Pore-Scale Network Modeling of Three-Phase Flow Based on Thermodynamically Consistent Threshold Capillary Pressures. I. Cusp Formation and Collapse

Abstract

We present a pore-scale network model of two- and three-phase flow in disordered porous media. The model reads three-dimensional pore networks representing the pore space in different porous materials. It simulates wide range of two- and three-phase pore-scale displacements in porous media with mixed-wet wettability. The networks are composed of pores and throats with circular and angular cross sections. The model allows the presence of multiple phases in each angular pore. It uses Helmholtz free energy balance and Mayer–Stowe–Princen (MSP) method to compute threshold capillary pressures for two- and three-phase displacements (fluid configuration changes) based on pore wettability, pore geometry, interfacial tension, and initial pore fluid occupancy. In particular, it generates thermodynamically consistent threshold capillary pressures for wetting and spreading fluid layers resulting from different displacement events. Threshold capillary pressure equations are presented for various possible fluid configuration changes. By solving the equations for the most favorable displacements, we show how threshold capillary pressures and final fluid configurations may vary with wettability, shape factor, and the maximum capillary pressure reached during preceding displacement processes. A new cusp pore fluid configuration is introduced to handle the connectivity of the intermediate wetting phase at low saturations and to improve model’s predictive capabilities. Based on energy balance and geometric equations, we show that, for instance, a gas-to-oil piston-like displacement in an angular pore can result in a pore fluid configuration with no oil, with oil layers, or with oil cusps. Oil layers can then collapse to form cusps. Cusps can shrink and disappear leaving no oil behind. Different displacement mechanisms for layer and cusp formation and collapse based on the MSP analysis are implemented in the model. We introduce four different layer collapse rules. A selected collapse rule may generate different corner configuration depending on fluid occupancies of the neighboring elements and capillary pressures. A new methodology based on the MSP method is introduced to handle newly created gas/water interfaces that eliminates inconsistencies in relation between capillary pressures and pore fluid occupancies. Minimization of Helmholtz free energy for each relevant displacement enables the model to accurately determine the most favorable displacement, and hence, improve its predictive capabilities for relative permeabilities, capillary pressures, and residual saturations. The results indicate that absence of oil cusps and the previously used geometric criterion for the collapse of oil layers could yield lower residual oil saturations than the experimentally measured values in two- and three-phase systems.

Appendices

Appendix A Cusp Interfacial Contact Lengths and Phase Areas

1.1 Fluid–Fluid and Fluid–Solid Contact Length Calculations

For an interface, the length of an arch can be determined by knowing its two end points and radius. Oil/water and gas/oil interfaces are stretched between P3P contact and $b_{ow}$, and P3P contact and $b_{go}$, respectively (see Fig. 2). By using Eq. (3) for P3P’s coordinates, the oil/water and gas/oil contact lengths in a given cusp configuration can be formulated as:

$$\begin{aligned} L_{12}=2 \, r_{12} \times \arcsin { \Bigg \lbrace { (r_{12})^{-1} \sqrt{ \begin{aligned}&\left[ b_{contact} \, \cos \left( \alpha \right) +H \sin \left( \alpha \right) -b_{12} \, \cos \left( \alpha \right) \right] ^2 \\ +&\left[ b_{contact} \, \sin \left( \alpha \right) -H \cos \left( \alpha \right) -b_{12} \, \sin \left( \alpha \right) \right] ^2 \end{aligned} } \Bigg \rbrace } } \end{aligned}$$

(40)

where subscript 12 represents ow or go interfaces, $b_{contact}$ is the b-value of the perpendicular line passing through P3P contact to the pore wall, and H is the cusp height. All parameters are clearly shown in Fig. 2. The multiplication by 2 in Eq. (40) is to account for two mirror-imaged interfaces between similar fluids in each corner. The gas/water contact length in a cusp is calculated by:

$$\begin{aligned} L_{gw}=2 \, r_{gw} \times \arcsin {\left( \frac{b_{contact} \, \sin \left( \alpha \right) -H \cos \left( \alpha \right) }{r_{gw}}\right) } \end{aligned}$$

(41)

where $r_{gw}$ is the gas/water curvature. The oil/solid contact length is simply the difference between $b_{go}$ and $b_{ow}$ accounting for cusp configurations on both pore walls. It is expressed as:

$$\begin{aligned} L_{os}=2 \, \left( b_{go} - b_{ow}\right) \end{aligned}$$

(42)

Since pore’s perimeter is constant, by having oil/solid and water/solid ($L_{ws}=2 \, b_{ow}$) contact lengths, the gas/solid contact length can be calculated by:

$$\begin{aligned} \begin{aligned} L_{gs}&=P_t-L_{os}-L_{ws}\\&= P_t-2 \, b_{go} \end{aligned} \end{aligned}$$

(43)

where $P_t$ is the perimeter of the pore.

1.2 Pore Fluid Area Calculations

In this section, we present equations for the calculation of pore cross-sectional areas occupied by water, oil, and gas. To calculate different phase areas, all the interfaces and points of contact should be mathematically defined for each corner of an angular pore. Oil area in a cusp configuration is calculated using:

(44)

One should note that each fluid–fluid interface on the pore cross section is modeled as part of a circle. $f_{go}$ and $f_{ow}$ in Eq. (44) are used to determine the sections to which the gas/oil and oil/water interfaces belong. They could hold $-1$ and 1 values representing the upper and lower semicircles, respectively. $e_{ow}$ and $e_{go}$ are the correcting terms for oil/water and gas/oil interfaces, respectively, when they are part of both lower and upper semicircles, otherwise, they are zero. Their values are calculated geometrically for any interface in a given cusp configuration that requires integration over two semicircles. The first and second terms in Eq. (44) are integrals of the gas/oil and oil/water interfaces, respectively. It is formulated based on the premise that gas/oil interface is below oil/water interface between $x_{P3P}$ and $x_{bgo}$ (see Fig. 2). Otherwise, gas/oil interface intersects oil/water interface for $x > x_{P3P}$, and hence, cusp configuration is not geometrically possible. Equation (44) is used only for geometrically feasible cusp configuration. Another criterion is to have a positive oil/solid contact length (i.e., $x_{bow}<x_{bgo}$). This is an important assumption for writing the third term in Eq. (44), which is the integral over the pore line between two intersections of oil/water and gas/oil interfaces (i.e., $x_{bow}$ and $x_{bgo}$). For each corner with cusp configuration, there are two cusps that are mirror image of one another with respect to the middle line. And therefore, a multiplication by 2 is used in Eq. (44).

Having three fluid phases, i.e., oil, water and gas, in a pore with a constant cross-sectional area requires calculation of two of the fluid phase areas. We select oil and water areas. Although the same concept of integration over two different interfaces could be used for the calculation of water area [as was used for the case of oil area in Eq. (44)], it would be complicated to present a general equation to cover all possible interface curvatures and P3P locations. Instead, gas area is calculated in a predetermined site, called base area, covering the entire cusp configuration in a corner. The water area is then calculated as the total base area minus gas and oil areas within the selected base. Below, we provide more details.

Fig. 10

Cross-sectional areas representing various terms given in Eq. (47) developed for calculation of gas area within the selected region. $A_{g, \, base}$ is shown by the dotted black area. Areas shaded by the green squares and red lines represent the second term, and the summation of third and fourth terms on the right-hand side of Eq. (47), respectively

The base area is chosen to encompass the entire cusp configuration. A line with the equation of $y=y_{base}$ is used to create a triangle by intersecting the pore wall and the middle line (see Fig. 10). The triangle creates the base site for the calculation of water area. The value of $y_{base}$ is defined by:

$$\begin{aligned} y_{base} = {\left\{ \begin{array}{ll} \max {\left( y_{cgw}, \, y_{bgo}, \, y_{P3P}\right) } &{} \text{ if } r_{gw} > 0 \\ \max {\left( y_{cgw}+|r_{gw}|, \, y_{bgo}\right) } &{} \text{ otherwise } \end{array}\right. } \end{aligned}$$

(45)

As an example, see Fig. 10, where the base line is drawn for a cusp configuration with positive gas/water capillary pressure, and gas/water center being beneath the P3P contact and gas/oil center. The total area of the base is written as:

$$\begin{aligned} A_{t, \, base}=\frac{1}{2} \, y_{base}^2 \tan \left( \alpha \right) \end{aligned}$$

(46)

Gas area within the base region ($A_{g, base}$), black dotted area in Fig. 10, can be written as the total area of the base minus the areas occupied by water and oil:

(47)

where $A_{t, \, base}$ is calculated using Eq. (46). The second term on the right-hand side excludes a triangle at the top of the base area, which is not covered by the third and fourth integrals (see the triangular area depicted by green squares in Fig. 10). The third term is the integral of gas/oil interface between $x_{P3P}$ and $x_{bgo}$, similar to the first term in Eq. (44). The fourth term is the pore wall’s integral. It covers the same integral bounds as of the previous term. Together, the third and fourth terms deduct the area shaded by the red lines in Fig. 10 from the base area. The sixth term removes an area of a rectangle over the gas/water interface where is mainly covered by the water phase. The last term in Eq. (47) corrects for the gas area that is removed by the rectangle in the previous term. Its value depends on the sign of gas/water curvature and the position of P3P contact with respect to the center of gas/water interface. For a positive curvature, i.e., $\mathrm {P_{cgw}>0}$, it is given by:

$$\begin{aligned} A_{g,int} = {\left\{ \begin{array}{ll} \dfrac{r_{gw}^2}{2} \times \arcsin { \left( \dfrac{x_{P3P}}{r_{gw}}\right) }-\dfrac{x_{P3P}^2}{2 \times \tan {\left[ \arcsin { \left( \dfrac{x_{P3P}}{r_{gw}}\right) }\right] }} &{} \text{ if } y_{P3P}<y_{cgw} \\ \dfrac{r_{gw}^2}{4} \times \pi &{} \text{ if } y_{P3P} = y_{cgw} \\ \begin{aligned} \dfrac{r_{gw}^2}{2} &{} \times \left[ \arcsin { \left( \dfrac{y_{P3P}-y_{cgw}}{r_{gw}}\right) }+\dfrac{\pi }{2}\right] \\ &{}+\dfrac{\left( y_{P3P}-y_{cgw}\right) ^2}{2 \times \tan {\left[ \arcsin { \left( \dfrac{y_{P3P}-y_{cgw}}{r_{gw}}\right) }\right] }} \end{aligned} &{} \text{ if } y_{P3P}>y_{cgw} \end{array}\right. } \end{aligned}$$

(48)

By using oil area from Eq. (44), total base area from Eq. (46), and gas area within the base from Eq. (47), water area in the cusp configuration is calculated by:

$$\begin{aligned} A_w=A_{t, \, base}-A_{g, \, base}-A_o \end{aligned}$$

(49)

Gas area is then calculated by knowing oil and water areas as well as the total cross-sectional area of the pore. At this stage, all the required parameters are determined for invoking the MSP analysis of a cusp formation displacement.

Appendix B Thermodynamically Consistent Threshold Capillary Pressures for Two-phase Flow

1.1 Displacements in Capillary Tubes With Regular Angular Cross Sections

In this section, we present the final equations pertaining to various PL and LC displacements in capillary tubes with regular angular cross sections. They are derived using the MSP method. Since capillary tubes with regular cross sections have the same corner half angels and hinging contact angles in all corners, final threshold capillary pressure equations are relatively simple compared to their counterparts in irregular triangles. One may have three different displacements in regular capillary tubes:

• PL displacement with the formation of wetting oil layers (configuration changes A $\rightarrow$ B in Fig. 1):

  $$\begin{aligned} 3 \, r_{ow}^2 \bigg \{ \frac{\pi }{2}+\alpha -\theta _{ow}^a + \cos (\theta _{ow}^a)\frac{\cos (\theta _{ow}^a-\alpha )}{\sin (\alpha )} \bigg \} - r_{ow} L_t \cos (\theta _{ow}^a)+ A_t=0 \end{aligned}$$

  (50)

• PL displacement without formation of any wetting oil layers (configuration changes A $\rightarrow$ C in Fig. 1):

  $$\begin{aligned}&r_{ow} L_t \cos (\theta _{ow}^a)-A_t + 3 \, r_{ow}^2 \bigg \{\frac{\pi }{2}-\alpha -\theta _{ow,\smallfrown _1}^h \nonumber \\&+\left[ \cos (\theta _{ow,\smallfrown _1}^h)-2\cos (\theta _{ow}^a)\right] \frac{\cos (\theta _{ow,\smallfrown _1}^h+\alpha )}{\sin (\alpha )} \bigg \}=0 \end{aligned}$$

  (51)

  One should note that subscript 1 in $\theta _{ow,\smallfrown _1}^h$ refers to interface one (counted from apex toward center). We do not specify the corner number because all the corners have similar corner half angles. Equations (50) and (51) can be used for the similar displacements in square cross-section pores by replacing the coefficient 3 with 4. Note that $\alpha =\pi /4$ for squares.

• PL displacement with the collapse of wetting oil layers (configuration change B $\rightarrow$ C in Fig. 1):

  $$\begin{aligned}&\theta _{ow}^a+\theta _{ow,\smallfrown _1}^h-\pi - \cos (\theta _{ow}^a)\frac{\cos (\theta _{ow}^a-\alpha )}{\sin (\alpha )} \nonumber \\&+\big [2\cos (\theta _{ow}^a)-\cos (\theta _{ow,\smallfrown _1}^h)\big ] \frac{\cos (\theta _{ow,\smallfrown _1}^h+\alpha )}{\sin (\alpha )} =0 \end{aligned}$$

  (52)

1.2 Displacements in Capillary Tubes With Irregular Triangular Cross Sections

Here we list the threshold capillary pressure equations for various displacements in capillary tubes that are irregular triangle in cross section. They are divided into two groups: 1) PL displacement of oil by water with and without oil layer formation and 2) oil LC displacements.

1.2.1 PL Displacements With and Without Oil Layer Formation

• PL displacement with the formation of one wetting oil layer in corner 1:

  $$\begin{aligned}&A_t - r_{ow} L_{t} \cos (\theta _{ow}^a)+ r_{ow}^2 \bigg \{\ \theta _{ow,\measuredangle _2,\smallfrown _1}^{h} \nonumber \\&\quad +\,\theta _{ow,\measuredangle _3,\smallfrown _1}^{h} -\theta _{ow}^a+\frac{\xi _1+\xi _2+\xi _3}{\sin (\alpha _1)\sin (\alpha _2)\sin (\alpha _3)} \bigg \}\ =0 \end{aligned}$$

  (53)

  where

  $$\begin{aligned} \xi _1= & {} \cos (\theta _{ow}^a)\sin (\alpha _2)\sin (\alpha _3)\cos (\theta _{ow}^a-\alpha _1) \end{aligned}$$

  (54)
  $$\begin{aligned} \xi _2= & {} \sin (\alpha _1)\sin (\alpha _3)\cos (\theta _{ow,\measuredangle _2,\smallfrown _1}^{h}+\alpha _2)\big [2\cos (\theta _{ow}^a) -\cos (\theta _{ow,\measuredangle _2,\smallfrown _1}^{h}) \big ] \end{aligned}$$

  (55)
  $$\begin{aligned} \xi _3= & {} \sin (\alpha _1)\sin (\alpha _2)\cos (\theta _{ow,\measuredangle _3,\smallfrown _1}^{h}+\alpha _3)\big [2\cos (\theta _{ow}^a) -\cos (\theta _{ow,\measuredangle _3,\smallfrown _1}^{h}) \big ] \end{aligned}$$

  (56)

  Subscript 1 does not necessarily refer to the sharp corner. It represent any corner in which the oil layer forms.

• PL displacement with the formation of two wetting oil layers in corners 1 and 2:

  $$\begin{aligned}&A_t - r_{ow} L_{t} \cos (\theta _{ow}^a)+ r_{ow}^2 \bigg \{\ \pi +\theta _{ow,\measuredangle _3,\smallfrown _1}^{h} -2\theta _{ow}^a \nonumber \\&\quad +\frac{\xi _1+\xi _2+\xi _3}{\sin (\alpha _1)\sin (\alpha _2)\sin (\alpha _3)} \bigg \}\ =0 \end{aligned}$$

  (57)

  where

  $$\begin{aligned} \xi _1= & {} \cos (\theta _{ow}^a)\sin (\alpha _2)\sin (\alpha _3)\cos (\theta _{ow}^a-\alpha _1) \end{aligned}$$

  (58)
  $$\begin{aligned} \xi _2= & {} \cos (\theta _{ow}^a)\sin (\alpha _1)\sin (\alpha _3)\cos (\theta _{ow}^a-\alpha _2) \end{aligned}$$

  (59)
  $$\begin{aligned} \xi _3= & {} \sin (\alpha _1)\sin (\alpha _2)\cos (\theta _{ow,\measuredangle _3,\smallfrown _1}^{h}+\alpha _3)\big [2\cos (\theta _{ow}^a) -\cos (\theta _{ow,\measuredangle _3,\smallfrown _1}^{h}) \big ] \end{aligned}$$

  (60)

  Subscripts 1 and 2 do not necessarily refer to the sharp and medium corners. They represent any pair of corners in which the oil layers form.

• PL displacement with the formation of three wetting oil layers (configuration changes A $\rightarrow$ B in Fig. 1): see Eqs. (21)–(24) in Sect. 4.

• PL displacement without formation of any wetting oil layers: see Eqs. (17)–(20) in Sect. 4.

1.2.2 Layer Collapse Events

• PL displacement with the simultaneous collapse of two wetting oil layers:

  $$\begin{aligned}&-\pi +\theta _{ow}^a-\cos (\theta _{ow}^a)\sin (\theta _{ow}^a)\nonumber \\&\quad -\cos (\theta _{ow}^a)\sin (\theta _{ow,\measuredangle _1,\smallfrown _1}^{h})+\frac{1}{2} \bigg [\theta _{ow,\measuredangle _1,\smallfrown _1}^{h}+\theta _{ow,\measuredangle _2,\smallfrown _1}^{h}\nonumber \\&\quad +\cos (\theta _{ow,\measuredangle _1,\smallfrown _1}^{h})\sin (\theta _{ow,\measuredangle _1,\smallfrown _1}^{h}) +\cos (\theta _{ow,\measuredangle _2,\smallfrown _1}^{h})\sin (\theta _{ow,\measuredangle _2,\smallfrown _1}^{h}) \bigg ] \nonumber \\&\quad -\cos (\theta _{ow}^a)\sin (\theta _{ow,\measuredangle _2,\smallfrown _1}^{h}) -\frac{\cos (\alpha _1) \big [\cos (\theta _{ow}^a)-\cos (\theta _{ow,\measuredangle _1,\smallfrown _1}^{h}) \big ]^2}{2\sin (\alpha _1)} \nonumber \\&\quad -\frac{\cos (\alpha _2) \big [\cos (\theta _{ow}^a)-\cos (\theta _{ow,\measuredangle _2,\smallfrown _1}^{h}) \big ]^2}{2\sin (\alpha _2)}=0\quad \quad \end{aligned}$$

  (61)

  Subscripts 1 and 2 do not necessarily refer to the wide and medium corners. They represent any pair of corners in which the oil layers collapse.

• PL displacement with the simultaneous collapse of three wetting oil layers (configuration changes B $\rightarrow$ C in Fig. 1):

  $$\begin{aligned}&-\pi +\theta _{ow}^a-\cos (\theta _{ow}^a)\sin (\theta _{ow}^a) \nonumber \\&\quad -\frac{2}{3}\cos (\theta _{ow}^a)\bigg [ \sin (\theta _{ow,\measuredangle _1,\smallfrown _1}^{h}) +\sin (\theta _{ow,\measuredangle _2,\smallfrown _1}^{h})+\sin (\theta _{ow,\measuredangle _3,\smallfrown _1}^{h}) \bigg ] \nonumber \\&\quad +\frac{1}{3} \bigg [\theta _{ow,\measuredangle _1,\smallfrown _1}^{h}+\theta _{ow,\measuredangle _2,\smallfrown _1}^{h} +\theta _{ow,\measuredangle _3,\smallfrown _1}^{h}+\cos (\theta _{ow,\measuredangle _1,\smallfrown _1}^{h})\sin (\theta _{ow,\measuredangle _1,\smallfrown _1}^{h}) \nonumber \\&\quad +\cos (\theta _{ow,\measuredangle _2,\smallfrown _1}^{h})\sin (\theta _{ow,\measuredangle _2,\smallfrown _1}^{h})+ \cos (\theta _{ow,\measuredangle _3,\smallfrown _1}^{h})\sin (\theta _{ow,\measuredangle _3,\smallfrown _1}^{h})\bigg ]\nonumber \\&\quad -\frac{\cos (\alpha _1) \big [\cos (\theta _{ow}^a)-\cos (\theta _{ow,\measuredangle _1,\smallfrown _1}^{h}) \big ]^2}{3\sin (\alpha _1)} \nonumber \\&\quad -\frac{\cos (\alpha _2) \big [\cos (\theta _{ow}^a)-\cos (\theta _{ow,\measuredangle _2,\smallfrown _1}^{h}) \big ]^2}{3\sin (\alpha _2)} \nonumber \\&\quad -\frac{\cos (\alpha _3) \big [\cos (\theta _{ow}^a)-\cos (\theta _{ow,\measuredangle _3,\smallfrown _1}^{h}) \big ]^2}{3\sin (\alpha _3)}=0\quad \quad \end{aligned}$$

  (62)

• PL displacement with the single collapse of a wetting oil layer: see Eq. (25) in Sect. 4.

Appendix C Thermodynamically Consistent Threshold Capillary Pressures for Three-phase Flow

Threshold capillary pressure equations for various three-phase displacements in irregular triangular cross sections are presented in this section. They are presented in two main groups: 1) PL displacement of oil by gas with and without oil layer formation and 2) oil LC displacements.

1.1 PL Displacements With and Without Oil Layer Formation

• PL displacement with the formation of one spreading oil layer in corner 1:

  $$\begin{aligned} r_{go}^3 C_1+r_{go}^2 C_2+r_{go} C_3+C_4=0 \end{aligned}$$

  (63)

  where $C_1$–$C_4$ are given by:

  $$\begin{aligned} C_1&=\frac{\sigma _{ow}\sigma _{go}}{r_{ow}}\bigg \{ \frac{\pi }{2}-\alpha _1-\theta _{go}^r - \cos \left( \theta _{go}^r\right) \frac{\cos (\theta _{go}^r+\alpha _1)}{\sin (\alpha _1)} \bigg \} \end{aligned}$$

  (64)
  $$\begin{aligned} C_2&=\frac{\sigma _{ow}}{r_{ow}}\left[ \begin{aligned}&\sigma _{ow}\bigg \{ \frac{\Delta A_{w,2}+\Delta A_{w,3}}{r_{ow}}+2 \, r_{ow} \left( \theta _{ow,2}+\theta _{ow,3}+\alpha _2+\alpha _3-\pi \right) \bigg \}\\&+\sigma _{go}\cos \left( \theta _{go}^r\right) \bigg \{L_t-2 \, r_{ow} \left( \frac{\cos \left( \theta _{ow,2}+\alpha _2\right) }{\sin \left( \alpha _2\right) } +\frac{\cos \left( \theta _{ow,3}+\alpha _3\right) }{\sin \left( \alpha _3\right) }\right) \bigg \} \end{aligned} \right] \nonumber \\&\quad +\sigma _{go}^2 \bigg \{ \frac{\pi }{2}-\alpha _1-\theta _{go}^r - \cos \left( \theta _{go}^r\right) \frac{\cos (\theta _{go}^r+\alpha _1)}{\sin (\alpha _1)} \bigg \}\nonumber \\&\quad +\sigma _{gw}\left[ \begin{aligned}&2 \, \sigma _{gw}\Bigg \lbrace { \begin{aligned}&\pi -\left( \theta _{gw,2}+\theta _{gw,3}+\alpha _2+\alpha _3\right) -\cos \left( \theta _{gw,2}^{r,orig/alt}\right) \frac{\cos (\theta _{gw,2}^r+\alpha _2)}{\sin (\alpha _2)}\\&-\cos \left( \theta _{gw,3}^{r,orig/alt}\right) \frac{\cos (\theta _{gw,3}^r+\alpha _3)}{\sin (\alpha _3)} \end{aligned} \Bigg \rbrace }\\&+\frac{\sigma _{ow}}{r_{ow}}\cos \left( \theta _{gw}^{r,alt}\right) \Bigg \lbrace { \begin{aligned}&2 \, r_{ow} \left( \frac{\cos \left( \theta _{ow,2}+\alpha _2\right) }{\sin \left( \alpha _2\right) } +\frac{\cos \left( \theta _{ow,3}+\alpha _3\right) }{\sin \left( \alpha _3\right) }\right) \\&-L_{s,2}^{orig}-L_{s,3}^{orig} \end{aligned} \Bigg \rbrace }\\&+\frac{\sigma _{ow}}{r_{ow}}\bigg \{L_{s,2}^{orig}\cos \left( \theta _{gw,2}^{r,orig/alt}\right) +L_{s,3}^{orig}\cos \left( \theta _{gw,3}^{r,orig/alt}\right) \bigg \} \end{aligned} \right] \end{aligned}$$

  (65)
  $$\begin{aligned} C_3&=\frac{\sigma _{ow}\sigma _{go}}{r_{ow}}\left[ \begin{aligned}&2 \left( \Delta A_{w,2}+\Delta A_{w,3}\right) -A_t\\&+r_{ow}^2 \Bigg \lbrace { \begin{aligned}&\theta _{ow,2}+\theta _{ow,3}+\alpha _2+\alpha _3-\pi +\cos \left( \theta _{ow,2}\right) \frac{\cos \left( \theta _{ow,2}+\alpha _2\right) }{\sin \left( \alpha _2\right) } \\&+\cos \left( \theta _{ow,3}\right) \frac{\cos \left( \theta _{ow,3}+\alpha _3\right) }{\sin \left( \alpha _3\right) } \end{aligned} \Bigg \rbrace } \end{aligned} \right] \nonumber \\&\quad +2 \, \sigma _{ow}\sigma _{go}r_{ow}\left( \theta _{ow,2}+\theta _{ow,3}+\alpha _2+\alpha _3-\pi \right) \nonumber \\&\quad +\sigma _{go}^2 \cos \left( \theta _{go}^r\right) \bigg \{L_t-2 \, r_{ow} \left( \frac{\cos \left( \theta _{ow,2}+\alpha _2\right) }{\sin \left( \alpha _2\right) } +\frac{\cos \left( \theta _{ow,3}+\alpha _3\right) }{\sin \left( \alpha _3\right) }\right) \bigg \} \nonumber \\&\quad +\sigma _{gw}\sigma _{go}\bigg \{L_{s,2}^{orig}\cos \left( \theta _{gw,2}^{r,orig/alt}\right) +L_{s,3}^{orig}\cos \left( \theta _{gw,3}^{r,orig/alt}\right) \bigg \} \nonumber \\&\quad +\sigma _{gw}\sigma _{go}\cos \left( \theta _{gw}^{r,alt}\right) \bigg \{ 2 \, r_{ow} \left( \frac{\cos \left( \theta _{ow,2}+\alpha _2\right) }{\sin \left( \alpha _2\right) } +\frac{\cos \left( \theta _{ow,3}+\alpha _3\right) }{\sin \left( \alpha _3\right) }\right) -L_{s,2}^{orig}-L_{s,3}^{orig} \bigg \} \end{aligned}$$

  (66)
  $$\begin{aligned} C_4&=\sigma _{go}^2 \left[ \begin{aligned}&r_{ow}^2 \Bigg \lbrace { \begin{aligned}&\theta _{ow,2}+\theta _{ow,3}+\alpha _2+\alpha _3-\pi \\&\cos \left( \theta _{ow,2}\right) \frac{\cos \left( \theta _{ow,2}+\alpha _2\right) }{\sin \left( \alpha _2\right) } +\cos \left( \theta _{ow,3}\right) \frac{\cos \left( \theta _{ow,3}+\alpha _3\right) }{\sin \left( \alpha _3\right) } \end{aligned} \Bigg \rbrace } \\&+\Delta A_{w,2}+\Delta A_{w,3}-A_t \end{aligned} \right] \end{aligned}$$

  (67)

• PL displacement with the formation of two spreading oil layers in corners 1 and 2:

  $$\begin{aligned} r_{go}^3 C_1+r_{go}^2 C_2+r_{go} C_3+C_4=0 \end{aligned}$$

  (68)

  where $C_1$–$C_4$ are calculated from:

  $$\begin{aligned} C_1=&\frac{\sigma _{ow}\sigma _{go}}{r_{ow}}\bigg \{ \pi -\alpha _1-\alpha _2-2 \, \theta _{go}^r - \cos \left( \theta _{go}^r\right) \left( \frac{\cos (\theta _{go}^r+\alpha _1)}{\sin (\alpha _1)}+\frac{\cos (\theta _{go}^r+\alpha _2)}{\sin (\alpha _2)}\right) \bigg \} \end{aligned}$$

  (69)
  $$\begin{aligned} C_2=&\frac{\sigma _{ow}}{r_{ow}}\left[ \begin{aligned}&\sigma _{ow}\bigg \{ \frac{\Delta A_{w,3}}{r_{ow}}+2 \, r_{ow} \left( \theta _{ow,3}+\alpha _3-\frac{\pi }{2} \right) \bigg \}\\&+\sigma _{go}\cos \left( \theta _{go}^r\right) \bigg \{L_t-2 \, r_{ow} \frac{\cos \left( \theta _{ow,3}+\alpha _3\right) }{\sin \left( \alpha _3\right) }\bigg \}\\&+\sigma _{gw} \bigg \{L_{s,3}^{orig}\cos \left( \theta _{gw,3}^{r,orig/alt}\right) +\cos \left( \theta _{gw}^{r,alt}\right) \left( 2 \, r_{ow} \frac{\cos \left( \theta _{ow,3}+\alpha _3\right) }{\sin \left( \alpha _3\right) }-L_{s,3}^{orig}\right) \bigg \} \end{aligned} \right] \nonumber \\&+\sigma _{go}^2 \bigg \{ \pi -\alpha _1-\alpha _2-2 \, \theta _{go}^r - \cos \left( \theta _{go}^r\right) \left( \frac{\cos (\theta _{go}^r+\alpha _1)}{\sin (\alpha _1)}+\frac{\cos (\theta _{go}^r+\alpha _2)}{\sin (\alpha _2)}\right) \bigg \}\nonumber \\&+2 \, \sigma _{gw}^2 \bigg \{ \frac{\pi }{2}-\theta _{gw,3}-\alpha _3 -\cos \left( \theta _{gw,3}^{r,orig/alt}\right) \frac{\cos (\theta _{gw,3}^r+\alpha _3)}{\sin (\alpha _3)} \bigg \} \end{aligned}$$

  (70)
  $$\begin{aligned} C_3=&\frac{\sigma _{ow}\sigma _{go}}{r_{ow}}\left[ 2 \, \Delta A_{w,2}-A_t+r_{ow}^2 \bigg \{\theta _{ow,3}+\alpha _3-\frac{\pi }{2} +\cos \left( \theta _{ow,3}\right) \frac{\cos \left( \theta _{ow,3}+\alpha _3\right) }{\sin \left( \alpha _3\right) }\bigg \} \right] \nonumber \\&+2 \, \sigma _{ow}\sigma _{go}r_{ow}\left( \theta _{ow,3}+\alpha _3-\frac{\pi }{2}\right) +\sigma _{go}^2 \cos \left( \theta _{go}^r\right) \bigg \{L_t-\frac{\cos \left( \theta _{ow,3}+\alpha _3\right) }{\sin \left( \alpha _3\right) }\bigg \} \nonumber \\&+\sigma _{gw}\sigma _{go}\Bigg \{L_{s,3}^{orig}\cos \left( \theta _{gw,3}^{r,orig/alt}\right) +\cos \left( \theta _{gw}^{r,alt}\right) \left( 2 \, r_{ow}\frac{\cos \left( \theta _{ow,3}+\alpha _3\right) }{\sin \left( \alpha _3\right) }-L_{s,3}^{orig}\right) \Bigg \} \nonumber \\ \end{aligned}$$

  (71)
  $$\begin{aligned} C_4=&\sigma _{go}^2 \left[ r_{ow}^2 \bigg \{\theta _{ow,3}+\alpha _3-\frac{\pi }{2} +\cos \left( \theta _{ow,3}\right) \frac{\cos \left( \theta _{ow,3}+\alpha _3\right) }{\sin \left( \alpha _3\right) } \bigg \} +\Delta A_{w,3}-A_t\right] \end{aligned}$$

  (72)

• PL displacement with the formation of three spreading oil layers: see Eq. (30) in Sect. 4.

• PL displacement without formation of any spreading oil layers: see Eqs. (26)–(29) in Sect. 4.

1.2 Three-Phase Layer Collapse Events

• PL displacement with the single collapse of a spreading oil layer in corner 3: see Eqs. (31)–(35) in Sect. 4.

• PL displacement with the simultaneous collapse of two spreading oil layers in corners 2 and 3:

  $$\begin{aligned} r_{gw}^3 \frac{\sigma _{ow}^2 C_1}{r_{ow}^2}+r_{gw}^2 \frac{2 \, \sigma _{ow}\sigma _{gw} C_2}{r_{ow}} +r_{gw} C_3+\sigma _{gw} C_4=0 \end{aligned}$$

  (73)

  where $C_1$–$C_4$ are expressed by:

  $$\begin{aligned} C_1&=\theta _{gw,2}+\theta _{gw,3}+\alpha _2+\alpha _3-\pi +\cos \left( \theta _{gw,2}\right) \frac{\cos (\theta _{gw,2}+\alpha _2)}{\sin (\alpha _2)} \nonumber \\&\quad +\,\cos \left( \theta _{gw,3}\right) \frac{\cos (\theta _{gw,3}+\alpha _3)}{\sin (\alpha _3)} \end{aligned}$$

  (74)
  $$\begin{aligned} C_2&=\frac{\cos \left( \theta _{gw,2}+\alpha _2\right) }{\sin \left( \alpha _2\right) } \left( \cos \left( \theta _{gw,2}^{r,orig/alt}\right) -\cos \left( \theta _{gw,2}\right) \right) \nonumber \\&\quad +\,\frac{\cos \left( \theta _{gw,3}+\alpha _3\right) }{\sin \left( \alpha _3\right) } \left( \cos \left( \theta _{gw,3}^{r,orig/alt}\right) -\cos \left( \theta _{gw,3}\right) \right) \end{aligned}$$

  (75)
  $$\begin{aligned} C_3&= \sigma _{ow}^2 \bigg \{2 \left( \pi -\theta _{ow,2}-\theta _{ow,3}-\alpha _2-\alpha _3\right) -\frac{\Delta A_{w,2}+\Delta A_{w,3}}{r_{ow}^2} \bigg \}\nonumber \\&\quad +\,2 \,\sigma _{ow}\sigma _{go} \cos \left( \theta _{go}^r\right) \left( \frac{\cos \left( \theta _{ow,2}+\alpha _2\right) }{\sin \left( \alpha _2\right) } +\frac{\cos \left( \theta _{ow,3}+\alpha _3\right) }{\sin \left( \alpha _3\right) }\right) \nonumber \\&\quad -\frac{\sigma _{ow}\sigma _{gw}}{r_{ow}} \left[ \begin{aligned}&L_{s,2}^{orig}\cos \left( \theta _{gw,2}^{r,orig/alt}\right) +L_{s,3}^{orig}\cos \left( \theta _{gw,3}^{r,orig/alt}\right) \\&\quad +\,\cos \left( \theta _{gw}^{r,alt}\right) \Bigg \lbrace { \begin{aligned}&2 \, r_{ow} \left( \frac{\cos \left( \theta _{ow,2}+\alpha _2\right) }{\sin \left( \alpha _2\right) } +\frac{\cos \left( \theta _{ow,3}+\alpha _3\right) }{\sin \left( \alpha _3\right) }\right) \\&-L_{s,2}^{orig}-L_{s,3}^{orig} \end{aligned} \Bigg \rbrace } \end{aligned} \right] \nonumber \\&\quad +\,\sigma _{go}^2 \left\{ 2 \, \theta _{go}^r+\alpha _2+\alpha _3-\pi +\cos \left( \theta _{go}^r\right) \left( \frac{\cos \left( \theta _{go}^r+\alpha _2\right) }{\sin \left( \alpha _2\right) }+\frac{\cos \left( \theta _{go}^r+\alpha _3\right) }{\sin \left( \alpha _3\right) }\right) \right\} \nonumber \\&\quad +\,\sigma _{gw}^2 \Bigg \lbrace { \begin{aligned}&\pi -\theta _{gw,2}-\alpha _2-\frac{\cos \left( \theta _{gw,2}+\alpha _2\right) }{\sin \left( \alpha _2\right) }\left( 2 \, \cos \left( \theta _{gw,2}^{r,orig/alt}\right) -\cos \left( \theta _{gw,2}\right) \right) \\&-\theta _{gw,3}-\alpha _3-\frac{\cos \left( \theta _{gw,3}+\alpha _3\right) }{\sin \left( \alpha _3\right) }\left( 2 \, \cos \left( \theta _{gw,3}^{r,orig/alt}\right) -\cos \left( \theta _{gw,3}\right) \right) \end{aligned} \Bigg \rbrace } \end{aligned}$$

  (76)
  $$\begin{aligned} C_4&=\sigma _{ow} \bigg \{\frac{\Delta A_{w,2}+\Delta A_{w,3}}{r_{ow}}+2 \, r_{ow}\left( \theta _{ow,2}+\theta _{ow,3}+\alpha _2+\alpha _3-\pi \right) \bigg \} \nonumber \\&\quad -\,2 \, \sigma _{go} r_{ow} \cos \left( \theta _{go}^r\right) \left( \frac{\cos \left( \theta _{ow,2}+\alpha _2\right) }{\sin \left( \alpha _2\right) } +\frac{\cos \left( \theta _{ow,3}+\alpha _3\right) }{\sin \left( \alpha _3\right) }\right) \nonumber \\&+\sigma _{gw}\left[ \begin{aligned}&L_{s,2}^{orig}\cos \left( \theta _{gw,2}^{r,orig/alt}\right) +L_{s,3}^{orig}\cos \left( \theta _{gw,3}^{r,orig/alt}\right) \\&\quad +\,\cos \left( \theta _{gw}^{r,alt}\right) \Bigg \lbrace { \begin{aligned}&2 \, r_{ow}\left( \frac{\cos \left( \theta _{ow,2}+\alpha _2\right) }{\sin \left( \alpha _2\right) } +\frac{\cos \left( \theta _{ow,3}+\alpha _3\right) }{\sin \left( \alpha _3\right) }\right) \\&-L_{s,2}^{orig}-L_{s,3}^{orig} \end{aligned} \Bigg \rbrace } \end{aligned} \right] \end{aligned}$$

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• PL displacement with the simultaneous collapse of three spreading oil layers (configuration changes D $\rightarrow$ F in Fig. 1):

  $$\begin{aligned} r_{gw}^3 \frac{\sigma _{ow}^2 C_1}{r_{ow}^2}+r_{gw}^2 \frac{2 \, \sigma _{ow}\sigma _{gw} C_2}{r_{ow}} +r_{gw} C_3+\sigma _{gw} C_4=0 \end{aligned}$$

  (78)

  where $C_1$–$C_4$ are given by:

  $$\begin{aligned} C_1&=\theta _{gw,1}+\theta _{gw,2}+\theta _{gw,3}-\pi +\cos \left( \theta _{gw,1}\right) \frac{\cos (\theta _{gw,1}+\alpha _1)}{\sin (\alpha _1)} \nonumber \\&\quad +\,\cos \left( \theta _{gw,2}\right) \frac{\cos (\theta _{gw,2}+\alpha _2)}{\sin (\alpha _2)}+\cos \left( \theta _{gw,3}\right) \frac{\cos (\theta _{gw,3}+\alpha _3)}{\sin (\alpha _3)} \end{aligned}$$

  (79)
  $$\begin{aligned} C_2&=\frac{\cos \left( \theta _{gw,1}+\alpha _1\right) }{\sin \left( \alpha _1\right) } \left( \cos \left( \theta _{gw,1}^{r,orig/alt}\right) -\cos \left( \theta _{gw,1}\right) \right) \nonumber \\&\quad +\,\frac{\cos \left( \theta _{gw,2}+\alpha _2\right) }{\sin \left( \alpha _2\right) } \left( \cos \left( \theta _{gw,2}^{r,orig/alt}\right) -\cos \left( \theta _{gw,2}\right) \right) \nonumber \\&\quad +\,\frac{\cos \left( \theta _{gw,3}+\alpha _3\right) }{\sin \left( \alpha _3\right) } \left( \cos \left( \theta _{gw,3}^{r,orig/alt}\right) -\cos \left( \theta _{gw,3}\right) \right) \end{aligned}$$

  (80)
  $$\begin{aligned} C_3&= \sigma _{ow}^2 \bigg \{2 \left( \pi -\theta _{ow,1}-\theta _{ow,2}-\theta _{ow,3}\right) -\frac{\Delta A_{w,1}+\Delta A_{w,2}+\Delta A_{w,3}}{r_{ow}^2} \bigg \}\nonumber \\&\quad +\,2 \,\sigma _{ow}\sigma _{go} \cos \left( \theta _{go}^r\right) \left( \frac{\cos \left( \theta _{ow,1}+\alpha _1\right) }{\sin \left( \alpha _1\right) } +\frac{\cos \left( \theta _{ow,2}+\alpha _2\right) }{\sin \left( \alpha _2\right) } +\frac{\cos \left( \theta _{ow,3}+\alpha _3\right) }{\sin \left( \alpha _3\right) }\right) \nonumber \\&-\quad \frac{\sigma _{ow}\sigma _{gw}}{r_{ow}} \left[ \begin{aligned}&L_{s,1}^{orig}\cos \left( \theta _{gw,1}^{r,orig/alt}\right) +L_{s,2}^{orig}\cos \left( \theta _{gw,2}^{r,orig/alt}\right) +L_{s,3}^{orig}\cos \left( \theta _{gw,3}^{r,orig/alt}\right) \\&+\cos \left( \theta _{gw}^{r,alt}\right) \Bigg \lbrace { \begin{aligned}&2 \, r_{ow} \left( \begin{aligned}&\frac{\cos \left( \theta _{ow,1}+\alpha _1\right) }{\sin \left( \alpha _1\right) } +\frac{\cos \left( \theta _{ow,2}+\alpha _2\right) }{\sin \left( \alpha _2\right) }\\&+\frac{\cos \left( \theta _{ow,3}+\alpha _3\right) }{\sin \left( \alpha _3\right) } \end{aligned} \right) \\&-L_{s,1}^{orig}-L_{s,2}^{orig}-L_{s,3}^{orig} \end{aligned} \Bigg \rbrace } \end{aligned} \right] \nonumber \\&\quad +\,\sigma _{go}^2 \Bigg \{ 3 \, \theta _{go}^r-\pi +\cos \left( \theta _{go}^r\right) \left( \frac{\cos \left( \theta _{go}^r+\alpha _1\right) }{\sin \left( \alpha _1\right) }+ \frac{\cos \left( \theta _{go}^r+\alpha _2\right) }{\sin \left( \alpha _2\right) }+\frac{\cos \left( \theta _{go}^r+\alpha _3\right) }{\sin \left( \alpha _3\right) }\right) \Bigg \} \nonumber \\&\quad +\,\sigma _{gw}^2 \Bigg \lbrace { \begin{aligned}&\pi -\theta _{gw,1}-\frac{\cos \left( \theta _{gw,1}+\alpha _1\right) }{\sin \left( \alpha _1\right) }\left( 2 \, \cos \left( \theta _{gw,1}^{r,orig/alt}\right) -\cos \left( \theta _{gw,1}\right) \right) \\&-\theta _{gw,2}-\frac{\cos \left( \theta _{gw,2}+\alpha _2\right) }{\sin \left( \alpha _2\right) }\left( 2 \, \cos \left( \theta _{gw,2}^{r,orig/alt}\right) -\cos \left( \theta _{gw,2}\right) \right) \\&-\theta _{gw,3}-\frac{\cos \left( \theta _{gw,3}+\alpha _3\right) }{\sin \left( \alpha _3\right) }\left( 2 \, \cos \left( \theta _{gw,3}^{r,orig/alt}\right) -\cos \left( \theta _{gw,3}\right) \right) \end{aligned} \Bigg \rbrace } \end{aligned}$$

  (81)
  $$\begin{aligned} C_4&=\sigma _{ow} \bigg \{\frac{\Delta A_{w,1}+\Delta A_{w,2}+\Delta A_{w,3}}{r_{ow}}+2 \, r_{ow}\left( \theta _{ow,1}+\theta _{ow,2}+\theta _{ow,3}-\pi \right) \bigg \} \nonumber \\&\quad -\,2 \, \sigma _{go} r_{ow} \cos \left( \theta _{go}^r\right) \left( \frac{\cos \left( \theta _{ow,1}+\alpha _1\right) }{\sin \left( \alpha _1\right) } +\frac{\cos \left( \theta _{ow,2}+\alpha _2\right) }{\sin \left( \alpha _2\right) }+\frac{\cos \left( \theta _{ow,3}+\alpha _3\right) }{\sin \left( \alpha _3\right) }\right) \nonumber \\&\quad +\,\sigma _{gw}\left[ \begin{aligned}&L_{s,1}^{orig}\cos \left( \theta _{gw,1}^{r,orig/alt}\right) +L_{s,2}^{orig}\cos \left( \theta _{gw,2}^{r,orig/alt}\right) +L_{s,3}^{orig}\cos \left( \theta _{gw,3}^{r,orig/alt}\right) \\&+\cos \left( \theta _{gw}^{r,alt}\right) \Bigg \lbrace { \begin{aligned}&2 \, r_{ow}\left( \begin{aligned}&\frac{\cos \left( \theta _{ow,1}+\alpha _1\right) }{\sin \left( \alpha _1\right) }+\frac{\cos \left( \theta _{ow,2}+\alpha _2\right) }{\sin \left( \alpha _2\right) } \\&+\frac{\cos \left( \theta _{ow,3}+\alpha _3\right) }{\sin \left( \alpha _3\right) } \end{aligned} \right) \\&-L_{s,1}^{orig}-L_{s,2}^{orig}-L_{s,3}^{orig} \end{aligned} \Bigg \rbrace } \end{aligned} \right] \end{aligned}$$

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