Analytical solutions by the hodograph method to hydrodynamic problems for oil and gas traps

Resting gas, oil and combined traps in equilibrium with moving water are studied by the methods of complex analysis (conformal mapping of hodograph circular polygons onto the strip in the complex potential plane). For a monocline dipping at an arbitrary angle and a one-phase trap two regimes (diffuser and confuser) are possible and explicit analytical solutions for the shape of a sharp interface between water and hydrocarbon are derived in terms of hypergeometric functions. The free surface is shown to coincide with the Saffman–Taylor finger in the Hele–Shaw apparatus if the monocline axis is vertical. For a gas–oil trap the interface consists of two branches along which the isobaric condition (constant pressure in the gas aloft) and the condition of a linear increase of pressure with depth (in the oil phase separated by a horizontal hydrostatic interface from the gas finger and by a hydrodynamic interface from water) are reduced to a standard hodograph representation through two touching circles. A critical regime is analyzed when the free surface does not have an inflexion point.