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MethodsX
MethodsX
MethodsX
2215-0161
Elsevier

S2215-0161(24)00351-0
10.1016/j.mex.2024.102899
102899
Earth and Planetary Science
Constant head-transient method using pressure and flow data for determining permeability and specific storage of tight rocks
Song Insun isong@kigam.re.kr

Climate Change Response Research Division, Korea Institute of Geoscience and Mineral Resources, Daejeon 34132, South Korea
06 8 2024
12 2024
06 8 2024
13 1028993 6 2024
31 7 2024
6 8 2024
© 2024 The Author
2024
https://creativecommons.org/licenses/by-nc/4.0/ This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
This paper describes a ‘constant head-transient method’ for estimating permeability and specific storage of tight rock samples, such as shale and crystalline rocks. Experimental tests are conducted using a cylindrical rock sample subjected to confining pressure, through which pressure diffusion occurs from a constant upstream pressure (or constant head) to a finite downstream storage. Unlike the pulse-transient method, the upstream fluid flow into the sample can be measured using a syringe pump because of no change in upstream pressure. By minimizing the downstream storage, the test time can be significantly reduced, but only the downstream pressure transient data do not yield accurate results on permeability and specific storage estimations. By combining the flow data with the pressure data, the proposed method aims at saving the test time and improving the accuracy of their estimations in extremely low permeability rock samples.• A constant head-transient method for measuring the hydraulic properties of tight rocks was developed with a boundary condition of constant upstream pressure and finite downstream storage.

• The test time can be saved by minimizing the downstream storage, and the upstream flow can be measured to improve the accuracy in measuring the hydraulic properties.

• Combining the flow and pressure objective functions yields the best curve fitting for both pressure and flow curves.

Graphical abstract

Image, graphical abstract

Keywords

Hydraulic property measurement
Pressure transient
Pressure and flow analysis
Objective function
Curve-fitting technique
Method name

Constant head-transient method using pressure and flow data for determining permeability and specific storage of tight rocks
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pmcSpecifications tableSubject area:	Earth and Planetary Sciences	
More specific subject area:	Rock mechanics	
Name of your method:	Constant head-transient method using pressure and flow data for determining permeability and specific storage of tight rocks	
Name and reference of original method:	Song I., 2016, Theoretical analysis of one-dimensional pressure diffusion from a constant upstream pressure to a constant downstream storage, Pure and Applied Geophysics 173, 1721–1731.	
Resource availability:	Not applicable	

Introduction

Darcy's method of measuring permeability is not convenient for extremely low permeability rocks because it takes a long period to establish a steady state [[1], [2]]. The most time-saving method for tight rocks such as shale and crystalline rocks is to use the pressure transient curves obtained prior to pore pressure stabilization [1,3]. The pulse-transient method is commonly used to determine the permeability k and the specific storage Ss from the pressure transient curves, where direct measurement of fluid flow through the rock sample is not possible. However, using only pressure transient data yield relatively poor results on permeability estimations, and even worse in determining the specific storage [4]. Specially, Ss is very sensitive to increment of fluid content in a sample [5]. Also k is defined from the relation between pore pressure gradient and flow rate. Thus, in addition to the pressure transient, the measurement of fluid flow is very important to improve the accuracy of the estimated k and Ss of the sample. In this paper, we propose a ‘constant head-transient method’ where the upstream pressure (head) is kept constant so that the upstream flow into the sample as well as the downstream pressure transient can be measured. Using both pressure and flow data gives more accurate results on the permeability and the specific storage of tight rock samples.

Method details

Nomenclature and theoretical background

The experimental configuration for the constant head-transient method is shown in Fig. 1. The nomenclature used in the method description is as follows:Fig. 1 Conceptual boundary conditions for theoretical review and experimental configuration.

Fig 1

p(x,t): the fluid pore pressure in the sample [Pa]

x: the distance along the sample from the downstream [m]

t: the time from the start of experiment [s]

A: the cross-sectional area of the sample [m2]

L: the length of the sample [m]

P0: the initial pore pressure equilibrated along the system [Pa]

Pu: the upstream pore pressure during the test [Pa]

qu(t): the upstream flow into the sample measured from the upstream pump [m3/s]

k: the hydraulic permeability of the sample [m2]

Ss: the specific storage of the sample [Pa-1]

μ: the dynamic viscosity of pore fluid [Pa s]

κ: the hydraulic diffusivity (κ=k/(μSs)) [m2/s]

Sd: the compressive storage capacity of the downstream reservoir [m3/Pa]

δ: the compressive storage capacity ratio of the sample to the downstream reservoir [-]

ϕm: the roots ϕ of the peoriodic function, tanϕ=1δϕ [-].

The governing equation of one-dimensional pore pressure diffusion in x-direction:(1) ∂2p(x,t)∂x2−μSsk∂p(x,t)∂t=0.

The initial condition (t=0):(2) p(x,0)=P0.

The boundary conditions:(3) μSdkA∂p(0,t)∂x−∂p(0,t)∂x=0atx=0,

(4) p(L,t)=Puatx=L.

The solutions of the governing Eq. (1) in terms of the downstream pressure p¯d(t) and the cumulative amount of upstream fluid imported into the sample q¯u(t) during the transient period are given by [6]:(5) p¯d(t)=ΔP[1−2∑m=1∞1(δϕm2−2)cosϕm+ϕm(3δ+1)sinϕmexp(−ϕm2τ)]+P0,

(6) q¯u(t)=2ΔPALSs∑m=1∞δ2ϕm2+1δ2ϕm4+δϕm2+ϕm2[1−exp(−ϕm2τ)],

where ΔP=Pu−P0, the differential pressure between the initial pore pressure and the upstream pressure during the test; τ=κt/L2, the dimensionless time; and δ=Sd/(ALSs), the compressive storage ratio of the downstream to the sample. The eigenvalues ϕm are roots of(7) tanϕ=1δϕ.

Some examples of theoretical curves from the solutions for downstream pressure and upstream flow for different δ, κ, k, and Ss are shown in Fig. 2. For very low permeability rock samples such as shale and tight crystalline rocks, the downstream storage (Sd) should be minimized for saving the test time as shown in Fig. 2a. If there is no downstream storage (Sd=0), Ss and k are not separatable in the solution in terms of pressure transient (Eq. (5)). If the flow data are used, no problem occurs in obtaining both values of Ss and k separately in Eq. (6).Fig. 2 Theoretical curves of downstream pressure transients for different (a) δ and (b) κ and of upstream fluid flow imported into the rock specimen for different (c) k and Ss.

Fig 2

Experimental test

In this experimental demonstration, a shale sample from a caprock formation of a potential CO2 geological storage in South Korea was used. The sample assembly consists of a cylindrical specimen, two porous steel disks, and two endplugs, in sequence from the middle outward (Fig. 3a). The two porous disks are used to facilitate more uniform fluid distribution at the ends of the sample. The void volume of the downstream disk is also used as a compressive storage (Sd). Details on downstream pressure measuring system is shown in Fig. 3b. The jacketed assembly is placed in a pressure cell for the application of confining pressure (Fig. 3a). After the application of confining pressure, the upstream pressure is quickly raised and kept constant. A sudden excitation of upstream pressure leads to one-dimensional pore pressure diffusion through the cylindrical sample. The cumulative amount of upstream fluid imported into the specimen and the downstream pressure variation are obtained during the transient period. It is possible to measure the upstream flow into the sample using the pump because of no change in upstream pressure. An example of experimental measurements during the transient period illustrates that the downstream pressure increases with decreasing rate and asymptotically approaches the constant upstream pressure (Fig. 4a). The upstream flow shows the similar fashion as the downstream pressure (Fig. 4b).Fig. 3 Schematic drawings of (a) experimental flow system and (b) details in the endplug for employing a miniature pressure transducer to minimize the downstream storage.

Fig 3

Fig. 4 An example of experimental results of (a) fluid pressures measured at upstream and downstream and (b) cumulative fluid flow measured at upstream pump during the transient period.

Fig 4

Objective function and curve fitting process

The hydraulic permeability and specific storage can be determined using the curve fitting technique that minimizes misfits between the experimental curves and the theoretical values. In this study, a grid searching algorithm was used to find the best fit curve. The objective function in terms of k and Ss is used for obtaining the best fit curves. In this proposed method, the average of all misfits between the experimental data and the theoretical values as a function of k and Ss is used to obtain the objective functions. The objective function for the downstream pressure is given by:(8) Zp(k,Ss)=1m∑i=1m|pd(i)−p¯d(i:k,Ss)|,

and for the upstream flow:(9) Zq(k,Ss)=1m∑i=1m|qu(i)−q¯u(i:k,Ss)|.

In order to combine the two objective functions, Zp(k,Ss) and Zq(k,Ss), they are normalized by the differential pressure, ΔP=Pu−P0, and the maximum value of qu(m), respectively. Fig. 5 shows the combined objective functions of pressure and flow data. In the example, the yellow dot indicates the coordinate (k,Ss) with the minimum misfits in the combined objective function. Thus, when k=2.66×10−19m2 and Ss=2.00×10−10Pa−1, we obtain the best fittings of both pressure and flow curves (Fig. 6).Fig. 5 An example of combined objective function with normalized misfits from flow and pressure analyses. The yellow dot with the least misfits indicates that k=2.66×10−19m2 and Ss=2.00×10−10Pa−1 for the tested sample.

Fig 5

Fig. 6 Comparison of experimental data (solid lines) and theoretical values (solid dots) with the best sets of hydraulic properties obtained from the combined objective functions (Fig. 5) showing well matching in (a) pressure curve and (c) flow curve.

Fig 6

Conclusion

A ‘constant head-transient method’ is proposed from theoretical analysis and experimental tests as a time-saving and accurate approach for measuring the hydraulic permeability k and specific storage Ss of extremely low permeability rocks, such as shale and crystalline rocks. The method involves maintaining a constant upstream pressure (head) and measuring both the upstream flow into the sample and the downstream pressure transient. The test time can be significantly reduced by minimizing the downstream compressive storage. However, in this case, only the pressure transient data is not enough to accurate results on the hydraulic properties, k and Ss. Integrating flow data with pressure data yields improved accuracy in their estimations.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Ethics statements

Not applicable

CRediT author statement

Insun Song: Conceptualization, Methodology, Software, Experimental tests, Data curation, Writing- Original draft preparation, Reviewing and Editing, Visualization, Investigation.

Acknowledgments

This research was supported by the Basic Research Project of the Korea Institute of Geoscience and Mineral Resources (KIGAM) funded by the Ministry of Science and ICT, and also by the Korea Institute of Energy Technology Evaluation and Planning (KETEP) and the Ministry of Trade, Industry & Energy (MOTIE) of Republic of Korea (20212010200020 ).

Related research article: Integrating transient pressure and flow data for determination of hydraulic permeability and specific storage of a shale sample: Experimental tests and sensitivity analysis (International Journal of Rock Mechanics & Mining Sciences 176, (2024) 105718).
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References

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