Question

In a drawdown test conducted on a well, \(p_D\), \(t_D\) and \(C_D\) denote the dimensionless pressure, the dimensionless time and the dimensionless wellbore storage coefficient, respectively. In the early time of the drawdown test, the fluid produced at the surface results purely from the unloading (expansion) of fluid already stored in the wellbore, and no fluid from the formation has reached the well yet. This period is called the pure wellbore storage period. For this period, which of the following relationships is CORRECT?

• A. \(t_D = p_D \, C_D\)
• B. \(C_D = p_D \, t_D\)
• C. \(p_D = t_D \, C_D\)
• D. \(t_D = p_D + C_D\)

Hint:

During pure wellbore storage the bottomhole pressure drop is directly proportional to elapsed time, which in dimensionless form gives the unit slope relation pD equals tD divided by CD.

Answer 1

The Correct Option is A

Step 1: Understand what pure wellbore storage means:
When a well is opened to flow, the pressure disturbance needs time to travel outward into the reservoir. Right at the start, almost none of the produced fluid actually comes from the formation, the sandface rate \(q_{sf}\) is nearly zero, and essentially the whole surface rate \(q\) is being supplied by the compressed or unloading fluid already sitting inside the wellbore. This early period, before the reservoir contributes meaningfully, is the pure wellbore storage period.
Step 2: Write the material balance for this period:
Since the wellbore itself is acting like a small compressible tank, the surface rate equals the rate at which fluid is expelled from wellbore storage. In field units this is written as \(q = 24\,C\left(\dfrac{dp_{wf}}{dt}\right)\), where \(C\) is the wellbore storage constant. Because \(q\) is essentially constant, this equation says the flowing bottomhole pressure changes linearly with elapsed time during this period.
Step 3: Convert this linear relationship into dimensionless form:
Using the standard definitions \(p_D = \dfrac{kh\,\Delta p}{141.2\,qB\mu}\), \(t_D = \dfrac{0.0002637\,k\,t}{\phi\mu c_t r_w^2}\) and \(C_D = \dfrac{0.8936\,C}{\phi c_t h r_w^2}\), substituting the linear \(\Delta p\) versus \(t\) relation from Step 2 into these definitions and simplifying algebraically cancels out all the reservoir rock and fluid properties, leaving the clean unit slope relation \(p_D = \dfrac{t_D}{C_D}\). Rearranging this expression for \(t_D\) gives \(t_D = p_D\,C_D\).
Step 4: Match against the given options:
Option A, \(t_D = p_D C_D\), is exactly the relation derived above. Option B inverts the roles of \(C_D\) and \(t_D\) incorrectly. Option C rearranges the relation wrongly, giving \(p_D = t_D C_D\) instead of \(p_D = t_D/C_D\). Option D proposes an additive relationship that has no basis in the wellbore storage material balance, since the true relationship between these dimensionless groups during this period is multiplicative, not additive.
Final Answer:
\[ \boxed{t_D = p_D\,C_D} \]