Derivation of pressure distribution models for horizontal well using source function

Abstract

In oil and gas operation, horizontal well technology has become an important technique because of its numerous advantages. Despite these advantages, determination of pressure distribution expression is a problem because it involve 3D transient-flow that need to be solved. In this work, ten (10) models for pressure distribution for horizontal well under different boundary variation were derived following these steps for each of the models:(i) choosing a boundary condition for each axis (ii) selecting the appropriate source function for each axis and (iii)applying Newman product rule to arrive at the pressure expression. In conclusion, the following were observed.(a) along the xaxis, seven of the models have infinite-slab sources in infinite-slab sources(b) five of the models derived, along the yaxis they have infinite-plane sources in infinite-slab sources while for the remaining five along the y- axis they have infinite sources(c) along z-axis, six of the models have infinite-plane sources in infinite-slab sources while four of the remaining one have infinite sources(d) Three types of boundaries are observed:(i) Completely bounded boundaries (ii) Mixed boundaries (iii) infinite-acting boundaries(e)Three instantaneous source functions are involved in the pressure distribution expression.

NOMENCLATURE

y_e= distance to the external boundary in y-direction

h_D= dimensionless height

x_e= distance to the external boundary in x-direction

x_D= arbitrary dimensionless distance along the x axis

z_e= distance to the external boundary in z-direction

y_D = arbitrary dimensionless distance along the y axis

z_w = Well coordinate in z-direction

z_D = arbitrary dimensionless distance along the z axis

y_w = Well coordinate in y-direction

τ= dummy integration dimensionless time variable

x_w = Well coordinate in x-direction

ɳ= diffusivity constant1

t = Time

Keywords: Source function, Horizontal well, Reservoir, Boundary