Exact analytical solutions for moving boundary problems of one-dimensional flow in semi-infinite porous media with consideration of threshold pressure gradient
By defining new dimensionless variables, nonlinear mathematical models for one-dimensional flow with unknown moving boundaries in semi-infinite porous media are modified to be solved analytically. The exact analytical solutions for both constant-rate and constant-pressure inner boundary constraint p...
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| Published in | Journal of hydrodynamics. Series B Vol. 27; no. 4; pp. 542 - 547 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Singapore
Elsevier Ltd
01.08.2015
Springer Singapore |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1001-6058 1878-0342 |
| DOI | 10.1016/S1001-6058(15)60514-5 |
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| Summary: | By defining new dimensionless variables, nonlinear mathematical models for one-dimensional flow with unknown moving boundaries in semi-infinite porous media are modified to be solved analytically. The exact analytical solutions for both constant-rate and constant-pressure inner boundary constraint problems are obtained by applying the Green＇s function. Two transcendental equations for moving boundary problems are obtained and solved using the Newton-Raphson iteration. The exact analytical solutions are then compared with the approximate solutions. The Pascal＇s approximate formula in reference is fairly accurate for the moving boundary development under the constant-rate condition. But another Pascal＇s approximate formula given in reference is not very robust for constant-pressure condition problems during the early production period, and could lead to false results at the maximum moving boundary distance. Our results also show that, in presence of larger TPG, more pressure drop is required to maintain a constant-rate production. Under the constant-pressure producing condition, the flow rate may decline dramatically due to a large TPG. What＇s more, there exists a maximum distance for a given TPG, beyond which the porous media is not disturbed. |
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| Bibliography: | 31-1563/T threshold pressure gradient, moving outer boundary, analytical solution, porous media, transient pressure analysis, flowrate WANG Xiao-dong , ZHU Guang-ya , WANG Lei （1. School of Energy, China University of Geosciences （Beijing）, Beijing 100083, China, 2. PetroChina Research Institute of Petroleum Exploration and Development, Beijing 100083, China） By defining new dimensionless variables, nonlinear mathematical models for one-dimensional flow with unknown moving boundaries in semi-infinite porous media are modified to be solved analytically. The exact analytical solutions for both constant-rate and constant-pressure inner boundary constraint problems are obtained by applying the Green＇s function. Two transcendental equations for moving boundary problems are obtained and solved using the Newton-Raphson iteration. The exact analytical solutions are then compared with the approximate solutions. The Pascal＇s approximate formula in reference is fairly accurate for the moving boundary development under the constant-rate condition. But another Pascal＇s approximate formula given in reference is not very robust for constant-pressure condition problems during the early production period, and could lead to false results at the maximum moving boundary distance. Our results also show that, in presence of larger TPG, more pressure drop is required to maintain a constant-rate production. Under the constant-pressure producing condition, the flow rate may decline dramatically due to a large TPG. What＇s more, there exists a maximum distance for a given TPG, beyond which the porous media is not disturbed. |
| ISSN: | 1001-6058 1878-0342 |
| DOI: | 10.1016/S1001-6058(15)60514-5 |