The test cases presented in this chapter show a good agreement of the non-isothermal DuMuX model with the results of experiments and the HYDROTHERM simulator.
However, the applicability of the model is restricted. Especially steam injection into dry as well as into liquid-saturated soils is a very hard benchmark example for which unconditional stability of the model can not be guaranteed at present state. This is mainly due to two phenomena which are inherent to the presented formulation. Both phenomena arise at the condensation front, i.e. at the control volume where steam flows in and condenses because the temperature lies below the boiling point. To ease the explanation, water and air are considered as components and when speaking of the boiling point, the one of water is meant. However, these considerations are general and can also be applied to other components.
Consider a control volume at a temperature below the boiling point which is fully saturated with gas. Steam flows into this control volume from an adjacent cell or
Figure 4.11: Simulation results of 3p3c soil vapor extraction after 13 hours. Left: in-jection of water saturated air. Right: inin-jection of pure air.
boundary or from a source. This means that water and a high amount of energy flow into the cell. When determining the volume derivatives for the pressure equation, the change in mass of water without regarding the change of internal energy is considered and vice versa. When the derivative of fluid volume with respect to water is determined, a certain amount of water is added to the current mixture and a flash calculation at constant internal energy is performed. This will predict that most of the water forms a liquid phase and that a small amount evaporates. The same procedure for increased energy and a dry control volume will predict increasing temperature and an expanded gas phase. Compared to the actual steam injection, these derivatives overestimate the change in temperature and underestimate the change in fluid volume (since too much water is considered to be in the dense, liquid phase). Consequently, this will lead to a wrong prediction of the volume change over the next time step and therefore to considerable volumetric errors. This effect is dampened by the heat capacity of the solid matrix. A high heat capacity leads to low temperature changes due to the internal energy of the steam and therefore to the condensation of most of the steam. Hence, the actual change of fluid volume approaches the volume derivatives for increasing heat capacities.
Consider now a control volume saturated with liquid water. Again, steam flows into this control volume and condenses causing the temperature to rise until it nearly reaches the boiling point. The transport equation in the pre-step predicts that enough energy will flow into the cell to reach the boiling point and vaporize some water. Accordingly, the derivative of fluid volume with respect to energy will be considerably high. This, in turn, means that liquid water must be displaced to fulfill the volume constraint (Equation (2.65)). This is achieved by an increased pressure which causes a higher fluid flux out of the cell. At a higher pressure, however, also the boiling temperature is
higher and therefore the energy added during the following time step may not be enough to heat the control volume to boiling conditions. Hence, water was pressed out of the control volume to produce space for a steam phase that does not develop which results in a negative volume error. In the next timestep, this error is compensated by a decreased pressure which will in turn result in more vaporized water than predicted and so on.
This effect is also observed when the control volume is only heated without adding any mass. In fact, the effect on the stability of the simulation is even worse since the system is not affected by a pressure gradient originating from this cell before the boiling starts. Especially this effect is the main reason for the oscillating pressures which were observed in all steam injection examples and which can be particularly seen in Figure 4.8, bottom. Attempts to solve this problem by iterating the pressure equation did not lead to success. The result then only jumps between two pressure distributions. The effects is, however, influenced by several parameters. The pressure influences the steam density and hence also the density contrast between steam and liquid. The contrast decreases with increasing absolute pressure as does the derivative of fluid volume with respect to internal energy at boiling conditions. Thus, high pressure levels make the model more stable at boiling conditions. High pressure gradients enforce a clear flow direction. At low pressure gradients, the volume errors may cause reversal flow. If large parts of the domain are at a high temperature, the pressure drop which causes reversal flow may cause a drop below the vapor pressure. This leads to a sudden (unphysical) vaporization of water inside large parts of the domain. The accordingly occurring high volume errors certainly cause an unsolvable system of equations in the next timestep.
The compressibility, however, has the highest influence on the volume errors occurring at the condensation front. A high compressibility causes a high influence of the first term of the pressure equation (3.67). For the limit of an infinite compressibility, the pressure equation becomes
pt=pt−∆t,
hence, to a constant pressure. Thus, a high compressibility dampens fast pressure changes and stabilizes the model. This explains the importance to determine the derivative ∂vt/∂pat boiling point conditions according to Equation (3.82). The value of the derivative according to this equation is typically two to four orders of magnitude higher than the compressibility of a gas.
In this section, two applications of the multiphysics models are presented. This is, on the one hand, to demonstrate the practical applicability of the models, and on the other, to quantify the benefits in using the multiphysics schemes at concrete examples.
First, the isothermal and second, the non-isothermal case will be considered.