Gas diffusion in a cylindrical coal sample – A general solution, approximation and error analyses Li Yaobin a,b , Xue Sheng a,c,d,⇑ , Wang Junfeng d , Wang Yucang c , Xie Jun c a Key Laboratory of Integrated Coal Exploitation and Gas Extraction, Anhui University of Science and Technology, Huainan 232001, China b School of Energy and Safety, Anhui University of Science and Technology, Huainan 232001, China c CSIRO Earth Science and Resource Engineering, Kenmore 4069, Australia d School of Mining Engineering, Taiyuan University of Science and Technology, Taiyuan 030024, China article info Article history: Received 12 May 2013 Received in revised form 15 June 2013 Accepted 20 July 2013 Available online 3 January 2014 Keywords: Gas content Lost gas Gas diffusion Cylindrical coal sample Approximation Error analysis abstract The analytical mathematical solutions of gas concentration and fractional gas loss for the diffusion of gas in a cylindrical coal sample were given with detailed mathematical derivations by assuming that the dif- fusion of gas through the coal matrix is concentration gradient-driven and obeys the Fick’s Second Law of Diffusion. The analytical solutions were approximated in case of small values of time and the error anal- yses associated with the approximation were also undertaken. The results indicate that the square root relationship of gas release in the early stage of desorption, which is widely used to provide a simple and fast estimation of the lost gas, is the first term of the approximation, and care must be taken in using the square root relationship as a significant error might be introduced with increase in the lost time and decrease in effective diameter of a cylindrical coal sample. Ó 2014 Published by Elsevier B.V. on behalf of China University of Mining & Technology. 1. Introduction Gas content in a coal seam is commonly used in coal mine safety such as gas emission control and gas outburst control as well as coal seam methane resource assessment and recovery applica- tions. The gas content is usually measured with either an indirect method or a direct method. The indirect method is based on empir- ical correlations or laboratory derived sorption isotherm gas stor- age capacity data. The direct method is based on observations of gas release from newly obtained samples, and it typically involves extracting a coal sample (often core sample), enclosing it in a sealed container and measuring the volume of gas released. As the direct method provides a fast in situ estimation of gas content, it is widely used in the coal industry and coal seam gas industry. With the direct method, the total gas content of a coal sample is made of three parts: lost gas, measurable gas, and residual gas [1–5]. The lost gas (Q 1 ) is the gas lost from the sample, subsequent to its being removed from its in situ position and prior to its containment in an airtight desorption canister. The measurable gas (Q 2 ) is the gas desorbed at atmospheric pressure from the non-pulverized coal sample. The residual gas (Q 3 ) is the gas still contained in coal at one atmospheric pressure. While Q 2 and Q 3 can be directly measured, Q 1 has to be estimated. The Q 1 estimation method was firstly described in a paper writ- ten by Bertard et al. [6]. It was stated in the paper that early in the desorption process the volume of gas released from coal was pro- portional to the square root of time, however no details were given as how the relationship was theoretically derived except mention- ing that it was based on kinetics of gas desorption from coal. Since then this square root relationship has been widely used as a standard lost gas estimation method, which is indicated by US Report of Investigation 7767 (1973), Australian Standard AS 3980-1999 (1999) and Standards of China (2009) [7–9]. However, the relationship has been found to be significantly dependent on a number of factors such as sample retrieval time, physical charac- ter of the sample, and the type of drilling fluid [10,11]. This raises the questions of how the relationship was theoretically derived, its validity, its applicable conditions, and its error. To answer some of the questions and improve the accuracy of estimation, this paper gives detailed derivations of a general math- ematical solution for the diffusion of gas in a cylindrical coal, an approximation solution, error analyses of the approximation and its application in the Q 1 estimation. 2095-2686/$ - see front matter Ó 2014 Published by Elsevier B.V. on behalf of China University of Mining & Technology. http://dx.doi.org/10.1016/j.ijmst.2013.12.012 ⇑ Corresponding author. Address: CSIRO Earth Science and Resource Engineering, Kenmore 4069, Australia. Tel.: +61 7 3327 4443. E-mail address: sheng.xue@csiro.au (S. Xue). International Journal of Mining Science and Technology 24 (2014) 69–73 Contents lists available at ScienceDirect International Journal of Mining Science and Technology journal homepage: www.elsevier.com/locate/ijmst