A mathematical model is presented and solved for determination of the saturation distribution and pressure in a radial gas-storage reservoir. The model consists basically of two parts:the growing gas-bubble core, andthe surrounding aquifer. Since the total pressure at the injection well is a function of the two-phase flow in the gas bubble and the unsteady single-phase flow in the aquifer, the resistance to flow in both zones was taken into consideration. The assumptions involved for both the radial equivalent of the Buckley-Leverett two-phase-flow equation and the injection-pressure equation are as follows:the geometry is radial,the gas bubble is free to expand or contract,compression or expansion of the gas within the bubble may occur at the beginning of a time step,the fluids are immiscible,water is incompressible within the gas-storage region whereas it is compressible outside of this region,a stable gas-water interface exists andgas injection occurs at a constant rate or a series of constant rates. The mathematical model was solved numerically using an IBM 650 computer. A comparison is presented between the predicted results of the model, the results assuming steady-state flow and the actual initial injection-pressure history of an operating reservoir. Using the initial field pressure for a basis, the average deviation between the predicted pressure and the actual field pressure was less than 4.3 per cent. Introduction Underground storage of natural gas in abandoned oil fields, in abandoned coal mines, in caverns and in aquifers has had varying degrees of success. The lack of these first three facilities in the vicinity of most major gas marketing areas is leading more and more to storage in virgin aquifers. For the most part, this storage in aquifers has led to the study upon which this paper is based. Several excellent papers have been published discussing the problems involved in the underground storage of natural gas; however, to these writers' knowledge, none of them considered the two-phase flow of fluids in the gas bubble and the fact that the gas bubble will grow or shrink in size depending upon the injection and withdrawal history of the reservoir. Not only will both of these factors influence the required injection pressure, but also they will have a definite bearing upon the amount of water production occurring upon withdrawal from the storage area. To solve this problem on an IBM 650 computer, several simplifying assumptions were made. Only the drainage portion of the relative-permeability curve was considered, thereby neglecting any hysteresis effects introduced during an imbibition cycle. The quantity of gas withdrawn during the period of the field study was only a small percentage of the total gas in place; therefore, this assumption is justified. For the problem studied in this case, the combined effects of capillarity and gravity were assumed to be negligible. This assumption becomes less valid as the formation thickness or the average pore diameter increases markedly. The justification of these assumptions should be studied in each case. THEORY This mathematical model for the prediction of saturation distribution and pressure for a radial gas-storage aquifer is based upon the equations for radial two-phase fluid flow and for radial, unsteady-state, single-phase fluid flow. The two-phase flow is considered to take place in a "core" whose radius is equal to the maximum radius that the gas zone will attain, as indicated in Fig. 1. The injection-withdrawal history is approximated by a series of constant flow rates. Within the "core", gas is assumed to behave as a semi-compressible fluid; that is, the gas is assumed to have a constant density based on the average gas-zone pressure for the flow period. Between each constant-rate time increment, the gas density is allowed to change. This involves an iterative solution of the pressure equation. The liquid phase in this zone is considered to be incompressible, whereas it behaves as a compressible liquid outside of this region. JPT P. 1389^

Thesis Approved:De~of the Graduate College ii PREFACE This dissertation is c<;>ncer:qed with the development of a nup,erical simulator for the three-dimensional flow of two immiscible, compressible :fluid phases in porous media, including the effects of gravity, capillarity, heterogeneity, and hysteresis in relative permeability and capillarity.The primary objective of the research project was the development of an evaluation-prediction tool for gas recovery from gas fields under active water drive, cyclic gas movement in aquifer gas storage and, in particular, to determine the quantity of residual gas remaining in an aquiferreservoir complex at abandonment.The secondary objective was the investigation of the qualitative and quantitative effects of rigorol.ls representation of hysteretic relative permeability and capillarity, on the numerically computed saturation and pressure distributions.A set of two, second-order, non-linear, parabolic partial differential equations governing two-phase, hysteretic flow were derived. An analogous system of difference equations were developed for numerical solution by an alternating direction implicit iterative method. Semiempirical r.elationships spanning the domain of relative permeability and capillarity data were derived from laboratory data and statistical-capillaric models of the porous media.iii Parametric ~ensitivity studies of the effects of relative permeability and capillarity on the numerical calculation of fluid saturation-pressure distributions were made by simulation of hypothetical systems representing typical prototype reservoirs.

Pre-set or off-depth composite plugs can cause significant non-productive time for a well operator. In the past, fracturing operations using a composite frac or bridge plug that has been pre-set or set off depth required a coiled tubing unit or workover rig to drill the plug out. Then, the well operator could resume the fracturing job or access the wellbore below the plug. However, as this paper demonstrates, composite plug milling via wireline using a tractor and a tractor-based milling tool is a faster, safer, and more cost-effective solution. In a shale well located in the northern panhandle of West Virginia, a composite frac plug was set off- depth. Prior to mobilizing the tractor-based solution to location, the operator attempted pumping approximately 60,000 pounds of sand to sand-cut the off-depth frac plug out of the well. The sand cutting, though, did not work because perforations above the frac plug took the sand. Other tubing-based solutions required more mobilization time and complex logistics for rigging down and/or moving equipment on location. Therefore, the operator chose a wireline-based method for ease of operation, reduced HSE risk, and cost savings. The tractor took 50 minutes to drive down 1718 ft in the lateral to the plug. The milling tool milled the top slips on the frac plug in approximately nine hours, and the tractor then pushed the plug 222 ft downhole on top of the previous frac plug. The total time rigged up on the well was 14 hours, and the total time on location was 18 hours. Although this wireline-based plug-milling method takes several hours to mill a plug, the rig-up and execution is simpler than conventional methods, and associated HSE risks on the wellsite are greatly reduced. The ability to effectively release plugs via wireline provides well operators with another option to complete their objectives, especially when tubing-based methods often take many days or weeks to mobilize at substantial cost to operators.

The extrapolation of oil well production decline data 'intothe future has long been accomplished through the use of semi-log and log-log plots of the data. These conmonly used methods encompass a family of hyperbolic decline curves. The object of such extrapolations is to determine the future production capacity of the oil producing unit.

In this paper, simulation studies together with actual field data are used to show that much water is unnecessarily confined in the interwell space as a result of the conventional multi-well injection operations in aquifer gas storage. Reasons are given for the universally observed difficulty in increasing the thickness of the gas bubbles.It is shown that, after many years of development and operation, the average gas saturation of the invaded pore space has limiting values closer to 50 percent rather than the 70-80 percent expected by undeserved reliance on the full gravity segregation of gas and water.To overcome the problems stated above, the authors recommend the use of pressure relief by water pumping during the development phase of aquifer storage in moderate-to-Iow permeability, relatively thin formations. The purpose is to: 1) increase early injection volumes for faster development, 2) reduce the inefficient invasion of the aquifer by the gas front, J) increase the gas saturation of the interwell space to optimize storage space per unit bulk rock, and 4) increase well deliverability.