Published in Petroleum Transactions, AIME, Volume 207, 1956, pages 240–245. Abstract By means of a finite difference expansion a fluid flow field in cylindrical coordinates with axial symmetry, is simulated by a network of electrical resistors. A series of DC analog computing units, connected to the network boundary corresponding to the free surface effectively adjust the network parameters to satisfy simultaneously Laplace's equation and the boundary conditions. The system automatically "relaxes" to the correct boundary shape in a fraction of a second. A detailed application of the method to the water coning problem in oil wells is presented. Introduction The water coning problem in petroleum production is a prime example of a class of problems known as free surface problems. A free surface arises in fluid flow regions when the shape of the boundary of a single phase flow region is affected in some manner by the pressure or potential gradients within the region. When there exists an interface of two fluids of different densities, the shape of this interface will be determined by the relative velocities of the two fluids and will therefore comprise a free surface. Other free surface problems of interest to petroleum engineers include the gas coning, water injection, and gravity drainage problems. In each case it is desired to predict the shape of the boundary of the oil zone as a function of the flow rate, the well penetration and the physical parameters and dimensions of the system. Such problems do not lend themselves to the classical analytical treatments of boundary value problems because in this case the boundary of the potential field and the potential distribution within the field are interdependent and must be calculated simultaneously. A further difficulty arises from geometric considerations. Since the fluid flow is of the radial type in most cases, such effective methods as conformal transformation and the hodograph are not applicable. It is the purpose of this paper to present a method whereby free surface problems of this type may be solved rapidly to the desired order of accuracy by means of a combination of a network of electrical resistors and conventional analog computer units. The water coning problem is treated in some detail. Then the procedure for extending the technique to other free surface problems is indicated.