### 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