Transient pressure for cases of a horizontal wellbore axis with any angle inclined to a fault or a constant pressure supply boundary in anisotropic formations were studied. This axis parallel to or perpendicular to a fault is only its a simple special case. Particularly, the cases of multi-branched horizontal wells and horizontal well in pinch angle (wedge angle) were developed.

The table of the analysis solutions of transient pressures for various arrangements of horizontals are provided. The wellbore pressures took into account wellbore storage coefficients and skin factor are handled by the help of Laplace transforms.


Transient pressure analysis for horizontal wells have presented in several publications, Refs. 1–20. These literatures considered the cases of horizontal wellbore axis parallel to a fault, a horizontal well in infinite formation and in rectangular bounded formations (box-shaped drainage, well parallel to or perpendicular to side faces), respectively.

In this paper, we first give the line source solutions with no wellbore storage and no skin for cases of horizontal wellbore axis parallel to or perpendicular to a fault or constant pressure supply boundary. Then we present the line source solution of multi-branched horizontal wells, i.e. drilling several horizontal wells with various directions from main vertical well. It is equivalent to a horizontal well in a pinch angle in certain conditions. The line source solutions for a definite problem are based on the use of source and Green's functions. Then we take into account wellbore storage and skin used Laplace transforms. A table of Green's functions of transient pressures was prepared in this study for multi-branched horizontal wells and horizontal well in pinch angle (wedge angle) as well as horizontal wells near a fault or constant pressure boundary.

Physical Model and Governing Equations

The model considered in this study is assumed as following

  1. The fluid is slightly compressible, its viscosity and compressibility c1 are constants.

  2. The formation is horizontal and equal in thickness h.

  3. The well is assumed to be parallel to the top and bottom which may be impermeable or constant pressures, to be locate at any location, zw, from the bottom, its length is 2L.

  4. The formation is anisotropic system with KH KV

  5. Gravity effects are considered to be negligible.

The equation governing flow is the diffusivity equation as derived from the continuity equation, an equation of state, and Darcy's Law


where p(x,y,z,t) is pressure, and


in Eq. (2) c is total compressibility.

Horizontal Wells near a Fault or constant Pressure Boundary

Horizontal wells parallel to a fault

For a horizontal well parallel to a linear fault, the x-coordinate direction is oriented along he horizontal wellbore axis, z-coordinate direction upward vertically. P. 285

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