Although there have been many analytical studies on pressure-transient behavior of hydraulic fracture systems, no single analytical solution capable of describing both vertical and horizontal fracture transient state behaviors has been developed. The purpose of this work is to develop a single analytical solution that is robust enough to fit this need. This paper presents a type curve solution for a well producing from a solid bar source in an infinite-acting reservoir with impermeable upper and lower boundaries. Computation of dimensionless pressure reveals that the pressure-transient behavior of any hydraulic fracture system is governed by two critical parameters (i) aspect ratio, m, and (ii) dimensionless length, LD. Analysis of a typical log-log plot of pwD vs. tDxf indicated the existence of four distinct flow periods (i) fracture fill-up period causing a typical storage dominated flow, (ii) vertical linear flow period, (iii) transition period, and (iv) radial flow period. As aspect ratio tends to zero, the fracture fill-up periods disappear resulting in typical fully/partially penetrating vertical fracture pressure response. This analytical solution reduces to the existing fully/partially penetrating vertical fracture solution developed by Raghavan et al1 as aspect ratio tends to zero, and a horizontal fracture solution is obtained as aspect ratio tends to unity. This new horizontal fracture solution yields superior early time (tDxf < 10–3) solution compared with the existing horizontal fracture solution developed by Gringarten and Ramey2, and shows excellent agreement for tDxf > 10–3. Introduction Hydraulically fractured wells and horizontal well completions are intended to provide a larger surface area for fluid withdrawal and thus, improve well productivity. This increase in well productivity is usually measured in terms of negative skin generated as a result of a particular completion type. Hydraulic fractures leading to horizontal or vertical fractures could produce the same negative skin effect as a horizontal well, but possibly different transient pressure response; hence, having a good understanding of the transient behavior of hydraulic fractures systems and horizontal well completion is very vital for accurate interpretation of well test data. The orientation of hydraulic fractures is dependent on stress distribution. The orientation of fracture plane should be normal to the direction of minimum stress. Since most producing formations are deep, the maximum principle stress is proportional to the overburden load. Thus, vertical fractures are more common than horizontal fractures. The only difference between a vertical and a horizontal fracture system is the orientation of the fracture plane; a vertical fracture can be viewed as parallelepiped with zero width, while a horizontal fracture, as a parallelepiped with zero fracture height. This same argument can be extended to horizontal well completions; a horizontal wellbore can be viewed as a parallelepiped with the height and width equal to the wellbore diameter. This configuration makes a horizontal well completion behavior like a coupled fracture system made up of both vertical and horizontal fracture systems. Considering the similarity in the physical models, one will expect a single analytical solution can be developed for hydraulically fractured (vertical and/or horizontal) well and horizontal well completions. The primary purpose of this work is to present a general analytical solution for describing the transient pressure behaviors of (i) vertical fracture system, (ii) horizontal fracture system, and (iii) horizontal well or drainhole. New physical insights into the critical variables that govern the performance of these completions are also provided.

Conventional horizontal well transient response models are generally based on the line source approximation of the partially penetrating vertical fracture solution[1].These models have three major limitations:wellbore pressure is computed at a finite radius outside the source; it is impossible to compute wellbore pressure within the source,it is difficult to conduct a realistic comparison between horizontal well and vertical fracture productivities, because wellbore pressures are not computed at the same point, andthe line source approximation may not be adequate for reservoirs with thin pay zones. This work attempts to overcome these limitations by developing a more flexible analytical solution using the solid bar approximation. A technique that permits the conversion of the pressure response of any horizontal well system into a physically equivalent vertical fracture response is also presented. A new type curve solution is developed for a horizontal well producing from a solid bar source in an infinite-acting reservoir by means of Newman's product solution[2]. Analysis of computed wellbore pressures reveals that error ranging from 5 to 20% was introduced by the line source assumption depending on the value of dimensionless radius (rwD). Computations show that for rwD = 10–4 the transient response of a horizontal well is identical to that of a partially penetrating vertical fracture system, and for rwD = 0.01 the transient response of a horizontal well is indistinguishable from that of a horizontal fracture system. Type-curve plots for the ranges 0.01 = dimensionless length (LD) = 10, and 10–4 =rwD = 1.0 are presented. A dimensionless rate function (ß-function) is introduced to convert the transient-response of a horizontal well into an equivalent vertical fracture response. A step-wise algorithm for the computation ofß-function is developed using Duhamel's principle. This provides an easier way of representing horizontal wells in numerical reservoir simulation without the rigor of employing complex formulations for the computation of effective well block radius. Introduction Conventional models for horizontal well test analysis were mostly developed during the 1980s. The rapid increase in the applications of horizontal-well-technology during this period led to a sudden need for the development of analytical models capable of evaluating the performance of horizontal wells. Ramey and Clonts[3] developed one of the earliest analytical solutions for horizontal well test analysis based on the line source approximation of the partially penetrating vertical fracture solution. Conventional models [4–16] assume that a horizontal well may be viewed as a well producing from a line source in an infinite-acting reservoir system. These models have three major limitations:wellbore pressure is computed at a finite radius outside the source; it is impossible to compute wellbore pressure within the source,it is difficult to conduct a realistic comparison between horizontal well and vertical fracture productivities, because, wellbore pressures are not computed at the same point,the line source approximation may not be adequate for reservoirs with thin pay zones. The increased complexity in the configuration of horizontal well completions and applications towards the end of the 1980s made us question the validity of the horizontal well models and the well-test concepts adopted from vertical fracture analogies. In the beginning of the 1990s a new wave of developing horizontal-well solutions[17–27] under more realistic conditions emerged. As a result, some contemporary models were developed to eliminate the limitations of the earlier horizontal well models. However, the basic assumptions and methodology employed in the development of these new set of solutions have remained relatively the same as those of the earlier models. Ozkan[28] presented one of the most compelling arguments for the fact that horizontal wells deserve genuine models and concepts that are robust enough to meet the increasingly challenging task of accurately evaluating horizontal well performance. Ozkan's work presented a critique of the conventional and contemporary horizontal-well-test analysis procedures with the aim of establishing a set of conditions when the conventional models will not be adequate and the margin of error associated with these situations.

The rising portion of plunger cycle makes use of some type of designed sealing mechanisms on the plunger. These sealing techniques reduce the amount of gas that bypasses (leakage) the surfacing plungers. When falling, many plungers have mechanisms designed to allow increased leakage or slippage enabling the plungers to fall faster. Modeling techniques, for leakage about a rising plunger, shown in this paper are shown as analogous to orifice type flow restriction. For fast falling plungers the model is developed is similar to objects experiencing drag in a field of gas velocity. Data collected for this type of modeling is presented from suspension tests and confirmed with dynamic test data. Model ratification is done with some dynamic test data. Special two-piece and conventional plungers are hereby modeled from suspension and dynamic testing. The results should help the operators to select specific plunger hardware for specific conditions and should assist in modeling plunger cycles. Introduction Plunger lift is a common artificial lift method of producing liquids from a gas well to improve gas flow, usually without the addition of any outside energy or extra gas. The need for plunger lift arises as the reservoir pressure decreases and lower gas velocity fails to lift liquid from the well. The objective of plunger lift is to keep the wellbore free of liquids and associated pressure drop by lifting liquids on an intermittent basis to the surface. The feasibility of plunger lift is widely discussed in the literature[1–5]. This paper chiefly will deal with modeling and predicted results for plunger rise, plunger fall and also the lifting of liquid slugs over the plunger. The direct outcome of these results is to help assist the operators on the plunger hardware to be selected and in the set-up of the duration of the plunger cycles. Gas slips upward around the plunger when it rises in the wellbore during the plunger cycle. The percentage of leakage5 compared to the gas production is relatively small. This leakage can gradually reduce the pressure under the plunger and reduce lifting energy if too much gas slips past. However, conventionally the plunger is designed with sealing contours or mechanisms to prevent the gas from underneath the plunger to leak to the liquid slug above it. Considered here also, is the modeling of the two-piece plunger and the conventional ‘sealing mechanism’ used with plungers for the rising and falling portion of the plunger cycle. In addition, modeling for lifting of the liquid slug is also provided showing how effects of the liquid slug size can affect the average rise velocity of the plunger along with other parameters. The details can be found in the following sub sections. These new considerations in modeling help provide a better understanding, of plunger cycles and operation. Data obtained from experimental runs from a test site was input into the model developed. The experimental data was obtained by suspension and dynamic testing. The results from the models were in the approximate range of measured data, thus validating the model. Two-Piece Plunger. The plunger consists of a hollow cylindrical piston and a ball below. The hollow cylindrical piston could be changed in length, material used, thickness, size and number of grooves depending upon usage but is usually a fixed configuration with various materials available. The two-piece plunger cycle[5,6] typically requires about 5 to 10 seconds of shut-in time and the well is producing even when the plunger components are falling to the bottom of the tubing. The model developed is presented in Appendix A. Fig. D-1 shows the various types of two-piece plungers currently being used in the industry and Fig. D-2 shows the mechanical components. The shifting rod seen in Fig. D-2 generally has a taper to it with large diameter towards the bottom of the rod. This helps facilitate holding the hollow cylinder at the top while the well is producing.

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TX 75083-3836 U.S.A., fax 01-972-952-9435. AbstractUsing adiabatic compression of gas in the pump, a model is shown that can be used to calculate the down-hole pump dynamometer card for various degrees of pump fill of liquids and gas at various degrees of pressure. The load release portion of the card is emphasized. It is shown how the lower pressure gas in the pump promotes what is commonly termed fluid pound or slap and higher pressure gas in the pump promotes what is termed gas interference.The equations that are needed to model these effects for inclusion into wave equation pump models are presented and example calculated pump cards are shown, calculated from wave equation simulations. Since even with so-called fluid pound, gas is first compressed before the plunger encounters the mostly incompressible fluid in the pump, the traveling valve always encounters compressed gas sufficient to open the valve before the plunger "hits" the fluid. This is true as long as gas is in the barrel when incomplete fluid fill occurs. Given this fact, the time step in wave equation solution is examined as the load release in the so called "fluid pound" can happen over a very short time and longer time steps in the wave equation simulation could mask forces from short term forces.The effects of rapid load release on forces in the rods are studied vs. what the forces might be due to the commonly supposed forces from impact with the fluid when "fluid pound" is modeled. Other parameters such as pump fillage, intake pressure and sinker bars are examined for their effects on calculated rod compression