than that of the sample pore volume corrected permeability. Both the intrinsic permeability vs. porosity and intrinsic permeability vs. the effective stress curves follow power laws, and the effective stress coefficient is higher than unity due to the clay filling and lining.

Highlights

• Steady-state and unsteady-state gas permeabilities are compared.

• Gas slippage and effective stress dominate the permeability variations.

• The transient flow approach is identical to the Brace’s approach.

• Stress sensitivity of the permeability is investigated.

Appendices

Appendix A: The Transient Flow Approach is Equal to the Brace’s Approach

Both transient flow approach and Brace’s approach assume Darcy’s law is valid and they both ignore the sample storage volume. Therefore, similar values are expected for these two approaches, which can be verified as follows.

For Brace’s approach in pulse decay test, Eq. (3) can be written as follows.

$$\alpha = - \frac{{{\text{d}}\ln (P_{{\text{u}}} - P_{{\text{d}}} )}}{{{\text{d}}t}} = - \frac{1}{{P_{{\text{u}}} - P_{{\text{d}}} }}\left( {\frac{{{\text{d}}P_{{\text{u}}} }}{{{\text{d}}t}} - \frac{{{\text{d}}P_{{\text{d}}} }}{{{\text{d}}t}}} \right).$$

(20)

If the sample pore volume is ignored, the amount of gas entering the sample at upstream is equal to that received at downstream.

$$\frac{{V_{{\text{u}}} {\text{d}}P_{{\text{u}}} }}{{{\text{d}}t}} = - \frac{{V_{{\text{d}}} {\text{d}}P_{{\text{d}}} }}{{{\text{d}}t}}.$$

(21)

Then Eq. (20) can be transformed to Eq. (22).

$$\alpha = \frac{1}{{P_{{\text{u}}} - P_{{\text{d}}} }}\left( {\frac{{V_{{\text{d}}} }}{{V_{{\text{u}}} }}\frac{{{\text{d}}P_{{\text{d}}} }}{{{\text{d}}t}} + \frac{{{\text{d}}P_{{\text{d}}} }}{{{\text{d}}t}}} \right) = \frac{1}{{P_{{\text{u}}} - P_{{\text{d}}} }}\left( {\frac{{V_{{\text{d}}} + V_{{\text{u}}} }}{{V_{{\text{u}}} }}} \right)\frac{{{\text{d}}P_{{\text{d}}} }}{{{\text{d}}t}}.$$

(22)

Substitute Eq. (22) into (4),

$$k = \frac{{\mu \beta LV_{{\text{d}}} }}{{A(P_{{\text{u}}} - P_{{\text{d}}} )}}\frac{{{\text{d}}P_{{\text{d}}} }}{{{\text{d}}t}}.$$

(23)

Under isothermal conditions,

$$\beta = \frac{1}{{P_{{{\text{average}}}} }} = \frac{2}{{P_{{\text{u}}} + P_{{\text{d}}} }}.$$

(24)

Substitute Eq. (24) into (23), the permeability based on Brace’s approach can be written as Eq. (25).

$$k = \frac{{2\mu LV_{{\text{d}}} }}{{A(P_{{\text{u}}}^{2} - P_{{\text{d}}}^{2} )}}\frac{{{\text{d}}P_{{\text{d}}} }}{{{\text{d}}t}}.$$

(25)

Equation (25) is exactly the same as Eq. (11), which is the permeability for the transient flow approach. Therefore, the permeability obtained by the transient flow approach is theoretically the same as that derived from Brace’s approach in pulse decay test. As the $P_{{\text{d}}}$ build-up test is a simplified form of the pulse decay test, permeabilities obtained by these two approaches should be the same as well.

Appendix B: ${{{\text{d}}P_{{\text{d}}} } \mathord{\left/ {\vphantom {{{\text{d}}P_{{\text{d}}} } {{\text{d}}t}}} \right. \kern-\nulldelimiterspace} {{\text{d}}t}}$ is a negative exponential function of time for the $P_{{\text{d}}}$ build-up test

For the $P_{{\text{d}}}$ build-up test at a constant $P_{{\text{u}}}$, Eq. (2) can be transformed to the following equations.

$$\frac{{P_{{\text{u}}} (0) - P_{{\text{d}}} (t)}}{{P_{{\text{u}}} (0) - P_{{\text{d}}} (0)}} = {\text{e}}^{ - \alpha t} ,$$

(26)

$$P_{{\text{d}}} (t) = P_{{\text{u}}} (0) - \Delta P_{0} {\text{e}}^{ - \alpha t} ,$$

(27)

$${{{\text{d}}P_{{\text{d}}} (t)} \mathord{\left/ {\vphantom {{{\text{d}}P_{{\text{d}}} (t)} {{\text{d}}t}}} \right. \kern-\nulldelimiterspace} {{\text{d}}t}} = \alpha \Delta P_{0} {\text{e}}^{ - \alpha t} ,$$

(28)

where $\Delta P_{0}$ is the initial differential pressure between $P_{{\text{u}}}$ and $P_{{\text{d}}}$.

As $\alpha \Delta P_{0} e^{ - \alpha t}$ is positive and monotonically decreases with time, ${\text{d}}P_{{\text{d}}} {(}t{\text{)/d}}t$ declines with time. The initial increasing period of ${\text{d}}P_{{\text{d}}} {\text{/d}}t$ can be explained by the pore filling process. Gas penetrates the sample through preferred flow path after some time delay resulting in the increase of $P_{{\text{d}}}$, while more flow paths will contribute to the $P_{{\text{d}}}$ increase after the breakthrough. Therefore, it can be inferred that the peak point on the ${\text{d}}P_{{\text{d}}} {\text{/d}}t$ vs. time curve is a compromise between the increase caused by the enhanced contribution of more flow paths and the decrease when the normalized differential pressure approaches a negative exponential relationship with time. In this regards, the late time solution should be used and the time duration for the permeability calculation should be after the time of the maximum ${\text{d}}P_{{\text{d}}} {\text{/d}}t$.

Appendix C: Quantitative Comparison Between $k_{{\text{B}}}$ and $k_{{\text{C}}}$

The derivation of $k_{{\text{B}}}$ and $k_{{\text{C}}}$ are shown in Eqs. (5) and (9) respectively for the $P_{{\text{d}}}$ build-up test. The ratio of $k_{{\text{C}}} {/}k_{{\text{B}}}$ is expressed in Eq. (29).

$$\frac{{k_{{\text{C}}} }}{{k_{{\text{B}}} }} = \frac{\phi }{{\theta^{2} }}\frac{LA}{{V_{{\text{d}}} }}.$$

(29)

Incorporating Eqs. (7) and (8) into Eq. (29), Eq. (30) is obtained.

$$\frac{{k_{{\text{C}}} }}{{k_{{\text{B}}} }} = \frac{\gamma }{{\theta^{2} }} = \frac{\tan \theta }{\theta }.$$

(30)

It shows $k_{{\text{C}}} {/}k_{{\text{B}}}$ is determined by $\gamma$, as $\theta$ is also dependent on $\gamma$ according to Eq. (8).

$\theta$ is the first positive root of Eq. (8), therefore the range of $\theta$ is $0 < \theta < \pi /2$, and $\tan \theta > \theta$ is valid for $0 < \theta < \pi /2$. Therefore $k_{C}$ is always higher than $k_{B}$.

By differentiating of Eq. (30) over $\theta$, the following Eq. (31) is derived.

$$\frac{{{\text{d}}\left( {\frac{\tan \theta }{\theta }} \right)}}{{{\text{d}}\theta }} = \frac{{\frac{\theta }{{\cos^{2} \theta }} - \tan \theta }}{{\theta^{2} }} = \frac{\theta - \sin \theta \cos \theta }{{\theta^{2} \cos^{2} \theta }} = \frac{2\theta - \sin 2\theta }{{2\theta^{2} \cos^{2} \theta }}.$$

(31)

For $0 < \theta < \pi /2$, ${2}\theta > \sin 2\theta$, and

$$\frac{{{\text{d}}\left( {\frac{\tan \theta }{\theta }} \right)}}{{{\text{d}}\theta }} > 0.$$

(32)

Therefore, ${{\tan \theta } \mathord{\left/ {\vphantom {{\tan \theta } \theta }} \right. \kern-\nulldelimiterspace} \theta }$ or $k_{{\text{C}}} {/}k_{{\text{B}}}$ is a monotonically increasing function of $\theta$.

The relationship between $\gamma$ and $\theta$ is also investigated. $\gamma$ can be written as a function of $\theta$ by referring to Eq. (8).

$$\gamma = \theta \tan \theta$$

(33)

By differentiating of $\gamma$ over $\theta$, Eq. (34) is obtained

$$\frac{{{\text{d}}\gamma }}{{{\text{d}}\theta }} = tan\theta + \frac{\theta }{{\cos^{2} \theta }} = \frac{\sin \theta \cos \theta + \theta }{{\cos^{2} \theta }} = \frac{\sin 2\theta + 2\theta }{{2\cos^{2} \theta }}.$$

(34)

For $0 < \theta < \pi /2$, $\sin 2\theta { + 2}\theta > {0}$ and $d\gamma {/}d\theta > 0$. Therefore $\gamma$ is also a monotonically increasing function of $\theta$. Since both $\gamma$ and ${{\tan \theta } \mathord{\left/ {\vphantom {{\tan \theta } \theta }} \right. \kern-\nulldelimiterspace} \theta }$ are monotonically increasing functions of $\theta$, ${{\tan \theta } \mathord{\left/ {\vphantom {{\tan \theta } \theta }} \right. \kern-\nulldelimiterspace} \theta }$ is also a monotonically increasing function of $\gamma$. It means higher $\gamma$ corresponds to larger $k_{{\text{C}}} {/}k_{{\text{B}}}$. As the downstream reservoir volume is normally constant, the increase in the sample pore volume results in the higher discrepancy between $k_{{\text{B}}}$ and $k_{{\text{C}}}$. A series of $\theta$ and $k_{{\text{C}}} {/}k_{{\text{B}}}$ values have been calculated for different $\gamma$ based on the above equations and plotted in Fig. 

Fig. 21

Variations of $\theta$ and $k_{{\text{C}}} {/}k_{{\text{B}}}$ with $\gamma$

21a, b) respectively.

As can be seen from Fig. 21, $\theta$ increases with $\gamma$ indicating a positive relationship between them. While $k_{{\text{C}}} {/}k_{{\text{B}}}$ stays almost constant at around unity for $\gamma$ below 0.1, and it shows a gentle increase from 1.03 to 1.35 for $\gamma$ from 0.1 to 1 which is followed by a more significant increase to 4.9 when $\gamma$ is 10. Therefore, the sample pore volume should be taken into consideration in the $P_{{\text{d}}}$ build-up test if $\gamma$ is higher than 0.1.