I would like to derive an expression that describes reservoir temperature change (as a function of time) during fluid injection inside the reservoir. I have used the basics of thermodynamics and get two ODE that describes the temporal changes for both reservoir fluids and injected one, but I am not sure with my resulted equations
Refaat,
Is the volume of fluid in the reservoir always a fixed amount, or does it change from zero to some limiting value according to the injected flow and duration?
Feel free to share your equations.
Actually, the fluid saturation / pressure inside reservoir will increase as a result of cold fluid (water) injection.
In its simplest form heat flow and temperature change are determined by Fourier's law (https://en.wikipedia.org/wiki/Thermal_conduction ), which is not an ODE since it also involves spatial dependence. If heat flow is coupled to other flows (pressure, density, velocity) as your answer to James indicates, you need to take that into account as well. These equations can usually only be solved numerically.
The information you give is not sufficient to make an educated guess about which direction you want to go in. You state you have two coupled ODE's, do they refer to average temperatures of some sort? DId you solve them and does the solution go to the correct final state (which will depend on the boundary conditions, is the reservoir kept at constant temperature, or is it insulated), the heat capacities of the various components, and probably some other things I don't think of immediately.
Can you give us the ODE's and your derivation to see if they make sense?
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