A method has been developed for using nonstatic pressure measurements directly in gas reservoir material balances composed of various energy mechanisms. Applying this method leads to simultaneous determinations of the reservoir ji history, gas in place, and other parameters relevant to water influx and effective compressibility.Well-known methods IA of determining average static pressure, p, have at least two shortcomings: (1) an estimation of reservoir shape and (2) an often-neglected implicit relationship between p and the viscositycompressibility product. Errors resulting from these deficiencies are minimized by the proposed method through a simple coupling of the well-known pseudosteady-state flow and material-balance equations. The solution of this coupling is obtained through nonlinear regression, and it allows simultaneous evaluations of gas initially in place, static pressure history, and several other reservoir parameters. These parameters can include the initial reservoir pressure, a stabilized gas-deliverability constant, the effective compressibility, aquifer diffusivity, and aquifer volume plus water-influx constants. The results of applying the method to six published cases are presented to illustrate the utility of the method.p determinations, and it provides simultaneous solutions of gas initially in place, ji history, water influx, and effective rock and connate water compressibilities. Other studies 7,8 have shown applications of nonlinear regression to solve water-influx, material-balance problems where the p history was a predetermined input requirement. An important result of Rossen's 8 work is a solution of the material-balance problem with cumulative gas production as the dependent variable. This formulation is also used in the method of this study, 209 G = gas initially in place, Mscf (10 3 std m 3 ) G p = cumulative gas production, Mscf (10 3 std m 3 ) h = net thickness, ft (m) J 0 = Bessel function of first kind, zero order J 1 = Bessel function of first kind, first order

Rodgers Jr., John S., Member AIME, This paper was prepared for the 41st Annual Fall Meeting of the Society of Petroleum Engineers of AIME, to be held in Dallas, Tex., Oct. 2–5, 1966. Permission to copy is restricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledgment of where and by whom the paper is presented. Publication elsewhere after publication in the JOURNAL OF PETROLEUM TECHNOLOGY or the SOCIETY OF PETROLEUM ENGINEERS JOURNAL is usually granted upon request to the Editor of the appropriate journal provided agreement to give proper credit is made. Discussion of this paper is invited. Three copies of any discussion should be sent to the Society of Petroleum Engineers office. Such discussion may be presented at the above meeting and, with the paper, may be considered for publication in one of the two SPE magazines. Abstract Pressure transient tests when properly conducted and interpreted, offer valuable information to any exploration, development or production venture. Determinations of hydrocarbon volume initially in place, stabilized reservoir pressure, distance to a reservoir rock or fluid discontinuity, distance to impermeable barriers, extent and orientation of a fracture system, porosity-thickness, completion efficiency, formation damage and permeability-capacity are often obtainable from one or more of the various types of pressure transient tests. Knowledge of these factors when considering alternate investment opportunities can enable prudent decisions. A pressure transient is a pressure gradient or disturbance created by altering the equilibrium of a reservoir system. This occurs not only when new wells are completed, but each time the flow rate of a producing well is altered. In pressure transient can be analyzed to reveal the above factors. The mathematical analysis can be facilitated if the transient is established under controlled conditions. Transient tests are usually designed to meet the requirements imposed by the mathematics describing fluid flow through a porous media for the reservoir system being considered. For the purposes of this paper, only radial systems are discussed because they have the most application. A pressure transient test in a hydrocarbon reservoir has been compared to throwing a stone in a pond of water. When a stone hits the water ripples move out radially at a fixed rate contingent on the environmental conditions. The size of the stone will affect the amplitude of the transient but amplitude will not alter the speed of its propagation. If the wave hits a barrier such as the edge of the pond, it will be reflected. A similar occurrence is observed in the reservoir when the rate of production from a well is altered. A pressure transient radiates from the wellbore at a velocity which is a function of both reservoir rock and fluid properties. As in the case of the pond, if the transient encounters an environmental alteration, it will be reflected to the well. The amplitude of the transient is not related to the radial velocity of the transient; therefore, the analysis of the transient is theoretically independent to its amplitude.

This paper was prepared for presentation at the 47th Annual Fall Meeting of the Society of Petroleum Engineers held in San Antonio, Tex., Oct. 8–11, 1972. Permission to copy is restricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledgment of where and by who the paper is presented. Publication elsewhere after publication in the JOURNAL paper is presented. Publication elsewhere after publication in the JOURNAL OF PETROLEUM TECHNOLOGY or the SOCIETY OF PETROLEUM ENGINEERS JOURNAL is usually granted upon request to the Editor of the appropriate journal provided agreement to give proper credit is made. provided agreement to give proper credit is made. Discussion of this paper is invited. Three copies of any discussion should be sent to the Society of Petroleum Engineers office. Such discussion may be presented at the above meeting and, with the paper, may be considered for publication in one of the two SPE magazines. Abstract An investigation is presented in regard to the influence of solution technique and instrument sensitivity on transient pressure analyses. One of the actual pressure buildup cases included illustrates the effects of two different pressure-gauge sensitivities on conventional pressure-gauge sensitivities on conventional evaluations of mobility, skin effect and extrapolated pressure. The results of these evaluations emphasize the importance of using sensitive pressure gauges in all types of transient pressure tests. Another area investigated includes the effect of solution technique and precision of pressure measurement on the analyses of pressure measurement on the analyses of transient data influenced by reservoir boundaries. Two theoretical pressure buildup cases and one actual pressure buildup case are analyzed to illustrate how solutions of distances to partially closing boundaries and boundary shape are obtained by an implicit technique. This method employs a formulation of the shut-in pressure in terms of the extrapolated pressure (at "infinite" shut-in time) and the pressure (at "infinite" shut-in time) and the effects of nonclosing boundaries. Application of this method leads to an implicit solution of the diffusion constant (in terms of boundary distance) by establishing a relationship between the precision of pressure measurement and an observed departure time. Theoretical buildup data are generated in terms of the extrapolated pressure, the diffusion constant and a selected pressure, the diffusion constant and a selected boundary shape and distances. Measured and theoretical buildup profiles are compared in a systematic variation of distances for each boundary shape selected. The minimum standard deviations for each shape are then compared to determine which shape(s) and respective boundary distances can be considered for a most probable reservoir description. For an actual case studied, it is shown that pressure data measured with insensitive gauges pressure data measured with insensitive gauges can have a large band of standard deviation which contains all the minimum or "best fit" deviations of the different boundary shapes. In these situations a reservoir description cannot be inferred from the analysis without supporting information from other sources.

Most of the published analytical studies on stabilized gas deliverability predictions are based on radial flow and a symmetric well location. This approximation gives erroneous results for lenticular drainage shapes or ones with the well located asymmetrically. The range of error expected from radial and symmetrical assumptions is investigated in this study with analytical solutions of a theoretical backpressure shift factor as a function of dimensionless time for various rectangular drainage shapes and well locations. For the theoretical reservoir studied, these solutions show that serious error is possible in determining stabilized deliverability from the assumption of a symmetric well location in a square drainage shape. Deliverability projections incorporating the backpressure shift factor are generated for two theoretical cases. Case I for a well in the center of a square-shaped reservoir demonstrated stabilization in six months Whereas stabilization occurred in four years for Case II, an 8 × 1 rectangular reservoir with the well located at an extreme end. All other data affecting hydrocarbon pore volume and petrophysical properties of these cases were held constant. When the theoretical cases are produced at capacity, 69 percent of the total gas-in-place is recovered in Case I, compared to 47.4 percent in Case II, during the first Eve years of production. A technique for calculating a backpressure shift factor as a function of time for a given drainage shape is presented in the paper. We conclude that reservoir geometry and well location should be taken into account for a realistic deliverability prediction. Introduction Since the publication by Mac Roberts in 1949, there have been numerous studies on determining stabilized deliverability of a gas well. Surprisingly, all of these investigations have focused on the deliverability of a well symmetrically located in a circular drainage area.