2.1 Dimensionless model and representative parameter values

By forming scaling relationships between the terms in (2.6)–(2.10), we can determine the intrinsic horizontal length and time scales

(2.11a,b ) $$\begin{eqnarray}{\mathcal{L}}\equiv \frac{UH^{2}}{Q_{1}},\quad {\mathcal{T}}\equiv \frac{\unicode[STIX]{x1D719}UH^{3}}{Q_{1}^{2}}\end{eqnarray}$$

respectively, which characterise the flow of a gravity current of height $h\sim H$ injected into a quiescent aquifer. We use these scales to form dimensionless variables according to

(2.12a-d ) $$\begin{eqnarray}x=\left(\frac{UH^{2}}{Q_{1}}\right)\hat{x},\quad t=\left(\frac{\unicode[STIX]{x1D719}UH^{3}}{Q_{1}^{2}}\right)\hat{t},\quad h=H{\hat{h}},\quad Q=Q_{1}\hat{Q}.\end{eqnarray}$$

This choice maximises the generality of solutions to the associated dimensionless model. Recasting (2.6) in terms of (2.12) and dropping the hats, we obtain

(2.13) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}t}=-\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}\left[h\left(\frac{MQ(x,t)-(1-h)\displaystyle \frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}x}}{Mh+(1-h)}\right)\right],\end{eqnarray}$$

where

(2.14a,b ) $$\begin{eqnarray}Q(x,t)=\left\{\begin{array}{@{}ll@{}}I(t)+B & (x>0),\\ B & (x<0),\end{array}\right.\quad \text{and}\quad I(t)=\left\{\begin{array}{@{}ll@{}}1 & (0\leqslant t\leqslant V),\\ 0 & (t>V).\end{array}\right.\end{eqnarray}$$

The input condition (2.7b ) becomes

(2.15) $$\begin{eqnarray}\left[\frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}x}\right]_{-}^{+}=\left(M-\frac{1}{h}\right)I(t)\quad \text{at}~x=0.\end{eqnarray}$$

The flow-front conditions (2.8) become

(2.16a,b ) $$\begin{eqnarray}\displaystyle h(x_{+},t)=0,\quad {\dot{x}}_{+}=M[I(t)+B]-\frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}x}(x_{+},t), & & \displaystyle\end{eqnarray}$$

(2.17a,b ) $$\begin{eqnarray}\displaystyle \hspace{-33.30006pt}h(x_{-},t)=0,\quad {\dot{x}}_{-}\hspace{-0.3pt}=MB-\frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}x}(x_{-},t). & & \displaystyle\end{eqnarray}$$

The system depends on three dimensionless parameters,

(2.18a-c ) $$\begin{eqnarray}M\equiv \frac{\unicode[STIX]{x1D707}_{2}}{\unicode[STIX]{x1D707}_{1}},\quad B\equiv \frac{Q_{2}}{Q_{1}},\quad V\equiv \frac{{\mathcal{V}}}{Q_{1}{\mathcal{T}}},\end{eqnarray}$$

which represent the ratio of the ambient viscosity to the injectate viscosity, the ratio of the background flux to the input flux, and the dimensionless total volume of injected fluid ( $t=V$ is the dimensionless time at which the input is stopped).

Representative parameter values for different physical situations are given as follows. The case $M\ll 1$ represents the case of a negligible ambient viscosity, as applies to a good approximation to hydrological problems involving the flow of water into an air-saturated aquifer, for which $M\approx 0.01$ . For flows involving the evolution of a solution of water charged with a contaminating solute such as sodium chloride or $\text{CO}_{2}$ , $M\approx 0.5$ –1. The converse case of a low-viscosity contaminant ( $M>1$ ) characterises the injection of supercritical $\text{CO}_{2}$ into a saline aquifer. As estimated in § 6, $M\approx 5$ –20 are typical values for $\text{CO}_{2}$ injection, and $B$ is of order unity given representative injection rates and background flow rates.

3.1 Injection into a quiescent asymmetrical reservoir

We begin with a dedicated analysis of the special long-term regimes that arise with no background flow, $B=0$ . For this case, our solution of figure 2(a) indicated that the majority of the injected fluid eventually propagates towards the permeable fault at long times, $hu_{1}\sim 1$ . Therefore, the rightwards flow for $B=0$ approaches the same long-term regimes as a one-sided injection into a confined aquifer, as detailed by Pegler et al. (Reference Pegler, Huppert and Neufeld2014a ) and Zheng et al. (Reference Zheng, Guo, Christov, Celia and Stone2015). As determined therein, the asymptotic position of the positive flow front for $M\leqslant 1$ is

(3.2) $$\begin{eqnarray}x_{+}\sim t\quad (t\rightarrow \infty ),\end{eqnarray}$$

which is shown by the dashed line in figure 3(a). The general possible regimes, including those for $M>1$ and $B>0$ , will be reviewed later in the text preceding (3.10). The result of (3.2) confirms the long-term prediction of our numerical solution. In the practical context, the asymptotic regime of (3.2) will apply until the rightwards flow front interacts with the downstream end of the aquifer, beyond which time a different regime, possibly involving leakage of the current through the permeable fault, will apply. While the rightwards-propagating regime described by (3.2) is asymptotically equivalent to that of a one-sided injection towards a permeable fault, a qualitative difference is that the interface never touches the top boundary of the medium as it does in that case. A gap must persist in the present case in order to provide a conduit for fluid to escape the confined region $x<0$ . The gap gradually narrows over time but never vanishes completely. It should be noted that this conduit would not necessarily occur for the related problem of a point-source injection because, in that case, ambient fluid can flow around the injectate.

Figure 4.

The solid curve shows the prefactor $\unicode[STIX]{x1D702}_{N}(M)$ to the frontal position $x_{-}\sim -\unicode[STIX]{x1D702}_{N}t^{1/2}$ describing the long-term propagation of the injected fluid in the direction of the sealed fault for the case of no background flow ( $B=0$ ) plotted against the viscosity ratio  $M$ , obtained from numerical solutions of (3.5)–(3.6). The plot shows the decrease of the rate of propagation with  $M$ . The horizontal dashed green line shows the asymptote $\unicode[STIX]{x1D702}_{N}\sim 1.62$ for $M\rightarrow 0$ , which represents the limit of an unsaturated aquifer. The curve of blue circles represents the asymptote for $M\rightarrow \infty$ given by (3.7), as predicted by the boundary-layer analysis of appendix B. The insets illustrate the self-similar profiles for $M=10^{-3}$ and $M=10^{3}$ , the latter showing the strongly concave profile that arises for large  $M$ .

3.1.1 The self-similar flow towards the sealing fault

Figure 3(a) indicates that the leftwards current grows in proportion to $t^{1/2}$ , a significantly slower rate than the rightward current (3.2). To understand this evolution, it should be noted first that, as the gap above the current in the region $x>0$ closes, the fluid above the injection point gradually approaches the top boundary,

(3.3) $$\begin{eqnarray}h(0)\sim 1\quad (t\rightarrow \infty ).\end{eqnarray}$$

The leftwards current is therefore fed by a region of fixed height asymptotically. The long-term flow to the left can therefore be described by (2.6), (2.8a ) and (3.3). No explicit time-independent horizontal length scale can be formed from scalings of these equations, indicating a similarity solution. We define the relevant similarity variables by

(3.4a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D702}=t^{-1/2}x,\quad h=h(\unicode[STIX]{x1D702}),\end{eqnarray}$$

with frontal evolution $x_{-}=-\unicode[STIX]{x1D702}_{N}t^{1/2}$ , where $\unicode[STIX]{x1D702}_{N}$ is a constant to be determined. In terms of (3.4), equation (2.13) becomes

(3.5) $$\begin{eqnarray}-\frac{1}{2}\unicode[STIX]{x1D702}h^{\prime }=\left[\frac{h(1-h)h^{\prime }}{Mh+(1-h)}\right]^{\prime },\end{eqnarray}$$

where we use a prime to denote $\text{d}/\text{d}\unicode[STIX]{x1D702}$ . The conditions (3.3) and (2.16a,b ) become

(3.6a-c ) $$\begin{eqnarray}h(0)=1,\quad h(-\unicode[STIX]{x1D702}_{N})=0,\quad h^{\prime }(-\unicode[STIX]{x1D702}_{N})={\textstyle \frac{1}{2}}\unicode[STIX]{x1D702}_{N}.\end{eqnarray}$$

We solved (3.5)–(3.6) numerically using a scheme in which (3.5) is integrated subject to (3.6b,c ) using the Matlab routine ode113. This integration was iterated in conjunction with a bisectional search which tunes $\unicode[STIX]{x1D702}_{N}$ to meet (3.6a ). The height profile and extent thus determined are shown as curves of green crosses in figures 2(a) and 3(a), and provide a close match with our numerical solution to the initial-value problem. The $t^{1/2}$ regime described here will cease once the current makes contact with the sealing fault, assumed to lie far in the negative $x$ direction; beyond that time, a no-flux condition will apply at the left-hand edge of the current, and the current will proceed to fill the left-hand section of the aquifer to long times.

Figure 5.

The boundary-layer structure underlying the self-similar propagation of a gravity current towards a closed fault in the limit of small injectate viscosity, $M\rightarrow \infty$ . The case $M=100$ is illustrated. The numerical solution to the exact equations (3.5)–(3.6) is shown as a solid curve, with the flow front indicated by a star. As detailed in appendix B, the flow involves a boundary layer of extent $\unicode[STIX]{x1D6FF}\sim (M\log M)^{-1/2}$ near the flow front $\unicode[STIX]{x1D702}_{N}\sim 2(M^{-1}\log M)^{1/2}$ , representing a transition from a region dominated by resistance to ambient flow to a region wherein the resistance to flow of the injectate becomes leading order (and ultimately constrains the tip propagation). The outer solution satisfying (B 4a ) is shown as a dotted red curve. The inner solution (B 7), which consistently matches the outer solution to the frontal conditions (3.6b,c ), is shown as a dashed blue curve.

By solving (3.5)–(3.6) over a range of viscosity ratios $M$ , we determine the universal relationship between $\unicode[STIX]{x1D702}_{N}$ and $M$ shown in figure 4. For $M\ll 1$ , it is found that $\unicode[STIX]{x1D702}_{N}\sim 1.62$ . This limiting value represents the self-similar flow of a gravity current fed from a constant pressure head into an unsaturated or semi-infinite porous medium (Pritchard Reference Pritchard2007; Neufeld, Vella & Huppert Reference Neufeld, Vella and Huppert2009). It is recovered here for $M\rightarrow 0$ because the far-field condition of the sealing fault has no effect when $M=0$ . As $M$ increases from zero, $\unicode[STIX]{x1D702}_{N}$ decreases. Perhaps counterintuitively, reducing the viscosity of the current therefore decreases its rate of propagation. It should be noted that the opposite of this conclusion holds for the analogous (fixed-headed input) similarity solution describing flow towards a permeable fault (Pegler, Huppert & Neufeld Reference Pegler, Huppert and Neufeld2014b ). The coefficient $\unicode[STIX]{x1D702}_{N}$ instead decreases with $M$ here because, with a sealing fault, the current must drive a return flow of ambient fluid in order to propagate. For larger  $M$ , the stresses associated with mobilising the return flow are larger, thus resisting the propagation of the current.

For $M\rightarrow \infty$ , as applies approximately to pure $\text{CO}_{2}$ injected into a saline aquifer, the resistance to ambient flow becomes a dominant effect. This limit produces a distinct asymptotic regime in which the evolution of the interface throughout the majority of the flow is resisted by the mobilisation of the ambient fluid. Remarkably, the viscosity of the injectate nevertheless remains important near the flow front within a small boundary layer of extent $\unicode[STIX]{x1D6FF}=O[(M\log M)^{-1/2}]$ , and provides the ultimate constraint on the overall rate of propagation. The mathematical analysis of this boundary-layer structure is provided in appendix B. The structure is illustrated in figure 5, where our numerical solution to the exact equations (3.5)–(3.6) for $M=100$ is shown alongside the inner and outer solutions determined in appendix B. The analysis reveals the asymptote

(3.7) $$\begin{eqnarray}\unicode[STIX]{x1D702}_{N}\sim 2(M^{-1}\log M)^{1/2}\quad (M\rightarrow \infty ),\end{eqnarray}$$

which is shown as a curve of circles in figure 4 and validates the numerically determined values for $M\gg 1$ .

3.2 The effects of background flow

The solution for $B=1$ and $M=0.5$ (figure 2 b) indicates several effects of background flow. Most evident are the development of an asymptotically steady horizontal interface in the region $x>0$ and the approach of the flow for $x<0$ towards a steady state.

To determine these asymptotic regimes, we consider the steady form of (2.6), namely $\unicode[STIX]{x2202}(hu_{1})/\unicode[STIX]{x2202}x=0$ . Since the long-term flow only propagates in the positive $x$ direction, $u_{1}>0$ (otherwise, the leftwards current would remain unsteady), it follows that the dimensionless flux of injected fluid $hu_{1}$ is equal to zero for $x<0$ and unity for $x>0$ . Using (2.5), we therefore obtain

(3.8) $$\begin{eqnarray}hu_{1}=h\left[\frac{MQ(x)-(1-h)h^{\prime }}{Mh+(1-h)}\right]=\left\{\begin{array}{@{}ll@{}}0 & (x<0),\\ 1 & (x>0),\end{array}\right.\end{eqnarray}$$

where, here, $h$ and $Q$ are functions of $x$ only and we have used a prime to denote $\text{d}/\text{d}x$ .

To seek a solution to (3.8) with a horizontal interface for $x>0$ , we set $h^{\prime }\equiv 0$ and $Q(x)=B+1$ , to yield

(3.9a,b ) $$\begin{eqnarray}\frac{M(B+1)h}{Mh+(1-h)}=1\quad \text{and hence}\quad h=\frac{1}{MB+1}\equiv h_{H}\end{eqnarray}$$

on making $h$ the subject. The height predicted by (3.9b ) is shown as a horizontal dashed line in figure 2(b,c) and matches a prevailing interior region of the numerical solution at long times. Two fluids driven simultaneously into a porous medium thus divide along a critical height $h_{H}$ that depends on the viscosity ratio $M$ and flux ratio $B$ between the fluids. For less viscous injectate ( $M$  larger) or larger background flux ( $B$  larger), (3.9b ) predicts that the height confining the lower fluid, $h_{H}$ , becomes smaller.

The sustained gap above the injected current effectively confines the current into a shallower quiescent aquifer of effective depth $h_{H}$ . Consequently, the rightwards current is asymptotically equivalent to a one-sided injection with no background flow adjusted to an effective aquifer depth $h_{H}=1/(MB+1)$ . To review the regimes arising for a one-sided injection with no background flow (Pegler et al. Reference Pegler, Huppert and Neufeld2014a ; Zheng et al. Reference Zheng, Guo, Christov, Celia and Stone2015), we recall that, in that case, the current generally grows vertically to span the depth of the medium fully. For $M<1$ , the two flow fronts along the top and bottom boundaries both propagate as $t$ at long times and form the ends of an asymptotically linear interface of constant length which is relatively short compared with the horizontal extent of the flow at long times. For $M=1$ , a similar regime arises except with the horizontal length of the interface growing as $t^{1/2}$ under the effect of gravity. For $M>1$ , the interface approaches a large-scale concave profile in which the front propagates at the faster rate of $Mt$ and the upper front at the slower rate of $M^{-1}t$ . With the results for the long-term frontal position adjusted to incorporate the effective aquifer height of (3.9b ), the leading position of the positive flow front with background flow can be determined as

(3.10a,b ) $$\begin{eqnarray}x_{+}\sim \left\{\begin{array}{@{}ll@{}}t/h_{H}=(1+MB)t, & \text{for}~M\leqslant 1,\\ Mt/h_{H}=M(1+MB)t, & \text{for}~M>1.\end{array}\right.\end{eqnarray}$$

The asymptotes (3.10) are shown as dashed black lines in figure 3(a–c) and agree with the large-time numerical predictions. Expression (3.10a ) shows that for $M\leqslant 1$ , the effect of background flow is to increase the rate of propagation of the flow front by a value $MB$ relative to its speed if $B=0$ . For $M>1$ , the speed of propagation is further enhanced by the extra factor of $M$ . For $M\gg 1$ , (3.10b ) has the limiting form $x_{+}\sim BM^{2}t$ . The value of the quadratic prefactor $M^{2}$ is considerable for $\text{CO}_{2}$ storage, with representative values lying in the range 25–400. Background flow can therefore advect injected fluid significantly faster than the background flow itself, an effect that has the potential to provide a dominant control on the migration of  $\text{CO}_{2}$ .

To analyse the steady state arising for $x<0$ , we set $Q(x)=B$ in (3.8). By integrating that equation analytically subject to the asymptotic condition $h(0)\sim h_{H}$ , we obtain the steady-state height profile and negative frontal position

(3.11a,b ) $$\begin{eqnarray}h=1-[(1-h_{H})^{2}-2MBx]^{1/2},\quad x_{-}=-\frac{1-(1-h_{H})^{2}}{2MB}\end{eqnarray}$$

respectively. The predicted steady profile of (3.11a ) is shown by the curve of red circles in figure 2(b) and agrees with the numerically determined profile at long times. The corresponding agreement with (3.11b ) is shown in figure 3(b,c). Expression (3.11b ) implies that $|x_{-}|$ is smaller for less viscous injectate ( $M$  larger), and hence the current propagates less far against the background flow when it is less viscous (confirmed by the comparison of the steady profiles arising for $x<0$ in figure 2 a,b). The regime of (3.11) is not just steady but completely static ( $u_{1}=0$ ). That is, fluid entering the upstream region $x<0$ never leaves it and is retained there to long times.

4.1 The leading-order rate of propagation of the parcel

To analyse these regimes, we begin by considering asymptotic simplifications of the model equations. It is evident from the numerical solutions that the parcel generally becomes increasingly slender with time. That is, $h\rightarrow 0$ and $\unicode[STIX]{x0394}x\rightarrow \infty$ as $t\rightarrow \infty$ , where $\unicode[STIX]{x0394}x\equiv x_{+}-x_{-}$ . Accordingly, the slope $\unicode[STIX]{x2202}h/\unicode[STIX]{x2202}x\sim h/\unicode[STIX]{x0394}x\rightarrow 0$ . In these limits, the gravitational contributions to the flow-front conditions (2.16b ) and (2.17b ) become asymptotically smaller than the order-unity contributions from the background flow. With regards to the leading position of the parcel, gravitational spreading can therefore be neglected. In the absence of those contributions, (2.16b ) and (2.17b ) each integrate to yield the long-term leading-order displacements of the flow fronts,

(4.1) $$\begin{eqnarray}x_{\pm }\sim MBt\quad (t\rightarrow \infty ).\end{eqnarray}$$

The parcel as a whole therefore advects at a rate of $M$ multiplied by the background flow rate. For geological carbon storage, the rate of advection of $\text{CO}_{2}$ by the background flow may be as high as $M=20$ times the ambient flow rate, implying an order-of-magnitude difference in velocity between the parcel and the ambient fluid. The potential for considerable differences in flow rates between the parcel and the ambient differs qualitatively from how a background flow typically affects a fluid body in non-porous fluid-mechanical configurations, for which interfacial viscous drag drives an immersed fluid parcel towards the same velocity as the background flow. The relative smallness of interfacial shear stresses in a porous medium renders that effect negligible, allowing instead for considerable velocity contrasts between the parcel and the ambient fluid.

Figure 7.

The deformation velocity $u_{D}(h)$ , representing the advective prefactor to the nonlinear wave equation (4.6) and the rate at which the background pressure gradient advects the interface in the frame of the parcel. The cases shown correspond to (a) a more viscous injectate, $M=0.5$ , (b) an equally viscous injectate, $M=1$ , and (c) a less viscous injectate, $M=2$ .

4.2 The deformation of the parcel

To explore how the parcel deforms in its moving reference frame, we recast (2.13)–(2.17) in terms of the moving coordinate system $(\unicode[STIX]{x1D709},t)$ , where

(4.2) $$\begin{eqnarray}\unicode[STIX]{x1D709}=x-(MB)t.\end{eqnarray}$$

In terms of these coordinates, equation (2.13) becomes

(4.3) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}t}=MB\left(\frac{[Mh+(1-h)]^{2}-1}{[Mh+(1-h)]^{2}}\right)\frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\left(\frac{h(1-h)}{Mh+(1-h)}\displaystyle \frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\right)\end{eqnarray}$$

and conditions (2.16) and (2.17) become

(4.4a,b ) $$\begin{eqnarray}h(\unicode[STIX]{x1D709}_{+})=0,\quad \dot{\unicode[STIX]{x1D709}}_{+}=-\frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}},\end{eqnarray}$$

(4.5a,b ) $$\begin{eqnarray}\hspace{-0.7pt}h(\unicode[STIX]{x1D709}_{-})=0,\quad \dot{\unicode[STIX]{x1D709}}_{-}=-\frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}.\end{eqnarray}$$

Our numerical solutions for $M=0.5$ and 2 (figure 2 a,c) indicated prevailing interior regions of the parcel in which $|\unicode[STIX]{x2202}h/\unicode[STIX]{x2202}\unicode[STIX]{x1D709}|\ll 1$ connected to short shock layers through which $\unicode[STIX]{x2202}h/\unicode[STIX]{x2202}\unicode[STIX]{x1D709}$ becomes large at either the front or back of the parcel respectively. To understand this structure, we begin by considering the simplified flow regimes arising from the decay $\unicode[STIX]{x2202}h/\unicode[STIX]{x2202}\unicode[STIX]{x1D709}\rightarrow 0$ . Assuming that the first term in (4.3) is non-zero ( $M\neq 1$ ), this term will dominate over the second gravitational term in the limit $\unicode[STIX]{x2202}h/\unicode[STIX]{x2202}\unicode[STIX]{x1D709}\rightarrow 0$ , yielding

(4.6) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}t}\sim MB\left(\frac{[Mh+(1-h)]^{2}-1}{[Mh+(1-h)]^{2}}\right)\frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\equiv -u_{D}(h)\frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}.\end{eqnarray}$$

This nonlinear advection equation governs the deformation of the interface caused by the relative rates of advection of the two fluid species by the background flow. It shows that the relative rate of advection is dependent on the height of the interface separating the fluids,  $h$ . The parcel thus evolves as a nonlinear wave with rate of advection $u_{D}(h)$ in the frame moving with the background flow, which we refer to as the deformation velocity. The function $u_{D}(h)$ is different for each value of $M$ , and is shown for (a)  $M=0.5$ , (b)  $M=1$ and (c)  $M=2$ in figure 7. For $M<1$ , $u_{D}$ is positive, implying that fluid migrates forwards in the frame of the flow fronts. This is consistent with the development of a shock at the downstream edge. For $M=1$ , $u_{D}$ is identically zero because the background flow advects both fluids at exactly the same rate and hence does not generate the velocity contrast necessary to deform the parcel. In this special case, the advective term in (4.3) (proportional to  $B$ ) is identically zero and its assumed dominance in deriving (4.6) above is uniquely invalid; we defer addressing this case to § 4.5. For $M>1$ , $u_{D}$ is instead negative and fluid migrates backwards in the frame of the parcel, consistent with the formation of a trailing shock layer.

Figure 8.

The magnitude of the stretching factor $|\unicode[STIX]{x1D6FA}|=2BM|1-M|$ , representing the coefficient in the nonlinear advection equation (4.7) describing the leading-order dynamics as $h\rightarrow 0$ and $M\neq 1$ . For small ambient viscosity ( $M\rightarrow 0$ ),  $\unicode[STIX]{x1D6FA}\sim 2M\rightarrow 0$ and the background flow plays a small role in deforming the particle to leading order. For $M=0.5$ , $\unicode[STIX]{x1D6FA}$ attains a local maximum (the green circle), implying that for $M<1$ a fluid parcel is stretched fastest if it is exactly twice as viscous as the ambient fluid. For $M=1$ , the lack of viscosity contrast abruptly removes any effect of the background flow on deforming the parcel ( $\unicode[STIX]{x1D6FA}=0$ ), and the purely gravity-driven flow (in the frame of the parcel) described by the theory of § 4.5 applies uniquely. For $M>1$ , $\unicode[STIX]{x1D6FA}$ increases relatively significantly as $2M^{2}$ , implying the potential for considerable deformation of a parcel that is less viscous than the ambient fluid.

4.3 The self-similar horizontal stretching of the parcel

At long times, the injectate layer thins, $h\rightarrow 0$ , and the deformation velocity $u_{D}(h)$ defined in (4.6) approaches the linear asymptote $u_{D}(h)\sim \unicode[STIX]{x1D6FA}h$ , where $\unicode[STIX]{x1D6FA}\equiv 2BM(1-M)$ is the stretching parameter. In this limit, equation (4.6) reduces asymptotically to

(4.7) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}t}\sim -\unicode[STIX]{x1D6FA}h\frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}},\end{eqnarray}$$

which is a simple nonlinear wave equation with prefactor proportional to $\unicode[STIX]{x1D6FA}$ (an inviscid Burgers equation). The absolute value $|\unicode[STIX]{x1D6FA}|$ measures the rate at which the background flow stretches the interface horizontally. The value of $|\unicode[STIX]{x1D6FA}/B|$ is plotted as a function of $M$ in figure 8, with the main plot showing log–log axes and the inset showing natural axes. For $M\ll 1$ , $\unicode[STIX]{x1D6FA}$ increases with $M$ (as  $2M$ ) because $M$ increases the background stress driving the deformation. However, at $M=0.5$ , the increase stops and $\unicode[STIX]{x1D6FA}$ attains a local maximum, shown as a green circle. While increasing $M$ still increases the ambient stress in the range $0.5<M<1$ , the increasing similarity between the rates at which the background flow advects the two fluid species individually as $M$ approaches unity causes the ambient stress to become less effective at deforming the parcel. At $M=1$ , $\unicode[STIX]{x1D6FA}=0$ because the background flow does not drive any one fluid faster than the other. The value $M=0.5$ is the critical value at which the increase in the background pressure gradient balances the reduction in the rate of deformation caused by the viscosities of the two fluids becoming similar. For $M>1$ , the viscosity contrast is reinstated and the magnitude of $\unicode[STIX]{x1D6FA}$ continues to increase. Ultimately, it grows as $2M^{2}$ , attaining values larger than those possible for $M<1$ .

We begin our analysis of (4.7) with the case of a relatively more viscous injected fluid, $M<1$ , for which $\unicode[STIX]{x1D6FA}>0$ . Motivated by the development of a shock layer near the front $x_{+}$ , we seek a solution to (4.7) subject to (2.17a ) and the volume constraint

(4.8) $$\begin{eqnarray}\int _{\unicode[STIX]{x1D709}_{-}(t)}^{\unicode[STIX]{x1D709}_{+}(t)}h(x,t)\,\text{d}x=V,\end{eqnarray}$$

with the anticipation that a conflict between the resulting solution to these equations and condition (2.16a ) produces the shock front (this will be confirmed in § 4.4 below). Since no horizontal length scale can be formed from scalings of (4.7), (2.17a ) and (4.8), one can anticipate that they support a similarity solution. To determine this, we recast these three equations in terms of the similarity variables

(4.9a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D701}=\frac{x}{(\unicode[STIX]{x1D6FA}Vt)^{1/2}},\quad h=\frac{Vf(\unicode[STIX]{x1D701})}{(\unicode[STIX]{x1D6FA}Vt)^{1/2}},\end{eqnarray}$$

which yield

(4.10a-c ) $$\begin{eqnarray}\frac{1}{2}(\unicode[STIX]{x1D701}f^{\prime })^{\prime }=f\,f^{\prime },\quad f(0)=0,\quad \int _{0}^{\unicode[STIX]{x0394}\unicode[STIX]{x1D701}}f(\unicode[STIX]{x1D701})\,\text{d}\unicode[STIX]{x1D701}=1,\end{eqnarray}$$

where $\unicode[STIX]{x0394}\unicode[STIX]{x1D701}\equiv \unicode[STIX]{x1D701}_{+}-\unicode[STIX]{x1D701}_{-}$ . On integrating (4.10a ) subject to (4.10b,c ), we obtain

(4.11a,b ) $$\begin{eqnarray}f=\unicode[STIX]{x1D701},\quad \unicode[STIX]{x0394}\unicode[STIX]{x1D701}=\sqrt{2}.\end{eqnarray}$$

This result shows that the leading-order flow approaches a linear similarity solution with a single shock front (cf. de Loubens & Ramakrishnan Reference de Loubens and Ramakrishnan2011b ). The approach is confirmed in figure 9(a), where (4.11) is plotted as a dashed blue line and the numerical solution is shown as a solid black curve for $M=0.5$ at $t=4\times 10^{3}$ . The convergence occurs fastest near the negative flow front because the gravitational dynamics is weakest furthest from the shock layer. The evolution of the horizontal extent implied by (4.11b ) is

(4.12) $$\begin{eqnarray}\unicode[STIX]{x0394}x\sim (2\unicode[STIX]{x1D6FA}Vt)^{1/2}=2[BVM(1-M)t]^{1/2}.\end{eqnarray}$$

This result shows that the extent of the parcel grows as $t^{1/2}$ with a prefactor that is proportional to $\sqrt{\unicode[STIX]{x1D6FA}}$ and $\sqrt{V}$ . The parcel therefore stretches horizontally faster for larger values of the parameter $\unicode[STIX]{x1D6FA}$ and volumes  $V$ .

Figure 9.

The long-term regimes arising for (a)  $M=0.5$ shown at time $t=4\times 10^{3}$ and (b)  $M=2$ shown at $t=200$ , as given by our numerical solution to the full equations (2.13)–(2.16). The self-similar asymptotes describing the prevailing triangular regions, given by (4.11) for $M<1$ and (4.20) for $M>1$ , are shown by the dashed blue lines in (a) and (b) respectively. The leading-order solutions describing the gravitational smoothing of the shock fronts, given by (4.17) for $M<1$ , are shown by the curves of circular markers.

4.4 The shock layer

While describing the prevailing interior of the parcel, the similarity solution (4.11) conflicts with the condition of vanishing thickness at the nose (2.16a ). This conflict indicates that the second-order (gravitational) term in (4.7) remains significant near $x=x_{+}$ to produce a narrow ‘shock layer’ in which gravity suppresses the overturning of the interface. To investigate its intervention, we begin by considering the direct simplification of (4.3) arising from the limit $h\rightarrow 0$ only, namely

(4.13) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}t}\sim -\unicode[STIX]{x1D6FA}h\frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\left(h\frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\right).\end{eqnarray}$$

To study the shock layer, we transform this equation into the frame of the shock and the similarity height variable (3.4b ) by defining the horizontal coordinate and scaled height,

(4.14a,b ) $$\begin{eqnarray}X=\unicode[STIX]{x1D709}-(2\unicode[STIX]{x1D6FA}Vt)^{1/2},\quad h=(\unicode[STIX]{x1D6FA}Vt)^{-1/2}F(X).\end{eqnarray}$$

In terms of these variables, (4.13) becomes

(4.15) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D6FA}}{V}\left(-\frac{F}{2t^{1/2}}+t^{1/2}F_{t}\right)-\frac{\unicode[STIX]{x1D6FA}}{\sqrt{2}}F_{X}=-\unicode[STIX]{x1D6FA}FF_{X}+(FF_{X})_{X}.\end{eqnarray}$$

To seek a solution that matches steadily to (4.11), we set $F_{t}=0$ . Since $F$ is of order unity, the first term in (4.15) decays as $t^{-1/2}$ and is negligible as $t\rightarrow \infty$ , leaving

(4.16) $$\begin{eqnarray}-\frac{1}{\sqrt{2}}\unicode[STIX]{x1D6FA}F_{X}=-\unicode[STIX]{x1D6FA}FF_{X}+(FF_{X})_{X}.\end{eqnarray}$$

By integrating this equation subject to the frontal condition (2.16a ), we obtain

(4.17) $$\begin{eqnarray}F\sim \sqrt{2}(1-\text{e}^{\unicode[STIX]{x1D6FA}(x-x_{+})/2}).\end{eqnarray}$$

This solution is shown as a curve of green circles in figure 9(a) and successfully describes the profile through the shock layer. Expression (4.17) shows that the extent of the shock is of $O(\unicode[STIX]{x1D6FA}^{-1})$ , implying, in particular, that a stronger background flow creates a shorter shock layer. This length scale remains constant in time but, since $\unicode[STIX]{x0394}x$ given by (4.12) increases with time, the shock layer becomes increasingly smaller relative to the horizontal extent of the parcel as a whole as $t\rightarrow \infty$ . Equation (4.17) predicts that $F\rightarrow \sqrt{2}$ as $x\rightarrow -\infty$ . This confirms a correct matching between (4.17) and the value of the outer solution (4.11) in the limit $\unicode[STIX]{x1D701}\rightarrow \sqrt{2}$ , in accordance with van Dyke’s matching rule.

4.5 Equally viscous input and ambient fluids

As noted above, for $M=1$ , the background flow advects the injected fluid at exactly the same rate as the background flow itself. Its effect in deforming the parcel, represented by the first term on the right-hand side of (4.3), is therefore identically zero. In this unique case, the assumption that the gravitational term in (4.3) becomes negligible as the slope of the parcel decreases is invalid. Instead, (4.3) simplifies to

(4.18) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}t}=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\left(h(1-h)\displaystyle \frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\right)\sim \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\left(h\frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\right)\quad (h\rightarrow 0),\end{eqnarray}$$

and describes a purely gravity-driven relaxation of the parcel in the frame of the background flow. In its moving frame, the parcel therefore relaxes exactly as if in a quiescent porous medium. As determined by Barenblatt (Reference Barenblatt1952), this equation, along with the volume constraint (4.8), supports the exact similarity solution

(4.19a,b ) $$\begin{eqnarray}h=\left(\frac{3}{32}\frac{V^{2}}{t}\right)^{1/3}\left[1-\left(\frac{2}{9Vt}\right)^{2/3}\unicode[STIX]{x1D709}^{2}\right],\quad \unicode[STIX]{x0394}x=(36Vt)^{1/3}.\end{eqnarray}$$

This dome-shaped profile is shown as a curve of purple circular markers in figure 6(b). It confirms the leading-order symmetrical relaxation of the parcel in a frame moving with the background flow. It should be noted that the rate of relaxation $\unicode[STIX]{x0394}x\propto t^{1/3}$ predicted by (4.19b ) is slower than the rate $\unicode[STIX]{x0394}x\propto t^{1/2}$ predicted by (4.12) for $M\neq 1$ . The effect of the background flow in directly stretching a released parcel for $M\neq 1$ thus drives a greater rate of horizontal extension than would occur by gravity acting independently.

4.6 Less viscous input fluid

For $M>1$ , the first term in (4.3) is non-zero, and hence, like the cases of $M<1$ , it becomes dominant at late times. Likewise, the resulting simplified equation (4.7) describes the leading-order dynamics. The only difference is that the stretching factor $\unicode[STIX]{x1D6FA}$ is negative if $M<1$ . That is, the background flow advects higher fluid backwards in the frame of the parcel. To address this case, we repeat the steps of the analysis of § 4.3 above, except with $\unicode[STIX]{x1D6FA}$ replaced by its absolute value $|\unicode[STIX]{x1D6FA}|$ in (4.9), and applying (2.16a ) instead of (2.17a ). The analysis determines the similarity solution

(4.20a,b ) $$\begin{eqnarray}f=-\unicode[STIX]{x1D701},\quad \unicode[STIX]{x0394}\unicode[STIX]{x1D701}=\sqrt{2}.\end{eqnarray}$$

Like (4.11), (4.20) describes a triangular profile. However, its gradient is opposite to the case $M<1$ and thus predicts a shock at the negative flow front instead of the positive flow front. The asymptotic approach of our numerical solution towards (4.20) is confirmed in figure 9(b), where (4.20a ) is shown as a dashed blue line at $t=200$ . The time of transition towards the similarity solution is faster for $M=2$ than $M=0.5$ because the magnitude of the stretching parameter $|\unicode[STIX]{x1D6FA}|=2$ is four times larger. For $M>1$ , there is no maximum value of $|\unicode[STIX]{x1D6FA}|$ , contrasting with the case $M<1$ , for which it was noted that there is a maximum at $M=0.5$ . Background flow thus has the potential to provide significantly greater rates of stretching when the parcel is less viscous than the ambient fluid compared with cases where it is more viscous.

5.1 Experimental results and theoretical comparisons

The evolution of a representative experiment (run 4) is shown as a time-lapse sequence in figure 12. The experiment is also shown in the supplementary movie available at https://doi.org/10.1017/jfm.2017.501. In agreement with the theory, the current is initially close to symmetrical and subsequently develops a shorter component flowing upstream against the background flow and a larger component flowing downstream with the background flow. An approximately sharp interface between the injected and ambient fluids is retained. Some slight fading at the downstream flow front indicates a stronger effect of hydrodynamic dispersion in smearing the interface for regions where the current displaces the ambient fluid. The theoretical height profiles given by our numerical solution to the model (2.13)–(2.17) and (5.2) are overlaid as white dashed curves; generally good agreement is observed.

Some discrepancies between the theory and the data occur mainly along the upstream interface, where the theory overpredicts the distance propagated by the negative flow front $|x_{-}|$ . This trend is illustrated for all of our continuous-input experiments in figure 13, where the positions of the positive and negative flow fronts measured from the digital images and predicted by our theory are shown together. The experiments with larger background flow tend to produce larger discrepancies. We conjecture that the discrepancy is caused by stresses associated with vertical flow weakening the (Dupuit) approximation of hydrostatic pressure; this hypothesis is considered in detail below. Since less fluid propagates upstream than predicted by the theory, correspondingly more fluid flows downstream. This explains the slight underprediction of the position of the downstream flow front observed for each experiment.

Figure 12.

Time-lapse sequence of photographs showing a representative experiment (run 4) at times (a) 80 s, (b) 200 s and (c) 800 s after the injection of the dyed salty water was initiated. The dyed fluid, introduced at $x=0$ , is advected through the bead pack by an ambient background flow of the saturating clear freshwater. The vertical aspect has been stretched by a factor of two. The theoretical prediction of the full numerical model given by (2.13)–(2.17) with (5.2) in place of (2.14) is overlaid as a white curve in each panel.

Figure 13.

The positions of the positive and negative flow fronts $x_{+}$ and $x_{-}$ (crosses) as measured from the experimental images for runs 1–6 involving a continuous input of fluid. Each is compared alongside the theoretical prediction of (2.13)–(2.17) and (5.2) shown as a curve.

Figure 14.

Time-lapse sequence showing a representative finite-volume release (run 7) at (a) the time $t=140$  s at which the input is terminated and (b)  $t=1800$  s. The vertical aspect has been stretched by a factor of two. The theoretical prediction of the full numerical model given by (2.13)–(2.17) with (5.2) in place of (2.14) is overlaid as a white curve in each panel.

Figure 15.

Comparisons between the experimental data (crosses) and theoretical predictions (solid curves) of the model (2.13)–(2.17) for the negative and positive flow fronts, $x_{-}$ and $x_{+}$ , measured for experimental runs 7 and 8. In these runs, the inputs were terminated at the times indicated by the vertical dashed lines to create a finite-volume release.

A time-lapse sequence of the finite-volume experiment (run 7) is shown in figure 14. The flow-front evolutions for both finite-volume experiments (runs 7 and 8) are shown in figure 15, where the times at which the input ceases are shown as vertical dashed lines. The plots show good agreement with the theoretical predictions. Again, some discrepancies occur near the negative flow front, which persist as the parcel advects downstream. For run 8, the negative flow front $x_{-}$ propagates relatively further downstream and is observed to leave small patches of the injected fluid behind its trailing edge. The patches remain for a short period of time before being swept away by the ambient fluid. The phenomenon is indicated by the step-like progression of $x_{-}$ that occurs for $t>600$  s in figure 15(b), which measures the distance furthest upstream at which a patch is observed and lags intermittently behind the bulk of the parcel. We anticipate that these patches arise as a consequence of localised regions of lower permeability in the bead pack, which intermittently trap patches of the parcel as it passes through.

5.2 The role of vertical stresses

Analysis of the runs involving a continuous input revealed a persistent, if relatively small, discrepancy between the data and the theory in the position of the negative flow fronts. We anticipate that this discrepancy is a consequence of a weakening of the assumption of predominantly horizontal flow (the Dupuit approximation) caused by a large interfacial slope at the negative flow front. In the extreme limit of a large background flow, the interaction between the ambient flow and the injected fluid would more closely approximate the flow around a locally vertical boundary created by the injected fluid, with the ambient fluid just upstream of the negative flow front $x_{-}$ being directed vertically in a stagnation-point flow. Stresses associated with this vertical flow weaken the assumption of hydrostatic pressure (2.1b ) underlying the theory.

In steady state, the flow along the upstream interface is tangential to the interface. Therefore, we can anticipate that the significance of the vertical flow is measured by the magnitude of the gradient of the interface $\unicode[STIX]{x2202}h/\unicode[STIX]{x2202}x$ at the negative flow front $x_{-}$ . Using (3.11b ), the model prediction for the magnitude of this gradient is

(5.3) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}x}=MB\frac{H}{{\mathcal{L}}}=\frac{\unicode[STIX]{x1D707}_{2}Q_{2}}{\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}gkH}\equiv S,\end{eqnarray}$$

where $S$ is a dimensionless number that we refer to as the background flow strength. The self-consistency of the model depends on $S\ll 1$ , with larger values of $S$ predicting a larger deviation from hydrostatic pressure. A similar dimensionless parameter to (5.3) was used by Pegler et al. (Reference Pegler, Kowal, Hasenclever and Worster2013b ) to explain discrepancies due to vertical stresses for the related problem of a transition from a point-like radial injection towards gravitational relaxation in a vertical Hele-Shaw cell. The value of $S$ is evaluated for runs 1–6 in the final row of table 1. In figure 16, we have plotted the relative discrepancy between the final position of the negative flow front between theory, $x_{-}^{T}$ , and experiment, $x_{-}^{E}$ , against  $S$ . The plot indicates a positive correlation between the value of (5.3) and the degree of discrepancy, indicating that the discrepancy is indeed due to a weakening of the assumption of hydrostatic pressure.

Figure 16.

Scatter plot showing the correlation between the background strength parameter  $S$ , as defined by (5.3), and the relative discrepancy between the final position of the negative flow front as predicted by the theory, $x_{-}^{T}$ , and the experimental data, $x_{-}^{E}$ , for runs 1–6. The positive correlation indicates that the weakening of the Dupuit approximation of predominantly horizontal flow, as measured by  $S$ , is responsible for the discrepancy.

6.1 Navajo Sandstone

As a case study, we consider the leaking of $\text{CO}_{2}$ -charged brine into the water-saturated Navajo Sandstone in North America. The brine originates from a much deeper formation saturated by $\text{CO}_{2}$ . The $\text{CO}_{2}$ -charged brine propagates from this aquifer through a permeable fault and, on its route to the surface, partially leaks into the Navajo Sandstone. The leaked fluid is subsequently advected tens of kilometres (e.g. Kampman et al. Reference Kampman, Bickle, Becker, Assayag and Chapman2009) through this sandstone by a background flow of saturating water. Fluid samples taken from the reservoir through the depth of the aquifer near the fault indicate that $\text{CO}_{2}$ -charged brine spans the depth of the aquifer with a significant concentration gradient (Kampman et al. Reference Kampman, Bickle, Maskell, Chapman, Evans, Purser, Zhou, Schaller, Gattacceca and Bertier2014). Measured or calculated parameters representing the aquifer and fluids are provided in table 2, with references provided in the caption.

Table 2.

Geophysical parameter estimates. The column for the Navajo case study lists specific values measured or calculated. The column for $\text{CO}_{2}$ storage lists generic representative values for reservoirs in which $\text{CO}_{2}$ sequestration may be applied.

a Kampman et al. (Reference Kampman, Bickle, Maskell, Chapman, Evans, Purser, Zhou, Schaller, Gattacceca and Bertier2014).

b Kampman et al. (Reference Kampman, Bickle, Becker, Assayag and Chapman2009).

c Using the concentrations measured by Kampman et al. (Reference Kampman, Bickle, Maskell, Chapman, Evans, Purser, Zhou, Schaller, Gattacceca and Bertier2014) in conjunction with the tables of Dubacq, Bickle & Evans (Reference Dubacq, Bickle and Evans2013).

d Maskell (Reference Maskell2017).

The dissolved $\text{CO}_{2}$ and NaCl create a density difference of $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}\approx 12~\text{kg m}^{-3}$ and a viscosity ratio of $M=\unicode[STIX]{x1D707}_{2}/\unicode[STIX]{x1D707}_{1}=0.99$ . The flux ratio is $B=Q_{2}/Q_{1}\approx 3.9$ , indicating a significant role of background flow. Assuming that the input and ambient fluxes remain approximately constant, we apply the theoretical results of § 3. According to the theory, the interface between the injected and ambient fluids approaches an approximately horizontal interface along a height (3.9b ) given dimensionally by

(6.1) $$\begin{eqnarray}h_{H}=\frac{H}{1+MB}.\end{eqnarray}$$

For the values given above, $h_{H}\approx 0.2H\approx 30$  m. The brine is therefore predicted to occupy a region spanning one-fifth of the depth of the aquifer.

The corresponding prediction for the interstitial flow rate of the brine is

(6.2) $$\begin{eqnarray}u_{1}=\frac{Q_{1}}{\unicode[STIX]{x1D719}h_{H}}=(1+MB)\frac{Q_{1}}{\unicode[STIX]{x1D719}H},\end{eqnarray}$$

which is evaluated as $u_{1}\approx 4.2\times 10^{-7}~\text{m s}^{-1}\approx 13~\text{m yr}^{-1}$ . Based on this, the injected fluid is estimated to span the 20 km length of the Navajo Sandstone in approximately 1500 years. The prediction for the distance that the flow extends upstream against the background flow, as given by (3.11b ), has the dimensional form

(6.3) $$\begin{eqnarray}|x_{-}|=\frac{1}{2}\left[1-\left(1-\frac{1}{1+MB}\right)^{2}\right]\frac{\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}gkH^{2}}{\unicode[STIX]{x1D707}_{2}Q_{2}}.\end{eqnarray}$$

This distance is evaluated as approximately 400 m. This value is consistent with the measurements of downhole fluid samples taken 100 m upstream of the fault (Kampman et al. Reference Kampman, Bickle, Maskell, Chapman, Evans, Purser, Zhou, Schaller, Gattacceca and Bertier2014), indicating a limit to the extent of propagation of the brine against the groundwater flow. The result is also consistent with observations of bleaching of the exposed Entrada Sandstone related to previous inputs of $\text{CO}_{2}$ -rich brines (Wigley et al. Reference Wigley, Kampman, Dubacq and Bickle2012). The bleaching occurs in the basal 20 m of the formation, extends less than 500 m upstream, and approximately 10 km downstream of the fault, which are consistent with our theoretical predictions.

6.2 $\text{CO}_{2}$ injection

Next, we apply the results to investigate the effect of background flow on pure $\text{CO}_{2}$ injected into the subsurface for geological carbon storage. For this, we consider generic ranges of representative values for typical sandstone aquifers where geological carbon storage may be applied. These values are listed in the final column of table 2. For typical ranges of reservoir temperatures and fluid compositions, the $\text{CO}_{2}$ will be $M=5$ –20 times less viscous than the ambient water in the host rock. Given a representative input rate per unit width of $Q_{1}=10^{-4}~\text{m}^{2}~\text{s}^{-1}$ and a typical background flow rate of $Q_{2}=10^{-4}$  m, we estimate that $B$ can be of order unity. The case of no background flow, $B=0$ , may be more applicable to marine aquifers (such as the Sleipner project in the North Sea), for which precipitation cannot generate long-range hydrostatic imbalances. However, background flow can be driven in that case by poroelastic deformation due to the weight of the overlying rock. Below, we determine the magnitude of such motion necessary for an appreciable effect on the migration of $\text{CO}_{2}$ .

In accordance with the result of (3.10), the leading front of the $\text{CO}_{2}$ current is predicted to propagate in the direction of the background flow at a rate of

(6.4) $$\begin{eqnarray}u_{1}=M(1+MB)\frac{Q_{1}}{\unicode[STIX]{x1D719}H}\approx \frac{M^{2}Q_{2}}{\unicode[STIX]{x1D719}H}\approx M^{2}u_{2},\end{eqnarray}$$

where the last two approximations apply for $MB\gg 1$ . The injected fluid thus moves approximately $M^{2}$ times faster than the ambient fluid. The significant quadratic dependence on $M^{2}$ stems from the dual effect of the viscosity ratio for $M>1$ in both reducing the effective height of confinement of the current (given by (6.1)) and increasing the horizontal length of the convex similarity solution describing the injected current. For geological carbon storage, the $\text{CO}_{2}$ could travel as much as $M^{2}=400$ times faster than the background flow. A relatively small background flow may therefore be sufficient to have a significant impact on $\text{CO}_{2}$ spreading. Using a characteristic flow rate of the injected fluid of $u_{1}\approx 1.9\times 10^{-6}~\text{m s}^{-1}\approx 60~\text{m yr}^{-1}$ obtained from measurements of the $\text{CO}_{2}$ plume at Sleipner (Boait et al. Reference Boait, White, Bickle, Chadwick, Neufeld and Huppert2012) and the viscosity ratio $M\approx 13$ , we determine that a background flow rate of just $u_{2}\approx u_{1}/M^{2}\approx 1.2\times 10^{-8}~\text{m s}^{-1}\approx 0.37~\text{m yr}^{-1}$ would be necessary to affect the migration of $\text{CO}_{2}$ significantly.

For the evolution of a parcel of $\text{CO}_{2}$ after its input ceases, we apply the predictions of the theory of § 4. The result of (4.1) predicts that the parcel of injected $\text{CO}_{2}$ is subsequently advected at the slower rate of a single factor of $M$ times the background flow rate, $u_{1}=Mu_{2}$ . The horizontal extent of the asymptotic similarity solution (4.11) arising under a background flow (4.12) has the dimensional form

(6.5) $$\begin{eqnarray}\unicode[STIX]{x0394}x(t)=\frac{2[M(M-1)Q_{2}{\mathcal{V}}t]^{1/2}}{\unicode[STIX]{x1D719}H},\end{eqnarray}$$

where ${\mathcal{V}}$ is the volume per unit width of fluid released. For illustrative values of $H=10$  m, $\unicode[STIX]{x1D719}=0.2$ , $M=10$ , $Q_{2}=10^{-5}~\text{m}^{2}~\text{s}^{-1}$ and ${\mathcal{V}}=10^{3}~\text{m}^{2}$ , (6.5) predicts that $\unicode[STIX]{x0394}x\approx 0.95\,t^{1/2}$ . This implies a horizontal extent of 280 m in 1 day, 5.3 km in 1 year and 170 km in $10^{3}$  years. It should be noted that this result assumes negligible residual trapping, an approximation that may apply in the case of a pure $\text{CO}_{2}$ parcel only if the aquifer contains an existing residual of $\text{CO}_{2}$ from a previous injection; a detailed evaluation of the effect of residual trapping is provided in the following subsection. The surface area of the interface between the injected $\text{CO}_{2}$ and the ambient water predicted by (6.5) may provide an ultimate constraint on the rate at which a parcel of $\text{CO}_{2}$ dissolves into the ambient water to become permanently secured.

6.3 Residual trapping

If the injectate is immiscible or only partially miscible in the ambient fluid, as applies to pure $\text{CO}_{2}$ injected into a saline aquifer, a certain proportion of fluid can be retained in the porous matrix as a residue trailing the parcel (Hesse et al. Reference Hesse, Orr and Tchelepi2008). The primary effect of this trapping is to reduce the mass of the parcel of $\text{CO}_{2}$ over time. Let $R$ denote the proportion of the porous matrix occupied by residual $\text{CO}_{2}$ in the wake of the parcel; a typical value is $R\approx 0.2$ . Let $V(t)$ denote the volume of the pure $\text{CO}_{2}$ parcel. To investigate the effects of residual trapping, we model the loss to the capillary trail as being proportional to the height of the current multiplied by its leading-order rate of horizontal propagation,

(6.6) $$\begin{eqnarray}\dot{V}(t)\sim -Rh_{max}(t){\dot{x}}_{-}(t),\end{eqnarray}$$

where $h_{max}(t)\equiv \max _{x}[h(x,t)]$ . Using the asymptotic solution (4.20) to evaluate $h_{max}$ and (4.1) to evaluate ${\dot{x}}_{-}$ , we obtain the ordinary differential equation

(6.7) $$\begin{eqnarray}\dot{V}\sim -RH\left(\frac{2V}{\unicode[STIX]{x1D6FA}t}\right)^{1/2}\left[\frac{MQ_{2}}{\unicode[STIX]{x1D719}H}\right].\end{eqnarray}$$

On integrating this equation, we obtain the evolution of the volume of the parcel,

(6.8) $$\begin{eqnarray}V(t)=\left\{1-\frac{R}{\unicode[STIX]{x1D719}}\left[\frac{MQ_{2}t}{|M-1|V_{0}}\right]^{1/2}\right\}^{2}V_{0},\end{eqnarray}$$

where $V_{0}\equiv V(0)$ is the total volume of $\text{CO}_{2}$ input. This analytical result yields an efficient means to assess the effect of capillary retention for mobilisation of a packet of sequestered $\text{CO}_{2}$ over time. Setting $V=0$ in (6.8) yields the total time $t_{res}$ and horizontal distance $x_{res}=x_{\pm }$ at which the $\text{CO}_{2}$ becomes fully residually trapped, given by

(6.9a,b ) $$\begin{eqnarray}t_{res}=\frac{|M-1|}{M}\frac{\unicode[STIX]{x1D719}^{2}}{R^{2}}\frac{V_{0}}{Q_{2}},\quad x_{res}=\frac{|M-1|\unicode[STIX]{x1D719}V_{0}}{R^{2}H}\end{eqnarray}$$

respectively. Using the values for our numerical example given in § 6.2 above, we evaluate $t_{res}\approx 3$  yr, which indicates that a packet of $\text{CO}_{2}$ will become fully trapped by capillary retention after a few years. It is interesting that the distance propagated by the parcel, $x_{res}$ , given by (6.9b ) does not depend on the background flux  $Q_{2}$ . The ‘favourable’ cancellation of $Q_{2}$ in deriving this result is caused by the coincidence that a faster background flow rate simultaneously increases both the rate of propagation of the parcel and the rate of mass loss. For our illustrative parameter values, the total distance propagated by the parcel is $x_{res}\approx 4.5$  km.

Residual trapping is likely to be a significant effect for a first-time finite (post-injection) release of $\text{CO}_{2}$ . It should be noted, however, that it does not occur in the case of a constant-flux input (§ 3), because, in that case, the current only grows ( $\unicode[STIX]{x2202}h/\unicode[STIX]{x2202}t>0$ ) and thus cannot leave a trail. This case may be of greater relevance to industrial-scale $\text{CO}_{2}$ , for which the aim would be to use the full aquifer for storage, as opposed to releasing a finite volume that occupies a relatively small proportion of the total available storage space. Residual trapping is also not relevant if the injectate is miscible ( $R=0$ ), as applies to the migration of dissolved $\text{CO}_{2}$ or contaminants through aquifers.

6.4 Model applicability

We end by summarising the applicability of our theoretical results. Three sources of additional physical effects not included in our model include those identified by our laboratory study, namely the possible role of vertical stresses, as measured by the parameter (5.3), the role of time-dependent variations in fluxes caused by the gradual accumulation of less viscous fluid in the reservoir, and the effect of localised patches of fluid left in the wake of the parcel inside local regions of low permeability. Perhaps the primary limitation of the solutions we describe is as a model for the localised continuous injection of $\text{CO}_{2}$ at a point source. In such cases, it is permissible for the $\text{CO}_{2}$ to expand to fill the depth of the aquifer at the injection site because the ambient fluid can flow laterally around the injection zone, a feature that cannot occur for the infinite line source we have assumed. The input of brine into the Navajo Sandstone is introduced through a linear fault along a length of approximately 2 km. We anticipate that the flow near the input may approximate that of a linear source, but, for scales much larger than the fault, the flow will approach an asymptotic regime that is effectively fed by a point source. Some lateral spreading of the flow will also occur under gravity in the direction perpendicular to the background flow, which is not addressed by our analysis. Other three-dimensional effects may stem from the fact that the fault is aligned at an angle to the prevailing background flow, as well as being partially sealing. For the finite-volume release, we anticipate that the asymptotic regimes we describe do also apply to leading order in the case of a three-dimensional parcel released into a linear background flow because the control of the flow deformation by the background flow will dominate at long times over lateral spreading of the parcel under gravity. An exploration of these three-dimensional phenomena may provide interesting directions for future analysis.