6. (10 points) A water pool contains one-eighteenth of its capacity. Two water pipes start filling the pool simultaneously. When the water level reaches two-ninths of the pool's capacity, the first pipe continues to fill the pool alone for 81 minutes, during which it injects an amount of water equal to what the second pipe has already injected. Then, the second pipe fills the pool alone for 49 minutes, at which point the total amount of water injected by both pipes is the same. After that, both pipes continue to fill the pool. How many more minutes do the two pipes need to fill the pool together?
【Analysis】We set the time to $\frac{2}{9}$ as $X$ minutes; the water volume of pipe A for 81 minutes = the water volume of pipe B for $X$ minutes (the water volume of pipe A for the last 81 minutes is equal to the water volume of pipe B for the previous period) the water volume of pipe A for $X$ minutes = the water volume of pipe B for 49 minutes (the total volume is the same, pipe B makes up for the part of pipe A in the front) In the subsequent separate water injection stage, it's just that the water volumes are exchanged, so it also fills $\frac{1}{6}$; set up the equation to find the time for the joint water injection is 63, the two pipes completed $\frac{2}{9}-\frac{1}{18}=\frac{1}{6}$ in 63 minutes, then see how many $\frac{1}{6}$ are in the remaining workload, which means there are several 63 minutes.
【Solution】Solution: Let the time to $\frac{2}{9}$ be $X$ minutes;
$$
\begin{array}{l}
81: X=X: 49 \\
X \times X=81 \times 49, \\
X \times X=9 \times 9 \times 7 \times 7, \\
X \times X=(9 \times 7) \times(9 \times 7), \\
X=9 \times 7, \\
X=63 ;
\end{array}
$$

That is, the two pipes completed $\frac{2}{9}-\frac{1}{18}=\frac{1}{6}$ in 63 minutes,
The remaining water volume to be injected: $1-\left(\frac{1}{18}+\frac{1}{6} \times 2\right)$,
$$
\begin{array}{l}
=1-\frac{7}{18}, \\
=\frac{11}{18} ;
\end{array}
$$

The required time is: $63 \times\left[\frac{11}{18} \div \frac{1}{6}\right]$
$$
\begin{array}{l}
=63 \times \frac{11}{3}, \\
=231 \text { (minutes) }
\end{array}
$$

Answer: The two water injection pipes still need to inject water together for 231 minutes.
The answer is: 231.