Session: 16-01-01: Government Agency Student Poster Competition

Paper Number: 150548

150548 - A Mathematical Model for Fast Determination of Permeabilities of Tight Reservoir Cores From the Initial Period of the Pulse Decay Test 

Permeability of rock has recently gained interest in three emerging energy related areas including fracing of gas- or oil-bearing shale formations, CO₂ sequestration in deep formations of ultramafic rocks, and rock fracturing for geothermal energy extraction. These technologies all deal with tight rocks such as shale and serpentinized peridotite, which usually possess very fine pores and low permeability. For this reason, measuring permeability of these rocks using conventional steady-state methods is not practical, as it requires a considerable amount of time to achieve a steady-state flow in the test specimen [1]. Additionally, measurement of fluid volume transmitted through the specimen often falls within the noise level of most high accuracy pumps that will introduce errors in the calculated flow rates. To eliminate the limitations of the steady-state flow techniques, the pulse decay method was introduced by Brace et al. [2], which has been widely used so far because it reduces the measurement time significantly. However, in the case of tight rocks, the experimental time using the pulse decay technique is still fairly long [3], reaching up to several hours or even days for rock samples with permeability in the nano-Darcy range. Therefore, there have been several attempts to derive solutions for interpretation of initial part (early-time) of the pulse decay test as it enables determination of the rock permeability in a more practical manner [4,5,6]. In this study, analytical solution with a single exponential decay is developed for the upstream pressure during the initial period of the pulse decay test both before and after communication between up- and downstream sides, allowing fast and accurate calculation of rock permeability using linear regression analysis. In our formulation, the pressure distribution inside the specimen is approximated using a parabolic profile, which simplifies the initial-boundary value problem of fluid flow in the specimen governed by partial differential equations to a system of ordinary differential equations that can be easily solved analytically. For cases where the pressure pulse has reached the downstream side, the proposed scheme is applicable to a modified pulse decay test (i.e. the One-Chamber Pressure Pulse Decay (OC-PPD) method proposed by Yang et al. [7] in which a pressure pulse is applied at the upstream side of the test specimen, while the downstream end of the specimen is sealed). In this setting, the pressure profile inside the specimen will maintain a zero slope at the downstream side after the pressure pulse has reached the downstream side, maintaining the validity of the parabolic approximation of the pore pressure. We first developed an approximate analytical solution for the upstream pressure before the pressure pulse reaches the downstream side. Analysis shows that obtaining permeability from this very initial period might be inaccurate because (1) the time it takes for the initial pressure pulse to reach the downstream side of the specimen can be very short depending on the parameters such as the length and the permeability of the specimen, and (2) the initial stage of the pulse decay test is susceptible to non-Darcy flow due to initial high pressure gradient. To improve the limitations of this solution, we further developed an approximate analytical solution for the upstream pressure after the pulse has reached the downstream side of the specimen. This allows us to have enough data points after the very initial period to evaluate the permeability of the test specimen, without having to wait for the system to reach equilibrium. After proper calibrations, the permeability obtained from linear regression analysis based on this approximate analytical solution agrees very well with the reported experiment data. Furthermore, the specific storage of the test specimen is also considered in our solutions, unlike in the semi-analytical solution by Brace et al. [2] which assumed zero compressive storage in the rock in their formulation. Such an assumption is reasonable for some crystalline rocks such as granite but is not applicable to rocks that have a significant porosity and compressive storage such as shales and argillites [4].

 

References

[1] Mallon, A. J., & Swarbrick, R. E. (2007). How should permeability be measured in fine‐grained lithologies? Evidence from the Chalk. Geofluids, 8(1), 35–45. https://doi.org/10.1111/j.1468-8123.2007.00203.x

[2] Brace, W. F., Walsh, J. B., & Frangos, W. T. (1968). Permeability of granite under high pressure. Journal of Geophysical Research, 73(6), 2225–2236. https://doi.org/10.1029/jb073i006p02225

[3] Jones, S. C. (1997). A technique for faster pulse-decay permeability measurements in tight rocks. SPE Formation Evaluation, 12(01), 19–25. https://doi.org/10.2118/28450-pa

[4] Hsieh, P.A., Tracy, J.V., Neuzil, C.E., Bredehoeft, J.D., Silliman, S.E. (1981) Transient laboratory method for determining the hydraulic properties of ‘tight’ rocks - I. Theory. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 18(5), 89. https://doi.org/10.1016/0148-9062(81)90090-5

[5] Crank, J. (1975). The Mathematics of Diffusion. Clarendon Press, Oxford, pp. 56-59.

[6] Bourbie, T., & Walls, J. (1982). Pulse decay permeability: Analytical Solution and Experimental Test. Society of Petroleum Engineers Journal, 22(05), 719–721. https://doi.org/10.2118/9744-pa

[7] Yang, Z., Sang, Q., Dong, M., Zhang, S., Li, Y., & Gong, H. (2015). A modified pressure-pulse decay method for determining permeabilities of tight reservoir cores. Journal of Natural Gas Science and Engineering, 27, 236–246. https://doi.org/10.1016/j.jngse.2015.08.058

Presenting Author: Anh Tay Nguyen Northwestern University

Presenting Author Biography: Anh is from Hanoi, Vietnam. He attended SUNY Korea in South Korea where he earned a BE/MS in Mechanical Engineering. At SUNY Korea, he studied configurational mechanics under the supervision of Dr. Y. Eugene Pak. He then joined Professor Zdeněk P. Bažant research group at Northwestern University in Spring 2022, pursuing a PhD also in Mechanical Engineering. His research here focuses on computational fracture mechanics of quasibrittle material and multiphysics modeling of CO2 sequestration processes using carbon mineralization.

Authors:

Anh Tay Nguyen Northwestern University
Pouyan Asem University of Minnesota Twin Cities
Yang Zhao Northwestern University
Zdenek Bazant Northwestern University