Transient temperature distribution at any

2.5. Evaluate shut-in temperature at another depth along the wellbore
For a deepwater well, the temperature gauge is usually installed at downhole close to the pay zone, but the interesting location maybe in the upper section of the well. Eq. (5) shows that G-1 the dimensionless temperature is a function of dimensionless shut-in time and time before well shut-in. Assuming the thermal properties of the surrounding formation and the wellbore heat transmissibility are constant, we have:equation(11)TD=Td(h6,ts,tc)−T∞1Tfss(h6,tc)−T∞1=Td(h6,ts,tc)−T∞2Tfss(h6,tc)−T∞2
Assuming point 1 is located at the temperature gauge location, hG, and the point 2 is negative feedback control in the point of interest, which is located above the temperature gauge, solving for wellbore temperature at point 2, Td(h6,ts,tc):equation(12)Td(h6,ts,tc)=Tfss(h6,tc)[Td(h6,ts,tc)−T∞1]−Td(h6,ts,tc)T∞2+T∞1T∞2Tfss(h6,tc)−T∞1+T∞2
To solve Eq. (12), we need to determine the fluid temperature Tfss at point 2 during the normal production period. Ramey (1962) proposed the following solution for a constant geothermal gradient GgGg:equation(13)Tfss(h6,tc)=T∞2−GgA+[Tfss(h6,tc)−T∞1+GgA]eΔhAwhere,equation(14)A=−qρfcpf[K+rtiUTf(tDc)]2πrtiUTKequation(15)f(tDc)=ln(1+DtDc)equation(16)D=1.5708+1tDc+4.9589