PET504E-Advanced Well Test AnalysisHW# 1

Mustafa Onur and Murat nar, ITU2013-2014 Spring Semester

Given Date: March 04, 2014Due Data: March 10, 2014

Problem 1 (50 pts). Consider a horizontal cylindrical reservoir with height h and assume thatwe have flow only in the radial direction, r, as its cross-sectional view shown in Fig. 1. Usingconservation of mass over the control volume shown in Fig. 1, derive the continuity equation,which will describe flow in the radial direction. Fluid velocity in the r-direction is vr [RB/(ft2-day)] and density of fluid is . Use field units. Do not consider source/sink term in the continuityequation.

Then derive the diffusivity equation for slightly compressible fluid of constant viscosity andhomogeneous reservoir in terms of pressure replacing the velocity in continuity equation byDarcys equations and using the assumptions of slightly compressible fluid and homogeneousreservoir.

r-direction

r-direction

r-r/2 r + r/2

Controlvolume

Fig. 1. Cross-sectional view of cylindrical reservoir.

Problem 2 (50 pts). Assume that we will produce fluid at a constant rate qsc from a line-sourcewell (i.e., rw approaches to zero) and the center of the well is located at r = 0 in above figure.Using Darcys equation (neglect gravity effect), derive the appropriate inner boundary conditionfor which a fully penetrating1 line-source well is producing a slightly compressible fluid ofconstant viscosity at a constant-rate from a homogeneous reservoir in terms of permeability andporosity. Hint: Your equation should relate pressure gradient with respect to r to flow rate qsc.

1 If the well is perforated through the entire thickness of the formation, then this well is referred to as thefully penetrating well.