Question

Matthews-Brons-Hazebroek (MBH) plots are used to calculate the Dietz shape factor (C_A) for reservoirs of various shapes. The MBH plot for a right-angled triangle reservoir with the well located at the centre shows a reading of dimensionless pressure P_D(MBH) = 3.0, at a modified dimensionless time t_DA = 1.0.
The Dietz shape factor of this right-angled triangle reservoir is ____________ (rounded off to one decimal place).

Hint:

At t_DA = 1.0 the logarithmic time term in the MBH relation vanishes, so P_D(MBH) reduces directly to ln(C_A).

Answer 1

Correct Answer: 20.1

Step 1: State the known readings again:
$P_{D(MBH)} = 3.0$ and $t_{DA} = 1.0$
Step 2: Since the time term vanishes at t_DA = 1.0, work directly with the relation $P_{D(MBH)} = \ln(C_A)$, but solve it this time using common logarithms instead of natural logarithms as an independent cross check:
Convert the natural log equation into base 10 form using the identity $\ln(x) = 2.302585 \times \log_{10}(x)$
Step 3: Rewrite the equation:
$3.0 = 2.302585 \times \log_{10}(C_A)$
Step 4: Solve for the base 10 logarithm of C_A:
$\log_{10}(C_A) = 3.0 / 2.302585$
$\log_{10}(C_A) = 1.30288$
Step 5: Convert back by raising 10 to this power:
$C_A = 10^{1.30288}$
Step 6: Split the exponent into an integer part and a fractional part:
$C_A = 10^{1} \times 10^{0.30288}$
$10^{0.30288}$ is approximately equal to $2.0086$
Step 7: Multiply out the final value:
$C_A = 10 \times 2.0086 = 20.086$, which rounds to 20.1
Step 8: Compare with the natural log method:
This base 10 calculation gives the same shape factor as the direct $C_A = e^{3.0}$ calculation, confirming the answer is correct.
Final Answer:
\[ \boxed{C_A = 20.1} \]