Interference Analysis for Anisotropic Formations - A Case History (includes associated paper 6406 )

Abstract

Many formations, such as channel sand, appear to exhibit simple ky-kxanisotropy. Directional permeability has an important effect onplanning fluid-injection oil recovery. The method discussed in thispaper uses data obtained during water injection to determine major andminor permeability axes and the orientation of the unknown permeability matrix. Introduction It has long been known that many formations appear toexhibit simple ky-kx anisotropy. This model also may beapplicable for formations containing trending fracturepatterns. Knowledge of directional permeabilityobviously would have an important effect on planningreservoir development, particularly for fluid injectionoperations. A recent study by Papadopulos outlines methods for interference analysis in anisotropic formations. Themethod is used here to analyze a field water-injection testto determine major and minor permeabilities and theorientation of the unknown permeability matrix. The method also may be extended to pressure falloff (orbuildup) interference analysis. The theory of flow of either fluid or heat through ananisotropic medium is well established. A brief reviewof this information is sufficient here. We consider that awell is produced at a constant volumetric rate in aninfinite, anisotropic medium. The formation has aconstant thickness and porosity, and the total systemeffective compressibility is constant. Collins has presentedperhaps the best-known solution to this problem in thearea of oil production technology. In the mid-1960's, aseries of papers appeared in the groundwater hydrologyliterature dealing with the problem of well testanalysis for anisotropic aquifers. Studies by Hantush, Papadopulos, and Walton are particularly notable. ThePapadopulos study appears to be well-suited for applicationto analysis of oil and gas well test data and serves asthe basis for this study. The purpose of this paper is topresent analysis of actual field data with the Papadopulosmethod in a form readily useable by petroleum engineers.The method is extended to falloff data. Theory The Collins solution for the pressure field caused by awell producing from an anisotropic reservoir is correct, but it assumes the directions of the major and minorpermeability axes are known and are aligned with thewell-location coordinate system. In the general well testanalysis situation, the direction of the major permeabilityaxis would be unknown. Fig. 1 shows the known welllocation x-y coordinate system with the unknown permeability axes, X-Y, oriented at some unknown angle 0.The pressure at (x, y, t) caused by a line-source well at theorigin was presented by Papadopulos: 2 h (pi - px, y, t)kxxkyy - kxy ----------------- =141.2 q B 1- Ei .....(1)2 kxx = 1/2{(kxx+kyy) + [(kxx-kyy)2+4kxy2] 1/2}....(2) kyy = 1/2{(kxx+kyy) − [(kxx-Kyy)2+4kxy2]1/2}.....(3) kxx − kxx= arctan ......................(4)kxy JPT P. 1290^