A reservoir simulator modified to include non-Darcy flow and fractureclosure was used to demonstrate the effects of non-Darcy gas flow in ahydraulic fracture on well performance. Results illustrate the effects onthe gas-well productivity index and on the analysis of pressure builduptests. Introduction Laminar flow of fluid through porous media can bedescribed using Darcy's law:(1) This equation indicates that if the resistance (mu/k)remains constant, the pressure gradient (delta p/delta L) isproportional to the velocity of the fluid (v). However, when thevelocity is increased such that the flow is not laminar, thepressure drop will increase more than the proportional increase in velocity. Fancher et al. recognized this behavior and publisheda paper in 1933 that gave an analogy between the flow offluids through porous media and the flow of fluidsthrough pipe. Several authors, including Brownell andKatz and Tek, have since published methods forpredicting the laminar and turbulent regions of flow inporous media based on correlations similar to the Reynoldsnumber for flow in pipe. The generalized equation for flow through porousmedia may be represented by the following equationsuggested by Forchheimer. (2) If the constant (a) or velocity (i,) approaches zero. thenthe second term can be ignored and Eq. 2 is equal toDarcy's law (Eq. 1). Cornell and Katz reformulated Eq. 2 as follows: (3) In Eq. 3, the constant (a) was replaced by the product ofthe fluid density (rho) and the beta factor, which is acharacteristic of the porous medium. Several authors havepublished empirical correlations of the beta factor with theporosity and permeability of the porous media. Geertsma pointed out that the analogy betweenlaminar and turbulent flow of fluids in porous media to theflow of fluids in pipes could be misleading. Geertsmastated that turbulence does not actually occur in the smallpore systems of reservoir rock, and the cause of theincreased pressure gradients at high fluid velocities isinertial resistance. Consequently, Geertsma defined theparameter beta as the coefficient of inertial resistance. Geertsma's paper, therefore, has created a bit ofcontroversy concerning the terminology of the parameter betain the Forchheimer equation. In reality, the excess pressure gradients at high fluidvelocities can be caused by either turbulence or inertialresistance, or by a combination of the two, depending onthe particular pore configuration of the reservoir rockbeing considered. In this paper, the parameter beta isreferred to as the beta factor. Regardless of its name, the beta factoris a value used to calculate the correct pressure gradientsunder nondarcy flow conditions. Cooke investigated nondarcy flow in packed, hydraulically induced fractures. He noted that beta factors forfractures packed with multiple layers of sand had notbeen reported in the petroleum literature and suggested the following equation for calculating beta factors: (4) JPT P. 1169^

Laboratory studies of several factors affecting measurements of relativepermeability were made using the three-section plastic-covered core technique.Results show that the core assembly, properly constructed, will perform as asingle unit, and that the testing technique will, under suitable conditions ofpressure gradient, gas expansion, and migration of partial water saturation, permit measurement of flow characteristics not affected by technique.Wettability equilibrium is readily established in cores exhibiting strongwetting preference to water or oil when initially saturated with water.Laboratory tests must be conducted so that saturation changes represent thosethat occur in the reservoir. Immediate implications of saturation history arethat the possibility exists of increasing the displacement efficiency ofsolution gas drive reservoirs over the natural process, andresidual gassaturations following water flooding in gas or gas condensate reservoirs willbe 15 to 50 per cent pore space rather than 1 to 11 per cent as generallybelieved. Introduction Solutions of petroleum reservoir problems pertaining to productionperformance require the use of true relative permeability characteristics. Thisrelationship of fluid conductivity and saturation has been obtained byreservoir engineers in four ways, namely:From past gross reservoir performance and the extrapolation of this databased on experience,By using published fluid flow relationships obtained in laboratory studieson general type porous materials,By attempting a mathematical derivation of flow behavior, using someexperimentally obtained characteristics of reservoir rocks, andBy laboratory flow tests using representative rock samples of areservoir. The first three methods listed above have shortcomings which make their uselimited or questionable. Production characteristics of only certain processesare obtained from field data and these are not available at the beginning of areservoir's producing life, at which time they are desirable. It is fortuitousif general fluid flow characteristics obtained experimentally have accurateapplication to specific field problems. Also, it is felt that at this timethere is not sufficient knowledge of the flow behavior of oil, water, and gasin porous materials to enable applicable analytical description of this to bemade based on other measured rock characteristics. Measurement of relative permeability in the laboratory offers the only directmethod subject to adequate checking for determination of flow characteristicsapplicable to field problems. Primarily, this paper deals with laboratoryexperiments to establish the effects of several factors on the measurement ofrelative permeability and the practical significance of this knowledge. T.P. 3053

Theoretical and experimental investigations of a constant pressure gravitydrainage system are reported. Experimental data are presented to show thatrecovery to gas breakthrough by gravity drainage is inversely proportional torate. The gravity drainage reference rate, which is numerically equal to theso-called "maximum theoretical rate of gravity drainage" is shown tohave no particular significance from a recovery standpoint. Before this ratecan be used as a basis of comparison for recoveries, it is necessary that therelative permeability and capillary pressure characteristics and displacingfluid viscosities be identical for the systems compared. A method is presented by which accurate prediction of the performance of agravity drainage system can be made. Close agreement between experimental andcalculated drainage performance shows that steady state relative permeabilityand static capillary pressure data can be used to describe fluid displacementbehavior. The very wide range of liquid recoveries before gas breakthroughwhich result from production rate variation alone demonstrates the importanceof this factor in planning depletion of a gravity drainage reservoir.Calculated results are presented which show that little additional recovery canbe expected from a high pressure gravity drainage system between the times ofgas breakthrough and attainment of such high gas/liquid ratios as to makefurther pressure maintenance impractical. Introduction It has been recognized for some time that gravity forces play an importantpart in the recovery of oil from some types of reservoirs. Field experience hasshown that under certain conditions, gravity drainage can result in very highoil recoveries. Qualitative reasoning has led most engineers to the generalconclusions that:Where gravity drainage is important, the reservoir pressure should bemaintained by gas injection at the crest of the structure to prevent shrinkageof the oil in place and to keep a low viscosity so the oil can drain at thefastest possible rate.Recovery by gravity drainage is rate sensitive. A survey of the literature indicates that while considerable work has beendone on the effects of gravity in oil production problems, no satisfactorymethod of calculating the performance of gravity drainage reservoirs has beenreported. In the absence of any proven method of calculating reservoirperformance, the level at which pressure should be maintained and the rate ofproduction for most efficient operation has been open to debate. T.P. 3199

Flow tests on small core plugs have indicated that a large amount of gas istrapped and not recovered by water flooding a gas sand. Instead of 1 to 15 percent pore space, as is usually assumed, the residual gas saturation is 15 to 50per cent pore space, and is thus of the same magnitude as residual oil afterwater flooding oil sands. A thorough investigation was made to ascertain that large amounts ofresidual gas actually remain in reservoirs after a water flood and that thiscondition is not merely a laboratory phenomenon. In field experiments, theamount of gas left in a watered-out gas sand was measured by use of a pressurecore barrel and the residual gas saturation of two watered-out gas sands wasdetermined by electric log evaluation. In the laboratory, an investigation wasmade of factors that could possibly cause the value of residual gas saturationas measured on small core plugs to differ from that in the reservoir, and theeffect of these factors on the amount of residual gas saturation was studied.The factors studied include flooding rate, static pressure; temperature, samplesize and saturation conditions before flooding. All evidence established that arelatively high gas saturation is trapped in water flooded gas sands and thatthis residual gas saturation can be measured in the laboratory by tests onsmall core plugs. Introduction There has been general agreement among engineers that very high recovery ofgas could be obtained from natural reservoirs by water displacement. Gasrecoveries of 80 to 95 per cent of the original gas in place have become thenormal expectation in water drive fields. The assumption of high recovery hasbeen based on:low density and viscosity of gas compared with water;the erroneous assumption that the flow relationships in a gas-liquid systemwhere gas is the displaced phase will be the same as when it is the displacingphase. It has long been recognized that gas can flow at very low gas saturations(in the range of 1 to 15 per cent pore space) in systems where liquid is beingdisplaced by gas. By assuming the reversibility of this process, the conclusionwas reached that the residual gas saturation following water flooding of a gasreservoir would be the same (l to 15 per cent) as that at which gas firstflowed continuously as a displacing phase. T.P. 3279

American Institute of Mining, Metallurgical, and Petroleum Engineers, Inc. This paper was prepared for the 49th Annual Fall Meeting of the Society of Petroleum Engineers of AIME, to be held in Houston, Texas, Oct. 6–9, 1974. 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 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 discussions may be presented at the above meeting and, with the paper, may be considered for publication in one of the two SPE magazines. Abstract A study has been made of the flow behavior of fractured oil reservoirs produced by water displacement. A two-dimensional numerical model capable of simulating flow of water and oil in the matrix blocks as well as in the fractures has been developed. The validity of the model has been cheeked against data from a laboratory experiment involving a matrix-fracture system. Good agreement was observed between the laboratory and simulation results. By means of numerical simulation, the effects of production rate and fracture flow capacity on the production history and ultimate oil recovery of a fractured system have been evaluated. Results are presented for a single matrix-block system where the block is surrounded by horizontal and vertical fractures. Production rates ranging from 0.05 to 5 times Production rates ranging from 0.05 to 5 times the gravity reference rate of the matrix, and fracture flow capacities ranging from 0.1 to 10 times the flow capacity of the matrix are included in the investigation. At production rates much lower than the gravity reference rat the system behaves essentially as a nonfractured reservoir. It is also observed that for fracture flow capacities of the order of one-tenth of the matrix flow capacity, the effect of the fractures is negligible. At higher fracture flow capacities the water-oil ratio performance of the system becomes increasingly more sensitive to production rate. Water production starts much earlier with high fracture flow capacities and high production rates than it does from a nonfractured reservoir, and a large portion of the oil is produced at high water-oil portion of the oil is produced at high water-oil ratios. However, if the additional water can be handled economically, no oil is lost by high rate production. It is demonstrated that for a given fracture flow capacity, the producing water-oil ratio is a unique function of oil remaining in place and present producing rate. Thus, a reservoir can be produced at a high rate until the water-oil ratio becomes too high to handle. Then, reducing the rate causes the water-oil ratio to decrease to the value it would have had if all the oil had been produced at this lower rate. Introduction A significant number of petroleum reservoirs exist where discontinuities such as fractures or joints in the porous rock matrix are the main paths for transmitting fluids to the producing wells. In naturally fractured producing wells. In naturally fractured reservoirs, the matrix rock generally has a low permeability and one or more well-developed permeability and one or more well-developed fracture systems are present.