Interference Between Gravity Wells—Steady State Flowa

Under steady state conditions of flow, the seepage toward a single gravity well is governed by the Laplace Equation which may be written in terms of either the hydraulic head, the pressure head or the velocity potential. Although this equation is linear, the principle of superposition cannot be applied to sum up the individual effects in the case of a multiple gravity well system due to the variation of the flow domain under the effect of one or several wells. A method is presented allowing the use of the superposition principle in a restricted form. The superposition of the decrements of the base pressure heads than the initial heads before pumping is valid. Also the decrements in the areas of the pressure head diagrams across specific vertical sections than the original areas can be summed up together. The limitations of Dupuit's well formula are explained. The validity of that formula has been proven on the basis of the analysis of the hydraulic forces within the flow medium, an approach which is different than that given by Hantush and Charney. Furthermore, the derived equation is written in terms of the areas of the pressure head diagrams across vertical sections and termed as the Unified Well Formula because it has been proven that the same formula is also valid for artesian wells. The analysis of the hydraulic forces leads to the development of an equation for the free surface. This equation is then solved numerically in one iterative cycle. Due to the lack of simple available solutions, only one case, previously solved by the relaxation techniques, is compared with the presented method. The maximum percent difference in the depth of saturation within 82% of the flow region does not exceed 3.2% whereas in the remainder 18% of the flow region around the well, the percent difference varies between 2.63% to 4.67%. Even these differences do not really indicate actual errors due to the approximation implied in the relaxation method itself using a coarser grid. This distribution of the hydraulic head across a vertical section is assumed parabolic. Although Polubarinova-Kochina presented a mathematical proof which leads to the same conclusion, yet for the reasons explained in the text, the writer preferred to use this type of distribution as a valid physical assumption. The results of the analysis of each single well are applied to determine the pattern of the interference between several gravity wells. By means of the presented approach, the resultant values of the depths of saturation can be obtained on the basis of the explained restricted procedure, of superposition. The hydraulic potential distribution within the flow medium of a multiple gravity well system can also be obtained. The assumption of the parabolic hydraulic head distribution is maintained in analyzing a group of wells. It is recommended to establish a proper computer program covering a grid system that encompasses all the wells and their individual influence regions in a certain well field. In the entire analysis, Dupuit's assumptions are eliminated. However, the two main assumptions in the given analysis are: (a) the parabolic distribution of the hydraulic head across a vertical section within the flow medium, and (b) the elimination of the circumferential velocities. These two introduced assumptions are -i in the writer's opinion – practically valid.