Abstract
For various rock engineering, injection of fluids into rock fractures through boreholes is quite common. It is of great significance to investigate the characteristics of radial flow (RF) in rock fractures for these activities. In this study, macroscopic and mesoscopic characteristics of RF in rough rock fractures were investigated and compared with those of unidirectional flow (UF) by theoretical analysis, tests and simulations. An equation for nonlinear RF was derived for rock fractures according to conservation law of mass and Izbash’s law. Four scanned rough rock fracture models were established and used to experimentally investigate the macroscopic flow characteristics in both UF and RF. Numerical simulations were performed to clarify the mesoscopic differences in fluid pressure distributions and the flowlines of RF and UF in rock fractures. The parameters of hydraulic aperture and equivalent width for RF were obtained and correlated to those for UF. A method to calculate fracture roughness coefficient of fractures for RF related to the flow direction was proposed. The characteristic parameters, i.e., critical Reynolds numbers for the flow transition from linear to nonlinear flow, effective hydraulic apertures and non-Darcy coefficients, were obtained for the UF and RF based on the test results. It was indicated that the fracture roughness plays a critical role in the macroscopic and mesoscopic characteristics of both RF and UF. According to the test results, the macroscopic characteristic parameters for RF are related to those for UF, and the nonlinearity of RF was stronger than that of UF at a specified flow rate, which was consistent with the mesoscopic characteristics observed in the simulation that the distribution of water pressure, flow velocity and the streamlines in RF were more non-uniform than that in UF. The study results were useful to describe the RF characteristic in rock fractures with the characteristic parameters for UF, which have been investigated extensively in literature.
Highlights
-
A nonlinear flow equation for radial flow in rock fractures was derived to describe the relationship between the hydraulic head and flow rate.
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The differences and relations between radial and unidirectional flow were studied from macroscopic and mesoscopic aspects.
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The parameters of hydraulic aperture and equivalent width for radial flow were obtained and correlated to those for unidirectional flow.
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The effect of fracture roughness on radial and unidirectional flow was related to the flow direction and was incorporated in the Forchheimer equation.
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Data Availability
The data that support the findings of this study are available on request from the corresponding author.
Abbreviations
- A, B :
-
Linear and nonlinear term coefficients of Forchheimer equation, respectively
- e :
-
Geometric aperture
- e h :
-
Hydraulic aperture
- e hu, e hr :
-
Hydraulic aperture for unidirectional and radial flow, respectively
- E :
-
Non-Darcy effect factor
- h :
-
Hydraulic head
- h(r,t):
-
Hydraulic head at the distance r and the time t
- FRC:
-
Fracture roughness coefficient
- FRCu, FRCr:
-
Fracture roughness coefficient of the unidirectional and radial flow
- K :
-
Hydraulic conductivity
- m:
-
Nonlinear index
- M :
-
Total number of sampling intervals
- q :
-
Flow velocity
- q(r,t):
-
Flow velocity at the distance r and the time t
- Q :
-
Flow rate
- r :
-
Radial coordinate
- r 0, r 1 :
-
Inner radius and outer radius of the radial flow model, respectively
- R 2 :
-
Fitting correlation coefficient
- Re:
-
Reynolds number
- Rec :
-
Critical Reynolds number
- Reu c, Rer c:
-
Critical Reynolds number under unidirectional and radial flow configurations, respectively.
- RF:
-
Radial flow
- S s :
-
Specific storage coefficient of fracture
- u :
-
Velocity vector
- UF:
-
Unidirectional flow
- V :
-
Fluid volume
- w :
-
Width of the fracture along the direction perpendicular to the flow
- w r :
-
Equivalent width of the fracture for radial flow
- Z 2 :
-
Root mean square of the first derivative of the profile
- z i :
-
Asperity height of the ith sampling point
- β :
-
Non-Darcy coefficient
- β u, β r :
-
Non-Darcy coefficient under unidirectional and radial flow, respectively
- μ :
-
Dynamic viscosity
- ρ :
-
Fluid density
- Δx :
-
Sampling interval with the value of 0.5 mm
- ∇P :
-
Pressure gradient
- ∇P u, ∇P r :
-
Pressure gradient under the unidirectional and radial flow, respectively
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Acknowledgements
This study was financially supported by the National Natural Science Foundation of China under contract Nos. 42177157 and 51779045, Liao Ning Revitalization Talents Program under contract No. XLYC1807029, the Fundamental Research Funds for the Central Universities under contract No. N2001026 and Liaoning Natural Science Foundation under contract No. 2019-YQ-02.
Funding
National Natural Science Foundation of China, 42177157, Liping Qiao, 51779045, Zhechao Wang, Liao Ning Revitalization Talents Program, XLYC1807029, Zhechao Wang, Fundamental Research Funds for the Central Universities, N2001026, Zhechao Wang, Liaoning Natural Science Foundation, 2019-YQ-02, Zhechao Wang.
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ZW: conceptualization, methodology, writing—review and editing; JL: investigation, experiments, data curation, writing—original draft; TZ: experiments, data curation, writing—original draft; LQ: Investigation, writing—review and editing; KL: assistance with experiments.
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Wang, Z., Liu, J., Zheng, T. et al. Macroscopic and Mesoscopic Characteristics of Radial Flow in Rough Rock Fractures. Rock Mech Rock Eng 56, 4881–4900 (2023). https://doi.org/10.1007/s00603-023-03312-4
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DOI: https://doi.org/10.1007/s00603-023-03312-4