Find. The new drilling rate once the pore pressure gradient jumps to 10.0 ppg, all other operating parameters (depth, weight-on-bit, rotary speed, bit wear, hydraulics) held fixed.
Approach. Write the ratio of new to old ROP as the exponential of the SUM of $a_j\Delta x_j$ over the 8 Bourgoyne–Young terms; recognise that any term whose input variable is unchanged between the two conditions contributes zero to the ratio (it cancels), leaving only the two terms that actually depend on the pore pressure gradient.
Identify which terms actually change. The full model is $dD/dt=\exp\!\big(a_1+\sum_{j=2}^{8}a_jx_j\big)$, where $x_2=D$ (compaction), $x_5,x_6,x_7,x_8$ depend on weight-on-bit, rotary speed, bit wear and hydraulics. Since depth and every operational parameter besides $g_p$ are stated unchanged, $x_2,x_5,x_6,x_7,x_8$ take the SAME value in both conditions and their $a_j x_j$ terms cancel exactly in the ratio — only the two terms that are explicit functions of $g_p$ survive: $x_3=D^{0.69}(g_p-9.0)$ (under-compaction/abnormal-pressure term) and $x_4=D(g_p-\rho_c)$ (overbalance / chip-hold-down term).
Evaluate $x_3$ old and new. Since the OLD gradient is exactly the 9.0 ppg normal-trend baseline, $x_{3,old}=8{,}000^{0.69}(9.0-9.0)=0$. New: $x_{3,new}=8{,}000^{0.69}(10.0-9.0)=8{,}000^{0.69}(1.0)$, and $8{,}000^{0.69}=492.4$, so $\boxed{\Delta x_3=492.4}$.
Evaluate $x_4$ old and new. $x_{4,old}=8{,}000(9.0-9.5)=-4{,}000$; $x_{4,new}=8{,}000(10.0-9.5)=+4{,}000$, so $\boxed{\Delta x_4=8{,}000}$ — the well swings from over-balanced ($g_p<\rho_c$) to under-balanced ($g_p>\rho_c$), which is exactly why drilling into the high-pressure zone speeds up penetration.
Combine into the ROP ratio. $$\ln\!\left(\frac{R_{new}}{R_0}\right)=a_3\Delta x_3+a_4\Delta x_4=0.0001(492.4)+0.00002(8{,}000)=0.0492+0.160\Rightarrow\boxed{\ln(R_{new}/R_0)=0.2093}$$
Solve for the new drilling rate. $$R_{new}=R_0\,e^{0.2093}=50\times1.2329\Rightarrow\boxed{R_{new}=61.6\ \text{ft/hr}}$$
Check: constants $a_2,a_5,a_6,a_7,a_8$ (compaction, weight-on-bit, rotary speed, bit wear, hydraulics) are given in the source but never enter the answer — every operating parameter besides pore pressure is stated as unchanged, so those terms are identical old vs. new and cancel exactly out of the ratio. This is a deliberate feature of the question, not a missing datum.
