Question 3 of 5: Weighting Up Drilling Fluid with Barite

Question 3: Weighting Up Drilling Fluid with Barite (equal value)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. $V_f=2{,}000$ bbl (fixed tank capacity); $\rho_o=10.0$ ppg; $\rho_f=11.0$ ppg; API Barite $SG=4.2$ (so $\rho_b=4.2(8.34)=35.03$ ppg); 1 sack $=100$ lb.

Find. The volume of old mud that must be discarded and the number of sacks of Barite required, so the final 2,000 bbl of mud is at 11.0 ppg.

Approach. Because the tank capacity is capped at the starting volume, adding barite (which occupies volume as well as adding weight) must be offset by first discarding some old mud. Write a combined mass balance and volume balance across the whole operation, in terms of the retained old-mud volume $V_r$ and the barite mass $m_b$, and solve them together.

1. Set up the mass and volume balances. Let $V_r$ = bbl of old mud retained (undiscarded) and $m_b$ = lb of barite added. Mass balance: $\rho_o(42V_r)+m_b=\rho_f(42V_f)$. Volume balance (barite is idealized as occupying a volume $m_b/\rho_b$): $42V_r+m_b/\rho_b=42V_f$.
2. Eliminate $V_r$ and solve for the barite mass. Substituting $42V_r=42V_f-m_b/\rho_b$ into the mass balance and solving for $m_b$: $$m_b=\dfrac{42\,V_f\,(\rho_f-\rho_o)\,\rho_b}{\rho_b-\rho_o}=\dfrac{42(2{,}000)(1.0)(35.03)}{35.03-10.0}$$ so $\boxed{m_b=117{,}562\ \text{lb}}$.
3. Convert to sacks. $m_b/100=1{,}175.6$ sacks; rounding up so the target density is at least met, $\boxed{1{,}176\ \text{sacks of Barite}}$.
4. Volume of old mud to discard. Barite's own idealized volume is $v_b=m_b/\rho_b=117{,}562/35.03=3{,}356.9$ gal $=79.9$ bbl. From the volume balance, $V_r=V_f-v_b=2{,}000-79.9=1{,}920.1$ bbl retained, so the volume discarded is $\boxed{V_d=V_f-V_r=79.9\ \text{bbl}}$ — exactly equal to the volume the barite itself occupies, which is what keeps the final volume at the 2,000 bbl tank limit.