SatuDict uses the Pore Morphology method to determine the distribution of the two phases inside the porous media.
The Pore Morphology Method (Hilpert and Miller) calculates the stationary distribution of wetting and non-wetting phases for a given capillary pressure and it is applicable when:
gravity and viscous forces are negligible compared to capillary forces,
the material is homogeneous, i.e. there exists a well-defined contact angle between material surface and phase boundary, and
only two-phase systems are considered where fluids do not mix.
For such systems, the pore space accessible to the non-wetting phase is given by the Young-Laplace equation,
(345) Young-Laplace equation
where is the surface tension, the contact angle, is the capillary pressure and defines the minimum radius of accessible pores. Thus, the problem is reduced to a purely geometrical problem. The contact angle can be different at each solid phase inside the porous medium (Schulz et. al. (2008)) and, thus, variable wettability can be incorporated.
Two versions of the pore morphology method are available in GeoDict: The Quasi-Static Pore Morphology Method and the Dynamic Pore Morphology Method.
In the Quasi-Static Pore Morphology Method, the capillary pressure is increased (or decreased) strictly monotonically. This has the drawback, that for a drainage simulation, the whole structure is filled instantly after the invading fluid passed the narrowest pore throat. For an imbibition simulation, the whole structure is filled instantly, if the invading wetting phase filled the biggest pore. Since GeoDict 2021, the Dynamic Pore–Morphology Method is therefore available in SatuDict. It allows for a dynamic simulation of the drainage and imbibition processes, even with non-monotonic capillary pressure curves. Like this, the capillary pressure can drop down during a drainage simulation when the non-wetting phase passes a pore-throat. During an imbibition simulation, the capillary pressure is not monotonically decreasing, but can rise again, when the wetting phase passes a big pore. Even without the option of non-monotonic capillary pressure curves selected, several intermediate steps are computed for the same pressure value, to avoid instant filling of the whole structure.
With this new method, thin wetting layers can also be considered for imbibition processes by means of modified connectivity checks. Wetting residuals near the invading wetting phase front are always treated as connected. The distance from the invading wetting front considered for this modification is user-defined and constant during the simulation.
In the following, an example of the resulting capillary pressure curve of a drainage simulation with all three methods is shown below.
With this method, the capillary pressure for a drainage simulation is strictly monotonically increasing. At each simulation step, the capillary pressure is increased by a fixed value. This leads to the effect that large pores beyond small pore throats are filled at once when the pressure is high enough to pass the pore throat.
For an imbibition simulation, the capillary pressure is strictly monotonically decreasing. At each simulation step, the capillary pressure is decreased by a fixed value. This leads to the effect, that small pores beyond big pore bodies are filled at once when the capillary pressure is small enough.
Both effects are visible in the capillary pressure curve and responsible for large saturation jumps. Thus, there are sometimes too less computed saturation points in the saturation range between 20% and 80%. In the example shown below, only six saturation points are computed between a saturation of 20% and 80%.
This method is useful for the prediction at which capillary pressure the breakthrough occurs and provides detailed information about fluid distributions for high and low saturated structures. If the detailed fluid distribution at a broad range of saturations is of interest, this method should not be used anymore. However, for structures with a layered distribution of pore sizes, the method provides detailed fluid distributions. E.g. if pores are large at the top and getting smaller toward the bottom and a drainage process is simulated from top to bottom.
For drainage simulations, the capillary pressure is monotonically increasing. At each simulation step, the capillary pressure may be increased or it can stay the same.
For imbibition simulations, the capillary pressure is monotonically decreasing.
This method avoids large saturation jumps and provides many intermediate saturation steps. The interface between the two fluids is displaced by a small (interface) step size, defined in the Solver parameters. The method tries to move the interface according to this parameter. In the example shown, also in the range between 20% and 80% saturation, some intermediate saturation steps are computed and the wetting fluid invades without big jumps.
This method predicts a more accurate fluid movement compared to the quasi-static pore morphology method. It should be used if a detailed fluid distribution at a broad range of saturations is of interest.
However, certain physical phenomena, e.g., formation of water droplets above a fiber structure with large pore space at the top, cannot be modelled with this method. This requires that the capillary pressure can have a non-monotonic behavior.
For drainage simulations, the capillary pressure is non-monotonically increasing. At each simulation step, the capillary pressure may be decreased or it stays the same, if possible, otherwise the pressure is increased.
For imbibition simulations, the capillary pressure is non-monotonically decreasing. At each simulation step, the capillary pressure may be increased or it stays the same if possible, otherwise the pressure is decreased.
If this method is selected, it provides two capillary pressure curves in the result file. One for the monotonic and one for the non-monotonic behavior, shown below for the example.
The non-monotonic method avoids large saturation jumps as well, but can predict many more intermediate saturation steps. It allows to model more complex physical phenomena, e.g. the formation of water droplets above a fiber structure with large pore space at the top. The method is more precise than the other two methods, but also the runtime is higher. The interface step size is used in the same way as for the monotonic method, described above.
Quasi-Static Pore Morphology Method
