Dynamic Pore-Network Models Development

Abstract

A two-pressure dynamic drainage algorithm is developed for three-dimensional unstructured network model. The impact of time step is discussed through drainage simulations. Dynamic effects in average phase pressure for fluid phases with different viscosity ratios are explored using the developed code as an upscaling tool. For cases where two fluids have significant viscosity differences, the viscous pressure drop within one fluid may be neglected. This dynamic algorithm can then be simplified into a single-pressure one. This simplification has been done for both drainage and imbibition. Saturation patterns during imbibition for different boundary pressure drops are studied. With the increase of boundary pressure, invasion becomes less capillary dominant with a sharper wetting front.

Appendices

Appendix A: Fluid Distribution Patterns During Primary Imbibition

In general, there are 18 possible fluid occupancy of pores, as shown in Figure 10, where blue color shows the wetting phase and red color shows the non-wetting phase. Here, no attempt is made to represent the real interface shape. However, because of assumptions in our primary imbibition algorithm, some of these configurations will not emerge and the rest ones are shown in Figure 11. To save computational time, we only search these liquid filling scenarios. We use A,B,C to denote three columns and 1–6 to denote the six rows. So, we have 18 possible distribution patterns A1 to C6 in Figure 10. Below, we explain which distribution pattern cannot be encountered and will be, therefore, excluded from considerations.

Fig. 10

All possible liquid fillings during imbibition

Fig. 11

Liquid fillings during primary imbibition

• A2: As we do not consider volume of pore throats, so once there is wetting phase in the pore throat, neighboring pore body cannot be still fully saturated with non-wetting phase.

• A5: See A2.

• A6: It is not possible to trap wetting phase during primary imbibition in our algorithm.

• C2: See A2.

• C3: See A6.

• C4: See A2.

Appendix B: Interfacial Area for A Cubic Pore Body

Information of interfacial areas of corner interfaces and main terminal menisci for cubic pore bodies can be given as [13]:

1.1 5 Corner Interfaces

For a pore body with inscribed radius R _i filled with wetting and non-wetting phase, non-wetting phase volume can be bigger or smaller than the inscribed sphere volume.

$$\displaystyle \begin{aligned} {R}_{i,eq}=\begin{cases} {R}_{i}(\frac{6}{\pi}(1-s_{i}^{w}))^{1/3}& s_{i}^{w}\geq 0.48 \\ {R}_{i}(1-exp(-6.83s_{i}^{w}))^{1/3}& s_{i}^{w}<0.48 \end{cases} \end{aligned} $$

(28)

$$\displaystyle \begin{aligned} {A}_{i}^{wn}=\begin{cases} 4\pi R_{i,eq}^{2}& s_{i}^{w}\geq 0.48 \\ 4\pi R_{i,eq}^{2}+6\pi R_{i,eq}(R_{i}-R_{i,eq}) & s_{i}^{w}<0.48 \end{cases} \end{aligned} $$

(29)

1.2 5 Main Terminal Menisci

For main terminal meniscus, namely the interface between a pore body and its neighboring pore throat when non-wetting phase pressure in the pore body is not high enough to invade the pore throat, we have the area for this meniscus as $8\pi (\frac {\sigma ^{wn}}{p_{i}^{c}})^2(1-\sqrt {1-(\frac {r_{ij}p_{i}^{c}}{2\sigma ^{wn}})^2})$.