Unsteady-State Pressure Distribution Created By a Directionally Drilled Well
Authors: Cinco H. , Miller F.G. , Ramey H.J. 
Published: December 2007

Analysis of a solution derived to study, the unsteady-state pressure distribution created by a directionally drilled well indicates that the slant of a fully penetrating well creates a negative skin effect that is a function of the angle of slant and the formation thickness. Calculation of this pseudoskin factor permits evaluation of the actual well condition.

Introduction
Many methods have been developed to analyze transient wellbore pressure data to determine the formation permeability, porosity, average pressure, and well permeability, porosity, average pressure, and well condition. These methods usually are based on solutions of unsteady-state flow problems that consider a fluid flowing toward a fully penetrating well that is perpendicular to the upper and lower formation boundary perpendicular to the upper and lower formation boundary planes. Actually, most wells do not penetrate the planes. Actually, most wells do not penetrate the producing formation perpendicularly. Instead, there is a producing formation perpendicularly. Instead, there is a certain angle between the normal to the formation plane and the well axis, such as when a vertical well penetrates a dipping formation or when a directionally drilled penetrates a dipping formation or when a directionally drilled well penetrates a horizontal formation. These kinds of wells are called "slanted wells." Although such wells are common, there appears to have been only one study of the performance of such completions.
Roemershauser and Hawkins studied steady-state flow in a reservoir producing through a fully penetrating, slanted well using an electrical model. They considered a circular reservoir of finite extent and concluded that the slant of a fully penetrating well causes an increase in the well productivity. The increase in well productivity results from the decrease in the resistance productivity results from the decrease in the resistance to flow around the wellbore caused by an increase in the producing-interval area exposed to flow. This increase producing-interval area exposed to flow. This increase in well productivity indicates that a fully penetrating, slanted well creates a negative skin effect. Roemershauser and Hawkins graphed the increase in well productivity vs the angle of slant of the well. There productivity vs the angle of slant of the well. There appears to have been no study of the unsteady-state performance of slanted wells. performance of slanted wells. Mathematical Derivation
The unsteady-state laminar flow of a slightly compressible fluid through an anisotropic, homogeneous, porous medium can be described after assuming small porous medium can be described after assuming small pressure gradients everywhere in the reservoir and pressure gradients everywhere in the reservoir and neglecting gravity effects:
+++=, ......(1)
wherekr= ------- = constant.ct
In Eq. 1, it is also assumed that the horizontal permeabilities kx and ky are equal and constant, thus permeabilities kx and ky are equal and constant, thus equaling kr. This assumption is not necessary, and the following results can be generalized to the case of simple anisotropy, where kx, ky, and kr are all constant but are not equal, by redefinition of the horizontal variables x and y. For example, we define z' as
z' = z kr/kz................................(2)
JPT
P. 1392