Both the Rawlins and Schellhardt and Houpeurt analysis techniques are presented in terms of pseudopressures. Flow-after-flow tests, sometimes called gas backpressure or four-point tests, are conducted by producing the well at a series of different stabilized flow rates and measuring the stabilized BHFP at the sandface. Each different flow rate is established in succession either with or without a very short intermediate shut-in period. Conventional flow-after-flow tests often are conducted with a sequence of increasing flow rates; however, if stabilized flow rates are attained, the rate sequence does not affect the test. Fig 1 illustrates a flow-after-flow test.

A wellhead choke controls the surface pressure and production rate from a well. Chokes usually are selected so that fluctuations in the line pressure downstream of the choke have no effect on the production rate. This requires that flow through the choke be at critical flow conditions. Under critical flow conditions, the flow rate is a function of the upstream or tubing pressure only. For this condition to occur, the downstream pressure must be approximately 0.55 or less of the tubing pressure.

Proper sizing and selection of an electrical submersible pump (ESP) system is essential to efficient and cost-effective performance. Selection and sizing of proper ESP equipment for a particular application should be based on a nine-step design procedure. This nine-step procedure helps the engineer design the appropriate submersible pumping system for a particular well. Each of the nine steps is explained below, including gas calculations and variable-speed operations. Specific examples are worked through in ESP design. The design of a submersible pumping unit, under most conditions, is not a difficult task, especially if reliable data are available.

In a dynamic calculation, there are two effects not considered in steady flow: fluid inertia and fluid accumulation. In steady-state mass conservation, flow of fluid into a volume was matched by an equivalent flow out of the volume. In the dynamic calculation, there may not be equal inflow and outflow, but fluid may accumulate within the volume. For fluid accumulation to occur, either the fluid must compress, or the wellbore must expand. When considering the momentum equation, the fluid at rest must be accelerated to its final flow rate.

Energy is the rate of doing work. A practical aspect of energy is that it can be transmitted or transformed from one form to another (e.g., from an electrical form to a mechanical form by a motor). A loss of energy always occurs during transformation or transmission. In drilling fluids, energy is called hydraulic energy or commonly hydraulic horsepower. Rig pumps are the source of hydraulic energy carried by drilling fluids.

One of the first mathematical tools a neophyte engineer learns is calculus. Many of the mathematical tools engineers use to evaluate and predict behavior, such as vibrations, require equations that have continuously varying terms. Often, there are many terms regarding the rate of change, or the rate of change of the rate of change, and so forth, with respect to some basis. For example, a velocity is the rate of change of distance with respect to time. Acceleration is the rate of change of the velocity, which makes it the rate of change of the rate of change of distance with respect to time.

To quantify formation damage and understand its impact on hydrocarbon production, one must have reasonable estimates of the flow efficiency or skin factor. Several methods have been proposed to evaluate these quantities for oil and gas wells. Multirate tests can be conducted on both oil and gas wells. In these tests, several stabilized flow rates, qi, are achieved at corresponding stabilized flowing bottomhole pressures, pwf. The simplest analysis considers two different stabilized rates and pressures.

This article summarizes the fundamental gas-flow equations, both theoretical and empirical, used to analyze deliverability tests in terms of pseudopressure. The four most common types of gas-well deliverability tests are discussed in separate articles: flow-after-flow, single-point, isochronal, and modified isochronal tests. Deliverability testing refers to the testing of a gas well to measure its production capabilities under specific conditions of reservoir and bottomhole flowing pressures (BHFPs). A common productivity indicator obtained from these tests is the absolute open flow (AOF) potential. The AOF is the maximum rate at which a well could flow against a theoretical atmospheric backpressure at the sandface.

Based on the results of many laboratories' research and various field experiences and observations, the following general guidelines are recommended.

The Laplace transform of the diffusion equation in radial coordinates yields a modified Bessel's equation, and its solutions are obtained in terms of modified Bessel functions. This page introduces Bessel functions and discusses some of their properties to the extent that they are encountered in the solutions of more common petroleum engineering problems. A solution of Bessel's equation of order v is called a Bessel function of order v. Of particular interest is the case in which λ ki so that Eq. 2 becomes Eq. 3 is called the modified Bessel's equation of order v. A solution of the modified Bessel's equation of order v is called a modified Bessel function of order v.