Reverse circulation well tool

A method, comprising:

• receiving, in a first flow chamber of a well crossover sub of a drill string disposed in a wellbore, a fluid pressure greater than a threshold fluid pressure from a fluid in an annulus of the wellbore, the fluid pressure acting in a downhole direction on an uphole end of a sleeve valve of the well crossover sub, the well crossover sub comprising a substantially tubular housing comprising the first flow chamber and a second flow chamber, the sleeve valve, and a sealing structure circumscribing a portion of the substantially tubular housing, wherein the first flow chamber fluidly connects the annulus of the wellbore uphole of the sealing structure to a central bore of the drill string downhole of the well flow crossover sub, and wherein the second flow chamber fluidly connects a central bore of the drill string uphole of the sealing structure to the annulus of the wellbore downhole of the sealing structure;
• moving, in response to receiving the fluid pressure greater than the threshold fluid pressure in the first flow chamber and in response to the fluid pressure acting in the downhole direction on the uphole end of the sleeve valve, the sleeve valve in the downhole direction from a first, closed position restricting fluid flow through the second flow chamber to a second, open position allowing fluid flow through the second flow chamber;
• flowing a first fluid from the annulus uphole of the sealing structure to the central bore of the drill string downhole of the well crossover sub through the first flow chamber;
• and
• flowing a second fluid from the annulus downhole of the sealing structure to the central bore of the drill string uphole of the sealing structure through the second flow chamber.