2.1. Governing equations

We study the dynamic interaction of heavier and lighter gravity currents in a confined porous layer in a Cartesian configuration, as shown in figure 1. We follow and extend the standard steps to study the dynamics of single gravity current injection into a confined porous layer (e.g. Huppert & Woods Reference Huppert and Woods1995; Pegler et al. Reference Pegler, Huppert and Neufeld2014; Zheng et al. Reference Zheng, Guo, Christov, Celia and Stone2015a) and the interaction of two unconfined gravity currents (e.g. Woods & Mason Reference Woods and Mason2000). In the model problem, fluids 1 and 3 are both injected into a homogeneous porous layer with finite thickness $h_0$, initially filled with ambient fluid 2. The density and viscosity of the fluids are denoted by $\rho _i$ and $\mu _i$, respectively, with $i = \{1,2,3\}$, and the permeability and porosity of the porous layer are denoted by $k$ and $\phi$. It is assumed that $\rho _1 > \rho _2 > \rho _3$ in the current work, such that two gravity currents are generated, including a lighter one that spreads along the top ($\kern0.7pt y=h_0$) and a heavier one that spreads along the base ($\kern0.7pt y=0$). We neglect the influence of fluids mixing and wetting and capillary forces such that sharp interfaces are maintained between different fluids. The profile shape of the heavier current is denoted by $h_1 (x,t)$ and that of the lighter one is denoted by $h_2 (x,t)$, or $\hat {h}_2 (x,t) \equiv h_0 - h_2 (x,t)$, equivalently. Meanwhile, the thickness of the displaced (ambient) layer can also be estimated by $h_2 (x,t) - h_1 (x,t) = h_0 - h_1(x,t) - \hat {h}_2 (x,t)$.

It is also assumed that the fluid–fluid interfaces are long and thin, such that the vertical component of the Darcy velocity is small compared with the horizontal component. This is seen directly from the continuity equation for incompressible flow $\partial u/\partial x + \partial v/\partial y = 0$, i.e. the characteristic velocity and length scales in horizontal and vertical directions must satisfy $u_c/x_c \sim v_c/y_c$. We can hence define an aspect ratio $\delta \equiv y_c/x_c$ of a gravity current, such that $v_c/u_c \sim \delta$. We must need $\delta \ll 1$ for the vertical velocity scale to be negligible, i.e. $v_c \ll u_c$. Accordingly, the fluid pressure follows hydro-static distribution within each layer and is given by

(2.1a)\begin{gather} p_1 (x,y,t) = p_0 (x,t) - \rho_1 g y, \quad 0 \leq y \leq h_1, \end{gather}

(2.1b)\begin{gather}p_2 (x,y,t) = p_0 (x,t) - \rho_1 g h_1 - \rho_2 g(y-h_1), \quad h_1 < y \leq h_2, \end{gather}

(2.1c)\begin{gather}p_3 (x,y,t) = p_0 (x,t) - \rho_1 g h_1 - \rho_2 g(h_2-h_1) - \rho_3 g(y-h_2), \quad h_2 < y \leq h_0, \end{gather}

where $p_i(x,y,t)$ denotes the pressure field within fluid layer $i = \{1,2,3\}$ and $p_0 (x,t)$ denotes the background pressure distribution along the base of the porous layer ($\kern0.7pt y=0$). Here, $g$ is gravitational acceleration. The horizontal pressure gradients that drive the flow can then be calculated as

(2.2a)\begin{gather} \frac{\partial p_1}{\partial x} = \frac{\partial p_0}{\partial x}, \end{gather}

(2.2b)\begin{gather}\frac{\partial p_2}{\partial x} = \frac{\partial p_0}{\partial x} - {\rm \Delta} \rho_1 g \frac{\partial h_1}{\partial x}, \end{gather}

(2.2c)\begin{gather}\frac{\partial p_3}{\partial x} = \frac{\partial p_0}{\partial x} - {\rm \Delta} \rho_1 g \frac{\partial h_1}{\partial x} - {\rm \Delta} \rho_2 g\frac{\partial h_2}{\partial x}, \end{gather}

where ${\rm \Delta} \rho _1 \equiv \rho _1 -\rho _2 > 0$ and ${\rm \Delta} \rho _2 \equiv \rho _2 -\rho _3 > 0$. Darcy's Law can now be applied to provide the horizontal velocity of the fluids within each layer

(2.3a)\begin{gather} u_1 ={-}\frac{k}{\mu_1} \frac{\partial p_1}{\partial x} ={-}\frac{k}{\mu_1} \frac{\partial p_0}{\partial x}, \end{gather}

(2.3b)\begin{gather}u_2 ={-}\frac{k}{\mu_2} \frac{\partial p_2}{\partial x} ={-}\frac{k}{\mu_2} \left( \frac{\partial p_0}{\partial x} - {\rm \Delta} \rho_1 g \frac{\partial h_1}{\partial x} \right), \end{gather}

(2.3c)\begin{gather}u_3 ={-}\frac{k}{\mu_3} \frac{\partial p_3}{\partial x} ={-}\frac{k}{\mu_3} \left( \frac{\partial p_0}{\partial x} - {\rm \Delta} \rho_1 g \frac{\partial h_1}{\partial x} - {\rm \Delta} \rho_2 g\frac{\partial h_2}{\partial x} \right). \end{gather}

It is already seen that, physically, a horizontal velocity for the fluid layers can be driven by both the background pressure gradient ($\partial p_0/\partial x$) and buoyancy ($\propto \partial h_1/\partial x$ or $\propto \partial h_2/\partial x$). We later show that $\partial p_0/\partial x$ is also under the influence of injection rate, in addition to the buoyancy of the injected fluids. It is also of interest to note that, based on (2.2), the pressure gradient $\partial p_i/\partial x$ within each fluid layer is independent of $y$. Accordingly, within the same fluid layer, at any time $t$ of interest, the Darcy velocity remains the same along the $y$ direction at the same horizontal position $x$.

In addition, local conservation of fluid volume (i.e. local continuity) in $[x, x + {\rm d} x]$ within each fluid layer requires that

(2.4a)\begin{gather} \phi \frac{\partial h_1}{\partial t} + \frac{\partial (h_1 u_1)}{\partial x} = 0, \end{gather}

(2.4b)\begin{gather}\phi \frac{\partial (h_2 - h_1)}{\partial t} + \frac{\partial [(h_2 - h_1) u_2]}{\partial x} =0, \end{gather}

(2.4c)\begin{gather}\phi \frac{\partial (h_0 - h_2)}{\partial t} + \frac{\partial [(h_0 - h_2) u_3]}{\partial x} =0. \end{gather}

Physically, this says that the shape evolution of a fluid layer is dependent on the net fluxes across the vertical boundaries of an infinitely thin control volume $[x, x + {\rm d} x]$. Then, by substituting (2.3a,c) into (2.4a,c), we arrive at the evolution equations for the interface shape of the heavier current $h_1 (x,t)$ and the lighter current $\hat {h}_2 (x,t) \equiv h_0 - h_2 (x,t)$:

(2.5a)\begin{gather} \phi \frac{\partial h_1}{\partial t} - \frac{k}{\mu_1} \frac{\partial}{\partial x} \left( h_1 \frac{\partial p_0}{\partial x} \right) = 0, \end{gather}

(2.5b)\begin{gather}\phi \frac{\partial \hat{h}_2}{\partial t} - \frac{k}{\mu_3} \frac{\partial}{\partial x} \left[ \hat{h}_2 \left( \frac{\partial p_0}{\partial x} - {\rm \Delta} \rho_1 g \frac{\partial h_1}{\partial x} + {\rm \Delta} \rho_2 g \frac{\partial \hat{h}_2}{\partial x} \right) \right] = 0, \end{gather}

including a background pressure gradient $\partial p_0 / \partial x$ along the base of the porous layer ($\kern0.7pt y=0$) that is yet to be determined.

Global conservation of the total volume of the injected fluids, meanwhile, provides that

(2.6a)\begin{gather} \phi \int_0^{x_{f1}(t)} h_1 (x,t) \,\mathrm{d} x = q_1 t, \end{gather}

(2.6b)\begin{gather}\phi \int_0^{x_{f2}(t)} \hat{h}_2 (x,t) \,\mathrm{d} x = q_2 t, \end{gather}

with $q_1$ and $q_2$ representing the area injection rates, and $x_{f1}(t)$ and $x_{f2}(t)$ representing the time-dependent locations of the propagating front of the heavier and lighter currents, respectively. The form of (2.6) indicates that the injection of the heavier and lighter fluids proceeds simultaneously at constant rates $q_1$ and $q_2$. However, we also note that the analysis can be extended to account for the influence of time-dependent injection rates.

Then, denoting the total injection rate as $q \equiv q_1+q_2$, equivalently, volume conservation requires that

(2.7)\begin{equation} q = h_1 u_1 + (h_0-h_1-\hat{h}_2) u_2 + \hat{h}_2 u_3, \end{equation}

considering the contribution of all three fluid layers. Then, by substituting (2.3a–c) into (2.7), we obtain an explicit expression for the background pressure gradient in terms of $h_1 (x,t)$ and $\hat {h}_2 (x,t)$ as

(2.8)\begin{equation} \frac{\partial p_0}{\partial x} = \frac{-\dfrac{\mu_1}{k} q + {\rm \Delta} \rho_1 g [ M_2(h_0-h_1-\hat{h}_2) + M_3\hat{h}_2] \dfrac{\partial h_1}{\partial x}- {\rm \Delta} \rho_2 g M_3 \hat{h}_2 \dfrac{\partial \hat{h}_2}{\partial x}}{h_1 + M_2 (h_0-h_1-\hat{h}_2) + M_3 \hat{h}_2}, \end{equation}

where two viscosity ratios $M_2$ and $M_3$ have been introduced as

(2.9a,b)\begin{equation} M_2 \equiv \frac{\mu_1}{\mu_2}\quad \mbox{and}\quad M_3 \equiv \frac{\mu_1}{\mu_3}. \end{equation}

Equation (2.8) indicates that there are three major contributions of the background pressure gradient $\partial p_0/\partial x$ and hence the flow: the pumping force ($\propto q$), the buoyancy force of the heavier current ($\propto {\rm \Delta} \rho _1 g$) and the buoyancy force of the lighter current ($\propto {\rm \Delta} \rho _2 g$). Meanwhile, (2.8) is ready to be substituted back into (2.5) to provide two coupled PDEs for the time evolution of the profile shape of the heavier and lighter currents $h_1 (x,t)$ and $\hat {h}_2 (x,t)$:

(2.10a)\begin{gather} \phi \frac{\partial h_1}{\partial t} \!+\! \frac{\partial}{\partial x} \left[ \frac{q h_1 \!-\! \dfrac{{\rm \Delta} \rho_1 g k}{\mu_1}[ M_2(h_0-h_1-\hat{h}_2)+M_3\hat{h}_2]h_1\dfrac{\partial h_1}{\partial x} + \dfrac{{\rm \Delta} \rho_2 g k}{\mu_3} h_1 \hat h_2 \dfrac{\partial \hat{h}_2}{\partial x} }{h_1+M_2(h_0-h_1-\hat{h}_2)+M_3\hat{h}_2} \right] \!=\! 0, \end{gather}

(2.10b)\begin{gather}\phi \frac{\partial \hat{h}_2}{\partial t} \!+\! \frac{\partial}{\partial x} \left[ \frac{M_3 q \hat h_2 + \dfrac{{\rm \Delta} \rho_1 g k}{\mu_1}M_3 h_1\hat h_2 \dfrac{\partial h_1}{\partial x} \!-\! \dfrac{{\rm \Delta} \rho_2 g k}{\mu_3}[ M_2(h_0-h_1\!-\!\hat{h}_2)\!+\!h_1]\hat h_2\dfrac{\partial \hat{h}_2}{\partial x}}{h_1\!+\!M_2(h_0\!-h_1\!-\hat{h}_2)\!+\!M_3\hat{h}_2} \right] \!=\! 0. \end{gather}

Similarly, (2.10a,b) indicate that, physically, the time evolution of the profile shape of the two currents is under the influence of the pumping force ($\propto q$), the buoyancy force of the heavier current ($\propto {\rm \Delta} \rho _1 g$) and the buoyancy force of the lighter current ($\propto {\rm \Delta} \rho _2 g$). The competition between each of them can possibly vary with space and time, and leads to many facets of dynamic behaviours of the interacting gravity currents.

Finally, to complete the problem, appropriate initial and boundary conditions (IBCs) are needed. In particular, we assume that, initially, there is only the ambient fluid 2 filling up the entire space of the porous layer, which provides a set of initial conditions:

(2.11a,b)\begin{equation} h_1 (x,0) = 0\quad \mbox{and}\quad \hat{h}_2 (x,0) = 0. \end{equation}

In addition, the frontal condition of viscous gravity currents leads to a set of Dirichlet boundary conditions at $x=x_{f1}(t)$ and $x=x_{f2}(t)$:

(2.12a,b)\begin{equation} h_1 (x_{f1}(t),t) = 0\quad \mbox{and}\quad \hat{h}_2 (x_{f2}(t),t) = 0. \end{equation}

By integrating the evolution equations (2.10a,b) from $x=0$ towards $x = \infty$ and applying the global conservation of fluid volume (2.6a,b), we obtain a set of flux boundary conditions at $x = 0$:

(2.13a)\begin{gather} \left.\frac{q h_1 - \dfrac{{\rm \Delta} \rho_1 g k}{\mu_1}[ M_2(h_0-h_1-\hat{h}_2)+M_3\hat{h}_2]h_1\dfrac{\partial h_1}{\partial x} + \dfrac{{\rm \Delta} \rho_2 g k}{\mu_3} h_1 \hat h_2 \dfrac{\partial \hat{h}_2}{\partial x} }{h_1+M_2(h_0-h_1-\hat{h}_2)+M_3\hat{h}_2} \right|_{x=0} = q_1, \end{gather}

(2.13b)\begin{gather}\left.\frac{M_3 q \hat h_2 + \dfrac{{\rm \Delta} \rho_1 g k}{\mu_1} M_3 h_1\hat h_2 \dfrac{\partial h_1}{\partial x} - \dfrac{{\rm \Delta} \rho_2 g k}{\mu_3}[ M_2(h_0-h_1-\hat{h}_2)+h_1]\hat h_2\dfrac{\partial \hat{h}_2}{\partial x}}{h_1+M_2(h_0-h_1-\hat{h}_2)+M_3\hat{h}_2} \right|_{x=0} = q_2. \end{gather}

It is important to note that to arrive at the flux conditions (2.13a,b), we have also assumed that there is no entrainment of the ambient fluid into the gravity currents at the location of the propagating fronts $x_{f1}(t)$ and $x_{f2}(t)$. That said, the fluxes at $x_{f1}(t)$ and $x_{f2}(t)$ are zero. This is a typical situation of viscous gravity current flows, with experimental evidence in many previous studies (e.g. Huppert & Woods Reference Huppert and Woods1995; Woods & Mason Reference Woods and Mason2000; Pegler et al. Reference Pegler, Huppert and Neufeld2014; Zheng, Christov & Stone Reference Zheng, Christov and Stone2014). At high $Re$ with significant inertial effects, however, flow entrainment can become important and the total volume of the current can increase with time (e.g. Marino et al. Reference Marino, Thomas and Linden2005; Linden Reference Linden2012; Sher & Woods Reference Sher and Woods2015). The coupled PDEs (2.10a,b) can now be solved numerically, subject to appropriate initial conditions (ICs) (2.11a,b) and boundary conditions (BCs) (2.12a,b and 2.13a,b).

2.2. Non-dimensionalisation

It is standard first to non-dimensionalise the governing PDEs and IBCs before looking for asymptotic and numerical solutions. We first define dimensionless variables

(2.14a–e)\begin{equation} H_1 \equiv \frac{h_1}{h_0},\quad \hat{H}_2 \equiv \frac{\hat{h}_2}{h_0},\quad X \equiv \frac{x}{x_c},\quad T \equiv \frac{t}{t_c}\quad \mathrm{and}\quad P_0 \equiv \frac{p_0}{p_c}, \end{equation}

based on the following characteristic scales for length, time and fluid pressure:

(2.15a–c)\begin{equation} x_c=\frac{{\rm \Delta} \rho_1 gk h_0^2 }{\mu_1 q},\quad t_c=\frac{{\rm \Delta} \rho_1 g k h_0^3 \phi}{\mu_1 q^2}\quad \mathrm{and}\quad p_c={\rm \Delta} \rho_1 gh_0.\end{equation}

The characteristic scales in (2.15) are chosen such that the unsteady, advective and diffusive terms in the coupled PDEs (2.10a,b) balance each other at $T = {O}(1)$. We then arrive at the dimensionless PDEs:

(2.16a)\begin{gather} \frac{\partial H_1}{\partial T} + \frac{\partial }{\partial X} \left[ \frac{H_1-[ M_2(1-H_1-\hat{H}_2)+M_3\hat{H}_2]H_1\dfrac{\partial H_1}{\partial X}+G M_3H_1\hat{H}_2\dfrac{\partial \hat{H}_2}{\partial X} }{H_1+M_2(1-H_1-\hat{H}_2)+M_3\hat{H}_2} \right] = 0, \end{gather}

(2.16b)\begin{gather}\frac{\partial \hat{H}_2}{\partial T} + \frac{\partial }{\partial X} \left[\frac{M_3\hat{H}_2-[ M_2(1-H_1-\hat{H}_2)+H_1]GM_3\hat{H}_2\dfrac{\partial \hat{H}_2}{\partial X}+ M_3H_1\hat{H}_2\dfrac{\partial H_1}{\partial X} }{H_1+M_2(1-H_1-\hat{H}_2)+M_3\hat{H}_2}\right] = 0, \end{gather}

where four dimensionless parameters $M_2$, $M_3$, $G$ and $Q$ have been introduced, representing the viscosity ratios, density difference and injection rates of the fluids. The viscosity ratios $M_2$ and $M_3$ have already been defined in (2.9a,b). The ratio of the density differences $G$ and partition of the area injection rates $Q$ are defined as

(2.17a,b)\begin{equation} Q \equiv \frac{q_1}{q}\quad \mathrm{and}\quad G \equiv \frac{{\rm \Delta} \rho_2}{{\rm \Delta} \rho_1}.\end{equation}

The definition and physical meaning of the four dimensionless parameters $M_2$, $M_3$, $G$ and $Q$ are summarised in table 1, the influence of which will be discussed later. Compared with the model problem of single fluid injection (e.g. Zheng et al. Reference Zheng, Guo, Christov, Celia and Stone2015a), as governed by a single dimensionless parameter $M_2 \equiv \mu _1/\mu _2$, three more control parameters $M_3$, $G$ and $Q$ appear in the current problem of gravity current interaction due to the introduction of a second current.

Table 1. Summary and physical meaning of the four dimensionless control parameters $M_2$, $M_3$, $G$ and $Q$ that impact the dynamic interaction of heavier and lighter gravity current in confined porous layers.

Meanwhile, the initial and boundary conditions can also be made dimensionless based on (2.14) and (2.15) as

(2.18a,b)\begin{equation} H_1(X,0)=0\quad \mathrm{and}\quad \hat{H}_2(X,0)=0, \end{equation}

and

(2.19a)\begin{gather} H_1(X_{f1}(T),T) = 0, \end{gather}

(2.19b)\begin{gather}\hat{H}_2(X_{f2}(T),T) = 0, \end{gather}

(2.19c)\begin{gather}\left.\frac{H_1-[ M_2(1-H_1-\hat{H}_2)+M_3\hat{H}_2]H_1\dfrac{\partial H_1}{\partial X}+G M_3H_1\hat{H}_2\dfrac{\partial \hat{H}_2}{\partial X} }{H_1+M_2(1-H_1-\hat{H}_2)+M_3\hat{H}_2} \right|_{X=0} = Q, \end{gather}

(2.19d)\begin{gather}\left.\frac{M_3\hat{H}_2-[ M_2(1-H_1-\hat{H}_2)+H_1]GM_3\hat{H}_2\dfrac{\partial \hat{H}_2}{\partial X}+ M_3H_1\hat{H}_2\dfrac{\partial H_1}{\partial X} }{H_1+M_2(1-H_1-\hat{H}_2)+M_3\hat{H}_2} \right|_{X=0} = 1-Q, \end{gather}

now with $X_{f1}(T)$ and $X_{f2}(T)$ representing the location of the propagating front of the invading fluids 1 and 3 in the rescaled coordinate system, respectively. The dimensionless PDEs (2.16a,b) are ready to be solved numerically, subject to IBCs (2.18) and (2.19). Finally, it is of interest to note again that, with flux conditions (2.19c,d) imposed, the requirement for global conservation of mass is also naturally satisfied:

(2.20a)\begin{gather} \int_0^{X_{f1}(T)} H_1 (X,T) \,\mathrm{d} X = QT, \end{gather}

(2.20b)\begin{gather}\int_0^{X_{f2}(T)} \hat{H}_2 (X,T) \,\mathrm{d} X = (1-Q)T. \end{gather}

Equations (2.20a,b) indicate also that the total volume of fluids 1 and 3 must also satisfy

(2.21)\begin{equation} \int_0^{X_{f1}(T)}H_1(X,T)\,\mathrm{d}X + \int_0^{X_{f2}(T)}\hat{H}_2(X,T)\,\mathrm{d}X =T. \end{equation}

A finite-volume scheme has been employed in the current work to solve the coupled PDEs (2.16a,b). The scheme is centred difference in space and explicit in time. Similar schemes have been employed in a series of earlier studies of single current propagation (e.g. Zheng et al. Reference Zheng, Guo, Christov, Celia and Stone2015a; Hinton & Woods Reference Hinton and Woods2018) and more generally to solve nonlinear advective-diffusive PDEs (Kurganov & Tadmor Reference Kurganov and Tadmor2000). The convergence of the scheme has been tested, such that the solutions are stable and no significant differences are observed upon grid refinement in both time and space for both the profile shape and frontal location of the gravity current. More details of the scheme are also provided in Appendix A.

2.3. Some key features of the model

2.3.1. Feature 1: symmetric currents

We can show that if and only if

(2.22a,b)\begin{equation} M_3=1 \quad \mbox{and}\quad \frac{1}{Q}-\frac{1}{G}=1,\end{equation}

the profile shapes $H_1(X,T)$ and $\hat {H}_2(X,T)$ are related through

(2.23)\begin{equation} \hat{H}_2=\lambda H_1, \end{equation}

where $\lambda =1/G=1/Q-1$. Physically, (2.23) indicates that by stretching the profile shape of $H_1$ vertically by a factor of $\lambda$, the heavier and lighter currents become exactly symmetric, as shown in figure 2(a). The upper and lower fronts of the two currents always remain at the same location, i.e. $X_{f1}(T) = X_{f2}(T)$, even when the currents attach to each other and interact dynamically at late times. Accordingly, the governing PDEs (2.10a,b) reduce to a single one

(2.24)\begin{equation} \frac{\partial H_1}{\partial T} + \frac{\partial}{\partial X} \left[\frac{H_1}{M_2 + (1-M_2)H_1/Q} \right] - \frac{\partial}{\partial X} \left[\frac{ M_2H_1(1-H_1/Q)}{M_2 + (1-M_2)H_1/Q}\frac{\partial H_1}{\partial X} \right]= 0, \end{equation}

which includes the influence of both $M_2$ and $G$, and is hence different from the governing PDE for single current injection (in which case $Q \to 1^-$) (e.g. Zheng et al. Reference Zheng, Guo, Christov, Celia and Stone2015a).

Figure 2. Example solutions of symmetric and asymmetric currents $H_1(X,T)$ and $\hat H_2(X,T)$ from numerically solving the coupled PDEs (2.16a,b): (a) symmetric currents with $G=2$, $Q=2/3$; (b) asymmetric currents with $G=2$, $Q=4/5$; (c) asymmetric currents with $G=4$, $Q=2/3$. We have imposed $M_2=1$, $M_3=1$ and $T=100$ in all cases.

More remarks can be provided on conditions (2.22a,b) of symmetric currents. First of all, $M_3=1$ means that the viscosity of the heaviest fluid 1 and that of the lightest fluid 3 must be equal. However, there is no constraint on the viscosity of the ambient fluid 2. Meanwhile, $1/Q-1/G=1$ implies that the pumping and gravitational forces of fluids 1 and 3 must be in balance. Any incremental change in $Q$ or $G$, for example, due to variations in injection rate or density of the fluids, will break such a balance and lead to different flow patterns. The role of conditions (2.22a,b) will be seen more clearly when we provide asymptotic solutions for the dynamic interaction of heavier and lighter currents in § 4.1 (when $M_2=M_3=1$ in regime 1) and § 4.2 (when $M_2 > M_3 = 1$ in regime 2).

For example, subject to an incremental change of $Q$ on the basis of $1/Q-1/G=1$, the balance between pumping and gravitational forces will be broken. For example, when the lightest fluid 3 is injected at a lower rate, $Q$ increases accordingly. Fluid 3 then pushes upwards the heaviest fluid 1 in the neighbourhood of the inlet ($X=0$), making the height of the heavier current $H_1(0,T)$ to be higher than that of the intersection point of the three fluids $H_*$, as shown in figure 2(b). Similarly, the corresponding result is shown in figure 2(c), subject to an incremental change of $G$ on the basis of $1/Q-1/G=1$.

2.3.2. Feature 2: reflected currents

The dynamic interaction of heavier and lighter gravity currents can also lead to flipped (reflected) solutions, as shown in figure 3. If we denote the solutions in figure 3(a) as $H_1(X,T)$ and $\hat {H}_2(X,T)$ with control parameters $(M_2,M_3,Q,G)$, while we denote solutions in figure 3(b) as $H_1^*(X^*,T^*)$ and $\hat {H}_2^*(X^*,T^*)$ with control parameters $(M_2^*,M_3^*,Q^*,G^*)$, the connection of the reflected solutions in figure 3(a,b) can be expressed as

(2.25a)\begin{gather} H_1(X,T) = \hat{H}^*_2(X^*,T^*), \end{gather}

(2.25b)\begin{gather}\hat{H}_2(X,T) = H^*_1(X^*,T^*). \end{gather}

We can further show that the reflected solutions (2.25a,b) appear if and only if the dimensionless control parameters satisfy

(2.26a–d)\begin{equation} M^*_2=\frac{M_2}{M_3}, \quad M^*_3=\frac{1}{M_3},\quad G^*=\frac{1}{G},\quad Q^*=1-Q,\end{equation}

and, at the same time, the variables are subject to stretching rules

(2.27a,b)\begin{equation} X^*=\frac{X}{GM_3}\quad \mbox{and}\quad T^*=\frac{T}{GM_3}. \end{equation}

Physically, the existence of such a symmetry is due originally to the nature of buoyancy. For example, for the injection of a single current, there is no fundamental difference whether buoyancy acts upwards or downwards, as density difference ${\rm \Delta} \rho$ provides the driving force rather than the specific density of each fluid. Nevertheless, in the current context of two current injection and interaction, additional balance is needed between the heavier and lighter currents, in addition to the buoyancy between the injecting and displaced fluids. Detailed analysis indeed shows that such a balance is possible, but additional constraints are placed, for a given buoyancy ratio of the two currents, on the partition of injection rates and viscosity of the fluids, as described by (2.26).

Figure 3. Example solutions of reflected currents: (a) $H_1(X,T)$ and $\hat H_2(X,T)$ with $M_2=1/2$, $M_3=2$, $G=3/5$, $Q=2/5$, $X_R=220$ and $T=50$; (b) $H^*_1(X^*,T^*)$ and $\hat H^*_2(X^*,T^*)$ with $M_2^*=1/4$, $M_3^*=1/2$, $G^*=5/3$, $Q^*=3/5$, $X_R^*=180$ and $T^*=125/3$, when the transforms (2.26) and (2.27) are satisfied. Here, $X_R$ and $X_R^*$ represent the domain length in panels (a) and (b).

An easier way to verify the existence of such a transform is to start from the flux conditions (2.19c,d). Substituting (2.25a,b) into (2.19c,d), we obtain

(2.28a)\begin{gather} \left.\frac{\hat{H}^*_2-[M_2(1-H^*_1-\hat{H}^*_2)+M_3H^*_1]\hat{H}^*_2\dfrac{\partial \hat{H}^*_2}{\partial X}+GM_3H^*_1\hat{H}^*_2\dfrac{\partial H^*_1}{\partial X}}{M_2+(M_3-M_2)H^*_1+(1-M_2)\hat{H}^*_2}\right|_{X=0} = Q, \end{gather}

(2.28b)\begin{gather}\left.\frac{H^*_1-[M_2(1-H^*_1-\hat{H}^*_2)+\hat{H}^*_2]GH^*_1\dfrac{\partial H^*_1}{\partial X}+H^*_1\hat{H}^*_2\dfrac{\partial \hat{H}^*_2}{\partial X}}{\dfrac{M_2}{M_3}+\left(1-\dfrac{M_2}{M_3}\right)H^*_1+ \left(\dfrac{1}{M_3}-\dfrac{M_2}{M_3}\right)\hat{H}^*_2} \right|_{X=0} = 1-Q, \end{gather}

which is to be compared with the original form of (2.19c,d) for the inlet fluxes of the heavier and lighter currents in figure 3(b):

(2.29a)\begin{gather} \left.\frac{H_1^*-[M_2^*(1-H_1^*-\hat{H}_2^*)+M_3^*\hat{H}_2^*]H_1^*\dfrac{\partial H_1^*}{\partial X^*}+G^*M_3^*H_1^*\hat{H}^*_2\dfrac{\partial \hat{H}^*_2}{\partial X^*}}{M_2^*+(1-M_2^*)H_1^*+(M_3^*-M_2^*)\hat H^*_2}\right|_{X^*=0} = Q^*, \end{gather}

(2.29b)\begin{gather}\left.\frac{\hat{H}^*_2-[M^*_2(1-H^*_1-\hat{H}^*_2)+H^*_1]G\hat{H}^*_2\dfrac{\partial \hat{H}^*_2}{\partial X^*}+H^*_1\hat{H}^*_2\dfrac{\partial H^*_1}{\partial X^*}}{\dfrac{M^*_2}{M^*_3}+\left(\dfrac{1}{M^*_3} -\dfrac{M^*_2}{M^*_3}\right)H^*_1+\left(1-\dfrac{M^*_2}{M^*_3}\right)\hat H^*_2} \right|_{X^*=0} = 1-Q^*. \end{gather}

A comparison of the coefficient of every term in (2.28a) and (2.29b), or the coefficient of every term in (2.28b) and (2.29a), immediately leads to the connection of dimensionless parameters (2.26) and the transform for space (2.27a). Then, imposing the same procedure for the full PDEs (2.16), we obtain $T^*/T = X^*/X$, which leads to the transform for time (2.27b).

An important implication of Feature 2 is that we do not need to cover the full range of the dimensionless parameters $(M_2,M_3,Q,G)$ to provide the basic flow patterns of gravity current interaction. For example, we only need to consider $M_3 \in (0,1]$. This is because when $M_3 \in (1, +\infty )$, one can obtain reflected solutions in the $(X^*,T^*)$ space for $M_3^* = 1/M_3 \in (0,1)$ based on transform (2.26b), with $M_2^*$, $G^*$, $Q^*$ chosen according to (2.26a,c,d) and $(X^*,T^*)$ stretched according to (2.27a,b). More remarks are provided in Appendix B. In § 3, we only consider $M_3 \in (0,1]$, which leads to eight distinct regimes of dynamic interaction as governed by $M_2$, $M_3$, $G$ and $Q$. Key features of each regime are discussed later in § 3 and summarised in table 2 and figure 4.

Table 2. Key feature of the eight regimes of gravity current interaction under simultaneous injections of two fluids: parameters, transforms, solutions and special points of dynamic interaction. The propagating speed is $C_2 = 1-(1-M_3)Q$.

Figure 4. Regime diagram of gravity current interaction at late times ($T \gg 1$), as governed by four dimensionless parameters $M_2$, $M_3$, $G$ and $Q$ (see table 1). We only need to consider $M_3\leq 1$ based on Feature 2. Key features of the eight regimes of gravity current interaction are also summarised in table 2.

2.3.3. Feature 3: single current

When $Q \to 1^-$ or $Q \to 0^+$, the current problem degenerates into that of single fluid injection (e.g. Zheng et al. Reference Zheng, Guo, Christov, Celia and Stone2015a), with experimental observations also documented by Pegler et al. (Reference Pegler, Huppert and Neufeld2014). For example, when $Q \to 1^-$, the height of the lighter current vanishes, i.e. $\hat {H}_2(X,T) \to 0^+$. In this case, the governing PDEs (2.16a,b) and boundary and initial conditions reduce to

(2.30)\begin{equation} \frac{\partial H_1}{\partial T} + \frac{\partial }{\partial X} \left[ \frac{H_1}{H_1+M_2(1-H_1)} \right]-\frac{\partial }{\partial X} \left[ \frac{M_2 H_1 (1-H_1)}{H_1+M_2(1-H_1)} \frac{\partial H_1}{\partial X} \right] = 0 \end{equation}

and

(2.31a)\begin{gather} H_1(X,0) = 0, \end{gather}

(2.31b)\begin{gather}H_1(X_{f1}(T),T) = 0, \end{gather}

(2.31c)\begin{gather}\left.\frac{H_1}{H_1+M_2(1-H_1)}-\frac{M_2 H_1 (1-H_1)}{H_1+M_2(1-H_1)} \frac{\partial H_1}{\partial X} \right|_{X=0} = 1. \end{gather}

By denoting $M \equiv 1/M_2$, we recover exactly the same descriptions as those of Zheng et al. (Reference Zheng, Guo, Christov, Celia and Stone2015a) for single fluid injection into a confined porous layer. When $M_2 = 1$, PDE (2.30) further reduces to the form provided by Huppert & Woods (Reference Huppert and Woods1995) for equal-viscosity displacement of buoyant flows. Both the early-time and late-time asymptotic behaviours have been studied and the time transition between them has been demonstrated by numerical solutions of PDE (2.30) that span a wide range of time and length scales by Zheng et al. (Reference Zheng, Guo, Christov, Celia and Stone2015a) and Pegler et al. (Reference Pegler, Huppert and Neufeld2014).

Specifically, four self-similar solutions are identified in the corresponding asymptotic limits, including an unconfined nonlinear diffusive solution at early times ($T \ll 1$) and three different branches of confined self-similar solutions at late times ($T \gg 1$), depending on the viscosity ratio ($M \equiv 1/M_2$): (i) a rarefaction solution when the injected fluid is less viscous than the displaced fluid ($M_2 < 1$); (ii) an inclined straight-line solution with time-dependent slope for equally viscous injecting and displaced fluids ($M_2 = 1$); and (iii) an inclined straight-line solution with constant slope when the injected fluid is more viscous than the displaced fluid ($M_2 > 1$).

3.1. Regimes 1a and 1b: $M_2 = M_3 = 1 (\mu _1 = \mu _2 = \mu _3)$

We first evaluate the limit when the invading and displaced fluids take the same viscosity, i.e. $M_2=M_3=1$. In this case, at late times ($T \gg 1$), the governing PDEs (2.16a,b) become

(3.5a) \begin{gather} \frac{\partial H_1}{\partial T} + \frac{\partial }{\partial X} \bigg[ H_1-(1-H_1)H_1\frac{\partial H_1}{\partial X}+GH_1\hat{H}_2\frac{\partial \hat{H}_2}{\partial X} \bigg] = 0, \end{gather}

(3.5b) \begin{gather}\frac{\partial \hat{H}_2}{\partial T} + \frac{\partial }{\partial X} \bigg[\hat{H}_2-(1-\hat{H}_2)G\hat{H}_2\frac{\partial \hat{H}_2}{\partial X}+ H_1\hat{H}_2\frac{\partial H_1}{\partial X} \bigg] = 0. \end{gather}

A similarity transformation can be defined as $\zeta \equiv (X-T)/T^{1/2}$, and the coupled PDEs (3.5a,b) are then transformed into two coupled ordinary differential equations (ODEs)

(3.6a)\begin{gather} \frac{\zeta}{2}\frac{\mathrm{d} H_1}{\mathrm{d}\zeta} + \frac{\mathrm{d}}{\mathrm{d}\zeta}\bigg[ (1-H_1)H_1\frac{\mathrm{d} H_1}{\mathrm{d}\zeta}-G H_1\hat{H}_2\frac{\mathrm{d} \hat H_2}{\mathrm{d} \zeta}\bigg] = 0, \end{gather}

(3.6b)\begin{gather}\frac{\zeta}{2}\frac{\mathrm{d} \hat{H}_2}{\mathrm{d}\zeta} + \frac{\mathrm{d}}{\mathrm{d}\zeta}\bigg[{-}H_1\hat{H}_2\frac{\mathrm{d} H_1}{\mathrm{d}\zeta}+G(1-\hat{H}_2)\hat{H}_2\frac{\mathrm{d} \widehat{H_2}}{\mathrm{d}\zeta}\bigg] = 0. \end{gather}

The coupled ODEs (3.6a,b) can then be solved subject to BCs (3.2a,b) to provide the self-similar solutions for the interface shape $H_1(\zeta )$ and $\hat H_2(\zeta )$. The form of the ODEs (3.6a,b) and BCs (3.2a,b) indicate that $H_1(\zeta )$ and $\hat H_2(\zeta )$ depend also on $G$ and $Q$, which leads to either symmetric or asymmetric currents.

3.1.1. Regime 1a: symmetric currents $(1/Q-1/G=1)$

When $(1/Q-1/G=1)$, the currents are symmetric based on Feature 1, and $\hat H_2(\zeta ) = H_1(\zeta )/G$. The coupled PDEs (3.6a,b) then reduce to a single ODE for $H_1(\zeta )$ in the form of

(3.7)\begin{equation} \frac{\zeta}{2}\frac{\mathrm{d} H_1}{\mathrm{d}\zeta} + \frac{\mathrm{d}}{\mathrm{d}\zeta} \left[ H_1 \left( 1-\frac{H_1}{Q} \right) \frac{\mathrm{d} H_1}{\mathrm{d}\zeta} \right] = 0. \end{equation}

At leading order, an explicit solution is available as

(3.8)\begin{equation} H_1 =\left\{ \begin{array}{@{}ll} Q, & \zeta \leq \zeta_*,\\ \displaystyle \dfrac{Q}{2}-\zeta\dfrac{Q^{1/2}}{2}, & \zeta_* < \zeta \leq \zeta_{f1},\\ \end{array} \right. \end{equation}

which shows a straight-line frontal structure with

(3.9a,b)\begin{equation} \zeta_{f1}=Q^{1/2}\quad \mbox{and}\quad \zeta_*={-}Q^{1/2}. \end{equation}

Based on symmetry, $\hat H_2 = H_1/G$ is also obtained and the front of the lighter current locates at $\zeta _{f2}=Q^{1/2}$ in the rescaled coordinate system.

We can also compare the self-similar solution (3.8) with numerical solutions of PDEs (2.16a,b), as shown in figure 5. It is demonstrated that after appropriate rescaling based on the transform $\zeta \equiv (X-T)/T^{1/2}$, the PDE numerical solutions at different times collapse onto universal curves. Meanwhile, a further comparison with solutions (3.8) provides very good agreement for the profile shape of both the heavier and lighter gravity currents. It is shown that the profile shape of the heavier or lighter current maintains a straight-line structure near the propagating front, which is also consistent with the prediction of solution (3.8).

Figure 5. Comparison between symmetric solution (3.8) and numerical solutions of PDEs (2.16a,b) in regime 1a, when $M_2=M_3=1$, $G=2/3$ and $Q=2/5$ (such that $1/Q-1/G = 1$): (a) time evolution of the fluid interfaces $H_1$ and $\hat H_2$ at $T=\{20,40,60,80,100\}$ and (b) rescaled interfaces $H_1$ and $\hat H_2$ with $\zeta =(X-T)/T^{1/2}$ and the self-similar solution (3.8), shown as the green lines.

3.2. Regimes 2a and 2b: $M_3 = 1 < M_2 (\mu _2 < \mu _1 = \mu _3)$

We next discuss the interaction regime when the ambient fluid that is being displaced is less viscous than the injected fluids, while the injected heavier and lighter fluids are equally viscous, i.e. when $M_3 = 1 < M_2 (\mu _2 < \mu _1 = \mu _3)$. By substituting $M_3 = 1$ into the governing PDEs (2.16a,b), we obtain a simplified form

(3.10a)\begin{gather} \frac{\partial H_1}{\partial T} + \frac{\partial }{\partial X} \left[ \frac{H_1-[ M_2(1-H_1-\hat{H}_2)+\hat{H}_2]H_1\dfrac{\partial H_1}{\partial X}+G H_1\hat{H}_2\dfrac{\partial \hat{H}_2}{\partial X} }{H_1+M_2(1-H_1-\hat{H}_2)+\hat{H}_2} \right] =0, \end{gather}

(3.10b)\begin{gather}\frac{\partial \hat{H}_2}{\partial T} + \frac{\partial }{\partial X} \left[ \frac{\hat{H}_2-[ M_2(1-H_1-\hat{H}_2)+H_1]G\hat{H}_2\dfrac{\partial \hat{H}_2}{\partial X}+ H_1\hat{H}_2\dfrac{\partial H_1}{\partial X} }{H_1+M_2(1-H_1-\hat{H}_2)+\hat{H}_2} \right] =0. \end{gather}

We next introduce the travelling-wave transform $\eta \equiv X - T$, and PDEs (3.10a,b) are then transformed into two coupled ODEs (after an integration):

(3.11a)\begin{gather} [M_2(1-H_1-\hat{H}_2)+\hat{H}_2]\frac{\mathrm{d}H_1}{\mathrm{d}\eta}- G\hat{H}_2\frac{\mathrm{d}\hat{H}_2}{\mathrm{d}\eta} = (1-M_2)(1-H_1-\hat{H}_2), \end{gather}

(3.11b)\begin{gather}-H_1\frac{\mathrm{d}H_1}{\mathrm{d}\eta}+[H_1+M_2(1-H_1-\hat{H}_2)]G \frac{\mathrm{d}\hat{H}_2}{\mathrm{d}\eta} = (1-M_2)(1-H_1-\hat{H}_2), \end{gather}

for the profile shapes $H_1(\eta )$ and $\hat H_2(\eta )$.

3.2.1. Regime 2a: symmetric currents $(1/Q-1/G=1)$

Again, the currents are symmetric when $1/Q-1/G = 1$ based on Feature 1, in which case $\hat H_2(\eta ) = H_1(\eta )/G$. ODEs (3.11a,b) then reduce to a single one

(3.12)\begin{equation} \left(1-\frac{H_1}{Q}\right)\left[(1-M_2)-M_2 \frac{\mathrm{d}H_1}{\mathrm{d}\eta}\right]= 0, \end{equation}

which admits an explicit solution

(3.13)\begin{equation} H_1 = \left\{ \begin{array}{@{}ll} \displaystyle Q & \eta \leq \eta_*, \\ \displaystyle \dfrac{Q}{2}-\left(1-\dfrac{1}{M_2}\right)\eta & \eta_* < \eta \leq \eta_{f1}, \end{array} \right. \end{equation}

where

(3.14a,b)\begin{equation} \eta_{f1}=\frac{M_2 Q}{2(M_2-1)}\quad \mbox{and}\quad \eta_*={-}\frac{M_2 Q}{2(M_2-1)}. \end{equation}

It is easy to verify that the inlet condition (3.2) and the global mass constraint (2.20a) are both satisfied. Accordingly, the profile shape of the lighter current $\hat H_2(\eta ) = H_1(\eta )/G$ is also available. The frontal location of the lighter current is also $\eta _{f2} = \eta _{f1} = M_2 Q/ 2(M_2-1)$ due to symmetry.

We can also verify the analytical solution (3.13) based on a comparison with the rescaled numerical solutions of PDEs (2.16a,b), as shown in figure 6. After appropriate rescaling of the raw solutions of PDEs (2.16a,b) at different times, the PDE solutions approach a universal curve, which is exactly the self-similar solution (3.13).

Figure 6. Comparison between symmetric solution (3.13) and numerical solutions of PDEs (2.16a,b) in regime 2a, when $M_2=6/5$, $M_3=1$, $G=2$ and $Q=2/3$ (such that $1/Q-1/G = 1$): (a) time evolution of the interfaces $H_1$ and $\hat H_2$ at $T=\{60,120,180,240,300\}$ from numerically solving PDEs (2.16a,b); (b) rescaled interfaces $H_1$ and $H_2$ with $\zeta = X-T$, and the green lines are solution (3.13).

3.2.2. Regime 2b: asymmetric currents $(1/Q-1/G\neq 1)$

The currents are asymmetric in regime 2b, when $1/Q-1/G \neq 1$. However, the flow behaviour is fundamentally different from that in regime 1b. One of the key features is that the vertical deviation of the point where three fluids meet becomes negligible at late times, i.e. $|H^* - H_{i1}| \to 0^+$ for $T \gg 1$, while in regime 1b, the deviation $|H^* - H_{i1}|$ remains finite and time independent. Meanwhile, the frontal structure of both currents can be approximated as straight lines at leading order, while in regime 1b, the profile shapes are significantly curved.

In the current regime of asymmetric currents, the governing PDEs for the profile shapes remain as (3.10a,b). The transform $\eta \equiv X-T$ continues to apply, which leads to ODEs (3.11a,b) for self-similar solutions. Indeed, the rescaled PDE numerical solutions at different times approach a universal curve, as shown in figure 7(b), which is exactly the self-similar solution we explore in this section. Nevertheless, (3.11a,b) do not decouple for asymmetric currents under consideration here. Similar to regime 1b, the PDE numerical solutions (figure 7) show that the profile shapes can still be divided into two parts: (i) $X \in [0,X_*(T)]$, where the heavier and lighter currents attach to each other ($H_1 + \hat H_2 = 1$); and (ii) $X \in [X_*(T),\infty )$, where the heavier and lighter currents do not attach to each other ($H_1 + \hat H_2 < 1$).

Figure 7. Comparison between asymmetric solutions (3.13) and (3.17), and numerical solutions of PDEs (2.16a,b) in regime 2b, when $M_2=6/5$, $M_3=1$, $G=1$ and $Q=2/3$ (such that $1/Q-1/G\neq 1$): (a) time evolution of the interfaces $H_1$ and $\hat H_2$ at $T=\{120,240,360,480,600\}$ from numerically solving PDEs (2.16a,b); (b) rescaled interfaces $H_1$ and $H_2$ with $\eta =X-T$, and the green lines are solutions (3.13) and (3.17).

For $X \in [0,X_*(T)]$, since $H_1 + \hat H_2 = 1$, ODEs (3.11a,b) reduce to a single one which admits a solution of constant thickness

(3.15)\begin{equation} H_1 = a_3. \end{equation}

We expect that $a_3 = H_* = H_{i1}$ at late times ($T \gg 1$). In the frontal region, in contrast, (3.11a,b) admit straight-line solutions for the profile shapes

(3.16a)\begin{gather} H_1 = a_4-\left(1-\frac{1}{M_2}\right)\eta, \end{gather}

(3.16b)\begin{gather}\hat{H}_2 = a_5-\frac{1}{G}\left(1-\frac{1}{M_2}\right)\eta, \end{gather}

where the constants $a_4$ and $a_5$ are yet unknown. It is of interest to note that the slopes of the fronts in (3.16a,b) remain exactly the same as those in regime 2a of symmetric currents when $1/Q-1/G = 1$. This is not surprising since they both come from solving ODEs (3.11a,b) directly and are independent on whether or not $1/Q-1/G = 1$.

Nevertheless, based on the global mass constraint (2.20a) for the heavier current and inlet conditions (3.2a,b) (i.e. $H_{i1} = Q$ and $\hat H_{i2} = 1-Q$ when $M_3 = 1$), the three constants $a_3,a_4,a_5$ can be determined conveniently, and solutions (3.15) and (3.16) can be obtained. It turns out that the profile shape of the heavier current $H_1(\eta )$ remains exactly the same as (3.13) when the currents are symmetric. In contrast, the profile shape of the lighter current $H_2(\eta )$ is no longer $H_2(\eta ) = H_1(\eta )/G$. Now $H_2(\eta )$ is given by

(3.17)\begin{equation} \hat{H}_2 = \left\{ \begin{array}{@{}ll} \displaystyle 1-Q, & \eta \leq \eta_*,\\ \displaystyle 1-Q\left(1+\dfrac{1}{2G}\right)-\dfrac{1}{G}\left(1-\dfrac{1}{M_2}\right)\eta, & \eta_* < \eta \leq \eta_{f2}, \end{array} \right. \end{equation}

such that $H_1$ and $\hat H_2$ connect at $(\eta _*, H_*)$. The location of the propagating fronts and the intersection point of the three fluids can also be determined as

(3.18a–c)\begin{align} \eta_{f1}=\frac{Q}{2(1-1/M_2)},\quad \eta_{f2}=\frac{(1-Q)(1+2G)-1}{2(1-1/M_2)} \quad \mbox{and}\quad \eta_*={-}\frac{Q}{2(1-1/M_2)}. \end{align}

As expected, $\eta _{f1}$ and $\eta _*$ remain the same as (3.14) when the currents are symmetric, but $\eta _{f2}$ is different and now depends also on $G$, the buoyancy ratio of the heavier and lighter currents. It is important to note that the approximate solution (3.17) does not satisfy exactly the global mass constraint (2.20b) of the lighter current. Nevertheless, it can be shown that the relative error in total volume introduced is at ${O}(T^{-1})$ and is hence negligible at late times ($T \gg 1$).

Equations (3.13) and (3.17) are both verified in figure 7(b) based on a comparison with the appropriately rescaled numerical solutions of PDEs (2.16a,b) at late times, imposing $M_2=6/5$, $M_3=1$, $G=1$ and $Q=2/3$ as an example (now $1/Q-1/G \neq 1$). Indeed the rescaled PDE solutions approach the explicit solutions (3.13) and (3.17) as time progresses.

3.3. Regime 3: $M_3 < 1 < M_2$ $(\mu _2 < \mu _1 < \mu _3)$

We next discuss the flow regimes when the injected lighter fluid 3 is more viscous than the heavier fluid 1, while both fluids 1 and 3 are more viscous than the ambient fluid 2, i.e. when $M_3 < 1 < M_2$ $(\mu _2 < \mu _1 < \mu _3)$. When both the heavier and lighter fluids are under injection ($Q > 0$), the PDE numerical solutions indicate that, eventually, the heavier fluid 1 will move ahead of the lighter fluid 3 at late times, as shown in figure 8(a). That said, fluid 3 will eventually be left behind fluid 1 and in contact only with fluid 1, while the ambient fluid 2 will only be in contact with fluid 1 at the front of the heavier current.

Figure 8. Comparison between the asymptotic and numerical solutions of PDEs (2.16a,b) in regime 3, with $M_2=6/5$, $M_3=1/2$, $G=2$ and $Q=3/5$: (a) time evolution of the fluid interfaces $H_1$ and $\hat H_2$ at $T=\{60,120,180,240,300\}$ from numerically solving PDEs (2.16a,b); (b) rescaled interfaces $H_1$ collapse onto a universal shape under transform $\eta = X-T$ and agree reasonably well with analytical solution (3.21); and (c) rescaled interfaces $\hat H_2$ also collapse onto a universal shape under transform $\theta \equiv X - C_2 T$, which agrees well with the implicit solution (3.25) and its asymptotic approximates at two ends.

In this case, to obtain the frontal structure of the heavier current, we can simply impose $\hat {H}_2=0$ in PDE (2.16), which leads to

(3.19)\begin{equation} \frac{\partial H_1}{\partial T}+\frac{\partial}{\partial X}\left[\frac{H_1}{H_1+M_2(1-H_1)}\right]-\frac{\partial}{\partial X}\left[\frac{M_2 H_1 (1-H_1)}{H_1+M_2(1-H_1)}\frac{\partial H_1}{\partial X}\right] = 0. \end{equation}

We can also introduce a transform $\eta = X-T$, and PDE (3.19) reduces further to an ODE

(3.20)\begin{equation} \frac{\mathrm{d}H_1}{\mathrm{d}\eta}-\frac{\mathrm{d}}{\mathrm{d}\eta} \left[\frac{H_1}{H_1+M_2(1-H_1)}\right]+\frac{\mathrm{d}}{\mathrm{d}\eta} \left[\frac{M_2 H_1 (1-H_1)}{H_1+M_2(1-H_1)}\frac{\mathrm{d}H_1}{\mathrm{d}\eta}\right] = 0. \end{equation}

It is convenient to verify that ODE (3.20) admits an explicit solution of straight-line structure

(3.21)\begin{equation} H_1=\frac{1}{2}-\left(1-\frac{1}{M_2}\right)\eta,\end{equation}

which attaches to the top and bottom boundaries at

(3.22a,b)\begin{equation} \eta_{f1}=\frac{1}{2(1-1/M_2)} \quad \mbox{and}\quad \eta_{f3}={-}\frac{1}{2(1-1/M_2)}. \end{equation}

Equation (3.21) also indicates that the slope of the front depends only on the viscosity ratio $M_2 \equiv \mu _1/\mu _2$ between the injected heavier fluid 1 and the ambient fluid 2.

To obtain the profile shape of the injected lighter fluid 2, we impose $H_1+\hat {H}_2=1$ in PDE (2.16) and arrive at

(3.23)\begin{equation} \frac{\partial\hat{H}_2}{\partial T}+\frac{\partial}{\partial X}\left[\frac{M_3\hat{H}_2}{1-(1-M_3)\hat{H}_2}\right]-\frac{\partial}{\partial X}\left[\frac{M_3(1+G)\hat{H}_2(1-\hat{H}_2)}{1-(1-M_3) \hat{H}_2}\frac{\partial\hat{H}_2}{\partial X}\right]=0. \end{equation}

Knowing that the lighter current propagates at speed $C_2$ at the inlet, we introduce a transform $\theta \equiv X - C_2 T$ and PDE (3.23) reduces further to an ODE

(3.24)\begin{equation} -C_2\frac{\mathrm{d}\hat{H}_2}{\mathrm{d}\theta}+ \frac{\mathrm{d}}{\mathrm{d}\theta}\left[\frac{M_3\hat{H}_2}{1-(1-M_3) \hat{H}_2}\right]-\frac{\mathrm{d}}{\mathrm{d}\theta} \left[\frac{M_3(1+G)\hat{H}_2(1-\hat{H}_2)}{1-(1-M_3)\hat{H}_2} \frac{\mathrm{d}\hat{H}_2}{\mathrm{d}\theta}\right]=0, \end{equation}

which admits an implicit solution

(3.25)\begin{equation} \theta=\frac{M_3(1+G)}{2C_2(1-M_3)}\left[2H_{i1}\left(1+\ln|1- \left.\frac{\hat{H}_2}{\hat{H}_{i2}}\right|\right)+(\hat{H}_{i2}-2\hat{H}_2)\right], \end{equation}

which satisfies the global mass constraint of the lighter current 3

(3.26)\begin{equation} \int_0^{\hat{H}_{i2}}X\mathrm{d}\hat{H}_2 = (1-Q)T. \end{equation}

Notice that (3.26) is equivalent to (2.20b), since $\hat {H}_2$ is monotonic in $X$, but (3.26) is more convenient to use here. Meanwhile, (3.25) indicates that the propagating front of the lighter current 2 locates at

(3.27)\begin{equation} \theta_{f2}=\frac{M_3^2(1+G)(1-C_2)}{C_2^2(1-M_3)^2}+ \frac{M_3(1+G)\hat{H}_{i2}}{2C_2(1-M_3)}. \end{equation}

To summarise, we have obtained self-similar solutions for the profile shape of both the heavier current $H_1$ and the lighter current $\hat {H}_2$ as

(3.28)\begin{equation} H_1 = \left\{ \begin{array}{@{}ll} \displaystyle 1-\mathcal{F}_1(\theta), & \theta \leq \theta_{f2},\\ \displaystyle 1, & \theta \geq \theta_{f2}\quad \mathrm{and}\quad \eta \leq \eta_{f3},\\ \displaystyle \dfrac{1}{2}-\left(1-\dfrac{1}{M_2}\right)\eta, & \eta_{f3} \leq \eta \leq \eta_{f1},\\ \end{array} \right. \end{equation}

and

(3.29)\begin{equation} \hat{H}_2 = \mathcal{F}_1(\theta) \quad \mbox{for}\ \theta \leq \theta_{f2}, \end{equation}

where $\mathcal {F}_1(\theta )$ is the inverse function of implicit solution (3.25). In addition, it can be shown that $\mathcal {F}_1(\theta )$ asymptotically approaches

(3.30)\begin{equation} \hat{H}_2 = \frac{M_3-C_2}{M_3(1+G)}(\theta-\theta_{f2}), \end{equation}

as $\hat {H}_2 \to 0^+$, and

(3.31)\begin{equation} \hat{H}_2 = \hat{H}_{i2}-\hat{H}_{i2} \exp\left[\frac{C_2^2(1-M_3)^2\theta}{M_3^2(1+G)(1-C_2)}\right], \end{equation}

as $\hat {H}_2 \to \hat {H}_{i2}^-$.

The self-similar solutions are also verified based on a comparison with numerical solutions of PDE (2.16a,b) at different times, as shown in figure 8, imposing $M_2 = 6/5$, $M_3 = 1/2$, $G=2$ and $Q=3/5$ as an example. In particular, the time-dependent PDE solutions are shown in figure 8(a), which are rescaled based on $\eta \equiv X-T$ in figure 8(b) and collapse reasonably well onto a universal curve for the profile shape of the heavier current. The straight-line solution (3.29) are further included in figure 8(b), which exhibits good agreement with the collapsed PDE solutions. Meanwhile, the time-dependent PDE numerical solutions are further rescaled based on $\theta \equiv X-C_2T$ in figure 8(c), which, in contrast, collapse well onto a universal curve for the profile shape of the lighter current. The collapsed PDE solutions are also found to agree well with the implicit solution (3.25).

3.4. Regime 4: $M_3 < M_2 = 1$ $(\mu _1 = \mu _2 < \mu _3)$

We now discuss a degenerated case of regime 3 when $M_3 < 1 = M_2$. Similarly, the injected heavier fluid 1 will move ahead of the lighter fluid 3 eventually and separate fluid 3 from the ambient fluid 2 at late times. The profile shapes of $H_1$ and $\hat H_2$ are also similar to that of regime 3. Nevertheless, there is a key difference with regards to the straight-line frontal structure of the heavier current 1: the slope $\partial H_1/\partial X$ is now time-dependent (when $M_2 = 1$), while the slope $\partial H_1/\partial X$ remains constant in regime 3 (when $M_2 > 1$).

To obtain the profile shape of the heavier current $H_1(X,T)$, we impose $\hat {H}_2 = 0$ and $M_2 = 1$ in PDEs (2.16a,b), which leads to

(3.32)\begin{equation} \frac{\partial H_1}{\partial T}+\frac{\partial H_1}{\partial X}- \frac{\partial}{\partial X}\left[H_1(1-H_1) \frac{\partial H_1}{\partial X}\right] = 0. \end{equation}

A similarity transform $\zeta \equiv (X-T)/T^{1/2}$ can then be employed and (3.32) reduces further to an ODE

(3.33)\begin{equation} \frac{\zeta}{2}\frac{\mathrm{d}H_1}{\mathrm{d}\zeta}+ \frac{\mathrm{d}}{\mathrm{d}\zeta}\left[H_1(1-H_1) \frac{\mathrm{d}H_1}{\mathrm{d}\zeta}\right]=0, \end{equation}

which admits an explicit solution

(3.34)\begin{equation} H_1=\tfrac{1}{2}(1-\zeta). \end{equation}

Accordingly, the interface attaches to the top and bottom boundaries at

(3.35a,b)\begin{equation} \zeta_{f3}={-}1\quad \mbox{and}\quad \zeta_{f1}=1, \end{equation}

based on the straight-line structure of (3.34). It can also be verified that solution (3.34) satisfies the global volume constraint for the injected fluid 1.

To obtain the interface shape between the injected fluids 1 and 3, we impose $H_1 + \hat {H}_2 = 1$ in PDEs (2.16a,b) and again arrive at PDE (3.23). Similarly, by defining $\theta \equiv X - C_2T$, analytical solutions for $H_1$ can be obtained as

(3.36)\begin{equation} H_1 = \left\{ \begin{array}{@{}ll} \displaystyle 1-\mathcal{F}_1(\theta), & \theta \leq \theta_{f2},\\ \displaystyle 1, & \theta \geq \theta_{f2}\ \mathrm{and}\ \zeta \leq{-}1,\\ \displaystyle \tfrac{1}{2}(1-\zeta), & -1 \leq \zeta \leq 1,\\ \end{array} \right. \end{equation}

which now includes solution (3.34), and $\mathcal {F}_1(\theta )$ is again the inverse function of the implicit solution (3.25). The only difference between solutions (3.36) and (3.28) is that the leading straight-line frontal structure of the heavier fluid is obtained under different similarity transforms. Correspondingly, in regime 3, the slope $\partial H_1/\partial X$ remains constant based on $\theta \equiv X - C_2T$, while in regime 4, the slope $\partial H_1/\partial X$ becomes time-dependent based on $\zeta \equiv (X-T)/T^{1/2}$.

We can also compare the self-similar solutions in regime 4 with the PDE numerical solutions, as shown in figure 9, imposing $M_2 = 1$, $M_3 = 1/2$, $G=2$ and $Q=3/5$ as an example. The time-dependent PDE solutions are shown in figure 9(a), which is rescaled according to $\zeta \equiv (X-T)/T^{1/2}$ in figure 9(b), and the profile shapes of the heavier current is found to collapse onto a straight line, which also agrees with the prediction of solution (3.34). Meanwhile, we can also rescale the PDE numerical solutions according to $\theta \equiv X - C_2T$, and the profile shapes of the lighter current are found to collapse onto a universal curve, which agrees well with solution (3.36).

Figure 9. Comparison between the asymptotic solutions and numerical solutions of PDEs (2.16a,b) in regime 4, with $M_2=1$, $M_3=1/2$, $G=2$ and $Q=3/5$: (a) time evolution of the fluid interfaces $H_1$ and $\hat H_2$ at $T=\{60,120,180,240,300\}$ from numerically solving PDEs (2.16a,b); (b) rescaled interfaces $H_1$ collapse onto a universal shape under transform $\zeta = (X-T)/T^{1/2}$ and agree reasonably well with straight-line solution (3.34); (c) rescaled interfaces $\hat H_2$ also collapse onto a universal shape under transform $\theta \equiv X - C_2 T$, which agrees well with the implicit solution (3.36).

3.5. Regime diagram: eight facets of gravity current interaction

We have thus identified eight regimes of gravity current interaction at late times ($T \gg 1$) from simultaneous injection of heavier and lighter fluids into a confined porous layer, as distinguished by four dimensionless control parameters $M_2$, $M_3$, $G$ and $Q$, the definition and physical meaning of which are summarised in table 1. These eight regimes are also allocated in a regime diagram in figure 4, with key features of the flow summarised also in table 2. The $(M_2,M_3)$ plane is divided into eight regions by three straight lines ($M_2=1$, $M_3=1$, $M_2/M_3=1$). Regions 1 and 2 further reduce to two sub-regions depending on whether or not $1/Q-1/G=1$. Region 5 also leads to two sub-regions depending on whether or not $Q<(1-M_2)/(1-M_3)$.

In particular, in four of the eight regimes of late-time interaction (regimes 2, 6–8), self-similar solutions can be constructed by combining appropriately the three basic solutions (i.e. shock, rarefaction and travelling wave solutions) identified during single fluid injection in confined porous layers (e.g. Zheng et al. Reference Zheng, Guo, Christov, Celia and Stone2015a. In the four other regimes (regimes 1, 3–5), implicit solutions with logarithm dependence and described by error functions are identified due to the influence of flow confinement and the interaction of gravity currents. This is a key novel feature of the present study of gravity current interaction and is fundamentally different from that of single current injection into confined porous layers.

3.6. Time transition between early-time and late-time self-similar solutions

To further illustrate the time transition between different early-time and late-time behaviours described in §§ 2 and 3, we now solve the governing PDEs (2.16a,b) numerically and track the location of propagating fronts $X_{f1}(T)$ and $X_{f2}(T)$, the height of interface at the inlet $X = 0$ of the reservoir $H_{f1}(T)$ and $\hat {H}_{f2}(T)$, and the location of the intersection ($X_*(T)$, $H_*(T)$) where the three fluids meet. We impose $M_2 = M_3 = 1$, $G = 2$ and $Q = 2/3$ for an example calculation, in which case $1/Q - 1/G = 1$ is satisfied and the heavier and lighter currents are symmetric based on Feature 1, and the asymptotic behaviour of the flow is described in regime 1 in § 3.1. The PDE numerical solutions are shown as the symbols in figure 10.

Figure 10. Propagating fronts $X_{f1}$, $X_{f2}$, vertical fronts $H_{f1}$, $\hat H_{f2}$, and the intersection ($X_*$, $H_*$) all undergo a time transition from an early-time self-similar behaviour to another late-time self-similar behaviour in regime 1. The data are taken from numerically solving PDEs (2.16a,b). In panel (a), during the early-time period, the propagation fronts obey a power-law form of $X_{f1} \propto T^{2/3}$ and $X_{f2} \propto T^{2/3}$, while the vertical fronts propagate as $H_{f1} \propto T^{1/3}$ and $\hat H_{f2} \propto T^{1/3}$. In panel (b), during the late-time period, we have $X_{f1}(T) \sim T$ and $X_{f2}(T) \sim T$, while $H_1$, $\hat {H}_2$ and $H_*$ all remain constant. The imposed parameters are $M_2=M_3=1$, $G = 2$ and $Q = 2/3$.

We next impose a comparison between the PDE numerical solutions and the self-similar solutions obtained in § 3. In particular, at early times, the interface shape evolves according to the nonlinear diffusion equation (C1), with the fronts and heights propagating in power-law forms

(3.37a)\begin{gather} X_{f1}(T) \sim 1.48 Q^{1/3} T^{2/3} \sim 1.29 T^{2/3}, \end{gather}

(3.37b)\begin{gather}X_{f2}(T) \sim 1.48 [GM_3(1-Q)]^{1/3} T^{2/3} \sim 1.29 T^{2/3}, \end{gather}

and

(3.38a)\begin{gather} H_{f1}(T) \sim 1.30 Q^{2/3} T^{1/3} \sim 0.99 T^{1/3}, \end{gather}

(3.38b)\begin{gather}\hat{H}_{f2}(T) \sim 1.30 (GM_3)^{{-}1/3} (1-Q)^{2/3} T^{1/3} \sim 0.49 T^{1/3}. \end{gather}

In contrast, at late times, the time evolution of the interface shape is now influenced by an additional advective term due to the influence of flow confinement. The propagating fronts of the symmetric currents now locate at

(3.39a)\begin{gather} X_{f1}(T) = X_{f2}(T) \sim T + (QT)^{1/2} \sim T + 0.82 T^{1/2}, \end{gather}

(3.39b)\begin{gather}X_*(T) \sim T - (QT)^{1/2} \sim T - 0.82 T^{1/2}, \end{gather}

based on the discussions in § 3.1.1, which are all linear in time $T$ at leading order ${O}(T)$. Meanwhile, the heights $H_{f1}(T)$, $\hat {H}_{f2}(T)$ and $H_*(T)$ all remain constant at late times and can be estimated conveniently based on the inlet flux condition (3.2a,b):

(3.40a)\begin{gather} H_{f1}(T) = H_*(T) = H_{i1} = 2/3, \end{gather}

(3.40b)\begin{gather}\hat H_{f2}(T) = \hat H_{i2} = 1/3, \end{gather}

when the flow is significantly influenced by the confinement effect.

The asymptotic solutions for the frontal location are also plotted in figure 10 as a comparison with numerical solutions of the governing PDEs (2.16a,b). Very good agreement is observed at both the early and late times of the flow. One can also look into the profile shape evolution in figure 5, which shows that the PDE numerical solutions approach and collapse eventually onto the self-similar solutions.

Analytical progress can also be made based on scaling analysis on how the flow parameters can impact the regime boundaries (i.e. the transition time) for the validity of the early-time and late-time asymptotic solutions, similar to the $T$–$M$ chart in figure 10 of Zheng et al. (Reference Zheng, Guo, Christov, Celia and Stone2015a) for single fluid injection. For example, at early times ($T \ll 1$), for the unconfined self-similar solution from solving (C2) to apply, one requires that $H_1 \ll {min} \{1, M_2 \}$ and $\hat H_2 \ll {min} \{1,M_2/M_3\}$ to arrive at (C1a,b) from (2.16a,b). Hence, we need $H_1 \sim 1.48 Q^{2/3} T^{1/3} \ll {min} \{1,M_2\}$ and $\hat H_2 \sim 1.30 (GM_3)^{-1/3} (1-Q)^{2/3} T^{1/3} \ll {min} \{1,M_2/M_3\}$. After some rearrangements, we arrive at $T \ll Q^{-2} {min} \{1,M_2^3\}$ and $T \ll GM_3 (1-Q)^{-2} {min} \{1,M_2^3/M_3^3\}$, both of which must be satisfied for the early-time self-similar solutions to apply. Notably, the validity range of the early-time unconfined self-similar solution depends on $\{M_2, M_3, G, Q\}$.

To further demonstrate the influence of $\{M_2, M_3, Q, G\}$ on the time transition between early-time and late-time self-similar solutions, we now include numerical solutions for the frontal location $X_{f1}(T)$ of the coupled PDEs (2.16a,b) in regimes 1 to 8, as shown in figure 11. It is shown that the location of the leading front follows $X_{f1}(T) \sim 1.48 Q^{1/3}T^{2/3}$ at early times in all regimes, while $X_{f1}(T) \propto T$ at late times and the prefactor depends on $\{M_2, M_3, Q, G\}$. The imposed parameters $\{M_2, M_3, G, Q\}$ are consistent with those in the example calculations in § 3, i.e. $\{1, 1, 2/5, 2/3\}$ in regime 1a, $\{6/5, 1, 2, 2/3\}$ in regime 2a, $\{6/5, 1/2, 2, 3/5\}$ in regime 3, $\{1, 1/2, 2, 3/5\}$ in regime 4, $\{4/5, 1/2, 2, 1/5\}$ in regime 5a, $\{1/2, 1/2, 2, 1/2\}$ in regime 6, $\{1/2, 4/5, 1/5, 2/5\}$ in regime 7 and $\{1/2, 1, 1/5, 1/6\}$ in regime 8. To explore the specific dependence of $\{M_2, M_3, Q, G\}$ at late times, we need to consider a broader range of the parameters in the numerical calculations, e.g. by orders of magnitude, which goes beyond the scope of the numerical scheme employed here. Nevertheless, presumably this can be done with an improved numerical scheme, following the same method for the $T$–$M$ chart in figure 10 of Zheng et al. (Reference Zheng, Guo, Christov, Celia and Stone2015a) for single fluid injection.

Figure 11. Time-dependent locations of the leading front $X_{f1}(T)$ in regimes 1 to 8 from numerically solving the coupled PDEs (2.16a,b). It is observed that $X_{f1}(T) \sim 1.48\,Q^{1/3}T^{2/3}$ at early times, while $X_{f1}(T) \propto T$ at late times in all regimes. The imposed parameters $\{M_2, M_3, G, Q\}$ are consistent with those in § 3, i.e. $\{1, 1, 2/5, 2/3\}$ in regime 1a, $\{6/5, 1, 2, 2/3\}$ in regime 2a, $\{6/5, 1/2, 2, 3/5\}$ in regime 3, $\{1, 1/2, 2, 3/5\}$ in regime 4, $\{4/5, 1/2, 2, 1/5\}$ in regime 5a, $\{1/2, 1/2, 2, 1/2\}$ in regime 6, $\{1/2, 4/5, 1/5, 2/5\}$ in regime 7, and $\{1/2, 1, 1/5, 1/6\}$ in regime 8. The solid line in panel (b) represents the analytical solution $X_{f1}(T) \sim 1.48\,Q^{1/3}T^{2/3}$.

4.1. Potential implications

We next briefly remark on the potential implications of the model. Three example calculations are summarised in table 3 in the practice of ${\rm CO}_2$-water co-flooding projects for enhancing oil recovery (EOR) and geological sequestration of ${\rm CO}_2$ simultaneously, imposing geophysical and operational parameters from practical projects and fluid properties at reservoir conditions. In particular, we consider the recovery of both light and heavy oils with viscosity variations by orders of magnitude, and we also cover the influence of permeability variations of natural porous rocks.

Table 3. Potential implications in ${\rm CO}_2$-water co-flooding projects for geological ${\rm CO}_2$ sequestration and enhancing oil recovery simultaneously. The geophysical and operational data of cases 1 and 2 are from Western Canada Sedimentary basin and East Texas Field, respectively.

Two ${\rm CO}_2$-water co-flooding projects are considered here that took place in the Western Canada Sedimentary basin in Canada (Meyer, Attanasi & Freeman Reference Meyer, Attanasi and Freeman2007) and East Texas Field in the United States (Foote, Massingill & Wells Reference Foote, Massingill and Wells1988). The value for the permeability, porosity, thickness of the porous layers, depth of the oil reservoir and density of oil are made available in the reports, and we assume that the permeability, porosity and thickness of the porous layer all remain constant and the reservoir is horizontal. We also assume that the temperature in the rock formation increases by 2.5 $^{\circ }$C per 100 m and the pressure increases by 2.5 MPa per 100 m, which allows us to obtain the viscosity and density of water and ${\rm CO}_2$ at reservoir conditions based on the data of the National Institute of Standards and Technology (NIST). Representative injection rate of ${\rm CO}_2$ ranges from 3 mm$^2$ s$^{-1}$ to 12 mm$^2$ s$^{-1}$ and that of H$_2$O is from 8 mm$^2$ s$^{-1}$ to 25 mm$^2$ s$^{-1}$ based on the reports we read (e.g. Dai et al. Reference Dai, Middleton, Viswanathan, Fessenden-Rahn, Bauman, Pawar, Lee and McPherson2014; IEA 2015). These area injection rates are calculated also based on the assumption that fluids are injected through horizontal wells of length 10 km in the East Texas Field. Meanwhile, the spreading of ${\rm CO}_2$ current is much faster than that of the H$_2$O current, so the breakthrough of ${\rm CO}_2$ is expected to run earlier than that of H$_2$O. Finally, we are able to obtain the dimensionless parameters $M_2$, $M_3$, $G$ and $Q$, as shown in table 3, together with the geophysical and operational data of these two projects.

It hence becomes clear that both cases 1 and 2 correspond to regime 7 of gravity current interaction, as discussed in Appendix D.4. An example calculation is also provided for case 2, where the length of the ${\rm CO}_2$ current reaches $x_{\text {CO}_2} \approx 1420$ m, while that of water is only $x_{\text {H}_2\text {O}} \approx 69$ m after 180 days of continuous operation, as shown in figure 12. In this case, the area covered by ${\rm CO}_2$ is $A_{\text {CO}_2} \approx 182$ m$^2$ and that of water is $A_{\text {H}_2\text {O}} \approx 120$ m$^2$.

Figure 12. A snapshot of the interacting ${\rm CO}_2$-water currents at day 180 of continuous operation based on the geophysical and operational parameters of case 2. The parameters imposed in this example calculation are $M_2=0.4$, $M_3=10$, $G=1.0$ and $Q=0.4$.

We can also define a sweep/displacement efficiency as the total volume of injected fluids divided by the rectangular area enclosed by the leading front of the currents ($\max \{X_{f1}(T),X_{f2} (T)\}$). Since the total volume of injected fluids is $QT+(1-Q)T=T$, and since $X_{f1}(T)$ is always greater than $X_{f2}(T)$ when $M_3 \le 1$, the sweep efficiency in this work is always $\psi = T/X_{f1}(T)$ in the current work. Accordingly, in the current context, the efficiency of ${\rm CO}_2$ sequestration is $\psi _{\text {CO}_2} \approx 4\,\%$, while that of oil recovery is $\psi _{\text {H}_2\text {O}} \approx 2\,\%$ up to day 180. It is also important to note that we have neglected the influence of fluid mixing and interfacial tension in the example calculations. Meanwhile, it is also entirely possible that gas and water are injected alternately rather than simultaneously in EOR practice (e.g. IEA 2015), which might lead to new flow patterns that are not covered by the current model. Nevertheless, such issues arising in practice might inspire future studies on the basis of the current work.

4.2. On the influence of the Saffman–Taylor instability

We would also like to remark briefly on the influence of the Saffman–Taylor instability (e.g. Saffman & Taylor Reference Saffman and Taylor1958; Chuoke, von Meurs & van der Poel Reference Chuoke, von Meurs and van der Poel1959; Lenormand, Touboul & Zarcone Reference Lenormand, Touboul and Zarcone1988), when an invading fluid is less viscous than the displaced fluid, which is related to regimes 5 to 8 of the present study. For single current propagation, there is experimental evidence that the main structure of a gravity current is not significantly influenced by the development of the viscous fingering instability in the singular limit of zero mixing and surface tension. The injected fluid still propagates along one boundary as a gravity current, maintaining a monotonic interface shape, as well described by a one-dimensional sharp-interface model, and this is due to the more dominant influence of buoyancy/gravity segregation (e.g. Pegler et al. Reference Pegler, Huppert and Neufeld2014; Zheng et al. Reference Zheng, Guo, Christov, Celia and Stone2015a). Nevertheless, it is also important to note that when buoyancy segregation is not strong enough, it is possible that the influence of the viscous fingering instability becomes non-negligible, and, in particular, the shape of the local region near the propagating front can be significantly impacted. In this case, one is recommended to solve the full two-dimensional or three-dimensional problem (e.g. Nijjer et al. Reference Nijjer, Hewitt and Neufeld2022), or to employ other transversely averaged models that are verified by experimental or numerical solutions, to more precisely describe the major fluid-displacement behaviours. We are currently working on a two-dimensional model of miscible displacement to provide more quantitative insights on the influence of mixing and the viscous-fingering instability during the dynamic interaction of gravity currents in a confined porous layer.

4.3. Summary

We have conducted a theoretical and numerical investigation on the interaction of two gravity currents when two fluids are injected simultaneously into a confined porous layer initially filled with a third one. The three fluids can possibly have distinct density and viscosity and under different injection rates, leading to a variety of interaction regimes. Our primary focus is on the time evolution of the interfaces between them and location of the propagating fronts.

Neglecting the influence of mixing and interfacial tension between the fluids, we first derived two coupled nonlinear advective-diffusive PDEs (2.16a,b) to describe the time evolution of the interface shapes. In addition to providing numerical solutions of (2.16a,b), we also conducted detailed asymptotic analysis at both the early and late times and identified a series of self-similar and travelling-wave solutions in both regimes. In particular, at early times, the governing PDEs reduce to the classic nonlinear diffusion equation (C1) that describes the propagation of unconfined gravity currents. At late times, the currents attach to and interact with each other, and our analysis leads to eight distinct regimes of gravity current interaction, with key features summarised in figure 4 and table 2. The interaction is under the influence of four dimensionless control parameters: the viscosity ratio $M_2$ of the injected fluid 1 over the ambient fluid 2, the viscosity ratio $M_3$ of the injected fluid 1 over the injected fluid 3, the ratio $G$ of buoyancy of the lighter and heavier gravity currents, and the partition $Q$ of the area injection rate of the heavier current. The self-similar solutions are also verified through a comparison with the appropriately rescaled numerical solutions of PDEs (2.16a,b). By tracking the time-dependent location of the propagating fronts and the interface shapes, the time transition from early-time unconfined to late-time confined self-similar flows is also illustrated.

It is of interest to note that in five of the eight regimes of late-time interaction (regimes 1, 2, 6–8), the self-similar profile shapes can be constructed by combining appropriately the three basic solutions (i.e. shock, rarefaction, travelling-wave solutions) obtained already during single fluid injection in confined porous layers. In the three other regimes of late-time interaction (regimes 3–5), implicit solutions with logarithm dependence are obtained due to the influence of flow confinement. Potential implications of the model and solutions are also briefly discussed in the practical context of geological ${\rm CO}_2$ sequestration in oil fields, when a certain amount of oil can also be recovered which offsets the cost of ${\rm CO}_2$ sequestration. In the future, the influence of fluids mixing and wetting and capillary forces in porous rock layers can also be considered on the basis of the current work.