For several years the authors have felt the need for a source from which reservoir engineers could obtain fundamental theory and data on the flow of fluids through permeable media in the unsteady state. The data on the unsteady state flow are composed of solutions of the equation O'P + ~ oP = oP or' r Or at Two sets of solutions of this equation are developed, namely, for "the constant terminal pressure ca;;e" and "the constant terminal rate case." In the constant terminal pressure case the pressure at the terminal boundary is lowered by unity at zero time, kept constant thereafter, and the cumulative amount of fluid flowing across the boundary is computed, as a function of the time. In the constant terminal rate case a unit rate of production is made to flow across the terminal boundary (from time zero onward) and the ensuing pressure drop is computed as a function of the time. Considerable effort has been made to compile complete tables from which curves can be constructed for the constant terminal pressure and constant terminal rate cases, both for finite and infinite reservoirs. These curves can be employed to reproduce the effect of any pressure or rate history encountered in practice.Most of the information is obtained by the help of the Laplace transformations, which proved to be extremely helpful for analyzing the problems encountered in fluid flow. Tht' application of this method simplifies the mOTe tedious mathematical analyses employed in the past. With the help of Laplace transformations some original developments were obtained (and presented) which could not have been easily foreseen by the earlier methods.2~K or]
Piezohydraulic actuation is the use of hydraulic fluid to rectify the high frequency, small stroke oscillation movement of the piezoceramic stack into a unidirectional movement at a specific combination of stroke and force. In recent years, piezohydraulic pumps have been studied and developed by several groups. This paper presents two steady flow models based on different approximations of the hydraulic fluid. In the first model we assume a fully developed incompressible viscous flow, and in the second model we incorporate the compressibility of hydraulic fluid into our model. These two models are derived based on the energy equations for hydraulic flow in a circular pipe. Major and minor losses are identified and incorporated into the models. Assumptions and approximations were made to minimize computation effort while achieving good accuracy. A piezohydraulic actuation system with active valves is then built and tested. Comparison with test results shows that the simulations accurately predict system performance under loads at which value leakage is not present. The results reveal that friction losses due to viscosity are a major limitation of performance for the current test setup when operating at higher frequencies. Timing studies of the active vales show the valve timing is important to the performance of the system.
This paper is an analysis of the unsteady flow of a compressible fluid flowing radially to a well in a sand formation. The phenomenon of unsteady flow occurs as a result of fluid expansion. When the pressure in the formation is lowered, the fluids therein expand and the increase in volume imparts motion to the fluid which flows towards the region of lowest pressure in the formation. This process is continuous in the reservoir and extends further away from the well with increased production. By the ``equation of continuity,'' the solution for two specific cases of unsteady flow are derived. The first of these is that in which the fluid from a sand reservoir of limited size flows to a well in which the pressure at the level of the producing sand always remains constant. The variation of the pressure gradient, the rate of production, and the cumulative production with respect to time are given in Eqs. (11), (24), (29) and (30). The second case is that in which the flow of fluid to a well is such that the rate of production at the well is always constant. This case is derived by the assumption that the cylindrical body of sand, which is influenced by the well, is subject to a steady depletion of fluid, and in order that the rate remain constant at the well, fluid must flow into the reservoir from some extraneous source in increasing amounts. Eventually, however, the rate of flow of fluid into the reservoir becomes equal to the rate at which fluid is withdrawn from the well and steady flow is established in the sand. The equations for the pressure gradient and the decline of pressure at the well with respect to time are (39), (52) and (56).
In calculating negative skin, treating the skin as a zone of infinitesimal thickness leads to mathematical difficulties. These can be overcome by assuming an enlarged wellbore radius and using the same equations that apply for positive skin. Introduction Because of drilling, completion, and workover practices, the permeability around a wellbore generally is different from the permeability of the formation. The zone with the altered permeability is called "skin" and its effect on the pressure or flow behavior of the well is called "skin effect". Hawkins has shown that the radius and the permeability of this zone are related to the skin by: (1) If the permeability in the skin zone is less than that of the formation, the skin is positive; if it is more than that of the formation, the skin is negative. If the two permeabilities are equal, s is zero; that is, there is no skin. Van Everdingen and Hurst, have given mathematical solutions to the case of a zone of reduced permeability around the wellbore. This skin effect is illustrated in Fig. 1. They treat the positive skin as a zone of reduced permeability of infinitesimal thickness around the wellbore. When applied to a well with negative skin, however, their solutions lead to the calculated flowing well pressure, which is smaller than the formation pressure, i.e., an injection situation. This is a physical contradiction. In this paper, it is shown that this mathematical difficulty can be overcome by assuming an effective wellbore radius larger than the actual wellbore. Existing solutions are modified to include the effect of an enlarged wellbore radius, and thus enable the engineer to deal with negative as well as positive skins. Theoretical Development Van Everdingen and Hurst, through the use of Laplace transformation, obtained the following expression for transient fluid influx into a wellbore: (2) where tD, dimensionless time, is given by: (3) By use of the superposition technique, Eq. 2 can be expressed as (4) The values for QtD, reported extensively by van Everdingen and Hurst, are obtained from the following integral:(5) JPT P. 1483ˆ
This is a presentation of the diffusivity theory for the calculation of the water drive on an oil reservoir in which the history of reservoir pressure with time are the essential parameters for the determination of the rate and cumulative water encroachment into a field. The main body of the paper is essentially the application of the principles underlined here to actual field data. It consists of the work plots of the radial-flow case, and includes a discussion of an illustrated problem for which the water-drive calculations are treated in conjunction with the volumetric-balance equation. The appendix is a mathematical treatment of the linear, radial, and spherical cases of water influx into a reservoir.
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