2. Formulation

We study the gravity-driven flow of a viscous non-Newtonian fluid having density ${\it\rho}$ and spreading along the horizontal impermeable base of an infinite homogeneous porous domain of depth $h_{0}$ , initially saturated with an ambient fluid of density ${\it\rho}-{\rm\Delta}{\it\rho}$ . The intruding fluid is described rheologically by a power-law model, which uses two parameters, flow behaviour index $n$ and consistency index $\widetilde{{\it\mu}}$ , to express the shear stress ${\it\tau}$ as a function of shear rate $\dot{{\it\gamma}}$ in the form ${\it\tau}=\widetilde{{\it\mu}}|\dot{{\it\gamma}}|^{n-1}\dot{{\it\gamma}}$ , valid for simple shear. The cases $n<1$ , $n=1$ , and $n>1$ correspond to shear-thinning, Newtonian, and shear-thickening behaviour, respectively. For $n=1$ , the consistency index $\widetilde{{\it\mu}}$ becomes the dynamic viscosity ${\it\mu}$ .

A finite mound of fluid is released in the half-plane $x>0$ and can drain freely at  $x=0$ , see figure 1. The space–time development of the current height $h(x,t)$ is determined under the assumption of a thin gravity current having a large ratio between horizontal and vertical length scales (Bear Reference Bear1972), and small height with respect to the thickness of the porous domain ( $h(x,t)\ll h_{0}$ ). This allows modelling of the propagation as one-dimensional (neglecting vertical velocities) and the motion of the displaced fluid may be disregarded. The pressure within the intruding fluid is hydrostatic and given by

where $p_{1}=p_{0}+({\it\rho}-{\rm\Delta}{\it\rho})gh_{0}$ is a constant, and $p_{0}$ is the constant pressure at $z=h_{0}$ . The sharp interface approximation is assumed to hold; we neglect mixing due to convection assuming that no interface instabilities develop, and mixing by molecular diffusion assuming that the time scale of molecular diffusion is much longer than the time scale of the process. Surface tension effects are also neglected; this may be a limitation for very low-permeability media, as it implies disregarding of the capillary entry pressure limiting the migration of the injected fluid. We further assume the validity of the modified Darcy’s equation for power-law fluid flow in porous media (Pascal Reference Pascal1983; Shenoy Reference Shenoy1995)

where $\boldsymbol{u}$ is the Darcy velocity, $p$ the pressure in the intruding fluid, $g$ the gravity, $z$ the vertical coordinate and ${\it\phi}$ and $k$ are the medium porosity and intrinsic permeability, respectively. Equation (2.2) derives from the capillary bundle analogy for power-law fluid flow, modified to take tortuosity effects into account via a factor $C_{t}>1$ , which has several possible expressions (for a review, see Shenoy Reference Shenoy1995). The expression implemented here in (2.3) (for the details, see Di Federico et al. Reference Di Federico, Archetti and Longo2012a ), is $C_{t}=(25/12)^{(n+1)/2}$ (Pascal Reference Pascal1983). As a whole, the prefactor on the left-hand side of (2.2) is akin to an effective fluid mobility, and depends on medium properties (porosity/permeability) in a nonlinear fashion based on the value of the flow behaviour index $n$ . For a Newtonian fluid, $C_{t}=(25/12)$ (Cristopher & Middleman Reference Cristopher and Middleman1965), the factor ${\it\Lambda}_{0}$ in (2.3) reduces to $1/{\it\mu}$ and (2.2) to Darcy’s law $\boldsymbol{u}=-(k/{\it\mu})\boldsymbol{{\rm\nabla}}(p+{\it\rho}gz)$ .

The permeability and porosity of the medium are assumed to vary along the vertical as

where $x^{\ast }$ is a reference length scale, $k_{0}$ and ${\it\phi}_{0}$ are the permeability and porosity at $z=x^{\ast }$ , respectively, and ${\it\omega}$ and ${\it\gamma}$ are constant numerical coefficients. ${\it\omega}>1~({\it\gamma}>1)$ or ${\it\omega}<1~({\it\gamma}<1)$ represent a permeability (porosity) increasing or decreasing with $z$ , respectively. We further set ${\it\gamma}>{\it\gamma}_{1}=0$ and ${\it\omega}>{\it\omega}_{1}={\it\gamma}(1-n)/(n+1)$ ; this ensures the validity of the velocity scale defined in the following. The lower bounds ${\it\gamma}_{1}$ and ${\it\omega}_{1}$ imply that porosity and permeability do not decrease faster than $z^{-1}$ with elevation. This constraint is of limited impact as in most real applications to porous and fractured media, the permeability tends to decrease with depth (Jiang, Wang & Wan Reference Jiang, Wang and Wan2010), hence the case ${\it\omega}>1$ is more common than ${\it\omega}<1$ .

Note that in real porous media, the two exponents ${\it\omega}-1$ and ${\it\gamma}-1$ are related, and their ratio typically varies between $2$ and $3$ (Phillips Reference Phillips1991; Dullien Reference Dullien1992), corresponding to two limiting cases of the mediums microstructure, i.e. media with tubular fluid paths and with an intersecting network of cracks or fissures, respectively.

For a mound of Newtonian fluid propagating in a semi-infinite homogeneous medium and draining at the origin, the volume decreases in time, but the first spatial moment $Q=\int _{0}^{x_{N}(t)}xh(x,t)\,\text{d}x$ , with $x_{N}$ the position of the current front, is time invariant. The constancy of $Q$ , also termed the dipole moment, was first shown by Barenblatt & Zel’dovich (Reference Barenblatt and Zel’dovich1957). The large-time behaviour of the solution is well represented by self-similar functions, which have been demonstrated to converge to the general solution of the dipole problem (see Barenblatt & Vazquez Reference Barenblatt and Vazquez1998). In the following, we generalize the solution of the dipole problem (King & Woods Reference King and Woods2003) to non-Newtonian power-law fluids and vertically variable permeability and porosity, described by (2.4a,b ).

Under the previous assumptions, the flow equation (2.2) simplifies to its one-dimensional version, which reads in terms of the horizontal velocity $u(x,z,t)$ as

taking (2.1) and (2.4a,b ) into account. The local mass balance equation for a current with velocity and porosity varying along the vertical is

Finally, the dipole moment, $Q$ , for non-homogeneous porosity is given by

where $x_{N}(t)$ is the horizontal extent of the current. Substituting (2.5) in (2.6)–(2.7) yields respectively

where

and

are the natural velocity and length scales, respectively. The mathematical statement of the problem is completed by the boundary condition at the moving boundary

Regarding the condition at $x=0$ , Barenblatt & Zel’dovich (Reference Barenblatt and Zel’dovich1957) showed for a Newtonian fluid ( $n=1$ ) that imposing the null depth at the origin is tantamount to conserving the dipole moment $Q$ , while the volume flux draining out of the system at $x=0$ takes a finite non-zero value. In the following, we demonstrate that for a power-law fluid, the dipole moment is conserved for a subclass of self-similar solutions of the form $h=At^{F_{3}}{\it\Phi}(x/t^{F_{2}})$ ( $A,F_{2},F_{3}$ being constants, ${\it\Phi}$ a shape function), namely with $F_{3}=-2F_{2}/{\it\gamma}$ .

The differential problem constituted by (2.8) and (2.9) and boundary condition (2.12) becomes in non-dimensional form

where the dimensionless variables are $T=t/t^{\ast },X=x/x^{\ast },X_{N}=x_{N}/x^{\ast },H=h/x^{\ast },U=u/u^{\ast }$ , the velocity and spatial scales are defined by (2.11), and the time scale is $t^{\ast }=x^{\ast }/u^{\ast }$ .

We observe that the balances of (2.13)–(2.14) require, respectively

Hence $X\sim T^{F_{2}}$ and $H\sim T^{F_{3}}$ , with

and $F_{3}=-2F_{2}/{\it\gamma}$ , a result independent of the nature of the fluid.

3. Self-similar solution

The dimensionless differential problem (2.13)–(2.15) admits a similarity solution of the form

where ${\it\eta}$ is the similarity variable, ${\it\eta}_{N}={\it\eta}_{N}(n,{\it\omega},{\it\gamma})$ its value at the current front, ${\it\zeta}$ the reduced similarity variable, and

Substituting (3.1) in (2.13)–(2.15) yields

where ${\it\Phi}({\it\zeta})$ is the shape function and the prime indicates $\text{d}/\text{d}{\it\zeta}$ . The position of the current front at any time is given by

and its velocity is equal to

Note that the restrictions previously set on the parameter values ( ${\it\gamma}>{\it\gamma}_{1}$ and ${\it\omega}>{\it\omega}_{1}$ ) ensure that the time exponent of the current extension $F_{2}$ is positive. Correspondingly, the time exponent of the current thickness $F_{3}$ is always negative, as the (time decreasing) volume of the mound is spread over a larger distance as time increases. Further, taking the partial derivatives of $F_{2}$ and $F_{3}$ with respect to $n$ , ${\it\omega}$ , and ${\it\gamma}$ elucidates their dependence on model parameters. Firstly, it is seen that $F_{3}$ increases and $F_{2}$ decreases with increasing ${\it\omega}$ . This is so because a larger ${\it\omega}$ implies an increase (for ${\it\omega}>1$ ), or a less significant decrease (for ${\it\omega}<1$ ) of permeability with elevation, allowing the injected mound to be thicker; correspondingly, the current extension is smaller because of mass balance. Secondly, an increase of ${\it\gamma}$ entails a corresponding increase of $F_{2}$ , as an increase (for ${\it\gamma}>1$ ), or a less significant decrease (for ${\it\gamma}<1$ ) of porosity with elevation implies that more volume is available within the porous domain, thus favouring the current spreading. Thirdly, $F_{2}$ increases and $F_{3}$ decreases with increasing $n$ . The physical explanation is that the average spatial pressure gradient, proportional to time raised to $F_{3}-F_{2}$ , diminishes with time as $F_{3}-F_{2}<0$ . Hence the shear applied to the fluid decreases over time; this in turn implies, for a shear-thinning fluid with $n<1$ , an increased resistance to flow as the apparent viscosity increases with decreasing shear stress.

The flux draining into the source $q_{0}(t)=h(\partial h/\partial x)|_{x=0}$ , the mound volume $v(t)=\int _{0}^{x_{N}}\int _{0}^{h}{\it\phi}(z)\,\text{d}z\,\text{d}x$ , and the volume of fluid drained $v_{out}(t)=\int _{0}^{t}q_{0}(t)\,\text{d}t$ , are all quantities of interest defined per unit width. Their dimensionless counterparts are equal to $R_{0}=q_{0}/(u^{\ast }x^{\ast })$ , $V=v/x^{\ast 2}$ , and $V_{out}=v_{out}/x^{\ast 2}$ respectively. After some algebra, they are derived as

where $R_{0}$ and $V_{out}$ are negative. In both expressions for $V$ and $V_{out}$ , a singularity is present for $T\rightarrow 0$ . For $n=1$ (Newtonian fluid) and ${\it\gamma}={\it\omega}=1$ (homogeneous medium), the governing equations reduce to ${\it\Phi}=({\it\zeta}^{1/2}-{\it\zeta}^{2})/6$ , ${\it\eta}_{N}=40^{1/4}$ , and $X_{N}={\it\eta}_{N}T^{1/4}$ (Barenblatt & Zel’dovich Reference Barenblatt and Zel’dovich1957); the dimensionless flux into the source and the mound volume per unit width become $R_{0}=-{\it\eta}_{N}^{3}T^{-5/4}/72$ and $V={\it\eta}_{N}^{3}T^{-1/4}/18$ , respectively (King & Woods Reference King and Woods2003, with the sign of $R_{0}$ changed for the different convention adopted).

The numerical integration of (3.3)–(3.5) is conveniently performed by replacing the integral condition with an extra boundary condition on the first derivative in ${\it\zeta}=1$ , obtained by expanding in series the shape function near the front end.

Figure 2.

Shape function ${\it\Phi}({\it\zeta})$ versus reduced similarity variable for $n=0.2,0.4,0.6,0.8,1$ , and (a) ${\it\omega}={\it\gamma}=1$ ; (b) ${\it\omega}=1.75,{\it\gamma}=1.25$ . The dots represent the analytical solution for a Newtonian fluid $n=1$ (Barenblatt & Zel’dovich Reference Barenblatt and Zel’dovich1957).

Numerical results for the shape function ${\it\Phi}({\it\zeta})$ are shown as solid curves in figure 2 for different values of $n$ representing shear-thinning fluids, and depicting cases with uniform permeability and porosity (figure 2 a) and with permeability and porosity increasing with $z$ (figure 2 b). It is seen that the shape function tends to increase as $n$ decreases; this effect is compounded when the medium properties increase with elevation. For $n=1$ , the numerical solution is coincident with the analytical one, which exhibits a null height at the origin. For $n<1$ , the shape function at $x=0$ is non-zero and finite. This behaviour can be interpreted as the intermediate asymptotic of a process depending on $n,{\it\omega},{\it\gamma}$ and having a virtual origin at negative values of  ${\it\zeta}$ ; the solution evolves from a Dirac pulse at the initial stage with a left front end starting from zero and receding to meet the virtual origin. From that moment, the non-zero height condition at ${\it\zeta}=0$ applies (see Barenblatt & Vazquez Reference Barenblatt and Vazquez1998). The value of the shape function at the origin ${\it\Phi}(0)$ for shear-thinning fluids increases as $n$ decreases, as illustrated in figure 3(a); the decrease is more evident for low values of $n$ , i.e. very strongly shear-thinning fluids. Figure 3(a) also plots the multiplicative constant ${\it\eta}_{N}$ as a function of $n$ evaluated from the numerical solution using (3.4); ${\it\eta}_{N}$ tends to increase with increasing values of $n$ . The actual behaviour of the current in terms of extension and height is best analysed in dimensional coordinates, as the scales adopted to obtain the dimensionless solution are themselves a function of model parameters.

Figure 3.

(a) Similarity variable at the current front ${\it\eta}_{N}$ and shape function in the origin ${\it\Phi}(0)$ versus $n$ for shear-thinning power-law fluids with different conditions for porosity and permeability vertical variations. (b) The flux draining from the current $R_{0}$ and the volume of the mound $V$ as a function of time for $n=0.2,0.4,0.6,0.8,1$ . The dots represent the analytical solution for a Newtonian fluid (Barenblatt & Zel’dovich Reference Barenblatt and Zel’dovich1957).

Figure 3(b) depicts the flux into the source $R_{0}$ , and the current volume $V$ , as a function of time for different values of $n$ . It is seen that both the flux at the origin and the mound volume decrease as the mound spreads. The flux from the current is smaller for low values of $n$ ; as a consequence, the mound volume and its height at the origin are larger. In both cases, the available analytical solution for Newtonian fluid flow in a homogeneous medium (Barenblatt & Zel’dovich Reference Barenblatt and Zel’dovich1957) is perfectly reproduced by the numerical results.

The solution for shear-thickening fluids ( $n>1$ ) indicates a detached origin on the left for positive values of ${\it\zeta}$ , a condition also encountered by Barenblatt & Vazquez (Reference Barenblatt and Vazquez1998) (namely in their Problem 2, Draining problem). The self-similar solution would exist for a infinite time and the left origin would move to the right proportionally to $T^{-F_{2}}$ with a flux expressed by (3.8), which should be zero since no drain can be provided. A preliminary test conducted with a shear-thickening fluid (cornstarch suspension) showed that there is no detachment of the left origin, hence the self-similar solution (3.1) yields physically meaningless results for $n>1$ and will not be considered further.

4. Analogue Hele-Shaw model

The Hele-Shaw approximation, a classical approach in the study of viscous flow problems, is valid for laminar flow between two plates separated by a small gap. Analogue Hele-Shaw models are traditionally used for the study of two-dimensional porous media given the identity of the governing equations. Newtonian flow in a uniform Hele-Shaw cell of thickness $b$ models Darcy flow in a homogeneous medium of permeability $k(b)=b^{2}/12$ . Newtonian flow in a V-shaped Hele-Shaw cell reproduces Darcy flow in a heterogeneous medium with both permeability and porosity power-law variations along the vertical direction (Zheng et al. Reference Zheng, Soh, Huppert and Stone2013); the ratio between the exponents of permeability and porosity is necessarily equal to 3, a value appropriate to represent a fissured medium. The approximation was extended to non-Newtonian power-law fluids (e.g. Aronsson & Janfalk Reference Aronsson and Janfalk1992; Kondic, Palffy-Muhoray & Shelley Reference Kondic, Palffy-Muhoray and Shelley1996; King Reference King2000). Based on the similarity between the modified Darcy’s law (2.2)–(2.3) and the laminar flow between two plates separated by a gap with half-width $W$ varying along the vertical as $W(z)=b_{0}z^{r}$ , the analogy between flow of a power-law fluid in a Hele-Shaw cell of variable gap (the ‘model’) and a two-dimensional heterogeneous porous medium (the ‘prototype’) can be developed. Mapping $H_{v}\rightarrow H^{p}$ , where $H_{v}$ is the non-dimensional thickness of the current in the Hele-Shaw cell and $p>1$ (see appendix B for the details), the analogy requires the following correspondence amongst the parameters:

or, in terms of $p$ and $r$ :

where $n_{v}$ is the behaviour index of the fluid used in the Hele-Shaw cell. It is seen that in all cases the analogy requires fluids with the same behaviour index in the prototype and in the model. For the other parameters, a parallel plate Hele-Shaw cell ( $r=0$ ) represents flow in a porous medium with a variation of properties described by $({\it\omega}-1)/({\it\gamma}-1)=(n+3)/(n+1)$ . This ratio: (i) tends to a value of  $3$ , typical of fissure networks, for very strongly shear-thinning fluids with $n\ll 1$ ; (ii) is equal to $2$ , typical of tubular flow paths, for $n=1$ ; and (iii) is lower than $2$ for shear-thickening fluids with $n>1$ . Mapping with $p=1$ can only reproduce a fixed ratio $({\it\omega}-1)/({\it\gamma}-1)=3$ , corresponding to fissure networks and irrespective of the fluid behaviour index $n$ ; for example, a cell with $r=0.5$ reproduces the case ${\it\omega}=2.5$ and ${\it\gamma}=1.5$ . Figure 4 shows the possible combinations of parameters to ensure the analogy for fixed values of $n$ corresponding to Newtonian and shear-thinning fluids ( $n=0.5$ ), considering: (i) positive values of $({\it\omega}-1)$ and $({\it\gamma}-1)$ , implying an increase of properties with elevation, the most plausible case in real porous media; (ii) a positive value of the mapping parameter $p$ , to avoid singularities at the front end, where $H=0$ ; and (iii) a shape parameter of the cell $0<r<1$ , equivalent to a moderate (less than linear) increase of the gap with elevation. Inspection of the figure 4 shows the combination of parameters satisfying the limitations $2\leqslant ({\it\omega}-1)/({\it\gamma}-1)\leqslant 3$ and $p>0$ ; the former constraint is more severe.

Figure 4.

Interplay among parameters for Hele-Shaw analogue of flow in heterogeneous porous media with power-law permeability $k\propto z^{{\it\omega}-1}$ and porosity ${\it\phi}\propto z^{{\it\gamma}-1}$ variations in the vertical direction (a) for Newtonian fluid ( $n=1$ ); (b) for shear-thinning fluid ( $n=0.5$ ). Dashed lines represent isovalues of $p$ , bold lines are isovalues of $r$ ; the isovalue $p=1$ is not affected by fluid behaviour index $n$ . The closed hatched area (yellow online) represents a positive ratio between permeability and porosity exponents in the range 2–3 with positive values of $p$ ; in the closed cross-hatched area (grey online) $p$ is also positive but $({\it\omega}-1)/({\it\gamma}-1)>3$ .

5. Experiments

A series of experiments were conducted to compare with the theoretical work. The fluids were obtained by mixing glycerol, water and Xanthan gum in different ratios. In particular, 60 % (vol) glycerol, 40 % (vol) water and 0.1 % (weight) Xantham gum resulted in a fluid with $n=0.42$ ; 70 % (vol) glycerol, 30 % (vol) water and 0.1 % (weight) Xantham gum resulted in a fluid with $n=0.47$ ; 70 % (vol) glycerol, 30 % (vol) water and 0.06 % (weight) Xantham gum resulted in a fluid with $n=0.57$ . The Newtonian fluid was 95 % glycerol and 5 % water. The components were mixed and added with ink to enable an easy visualization of the current. The mixture was filtered in order to eliminate lumps and was stored in closed containers to avoid absorbing moisture. In general, the elimination of the lumps reduces the concentration of the Xantham gum with respect to the nominal mass fraction originally added. It is a source of variability in $n$ between different mixtures with the same nominal mass fraction of Xanthan gum.

The rheological parameters were measured independently with a strain controlled parallel plate rheometer (Dynamic Shear Rheometer Anton Paar Physica MCR 101) in the low shear rate range ( $\dot{{\it\gamma}}<5~\text{s}^{-1}$ ), with associated uncertainty equal to 2.9 % and 3.1 % for the fluid behaviour and consistency indexes, respectively. The error bounds are the prediction intervals at 95 % confidence level.

Figure 5.

The apparent viscosity ${\it\mu}\equiv \widetilde{{\it\mu}}|\dot{{\it\gamma}}|^{n-1}$ as a function of shear rate; the rheometric test refers to the fluid used in Experiments 8–10. The bold line indicates the least squares regression for the low shear rate range, the dashed curves represent the prediction limit at 95 % of confidence level.

Figure 5 shows the experimental data (apparent viscosity as a function of shear rate) and the fitting curve with the prediction interval for the fluid used in Experiments 8–10. A discussion on the methodology adopted for assessing the rheological parameters of the fluids is reported in Longo et al. (Reference Longo, Di Federico, Archetti, Chiapponi, Ciriello and Ungarish2013a ). The mass density was measured by a hydrometer (STV3500/23 Salmoiraghi) with an accuracy of 0.5 %.

Two different Hele-Shaw cells were used: the first had a constant gap of 3 mm or 3.2 mm; the second cell was V-shaped; the linearly varying gap reproduced a porous medium with a cubic variation of permeability and a linear variation of porosity along the vertical ( ${\it\omega}=4,{\it\gamma}=2$ ). Two vertical lock gates were used to confine the liquid. The lock gates were removed fast and the liquid spread along the cell and drained from the open end of the cell. The geometry of the current at different times was recorded by image analysis of frames recorded by a high-resolution video camera (1980 $\times$ 1080 pixels) recording at 25 frames $\text{s}^{-1}$ and with a spatial resolution of 3 pixel $\text{mm}^{-1}$ . The images were processed with a proprietary software in order to obtain a planar restitution and to extract the coordinates of the boundary between the dark current and the white background. The overall uncertainty was of $\pm 1~\text{mm}$ .

The parameters for all experiments are listed in table 1. Two experiments (no. 2 and no. 7) were carried out with a Newtonian fluid to calibrate the two cells. The front position as function of the time for all experiments is shown in figure 6(a). The relevant differences between theory and experiment during the initial stage of propagation were mainly due to the finite size of the mound at $t=0$ ; the slumping phase is relatively short due to the rapid dominance of viscous effects over inertial ones. At intermediate times, experimental data collapse onto the theoretical line. At long times, the current becomes thinner and the effect of the friction at the bottom of the cell becomes non-negligible; this is reflected in a reduction of the speed of the current, with a consequent overestimation of the theoretical speed. For most tests, the intermediate times are a multiple of the time scale (the time scale is generally less than 0.5 second), hence self-similarity develops after a few seconds from the opening of the locks. The long times depend on the factors determining the discrepancies between the model and the experiment. For the present tests, long times mean several tens of seconds.

For a more detailed analysis of the uncertainties in the results of the model as a function of the uncertainties in the parameters, the methodology described in Di Federico et al. (Reference Di Federico, Longo, Chiapponi, Archetti and Ciriello2014) can be applied. Figure 6(b) shows the dipole moment as function of time for three selected experiments (numbers 8, 9, and 10), characterised by initial aspect ratios (ratio of the initial height to initial length of the fluid behind the lock gates) equal to $0.8,1.5$ and $0.72$ , respectively. After an initial slumping phase, the moment tends to a constant, as predicted by the theory. The moment was estimated by fitting the experimental points representing the boundary of the current (detected with the video image analysis) and then integrating the fitting function. The uncertainty in the dipole moment has been evaluated with a Monte Carlo simulation, by (i) generating several samples of points ( $10\,000$ samples) representing the boundary, by assuming a Gaussian distribution with variance equal to 1 mm around the experimental value; (ii) fitting the synthetic samples and integrating the fitting curves; and (iii) computing the variance of the samples of the results of the integration. A comparison of the experimental profile with theory is shown in figure 7(a). Near the origin, the height of the current is non-zero, as predicted by the theory. Near the front, capillarity effects not included in the theory produce an underestimation of the height and of the front position. Elsewhere, the agreement between theory and experiments is remarkably good at intermediate and late times, indicating that the current converges to the self-similar regime. A balance between two counteracting effects not included in the model, i.e. the depression of the current associated with the vertical velocity and capillarity, probably contributes to the excellent fit.

Table 1.

Experimental parameters for all Experiments in Hele-Shaw cells. The last three Experiments are in a V shaped Hele-Shaw cell, reproducing flow in a porous medium with a cubic variation of permeability and a linear variation of porosity along the vertical. A power of the dipole moment (dimensions [ $L^{{\it\gamma}+2}$ ]) is listed. The nominal value of the dipole moment $Q_{0}$ was computed from the initial state at rest, by digitizing the images before opening the locks.

Figure 6.

(a) Experimental front positions (symbols) and theoretical predictions (solid lines) $X_{N}$ against time $T$ . The variables $X_{N}$ and $T$ are non-dimensional and $X_{N}$ is scaled according to $f(X_{N})=(X_{N}/{\it\eta}_{N})^{1/[{\it\gamma}n/({\it\omega}(n+1)+2)]}$ . The data for experiments in the V-shaped cell are multiplied by a factor 2 in order to be separated from the data for the constant gap cell. For clarity, one point out of three is plotted. (b) Experimental dipole moment for three experiments with aspect ratio $0.8,1.5$ , and $0.72$ , respectively. $Q_{0}$ is the nominal dipole moment at time $t=0$ . The error bar shown indicates a single standard deviation.

Figure 7.

Experiment no. 16 (a) Photographs showing the current profiles at various times since release (10, 30, 50, 70 and 90 s) in a V-shaped Hele-Shaw cell. The curves with symbols represent the theoretical profile. The vertical dashed lines are 10 cm apart. (b) The dimensionless volume of fluid drained at the origin $V_{out}$ (defined by (3.10)) plotted against $T^{-F_{2}}\equiv T^{-0.119}$ (filled symbols) and the dimensionless (scaled with $x_{\ast }$ ) height of the current in the origin plotted against $T^{F_{3}}\equiv T^{-F_{2}}$ (empty symbols). The time increases toward the origin. The error bars indicate a single standard deviation.

Finally, we investigate the current behaviour at the origin by plotting the volume of fluid drained $V_{out}$ , and the height $H_{0}$ against time for Experiment no. 16 (figure 7 b). Theoretically, $V_{out}$ and $H_{0}$ are proportional to $T^{F_{2}}$ and $T^{-F_{3}}$ respectively. In the selected case, $F_{3}=-F_{2}$ as ${\it\gamma}=2$ , hence the same exponent is expected experimentally for the two plots except for the sign. Inspection of figure 7(b) confirms the theory for intermediate and late times. Similar plots are obtained for the other experiments.