Well Test Analysis in Naturally Fractured Reservoirs using Elliptical Flow
Authors: Igbokoyi Alpheus Olorunwa , Tiab Djebbar 
Published: February 2016

Abstract
Some naturally fractured reservoirs can exhibit linear flow in the beginning as if it were hydraulically fractured. The linear flow may have been imparted as a result of limited connectivity in the well's drainage area, thus creating formation anisotropy and/or heterogeneity. The primary purpose of this study is to develop a welltest interpretation method to quantify anisotropy in naturally fractured reservoirs. The method, which is based on an elliptical flow model, is extended to hydraulic fractures with infinite conductivity.
Prior to this study, the half focal length in the elliptic model has been used as fracture's half length in hydraulic fracture in a vertical well. The same elliptic characteristic length also exists in a vertical well in naturally fractured reservoirs that is not hydraulically fractured. When a well is not hydraulically fractured, and it exhibits elliptical flow, the half focal length is referred to as the characteristic length or equivalent hydraulic length.
A reservoir with uniform characteristic length from various wells indicates homogeneus anisotropy. A reservoir with varying characteristic length from various wells indicates heterogeneuity. Relatively high characteristic length is an indication of high permeability. Therefore contour of characteristic length can indicate the rock quality domain in a naturally fractured reservoir.
The Warren and Root matrix pseudo-steady state flow model is assumed in this study. Through the analytical solution obtained in the elliptical coordinate; the linear and radial flow regimes of the fractures are observed. The transition/matrix flow regime and the total system radial flow regime are also observed.
Application was made to three field and one simulated cases. The estimation of formation properties is carried out by the use of type curve matching and Tiab direct synthesis technique (TDS). The results obtained are in good agreement

Literature survey
In 1959, Coats K. H.'s work1 on porous media having elliptical boundaries reveals error in making use of radial system to estimate production in an elliptical system. According to Coat, this error is inversely proportional to the magnitude of time. This suggests that the flow may not be radial from the beginning.
In 1961, Prats M. et al2 made use of elliptical flow to describe the flow in the vertical fracture in a vertical well. He plotted the pressure distribution around the well for various values of the ratio (Equation).k formation permeability.w fracture widthkf fracture permeabilityLf fracture half length
The flow around the wellbore is elliptical when this ratio is less than one as in the hydraulic fracture in a low permeability formation, and become circular for a ratio greater than one. The elliptical flow transits into radial flow at distances far away from the wellbore.
In 1964, Russel and Truit3 used numerical method to evaluate pressure drop at the wellbore of a vertically fractured well. The results obtained from the two-dimensional model did not yield linear or semi-log slope of 1.1515. Hence, the result is not useful in well test analysis.
In 1971, van Everdingen and Meyer4 made use of elliptical flow to solve the flow problem in vertical well with vertical fracture. Their result showed early linear but the long time solution gave semi-log slope of 0.576 instead of 1.1515. This then raises question about the validity of their solution.
In 1979, Kucuk F. et al5 presented an analytical solution to transient flow problem in an elliptical homogeneous reservoir. This work shows early linear flow followed by radial flow with semi-log slope of 1.1515. The duration of the early linear flow decreases with increase in the anisotropic factor. However, no application was made to welltest analysis.
Hale B. W.6 shows that elliptical flow equation can properly model linear, radial or transitional flow between linear and radial in a vertical fractured well. No detailed application was made to welltest analysis.