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In Alberta and British Columbia, a huge amount of tight gas is trapped inrelatively low-permeability rock formations. Physical fracturing of theseformations could enhance the overall formation permeability and thus improvetight gas extraction. One of the outstanding issues in rock fracturing is todetermine the magnitude of applied effective stress. The generaleffective-stress law is defined as seff =sc - asp, where sc andsp are total confining stress and fluid pore pressure,respectively. Each physical quantity of rock responds to total stress and porepressure in a different way, and thus each quantity has its own unique Biot'seffective-stress coefficient. The main objective of this study is toexperimentally determine the Biot's coefficient for permeability of Nikanassinsandstone. A series of permeability measurements was conducted on Nikanassinsandstone core samples from the Lick Creek region in British Columbia undervarious combinations of confining stress and pore pressure. In addition,permeability values were measured both along and across bedding planes toinvestigate any anisotropy in the Biot's coefficient.
The foundational paper by Klinkenberg contains a very rich dataset for gas flow in porous samples over a flange of mean pressures from 1 to 2,000 kPa. Based on his data, Klinkenberg proposed a correlation between pressure drop and flow rate that depends on both the Darcy permeability (the permeability at infinite mean pressure) and the ratio of a coefficient, now generally termed the Klinkenberg coefficient, and the mean pressure. Klinkenberg’s approach to analyze his data was to determine the Darcy permeability at a high mean pressure, then calculate Klinkenberg coefficients at lower values of mean pressures. He found that values of the calculated Klinkenberg coefficient remained constant for a certain range of mean pressures, but changed significantly at low mean pressures. Klinkenberg clearly stated that his results did not show a strictly linear function of effective permeability with the inverse of mean pressure—this observation has never been studied in detail. Based on an approach published by Arabjamaloei and Ruth, Klinkenberg’s data were reanalyzed using three methods: by optimizing the Darcy permeability and the Klinkenberg coefficient simultaneously; by holding the Darcy permeability constant but optimizing the value of the Klinkenberg coefficient to obtain a single value for all mean pressures; by optimizing Darcy permeability, the Klinkenberg coefficient, and a second Klinkenberg coefficient divided by mean-pressure-squared. The last approach is successful in correlating all of Klinkenberg’s data to within 5%. However, the improvements due to the modified Klinkenberg equation are marginal and do not explain all the disagreement. For this reason, a second dataset, published by Ash and Grove, was explored. This dataset, which has been largely ignored in the literature, provides convincing evidence for Klinkenberg’s ideas,once the data are reanalyzed to account for shortcomings in the ranges of experimental pressures. Based on ideas documented by Carman for mixed viscous/ diffusive flows, the results are used to derive estimates of an effective pore diameter and the tortuosity.
Tian, Jianwei (The University of Western Australia) | Liu, Jishan (The University of Western Australia) | Elsworth, Derek (The Pennsylvania State University) | Leong, Yee-Kwong (The University of Western Australia) | Li, Wai (The University of Western Australia) | Zeng, Jie (The University of Western Australia)
Abstract Heterogeneous pore structure is critically important for unconventional gas recovery. In this paper, a dynamic fractal permeability model is proposed to investigate the interplay between heterogeneous pore structures and gas transport for coal seam gas reservoir. In this model, pore diameter and fractal dimension of pore size distribution are dynamically changing as a result of the variation of effective stress. Besides, based on fractal approach, a new Klinkenberg coefficient that dynamically changes with pore pressure is employed to incorporate the non-Darcy effect. This dynamic permeability model is applied to couple Multiphysics in coal seam gas recovery process. The impacts of these fractal parameters on permeability evolution are explored through a benchmark reservoir simulation. The numerical results exhibit good agreements with experimental data. The simulation results indicate that: (1) the dynamic permeability model matches better with experimental data than other homogeneous models, especially in low-pressure stage; (2) reservoir with larger initial fractal dimension is more sensitive to pressure depletion; (3) fractal dimension would change more dramatically when initial porosity is relatively smaller; (4) Klinkenberg coefficient increases with the decreasing of reservoir pore pressure during gas depletion. In summary, the dynamic permeability model predicts permeability evolution well in gas production process and provide some fundamental insights into the implications of reservoir heterogeneity on gas transport in reservoir simulation. Introduction With the natural gas depletion, there is an increasing need for the exploration of unconventional natural gas, unconventional gas is typically regarded as a substitute that can relieve energy supply shortage. Permeability is the dominant factor that controls unconventional natural gas production. Therefore, it is significant to understand the mechanisms of permeability evolution and the associated influential factors. Notably, coal reservoir exhibits multi-scale heterogeneity, and pore size spans from micrometer to nanometer, which affects gas transport and storage capability substantially. The heterogeneous pore structure of coal reservoir is characterized by multiscale pore size distribution (PSD) and the tortuous flow channel. Knudsen number ( Kn ) is defined as the ratio between the molecular free path and characteristic length, which is usually applied to describe flow regimes. The gas flow regimes include viscous flow ( Kn < 0.001 ), slip flow ( 0.001 < Kn 0.1 ), transitional flow ( 0.1 < Kn < 10 ) and free molecular flow ( Kn > 10 ). According to the definition of Knudsen number, pore size distribution determines the flow regimes in micropores when pore pressure remains constant. Therefore, the pore structure of coal has a significant impact on the apparent permeability of coal matrix. Different distribution functions have been employed to study the effect of PSD on apparent permeability, demonstrating that permeability is highly sensitive to the variation of the distribution function (Tian et al. 2017, Civan 2002). When the proportion of micropores is larger, the specific surface will increase, which will provide much more adsorption volume for coal seam gas (Tian et al. 2017). The original gas in place (OGIP) and corresponding sorption-induced swelling can be influenced substantially. For coal seam at different depths, coal swells or contracts greatly depend on PSD (Yang et al. 2010). Except for porosity, tortuosity of pore structure is an essential parameter for permeability prediction, which reflects the ratio between actual flow length and characteristic length of coal sample. According to the Kozeny-Carman model, there is a negative correlation between permeability and tortuosity(Walsh and Brace 1984). The theoretical investigation indicates that large tortuosity can increase the resistance of gas transport (Wang et al. 2017).
Rock properties are controlled by a combination of confining and pore pressure as defined by the effective stress law. The existence of effective pressure coefficients is very well known but rarely accounted for. A rock under confining pressure and with a fluid inside will respond to changes of the pressure /stress conditions depending on the frame structure of the rock, porosity, and compressibility of the pore fluid, among other critical components.