A well tubing system includes a well tubing extending from a wellhead and disposed in a wellbore, and multiple capillary tubes coupled to the wellhead and extending from the wellhead into an annulus formed between an outer surface of the well tubing and a wall of the wellbore. The multiple capillary tubes provide a sealing fluid to the annulus, and extend to a respective multiple depths in the wellbore, where the multiple depths are different from each other. The capillary tubes guide the sealing fluid from the wellhead into the annulus, and provide the sealing fluid to the annulus.