On identifying the formation pressure and filtration coefficients of two gaz-bearing formations - Markovskii, Anatolii (Eastern-European Journal of Enterprise Technologies, 2016)
In the practice of gas field exploitation, there arises a problem of the production rates calculation for two gas-bearing formations opened by a single well. Its solution requires knowledge of the formation pressures and flow coefficients. While solving the problem, an important concept of the turning points has been introduced. They play a key role in developing a system of equations. In practice, no information on the formation flow coefficients and pressures is available; therefore, a question now arises of how they can be determined. It is possible to use downhole measurements, although doing this appears to be technically challenging. Wellhead measurements are simpler, but they provide only total production rate values under a fixed wellhead pressure. In [9], the authors are of the opinion that “measuring the flow coefficients of two gas-bearing formations opened by a single well without downhole measurements is currently impossible.” The problem under studying may have 13 different variants of setting up, depending on the placement of the wellhead pressure that is measured with respect to the turning points. It is unknown in advance which of the aforementioned variants is the case while measuring; therefore, it is necessary to study each of them individually. One of the most difficult cases is considered in the present article. Mathematically, the problem is reduced to solving a system of equations with 30 unknown quantities, and only 6 out of these equations are linear. Among the unknowns, there are the formation pressure and flow coefficients 3+3=6 of the upper and lower formations as well as 24 unknowns at each formation flow rate measuring: 6+6=12 and 6+6=12 wellbore pressures. It is reasonable to take the upper formation flow rates as the main unknowns in 1, 3, and 5 measurements, in which case the unknown formation pressures and flow coefficients are determined: they turn out to be the functions of the main unknowns, which in their turn satisfy a system of a three nonlinear equations’ polynomial over the first unknown of the seventh degree. Therewith, it is shown that two other main parameters are changed within the bounded domain, the boundary of which is explicitly described. This fact significantly simplifies solving the problem, whereas the desired formation pressures and filtration coefficients are a priori unbounded.