Stochastic Modeling of Fractured Gas Wells Completed in Low Permeability Noncontinuous Linticular Reservoirs (1981)

A mathematical model is presented which forms the basis for a reservoir simulator that can be used to assist in interpretation and prediction of the performance of hydraulically fractured gas wells performance of hydraulically fractured gas wells completed in noncontinuous sand lense formations. Equations are derived to simulate both reservoir and well performance. They are general in that the following effects are considered:noncontinuous sand lensesthree-dimensional flow in the reservoir and one-dimensional flow in the fracture.non-Darcy flow in the reservoir and fracturewellbore and fracture storageformation damage on the fracture facefrictional pressure drop in the production string andKlinkenberg effect.An operational two dimensional gas reservoir simulator has been developed which includes these effects and has been verified for some selected test cases.Introduction. Considerable research has been reported in the literature concerning gas reservoir simulator development. The pertinent papers on this subject are listed in the reference. In assessing the merits of the research reported it was evident that the greater majority of the work was concerned with the modeling and simulation of the performance of blanket sand reservoirs. However, the work of Knutson is of particular significance since it geologically treats the particular significance since it geologically treats the gas bearing formations as a noncontinuous entity. Other work found in the literature which considered noncontinuous sand lenses was that of Gidley, et al, who used Knutson's work as a guide for manipulating a conventional reservoir simulator to obtain a history match for performance data from wells completed in low-permeability gas formations stimulated by massive hydraulic fracturing techniques. To obtain a reasonable history match, the net sand thickness was reduced to 25 percent of the actual value, and the conclusion was made that this was a reasonable correlation to the 30 to 35 percent sand lense content in the western tight sands region, as reported by Knutson Kennedy and McElhiney applied a continuous reservoir simulator to analyze and predict the performance of gas wells completed in the western tight sands region. These producing formations are noncontinuous in nature, hence Kennedy and McElhiney reported discrepancies in their simulator results.Since considerable quantities of natural gas are contained in noncontinuous lenticular reservoirs, the research reported in this paper represents an extension of existing stat-of-the-art reservoir/well ulator models to this very important class of gas reservoir engineering problems.Additional considerations that are included in this work and to the authors knowledge have not been reported previously is non-Darcy flow in the formation damaged region near the fracture and fracture storage effects. Although the former may not be important, the latter effect may prove to be significant in low permeability lenticular formations. permeability lenticular formations.THEORETICAL CONSIDERATIONS GOVERNING SINULATOR EQUATIONS. The assumption and mathematical equations which describe flow in the reservoir, fracture and productio string are as follow:Noncontinuous Sand Lense Effects. To include the noncontinuous nature of the gas-bearing sand lenses, a statistical analysis of the geology of the producting formations was conducted. Using random variable theory and the fact that the gas bearing sand lenses are randomly distributed within the area of interest, a reservoir fracture model was developed, which gives an effective fracture length XFP that is proportional to the actual fracture length XF. The constant of proportionality is equal to the sand shale ratio on an areal basis in accordance with the work of Knutson. In arriving at XFP it was assumed that any fracture artifically induced by hydrauli fracturing technology would create a fracture whose axis was orientated N 70W in accordance with the published results of Bredenhoeft, et al. published results of Bredenhoeft, et al. p. 483