Question

The following equations describe the transient fluid flow in a typical petroleum reservoir system. Here, \( p \) is pressure, \( x \) and \( r \) are the spatial coordinates in rectangular and cylindrical systems respectively, and \( t \) is time. Also, \( \phi \) (porosity), \( \mu \) (viscosity), \( c_f \) (formation compressibility), \( c_t \) (total compressibility) and \( k \) (permeability) are constant coefficients. 
Match the equations (GROUP I) with their corresponding descriptions (GROUP II). 
 

• A. P – IV, Q – I, R – II, S – III
• B. P – IV, Q – III, R – II, S – I
• C. P – III, Q – IV, R – II, S – I
• D. P – III, Q – I, R – I, S – II

Hint:

To match equations with their descriptions, focus on the coordinates used (Cartesian or cylindrical) and whether the fluid is compressible or incompressible.

Answer 1

The Correct Option is A

Step 1: Equation (P).
Equation (P) represents a differential equation involving the radial coordinate \( r \) and its time derivative \( \frac{\partial p}{\partial t} \). This is used to describe incompressible fluid flow in cylindrical coordinates, making it corresponding to (IV).

Step 2: Equation (Q).
Equation (Q) involves the second derivative of pressure with respect to the spatial coordinate \( x \), indicating that this equation is in Cartesian coordinates. It is used for slightly compressible fluid flow, making it corresponding to (I).

Step 3: Equation (R).
Equation (R) includes the second derivative of pressure in terms of \( x \) and \( r \), with the term \( c_f \), which makes this equation describe slightly compressible fluid flow in cylindrical coordinates, corresponding to (II).

Step 4: Equation (S).
Equation (S) includes both \( r \) and \( t \) derivatives with the compressibility term \( c_f \), representing slightly compressible fluid flow in cylindrical coordinates, making it corresponding to (III).
Conclusion:
Thus, the correct matching is: P – IV, Q – I, R – II, S – III.