Improvement of fully-implicit two-phase pore-network models by employing generalized flux functions with additional throat variables

Abstract

In fully-implicit two-phase pore-network models, developing a well-converged scheme remains a major challenge, primarily due to the discontinuities in the phase conductivities. This paper addresses these numerical issues by proposing a generalized flux function that establishes a continuous flux expression for two-phase flows by introducing an additional throat variable $Θ$. Two approaches for expressing this additional throat variable are introduced: the first applies regularization strategies, while the second constructs an additional residual constraint equation. It is shown that this approach significantly improves accuracy and ensures the temporal convergence, as demonstrated through various numerical examples.

Figures (9)

• Figure 1: Schematic illustration of the pore-network model concepts. The pore-body radius ($r_i$) and pore-throat radius ($r_{ij}$) are shown, assuming a cubic pore body and a circular throat cross-section and $\Delta p_{c,ij} = \max(p_{c,i},p_{c,j}) - p_{ce,ij}$, where $p_{ce,ij}$ is the entry capillary pressure: (a) when the pore throat is not invaded ($\Delta p_{c,ij} < 0$) at time $t^k$ and (b) when it is invaded ($\Delta p_{c,ij} \geq 0$) by the non-wetting phase at time $t^{k+1}$.
• Figure 2: Hysteresis in non-wetting phase conductivity. The orange curve represents the non-wetting phase conductivity of pore throat during a throat local drainage process, where an invasion event occurs when $p_{c,ij} > p_{ce}$. In contrast, the blue curve depicts the conductivity during an imbibition process, where a snap-off event occurs when $p_{c,ij} < p_{c,s}$.
• Figure 3: Schematic illustration of the throat invasion during the time interval $[t^k,t^{k+1}]$. The throat gets invaded at time $t^{k+1}_*$, at which $\Delta p_{c,ij}$ changes sign such that the Heaviside function transitions from zero (before invasion) to 1 (after invasion).
• Figure 4: One-dimensional test case.$L^2$-errors (left), prediction errors of invasion events \ref{['eq:errorinvasions']} (middle), and total number of Newton iterations (right) plotted over the average time-step sizes.
• Figure 5: Setup for the bifurcated case. On the left bottom, non-wetting phase is injected with a fixed rate of $Q_n =$5e-10kgs. At the four outlet pores on the top, a pressure of $p_\mathrm{w} =$ 1e5Pa is set and the non-wetting fluid can flow out of the domain.

Theorems & Definitions (5)

• Definition 1: Network discretization
• Remark 1
• Remark 2
• Remark 3
• Remark 4