axisymmetric flows from fluid injection into a confined

PHYSICS OF FLUIDS 28, 022107 (2016)

Axisymmetric flows from fluid injection into a confinedporous medium

Bo Guo,1,a) Zhong Zheng,2,a) Michael A. Celia,1 and Howard A. Stone2,b)1Department of Civil and Environmental Engineering, Princeton University,Princeton, New Jersey 08544, USA2Department of Mechanical and Aerospace Engineering, Princeton University,Princeton, New Jersey 08544, USA

(Received 22 June 2015; accepted 20 January 2016; published online 25 February 2016)

We study the axisymmetric flows generated from fluid injection into a horizontalconfined porous medium that is originally saturated with another fluid of differentdensity and viscosity. Neglecting the effects of surface tension and fluid mixing, weuse the lubrication approximation to obtain a nonlinear advection-diffusion equationthat describes the time evolution of the sharp fluid-fluid interface. The flow behaviorsare controlled by two dimensionless groups: M , the viscosity ratio of displacedfluid relative to injected fluid, and Γ, which measures the relative importance ofbuoyancy and fluid injection. For this axisymmetric geometry, the similarity solu-tion involving R2/T (where R is the dimensionless radial coordinate and T is thedimensionless time) is an exact solution to the nonlinear governing equation forall times. Four analytical expressions are identified as asymptotic approximations(two of which are new solutions): (i) injection-driven flow with the injected fluidbeing more viscous than the displaced fluid (Γ ≪ 1 and M < 1) where we identifya self-similar solution that indicates a parabolic interface shape; (ii) injection-drivenflow with injected and displaced fluids of equal viscosity (Γ ≪ 1 and M = 1), wherewe find a self-similar solution that predicts a distinct parabolic interface shape; (iii)injection-driven flow with a less viscous injected fluid (Γ ≪ 1 and M > 1) for whichthere is a rarefaction wave solution, assuming that the Saffman-Taylor instability doesnot occur at the reservoir scale; and (iv) buoyancy-driven flow (Γ ≫ 1) for whichthere is a well-known self-similar solution corresponding to gravity currents in anunconfined porous medium [S. Lyle et al. “Axisymmetric gravity currents in a porousmedium,” J. Fluid Mech. 543, 293–302 (2005)]. The various axisymmetric flows aresummarized in a Γ-M regime diagram with five distinct dynamic behaviors includingthe four asymptotic regimes and an intermediate regime. The implications of theregime diagram are discussed using practical engineering projects of geological CO2sequestration, enhanced oil recovery, and underground waste disposal. C 2016 AIPPublishing LLC. [http://dx.doi.org/10.1063/1.4941400]

I. INTRODUCTION

The propagation of viscous gravity currents occurs in a variety of geophysical and indus-trial contexts, for example, the propagation of magma, geological CO2 sequestration, undergroundthermal energy storage, enhanced oil recovery, and underground waste disposal.2–6 There havebeen numerous studies on the flow of viscous gravity currents with geometries and driving forcesinspired by these applications. It is common to simplify the system by assuming that the injectedand displaced fluids are segregated due to strong buoyancy effects and reach equilibrium in the ver-tical direction. Here, we invoke this vertical equilibrium assumption and focus on the axisymmetric

a)B. Guo and Z. Zheng contributed equally to this work.b)Electronic mail: [email protected].

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022107-2 Guo et al. Phys. Fluids 28, 022107 (2016)

TABLE I. Summary of axisymmetric viscous gravity currents in unconfined and confined geometries. The front propagationlaws are also included in this table with r representing the length of the current and t representing time.

Flow conditions Front propagation laws Experiments References

In atmosphere, unconfined,outward flow

r ∝ t1/8 (constant volume) Yes Huppert,7

r ∝ t1/2 (constant injection) Yes Didden and Maxworthy8

In atmosphere, unconfined,outward flow

r ∝ t1/2 (at early times)r ∝ t1/4 (at late times) Yes Zheng, Griffiths, and Stone9

(Over elastic substrate)

In atmosphere, unconfined,inward flow

r ∝ (tc− t)0.762. . .a Yes Gratton and Minotti,10

(Second kind self-similarity) Diez, Gratton, and Gratton11

In atmosphere, unconfined,inward flow

r ∝ (τc−τ)0.762. . .b Yes Zheng, Shin, and Stone12

(Second kind self-similarity)(With slow vertical drainage)

Porous media, unconfined,outward flow

r ∝ t1/4 (constant volume) No Lyle et al.1

r ∝ t1/2 (constant injection) Yes

Porous media, unconfined,inward flow

(Second kind self-similarity) No Zheng, Christov, and Stone13


r ∝ t1/2 (constant injection) No Golding, Huppert, and Neufeld14

(Two-phase gravity currents)


r → finite maximum length Yes Pritchard, Woods, and Hogg,15

(With slow vertical drainage) Spannuth et al.16

Porous media, confined,outward flow

r ∝ t1/2 (constant injection) No Nordbotten and Celia17

(Rarefaction wave solution)

Porous media, confined,outward flow

r ∝ t1/2 (constant injection) No Current paper(Five distinct flow regimes)

aFor an inward spreading viscous gravity current, a second-kind self-similar solution is obtained with tc denoting the time forthe front to reach the origin; the scaling exponent is an irrational number obtained by solving a nonlinear eigenvalue problem.bWith vertical fluid drainage, a mathematical transform from t to τ is introduced.12,15,18

propagation that is of practical interest, especially in reservoir engineering where various fluidsare frequently injected underground through vertical wells; approximately, axisymmetric flows areexpected to be generated following fluid injection. A summary of previous studies on axisymmetricflows driven by buoyancy and/or fluid injection is provided in Table I, including characterization ofspreading rates in terms of similarity solutions of the first kind, where spreading occurs outward,and the second kind, where spreading occurs inward.

When a viscous dense gravity current spreads axisymmetrically above a horizontal imperme-able substrate, it is common to analyze the flow using the lubrication approximation, which leads toa nonlinear diffusion equation that describes the time evolution of the free interface.7,8 Self-similarsolutions have been obtained to describe the fluid flow subject to both constant total volume andconstant injection rate, and laboratory experiments have been conducted to verify the applicabilityof the self-similar solutions. Axisymmetric viscous gravity currents spreading over an elastic mem-brane have also been studied; a transition from an early-time similarity to a late-time similaritywas identified to characterize the spreading dynamics.9 An analogous situation is the axisymmetricspreading of a gravity current in an unconfined porous medium, and self-similar solutions havebeen obtained and verified using experiments in a porous medium of packed beads.1 A two-phasemodel has also been proposed to describe the axisymmetric spreading of a viscous gravity cur-rent,14 including the effect of interfacial tension. In addition, a second-kind self-similar solution hasbeen obtained for the inward spreading of an axisymmetric gravity current,10,11 where the scalingexponents are determined by solving a nonlinear eigenvalue problem.3,19

When the substrate is permeable, vertical drainage occurs during the horizontal spreading ofan axisymmetric gravity current. In this situation, the horizontal spreading of a gravity currentapproaches a steady state under constant fluid injection and the fluid-fluid interface reaches a finite


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FIG. 1. Axisymmetric cross section of fluid injection into a horizontal, confined porous medium uniform of permeability k

and porosity φ initially saturated with another fluid of different density and viscosity. The injection point is at the top leftcorner. This figure shows the case of injected fluid being less dense than the displaced fluid, i.e., ∆ρ = ρd−ρi > 0.

maximum horizontal extent; without fluid injection (constant total volume), the length of a gravitycurrent first reaches a maximum value and eventually decreases because of fluid loss through thepermeable substrate.15,16 For inward spreading of gravity currents on thin permeable substrates, asecond-kind self-similar solution can be obtained by introducing a mathematical transform to mapthe problem to the analogous flow situation without fluid drainage.12 If the horizontal substrate issubjected to sites of local drainage, for example, a fault, the gravity current no longer spreads in anaxisymmetric pattern because of the localized fluid loss.20,21 Breaking of symmetry can also occur ifa gravity current spreads above a substrate that is tilted from the horizontal direction.22,23

For confined geometries, there are fewer studies on the propagation of axisymmetric gravitycurrents from fluid injection, although there have been many studies in confined Cartesian geome-tries.2,5,24–36 For an axisymmetric flow, considering the movement of the displaced fluid that initiallysaturates the porous medium, a nonlinear advection-diffusion equation describes the dynamics ofthe fluid-fluid interface under constant fluid injection, and a rarefaction wave solution has beenobtained when the injected fluid is less viscous than the displaced fluid.17,37 In this paper, we extendthe previous work of Nordbotten and Celia17 on the axisymmetric flow from constant fluid injectioninto a confined porous medium and provide new approximate analytical solutions when the injectedfluid is equally viscous or more viscous, compared with the displaced fluid.

In structuring this paper, we begin in Section II with the derivation, in cylindrical coordinates,of the nonlinear advection-diffusion equation that describes the time evolution of the axisym-metric fluid-fluid interface. In Section III, we study the injection-dominated regimes and showthree different analytical expressions for the spreading dynamics, which depend on the viscosityratio of the injected and displaced fluids. In Section IV, we study the buoyancy-dominated regime,which recovers the flow behaviors in an unconfined porous medium. We summarize in Section Vour major findings in a regime diagram of five distinct flow behaviors that link the well-knownanalytical results from previous studies1,17 and new analytical solutions from the current study. Weinvestigate in Section VI the transition process to develop the various self-similar solutions, and wediscuss the implications for practical engineering applications in Section VII. The paper concludesin Section VIII with a summary of major findings and more discussion of the model assumptions.

II. THEORETICAL MODEL

We study the axisymmetric flow as fluid injection occurs in a horizontal confined porous me-dium of thickness h0, where both the upper and lower boundaries are impermeable, see Figure 1.The porous medium is assumed to be isotropic and homogeneous with constant permeability k andporosity φ. Also, the flow is assumed to be axisymmetric; hence, we use cylindrical coordinates(r, z) with r = 0 corresponding to the location of the injection well, and we assume that the radiusof the well is negligibly small. We assume that the two fluids are immiscible and we neglect theeffect of interfacial tension. Then, the two fluid phases are separated by a sharp interface and wedenote the thickness of the injected fluid by h(r, t). The injected fluid has density ρi and viscosityµi; the displaced fluid has density ρd and viscosity µd. Without loss of generality, we assume that


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the injected fluid is less dense than the displaced fluid, and we define the fluid density difference as∆ρ = ρd − ρi > 0. A schematic of the flow system is shown in Figure 1.

When the aspect ratio (height relative to length) of the fluid-fluid interface is small, the flow ismainly horizontal, and it is natural to analyze the flow using the lubrication approximation. Then,the vertical velocity is negligible, and the fluid pressure exhibits a hydrostatic distribution,

pi(r, z, t) = p0(r, t) + ρigz, for 0 ≤ z ≤ h(r, t), (1a)pd(r, z, t) = p0(r, t) + ρigh(r, t) + ρdg(z − h(r, t)), for h(r, t) < z ≤ h0, (1b)

where p0(r, t) is the pressure at the top of the porous medium, and z is positive downward.The horizontal (radial) velocities in the two fluids can be calculated using Darcy’s law,

ui = −kµi

∂pi∂r

(2a)

and

ud = −kµd

∂pd

∂r. (2b)

Also, the local continuity equations are given by

φ∂h∂t+

1r∂(rhui)∂r

= 0, (3a)

φ∂(h0 − h)

∂t+

1r∂(r(h0 − h)ud)

∂r= 0, (3b)

which can be summed and integrated radially to obtain

hui + (h0 − h)ud =q

2πr, (4)

where q is the volumetric injection rate through the injection well. In this paper, we only considerthe case of constant fluid injection, i.e., q is a constant.

In addition, the global mass constraint then requires

2πφ rN1(t)

0rh(r, t)dr = qt, (5)

where rN1(t) denotes the furthest radial extent of the front (see Figure 1). Using Equation (3), it canbe shown that Equation (5) is equivalent to Equation (4).

Combining Equations (1)–(4), we obtain a nonlinear advection-diffusion equation that governsthe space-time evolution of the fluid-fluid interface h(r, t),

∂h∂t=

1φr

∂

∂r

(∆ρgkh(h0 − h)rµdh + µi(h0 − h)

∂h∂r+

qµi(h0 − h)2π(µdh + µi(h0 − h))

). (6)

We note that a similar equation has been derived by Nordbotten and Celia,17 who also consideredthe effect of constant residual saturation. The analogous equation for the case of constant fluidinjection in Cartesian geometries was presented and analyzed by Pegler, Huppert, and Neufeld34

and Zheng et al.35

Appropriate boundary and initial conditions are needed to complete the problem statement. Weassume that fluid injection begins at t = 0 at the origin, and prior to that the domain is filled with theresident (dense) fluid. The initial condition for Equation (6) is then given by

h(r,0) = 0. (7)

The boundary condition at the front r = rN1(t) of the interface requires

h(rN1(t), t) = 0. (8)

We can obtain a second boundary condition at r = 0, assuming that the radius of the injection wellis negligibly small. To do this, we first multiply r on both sides of Equation (6) and integrate fromr = 0 to r = rN1(t),


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rN1(t)

0r∂h∂t

dr =1φ

∆ρgkh(h0 − h)rµdh + µi(h0 − h)

∂h∂r+


��

rN1(t)

0. (9)

Then, we multiply both sides of Equation (9) by 2πφ and substitute in global mass constraint (5).Also, we assume that there is no fluid entrainment at the front of the interface when we evaluate theterm on the right-hand side as r → rN1(t), i.e., h ∂h

∂r

�r→ rN1(t)

= 0. Then, we obtain

∆ρgkh(h0 − h)rµdh + µi(h0 − h)

∂h∂r+


��r=0=

q2π

. (10)

The complete problem statement is then to solve Equation (6) subject to initial condition (7) andboundary conditions (8) and (10) and so obtain h(r, t) and the front location rN1(t).

A. Non-dimensionalization

We can rescale Equation (6) by defining dimensionless variables H ≡ h/h0, R ≡ r/h0, andT ≡ t/(2πφh3

0/q). Then, Equation (6) in dimensionless form is

∂H∂T=

1R

∂

∂R

(ΓMH(1 − H)R1 + (M − 1)H

∂H∂R+

1 − H1 + (M − 1)H

), (11)

where M and Γ are two dimensionless groups defined as

M ≡ µd

µi(12a)

and

Γ ≡2π∆ρgkh2

0

µdq. (12b)

By definition, M is the viscosity ratio of the displaced fluid relative to the injected fluid, and Γmeasures the importance of buoyancy effects relative to fluid injection in driving the axisymmetricflow.

In addition, the dimensionless global mass conservation equation becomes RN1(T )

0RH(R,T) dR = T, (13)

where RN1(T) ≡ rN1(t)/h0 denotes the maximum radial extent of the front. The initial and boundaryconditions can also be rewritten in dimensionless form,

H(R,0) = 0, (14a)H (RN1(T),T) = 0, (14b)

ΓMH(1 − H)R1 + (M − 1)H

∂H∂R+

1 − H1 + (M − 1)H

��R=0= 1. (14c)

Equation (11) with initial and boundary conditions (14) can be solved numerically to provide thetime evolution of the fluid-fluid interface. We have implemented a central-difference scheme tosolve this problem,35,38 see also Appendix A for more details. From the numerical results, wenote that there may be a second radial location RN2(t) ≡ rN2(t)/h0, where for all R ≤ RN2(t),H(RN2(t)) = 1 (see sketch in Figure 1). Typical numerical solutions are provided in Figure 2. Thenumerical solutions motivate us to seek different approximate solutions in asymptotic limits basedon the two dimensionless groups M and Γ, as defined in (12). We have also plotted in Figure 2 theanalytical solutions obtained in Sections III and IV, and we have observed very good agreementbetween the analytical and the numerical solutions. These approximate analytical solutions willbe discussed below in more detail. We note that the fluid-fluid interfaces in Figures 2(a) and 2(c)have a large slope, where the lubrication approximation can fail. Nevertheless, we will show (insubsections A and B of Section III) that the slope decays with time as 1/

√T , so the lubrication

approximation is valid at late times.


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FIG. 2. Typical numerical solutions of the time evolution of the fluid-fluid interface, from solving Equation (11) subject toboundary and initial conditions (14), at different times T = 1,10 for different values of Γ and M . We chose Γ = 1/10,100and M = 1/2,1,5 to demonstrate the different types of solutions. (a), (c), and (e) Γ = 1/10 with M = 1/2,1,5, respectively.(b), (d), and (f) Γ = 100 with M = 1/2,1,5, respectively. The analytical solutions derived in Sections III and IV are alsoshown: see Section III A for (a), Section III B for (c), Section III C for (e), and Section IV for (b), (d), and (f). Note that, forsimplicity, black solid curves are used for analytical solutions at both T = 1 and T = 10 without confusion.

B. Similarity transform

Let us now define a similarity variable η = R2/T . Then, Equation (11) can be rewritten in termsof η only,

ηdHdη+ 2

ddη

(2ΓMH(1 − H)η

1 + (M − 1)HdHdη+

1 − H1 + (M − 1)H

)= 0. (15)

We can further obtain two boundary conditions at the propagating front RN1(T),H�ηN1

�= 0, (16a)

dHdη

��η→η−N1

=1

2ηN1Γ− 1

4ΓM, (16b)

where the constant ηN1 ≡ R2N1(T)/T represents the location of the propagating front in the (η,H)

space. From this point forward, whenever we use the term “front,” it means the frontal tips at thetop or bottom boundaries. Boundary condition (16b) is derived by evaluating Equation (15) forη → η−N1

, where H(ηN1) = 0 (see Appendix B for more details).A shooting scheme is used to solve Equation (15) subject to boundary conditions (16a) and

(16b). In order to determine ηN1, an iterative procedure is needed such that the global mass conser-vation condition is satisfied, ηN1

0H(η) dη = 2. (17)


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As mentioned in Section II A, the fluid-fluid interface may intersect with both boundaries andcontains a fast front RN1(T) with H (RN1(T),T) = 0 and a slow front RN2(T) with H (RN1(T),T) = 1(see sketch in Figure 1 and numerical results in Figure 2). Equivalently, the numerical shootingprocedure here may generate a second front in the (η,H) space, which corresponds to ηN2 = R2

N2/Twith H

�ηN2

�= 1. The initial condition is not necessary since we are looking for a self-similar

solution, which represents the intermediate asymptotic behavior with the initial condition eventuallyforgotten.3

We note that Equation (15) is analogous to Equation (11) in the paper of Nordbotten andCelia.17 In addition, for the limit of Γ ≪ 1, i.e., when fluid injection effect is much more importantthan buoyancy effect in driving the fluid flow, by neglecting the second-order term in Equation (15),Nordbotten and Celia17 obtained a rarefaction wave solution for M > 1. In this paper, we providetwo additional approximate solutions in Section III for M = 1 and M < 1, respectively, when theflow is mainly driven by injection (Γ ≪ 1). In Section IV, we show that when Γ ≫ 1, i.e., whenthe flow is mainly driven by buoyancy, for the majority part of the fluid-fluid interface awayfrom the narrow region near the injection site, the well-known nonlinear diffusion equation thatdescribes the propagation of a gravity current in an unconfined porous medium can be used todescribe the flow behavior.1 We also numerically calculate the boundaries when each analyticalapproximation provides good estimates and summarize the flow behaviors in a Γ-M regime dia-gram in Section V.

III. INJECTION-DRIVEN REGIMES: Γ ≪ 1

When Γ ≪ 1, the fluid flow is mainly driven by injection. In this case, the flow is confined,and the fluid-fluid interface intersects with both the top and bottom boundaries. In Sections III A–III C, we present three analytical solutions for M < 1, M = 1, and M > 1, respectively, in the limitof Γ ≪ 1. Physically, these three approximate solutions correspond to the injection-driven flowswith less viscous, equally viscous, and more viscous displaced fluids, respectively, compared withthe injected fluid.

A. Less viscous displaced fluid: M < 1

When M < 1, the injected fluid is more viscous than the displaced fluid. When Γ ≪ 1, numer-ical simulations of Equation (15) subject to boundary conditions (16) and global mass constraint(17) indicate that the fluid-fluid interface in the (η,H) space appears as a straight line with a slopedepending on the value of M . We first note that if the fluid-fluid interface propagates as a shock(vertical interface), the interface location corresponds to η = 2 in the (η,H) space, according toglobal mass constraint (17). Motivated by the numerical observation, we now define a new variableζ ≡ (η − 2)/Γα, substitute it into Equation (15), and we obtain

(Γαζ + 2)dHdζ+ 2

ddζ

(2M(Γαζ + 2)Γ1−αH(1 − H)

1 + (M − 1)HdHdζ+

1 − H1 + (M − 1)H

)= 0. (18)

To balance the terms associated with diffusion and advection, i.e., the two terms within the bracketof the differentiation in (18), we need α = 1. Neglecting the O(Γ) terms in Equation (18), we obtain

dHdζ+

ddζ

(4MH(1 − H)1 + (M − 1)H

dHdζ+

1 − H1 + (M − 1)H

)= 0. (19)

Since the flow is confined, we have two front conditions,

H�ζN1

�= 0 (20a)

and

H�ζN2

�= 1, (20b)


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where ζN1 ≡ (ηN1 − 2)/Γ and ζN2 ≡ (ηN2 − 2)/Γ ≡ (R2N2(T)/T − 2)/Γ, which refer to the intersec-

tions with the top and bottom boundaries, respectively (e.g., see Figure 1). Global mass conserva-tion equation (17) in this case can be rewritten as

ζN2 +

ζN1

ζN2

H(ζ) dζ = 0. (21)

We first integrate Equation (19) once and obtain

H +4MH(1 − H)1 + (M − 1)H

dHdζ+

1 − H1 + (M − 1)H = 1, (22)

where the integration constant is obtained from front conditions (20). Equation (22) can be reorga-nized as

(1 − H)H(

dHdζ− M − 1

4M

)= 0, (23)

which immediately suggests that H = 0, H = 1, or H has a linear structure,

dHdζ=

M − 14M

. (24)

Integrating Equation (24) once more, we obtain

H(ζ) = (M − 1)4M

ζ +12, (25)

where the integration constant is chosen such that global mass conservation equation (21) is satis-fied. The locations of the two fronts are also determined,

ζN1 =2M

1 − M(26a)

and

ζN2 = −2M

1 − M, (26b)

or equivalently,

ηN1 = 2 +2M

1 − MΓ (27a)

and

ηN2 = 2 − 2M1 − M

Γ. (27b)

Thus, we have obtained an approximate solution for M < 1 and Γ ≪ 1,

H(η) =

1, 0 ≤ η ≤ 2 − 2M1 − M

Γ,

M − 14MΓ

(η − 2) + 12, 2 − 2M

1 − MΓ < η ≤ 2 +

2M1 − M

Γ,

0, η > 2 +2M

1 − MΓ.

(28)

The fluid-fluid interface has a linear structure in the (η,H) space and a parabolic structure in the(R,H) space for a fixed time T . We note that the slope of the fluid-fluid interface is constant inthe (η,H) space and decays with time as 1/

√T in the (R,H) space, which supports the lubri-

cation approximation at late times. To verify this solution, we numerically solve Equation (15)subject to boundary conditions (16) and global mass constraint (17), and we compare this numericalsolution with analytical solution (28). The comparison is plotted in Figure 3 with M = 1/2 andΓ = 0.01,0.05,0.1 as examples, and we have observed very good agreement.


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FIG. 3. A comparison of the numerical solutions of Equation (15) subject to boundary conditions (16) and global massconstraint (17) and the analytical solution (28) for Γ = 0.01,0.05,0.1 and M = 1/2. (a) H as a function of η =R2/T . Notethat, for simplicity, black solid lines are used for analytical solutions for all Γ values without confusion. (b) H as a functionof the rescaled variable ζ ≡ (R2/T −2)/Γ.

B. Equally viscous displaced fluid: M = 1

When M = 1, the injected and displaced fluids are equally viscous. Again, we note that if thefluid-fluid interface propagates as a vertical interface (shock), the location of this interface corre-sponds to η = 2 in the (η,H) space, according to global mass constraint (17). Motivated by ournumerical results, we now define ζe ≡ (η − 2)/Γβ and substitute it into Equation (15) with M = 1 toobtain

ζeΓβ dH

dζe+ 4

ddζe

(Γ1−β(1 − H)H �

Γβζe + 2� dH

dζe

)= 0. (29)

The first term in (29) corresponds to both the unsteady and advective terms in partial differentialequation (11), while the second term is related to the diffusive term in (11). We note that, whenΓ ≪ 1, β = 1/2 is required for a balance of the advective and diffusive terms. Then, we obtain

ζedHdζe+ 4

ddζe

((1 − H)H �

Γ1/2ζe + 2� dH

dζe

)= 0. (30)

Next, we neglect the O(Γ1/2) terms in (30) and obtain

ζedHdζe+ 8

ddζe

((1 − H)H dH

dζe

)= 0. (31)

The front conditions are analogous to (20),

H(ζeN1

)= 0, (32a)

H(ζeN2

)= 1, (32b)


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with ζeN1≡ (ηN1 − 2)/Γ1/2 and ζeN2

≡ (ηN2 − 2)/Γ1/2 representing the locations of the fast andslow fronts. Global mass conservation equation (17) has the same form as (21),

ζeN2+

ζeN1

ζeN2

H(ζe)dζe = 0. (33)

Equation (31) has an analytical solution, which satisfies both front conditions (32) and global massconservation equation (33),

H = −14ζe +

12. (34)

The locations of the fronts are also determined from (32) and (34),

ζeN1= 2 (35a)

and

ζeN2= −2, (35b)

or equivalently,

ηN1 = 2 + 2√Γ (36a)

and

ηN2 = 2 − 2√Γ. (36b)

To summarize, we have obtained an approximate solution for M = 1 and Γ ≪ 1,

H(η) =

1, 0 ≤ η ≤ 2 − 2√Γ,

−14η − 2√Γ+

12, 2 − 2

√Γ < η ≤ 2 + 2

√Γ,

0, η > 2 + 2√Γ.

(37)

Solution (37) displays a very good agreement with the numerical solution to Equation (15) subjectto boundary conditions (16) and global mass constraint (17) with Γ ≪ 1 and M = 1, as shown inFigure 4. We have chosen Γ = 0.01,0.05,0.1 as examples. The fluid-fluid interface has a linearstructure in the (η,H) space and a parabolic shape in the (R,H) space for a fixed time T . The slopeof the fluid-fluid interface decays with time as 1/

√T in the (R,H) space; thus, the validity of the

lubrication approximation is supported at late times.

C. More viscous displaced fluid: M > 1

When M > 1, the displaced fluid is more viscous than the injected fluid. Similar to the anal-ogous problem in Cartesian geometries,34,35 when Γ ≪ 1, we neglect the second-order derivativeterm in Equation (15) that is associated with buoyancy. Then, Equation (15) is reduced to

ηdHdη+ 2

ddη

(1 − H

1 + (M − 1)H)= 0. (38)

The solution to (38) should satisfy both global mass constraint (17) and the front conditions,

H�ηN1

�= 0, (39a)

H�ηN2

�= 1. (39b)

We can obtain an analytical solution (a rarefaction solution),

H(η) =

1, 0 ≤ η ≤ 2/M,(2M/η − 1

)/(M − 1), 2/M < η ≤ 2M,

0, η > 2M,

(40)


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FIG. 4. A comparison of numerical solutions of Equation (15) subject to (16) and (17) and analytical solution (37) forΓ = 0.01,0.05,0.1 and M = 1. (a) H as a function of η =R2/T . Note that, for simplicity, black solid lines are used foranalytical solutions for all Γ values without confusion. (b) H as a function of the rescaled variable ζe ≡ (R2/T −2)/Γ1/2.

which has been reported previously,17 and the locations of the fronts are

ηN1 = 2M, (41a)

ηN2 = 2/M. (41b)

We can compare rarefaction solution (40) with the numerical solution to Equation (15) subject toboundary conditions (16) and global mass constraint (17). We observe good agreement when Γ issmall, as shown in Figure 5, with Γ = 0.01,0.05,0.1 and M = 5 as examples. In addition, the mostsignificant deviation is found near the bottom boundary, where the buoyancy effect is the strongestalong the fluid-fluid interface. Similar behaviors have also been reported in the analogous problemin Cartesian coordinates.34,35

IV. BUOYANCY-DRIVEN REGIME: Γ ≫ 1

When Γ ≫ 1, numerical simulation indicates that for the majority part of the interface (exceptthe narrow region near the origin), the thickness of the interface is small compared with thethickness of the porous medium (see, e.g., Figures 2(b), 2(d), and 2(f)). Thus, the flow is mainlyunconfined, buoyancy is the main driving force for the propagation of the injected fluid, and weneglect the advective term in Equation (11) that is associated with fluid injection. In addition, sincethe thickness is small, both H ≪ 1 and |(M − 1)H | ≪ 1 hold for the majority part of the interface.Then, Equation (11) reduces to

∂H∂T=

ΓMR

∂

∂R

(RH

∂H∂R

), (42)


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FIG. 5. A comparison of the numerical solutions of Equation (15) subject to boundary conditions (16) and global massconstraint (17) and analytical solution (40) for Γ = 0.01,0.05,0.1 and M = 5.

which is the well-known nonlinear diffusion equation for the propagation of a gravity current in anunconfined porous medium.1,2 Together with global mass conservation equation (5), a self-similarsolution can be obtained for the time evolution of the fluid-fluid interface that is away from thenarrow region near the injection site.

As discussed in Section II, we define the similarity variable as η ≡ R2/T . Then, Equation (42)can be transformed to

ηdHdη+ 4ΓM

ddη

(ηH

dHdη

)= 0. (43)

Note that Equation (43) can also be derived from Equation (15). We further define y ≡ η/ηN1 andf ≡ ΓMH/ηN1. Then, Equation (42) can be rewritten as

4(y f f ′)′ + y f ′ = 0. (44)

Two boundary conditions are needed to solve (44), which can be derived from Equations (16a) and(16b),

f |y→1− = 0, (45a)

f ′|y→1− = −1/4. (45b)

In addition, global mass conservation equation (13) can be used to calculate the location of the fastfront ηN1,

ηN1 =

(2ΓM

1

0f (y)dy

) 12

. (46)

Note that we have neglected the influence of the slow front along the bottom boundary since it isvery close to the origin, and the effect on the location of the fast front is negligibly small. For moredetails about this approximation, see Appendix C.

Equation (44) can be solved numerically with boundary conditions (45a) and (45b), which isshown in Figure 6(b); then, Equation (46) can be used to find the location of the front ηN1, whichintersects with the top boundary. We can also solve nonlinear advection-diffusion equation (15)numerically subject to boundary conditions (16) and global mass constraint (17). A comparison ofthis numerical solution and the self-similar solution by solving Equation (44) with (45a) and (45b)is provided in Figure 6. We observe very good agreement between the solutions for large valuesof Γ, which confirms that for the majority part of the fluid-fluid interface that is away from theinjection site, the effect of the slow front is negligible, and the assumption of unconfined flows isappropriate.


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FIG. 6. A comparison of the solution to Equation (15) subject to boundary conditions (16) and global mass constraint (17)and the solution to (44) subject to (45). (a) H as a function of R2/T . Note that, for simplicity, black solid curves are usedfor analytical solutions for all Γ values without confusion. (b) f ≡ ΓMH/ηN1 as a function of y ≡η/ηN1. We have chosenΓ = 10,100,500 and M = 1 as examples.

V. INJECTION REGIMES

A. Injection regimes

Five distinct self-similar flow regimes have been identified and summarized in a phase-typediagram, as shown in Figure 7, with respect to two dimensionless groups Γ and M . Recall that Mdenotes the viscosity ratio of the displaced to the injected fluids, and Γ measures the importanceof buoyancy relative to fluid injection effects. The five regimes include four regimes representingthe different asymptotic limits and an intermediate regime. The boundaries between each individualregime are calculated based on a 10% difference of the top front location between the predictionof each asymptotic solution and the prediction of the direct numerical solution to Equation (15), asdiscussed in Section II B. The five individual flow regimes are as follows:

• Regime I (Γ ≪ 1 and M < 1): The axisymmetric flow is mainly driven by fluid injection withthe injected fluid being more viscous than the displaced fluid. A new approximate solution (28)is identified in this regime, which represents a parabolic shape for the fluid-fluid interface, asdiscussed in Section III A. The interface intersects with both the top and bottom boundaries,and the intersections represent the horizontal extent of the interface.

• Regime II (Γ ≪ 1 and M = 1): The flow is mainly driven by fluid injection with the injectedand displaced fluids having the same viscosity. Another new approximate analytical solution(37) is obtained in this regime, as discussed in Section III B. Solution (37) represents a para-bolic shape for the fluid-fluid interface, which contains intersections with both the top andbottom boundaries.

• Regime III (Γ ≪ 1 and M > 1): The flow is mainly driven by fluid injection with the injectedfluid less viscous than the displaced fluid. A rarefaction solution (40) can be used to describe


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FIG. 7. Regime diagram for the axisymmetric flows generated by fluid injection into a horizontal confined porous mediuminitially saturated with another immiscible fluid of different density and viscosity. Five distinct self-similar regimes,summarized in Section V A, are identified based on two dimensionless control parameters: Γ, representing the relativeimportance of buoyancy effect over injection effect, and M , the viscosity ratio of the displaced to the injected fluids. In regimeI, approximate analytical solution (28) is obtained that indicates a parabolic interface shape. In regime II, distinct approximateanalytical solution (37) is obtained to characterize the parabolic interface shape. In regime III, rarefaction solution (40) isobtained to describe the dynamics of the fluid-fluid interface that attaches to both boundaries. In regime IV, buoyancy is themajor driving force; the majority of the fluid-fluid interface away from the injection point is unconfined with a well-knownself-similar solution (Section IV). There is intermediate regime V that describes the flow behavior when both injection andbuoyancy effects are important. More discussions on the regime boundaries (the dashed curves) are provided in Section V B.

the dynamics of the interface shape,17 as discussed in Section III C. Rarefaction solution (40)also predicts intersections with both the top and bottom boundaries.

• Regime IV (Γ ≫ 1): The flow is mainly buoyancy-driven. For the majority part of the inter-face (except the region close to the origin), the interface is far away from the bottom boundary,and governing equation (11) for the fluid-fluid interface reduces to the well-known nonlineardiffusion equation (42) that describes the spreading of gravity currents in an unconfined porousmedium. A self-similar solution can be obtained that is independent of the value of theviscosity ratio M ,1 as discussed in Section IV.

• Regime V: This regime is intermediate between each of the individual asymptotic limits, andthe axisymmetric flow is due to both injection and buoyancy effects. Direct numerical simula-tion of Equation (15) is necessary to obtain the time evolution of the fluid-fluid interface andthe location of the propagating fronts.

B. Regime boundaries

The regime boundary M = 1 is obvious, with M representing the viscosity ratio of the dis-placed to the injected fluids. We now explain the asymptotic behavior of the regime boundariesusing scaling arguments.

• M ≪ 1 and Γ ≫ 1: Numerical observation indicates that the boundary of regimes IV and Vhas a slope of −1 when M ≪ 1 in the log-log plot of Γ versus M . This slope is related to thecondition H ≪ 1 as M ≪ 1, under which the confinement effect is negligible for the majoritypart of the interface away from origin, and Equation (11) reduces to (42), as discussed inSection IV. Since H ≈ (ΓM)−1/2 in regime IV, as an a priori estimate, we obtain a crossovercondition (ΓM)−1/2 ≈ 1, which indicates a slope of −1 for the regime boundary.


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• M ≫ 1 and Γ ≫ 1: From numerical simulation, the boundary of regimes IV and V has a slopeof 1 when M ≫ 1 in the log-log plot of Γ versus M . This behavior is associated with thecondition |(M − 1)H | ≪ 1 as M ≫ 1, under which Equation (11) reduces to (42), as discussedin Section IV. Again, since H ≈ (ΓM)−1/2 in regime IV, as an a priori estimate, we obtain thecrossover condition (Γ/M)−1/2 ≈ 1, which indicates a slope of 1 for the regime boundary.

• M ≫ 1 and Γ ≪ 1: The boundary between regimes III and V is related to the condition underwhich Equation (15) reduces to (38), i.e.,

ddη

(2ΓMH(1 − H)η

1 + (M − 1)HdHdη

) ddη

(1 − H

1 + (M − 1)H)≪ 1. (47)

As an a priori estimate, we substitute solution (40) into (47), and, after some algebra, weobtain Γ/(M − 1) ≪ 1. Thus, the crossover condition Γ/(M − 1) ≈ 1 can be used to describethe qualitative behavior of this regime boundary. In particular, for M ≫ 1, we obtain Γ/M ≈ 1,which represents a slope of 1 in the log-log plot of Γ versus M , and this slope agrees withthe numerical observation; for M → 1+, Γ/(M − 1) ≈ 1 indicates a singular behavior that alsoappears in our numerical simulations.

• M ≪ 1 and Γ ≪ 1: The boundary between regimes I and V is related to the emergence of theslow front along the bottom boundary, i.e., 2 − 2MΓ/(1 − M) > 0. The qualitative behavior ofthe regime boundary is related to the crossover condition 2 − 2MΓ/(1 − M) = 0. For M ≪ 1,this crossover condition reduces to MΓ ≈ 1, which indicates a slope of −1 in the log-log plotof Γ versus M; for M → 1−, we obtain Γ/(1 − M) ≈ 1, which indicates a singular behavior.Both predictions as M ≪ 1 and M → 1− have been observed in our numerical simulations.

VI. TRANSITION TO SELF-SIMILAR SOLUTIONS

Given an initial condition, in general, there exists a transition time period for the self-similarsolutions to be fully developed, since the self-similar solutions do not contain information of theinitial condition. The approximate analytical solutions we obtained in Sections III and IV, andsummarized in the regime diagram, Figure 7, are all self-similar solutions that hold after theinitial transition period. To study this transition behavior from an initial condition to a self-similarsolution, we numerically solve partial differential equation (11), subject to initial and boundaryconditions (14), and obtain the time evolution of the front locations and the shape of the fluid-fluidinterface over a wide range of time scales for different values of M and Γ. Then, for both the timeevolution of the front locations and the fluid-fluid interface, we compare the predictions from thenumerical solutions with those from the self-similar solutions we obtained in Sections III and IV.

A. Location of the propagating fronts

The propagation laws of the fast (RN1) and slow (RN2) fronts are shown in Figures 8(a) and8(b), respectively, for a wide range of time T and representative values of Γ and M . The predictionsfrom various self-similar solutions are shown as the dashed lines, while the numerical results areplotted as the symbols. The T1/2 power-law behavior for the propagating fronts has been observedfrom the numerical results. Within the time range we considered, the self-similar solutions providevery good approximations for the location of both the fast and slow fronts.

B. Shape of the fluid-fluid interface

We show in this section the transition behavior of the fluid-fluid interface. In particular, inFigure 9, we show the interface shapes in the (η,H) space at T = {10−4,10−3,10−2,10−1,100,101}for different representative values of M and Γ. We have chosen Γ = 0.1 for regimes I–III, i.e., theinjection-driven regimes, and Γ = 100 for the buoyancy-driven regime IV. Within the time rangewe consider, very good agreements have been observed between the predictions of the numericalresults and the self-similar solutions.


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FIG. 8. Time evolution of the location of the fast and slow fronts: (a) the fast front and (b) the slow front. Different valuesof M and Γ have been chosen to demonstrate the transition behaviors in different regimes. The dashed lines represent thedistinct approximate analytical solutions obtained in Sections III and IV; the symbols are numerical solutions to partialdifferential equation (11) subject to boundary and initial conditions (14).

In addition, within this time range, we observe that the numerical curves for the interfaceshape collapse with each other, i.e., the transition process to develop the self-similar solutions isnot obvious in our numerical simulation. We note that T → 0+ maps to η → +∞ by definition, andfrom boundary condition (16a), we have H(+∞) = 0. Thus, initial condition (14a), i.e., H(X,0) = 0,is in the solution space of the self-similar solution of Equation (15). Global mass constraint (13) isalso satisfied by H(X,0) = 0 as T → 0+. Therefore, the similarity solution to ordinary differentialequation (ODE) (15) is an exact solution (for all time) to partial differential equation (PDE) (11)subject to boundary and initial conditions (14), and no transition period is necessary to developthe appropriate analytical solutions (self-similar solutions) we obtained in Sections III and IV, andthese solutions hold from T = 0.

VII. PRACTICAL IMPLICATIONS

We briefly discuss the practical implications of the approximate analytical solutions, as summa-rized in the self-similar regime diagram (Figure 7) in this section. We consider different engineeringapplications such as enhanced oil recovery, geological CO2 sequestration, and underground waste


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FIG. 9. Time evolution of the shape of the fluid-fluid interface in the (η,H ) space at T = {10−4,10−3,10−2,10−1,100,101}.The solid curves are generated from numerically solving partial differential equation (11), subject to initial and boundaryconditions (14). The dashed curves are from the approximate analytical solutions we obtained in Sections III and IV. Wehave chosen Γ = 0.1 as an example to demonstrate the flow behaviors in regimes I–III, the injection-driven regimes, andΓ = 100 as an example for buoyancy-driven regime IV.

disposal projects. We note that the axisymmetric assumption holds for horizontal geological forma-tions. It has also been shown to be a good assumption for inclined formations with a slope up to1◦, as long as the force from injection is the dominant driving force.39 The dimensional physicalproperties and the value for the dimensionless parameters (Γ and M) we calculated based on (12)for each individual project are listed in Table II.

• Case 1: The first example comes from a large-scale CO2-WAG (Water-alternating-gas)enhanced oil recovery project at the Kelly-Snyder oil fields in Texas.40 The values of thedimensionless control parameters are Γ ≈ 0.26 and M ≈ 0.14. Thus, the axisymmetric flow,

TABLE II. Examples of practical fluid injection projects, including CO2-WAG enhanced oil recovery (EOR, case 1),geological CO2 storage (cases 2 and 3), and underground waste disposal (case 4). The value of the dimensionless groups M

and Γ is calculated based on (12) and indicates the behavior of the axisymmetric flows and the corresponding approximatesolutions.

Parameter EOR CO2 storage CO2 storage Waste disposalDescription (units) (case 1) (case 2) (case 3) (case 4)

Permeability k (m2) 1.94 × 10−14 2 × 10−12 2 × 10−12 6.9 × 10−14

Porosity φ 0.04 0.36 0.36 0.07Thickness h0 (m) 81 20 20 15.24Injected fluid density ρi (kg m−3) 994 760 760 1006Displaced fluid density ρd (kg m−3) 670 1020 1020 1094Injected fluid viscosity µi (Pa s) 4.45 × 10−4 6 × 10−5 6 × 10−5 6.68 × 10−4

Displaced fluid viscosity µd (Pa s) 6.20 × 10−5 8 × 10−4 8 × 10−4 8.93 × 10−4

Injection rate q (m3 s−1) 0.16 0.04 8 × 10−4 6.3 × 10−3

Dimensionless Γ 0.26 0.4 20 0.02Dimensionless M 0.14 13 13 1.3


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for example, from water injection (displacing CO2 that was previously injected into the reser-voir) in this project lies in regime I, and approximate analytical solution (28) can be used tocharacterize the time evolution of the interface between water and CO2. We recognize that theCO2-WAG process is complicated as it involves multiple interfaces between different fluids(water-CO2, CO2-oil). Here, we only focus on the water-CO2 interface that is formed whenwater displaces the previously injected CO2.

• Case 2: The second example is the world’s first and longest running industrial-scale geolog-ical CO2 sequestration project in a saline aquifer at Sleipner in the North Sea.41 The targetformation for CO2 storage at the Sleipner site has very high permeability in a range from1 × 10−12 m2 to 5 × 10−12 m2. This formation contains a number of thin intra-formational shalelayers (approximately 0.5–2 m thick) that separate the formation into nine sand layers. Mostof the modeling efforts in the literature focused on the ninth sand layer (the uppermost layer)because this layer has the most available geophysical data. In addition, this layer is relativelyhomogeneous (permeability: ≈2 × 10−12 m2), and it has been shown that the vertical equilib-rium models are applicable.42 The representative values of the related parameters in Table IIare taken from the 2010 benchmark problem.41 The values of the dimensionless control param-eters are Γ ≈ 0.4 and M ≈ 13. Our calculation indicates that the flow behavior in the Sleipnerproject lies in regime V. Therefore, rarefaction wave solution (40) can be used to characterizethe propagation of the interface between supercritical CO2 and brine that originally saturatesthe aquifer.

• Case 3: We now decrease the injection rate to 2% of that in the benchmark study of theSleipner CO2 sequestration project, while we keep the other parameters unchanged. The valuesof the control parameters are Γ ≈ 20 and M ≈ 13. Thus, the flow behavior lies in regimeIV, and buoyancy is the major driving force for the spreading of the supercritical CO2. Thewell-known self-similar solution to nonlinear diffusion equation (44) can be used to describethe propagation of the CO2-brine interface, as discussed in Section IV.

• Case 4: The fourth example is an underground liquid waste disposal project in a salineaquifer.43 The values of the dimensionless control parameters are Γ ≈ 0.02 and M ≈ 1.3.Thus, the axisymmetric flow in this project lies in regime III, and rarefaction solution (40) canbe used to characterize the time evolution of the shape of the interface between the injectedliquid waste and brine that originally saturates the formation.

VIII. FINAL REMARKS

A. Summary of major findings

Motivated by underground fluid injection processes through vertical wells, we study the axisym-metric flows generated from fluid injection into a horizontal confined porous medium that is orig-inally saturated with another fluid of different density and viscosity. We have obtained a nonlinearadvection-diffusion equation to describe the time evolution of the fluid-fluid interface by neglectingthe effects of fluid mixing and interfacial tension. Two dimensionless groups are identified whichcontrol the fluid flow: M , the viscosity ratio of the displaced fluid over the injected fluid, and Γ,the relative importance of buoyancy effect compared to fluid injection effect. We have obtained fourapproximate analytical solutions (self-similar solutions) in asymptotic limits involving Γ and M: (i)analytical solution (28) that indicates a parabolic interface shape for the injection-driven flow witha more viscous injected fluid than the displaced fluid (regime I, Γ ≪ 1 and M < 1); (ii) analyticalsolution (37) that describes the injection-driven flow with equally viscous injected and displaced fluids(regime II, Γ ≪ 1 and M = 1); (iii) rarefaction solution (40) that describes the injection-driven flowwith a less viscous injected fluid than the displaced fluid (regime III, Γ ≪ 1 and M > 1); and (iv)the well-known self-similar solution that characterizes the buoyancy-driven flow in an unconfinedporous medium (regime IV, Γ ≫ 1). We have obtained a regime diagram (Figure 7) to summarizethe various flow behaviors. The regime diagram includes five distinct self-similar behaviors: the fourasymptotic regimes and an intermediate regime. We have also briefly discussed the implications of


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the approximate analytical solutions and the regime diagram to practical projects such as enhancedoil recovery, geological CO2 sequestration, and underground waste disposal projects.

B. Vertical equilibrium assumption and lubrication approximation

This validity of nonlinear equation (11) relies on two assumptions. First, the vertical equi-librium assumption indicates that the injected and displaced fluids are segregated due to strongbuoyancy effects and form a sharp interface, i.e., the time scale we consider is larger than thetime scale for the two fluids to segregate. Second, the assumption of the lubrication approximationrequires that the gravity current is long and thin (the ratio of the height relative to length is small)and the slope of the fluid-fluid interface is small (|∂H/∂R| ≪ 1). These two assumptions should besatisfied for the results in this paper to be applicable.

The vertical equilibrium assumption is considered reasonable and has been verified using fullmultidimensional models in the context of geological CO2 sequestration.44–46 Detailed discussionson the validity of the vertical equilibrium assumption can also be found in other references.47,48 Inour study, the slope of the fluid-fluid interface decays with time as 1/

√T , as predicted from the

self-similar solutions; thus, the validity of the lubrication approximation assumption is supportedat late times. We also note that the slope of the fluid-fluid interface can be large in the region veryclose to the injection point (e.g., in regime IV); nevertheless, the self-similar solution successfullydescribes the bulk part of the fluid-fluid interface and the location of the propagating front, which isverified by laboratory experiments.1

C. Saffman-Taylor instability

We note that when the injected fluid is less viscous than the displaced fluid (M > 1), the flowsituation corresponds to the condition when the Saffman-Taylor instability occurs.49,50 In the injec-tion dominant regime (regime III), we obtained a rarefaction solution, which was previously studiedby Nordbotten and Celia.17 Similar behavior has also been identified for the analogous problem inCartesian coordinates for fluid injection through a horizontal well into a confined porous mediuminitially saturated with another fluid of different density and larger viscosity.34,35 Experiments havebeen conducted in Hele-Shaw cells filled with glass beads in Pegler, Huppert, and Neufeld,34 and itwas observed that the Saffman-Taylor instability does not occur at the length scale of the Hele-Shawcells. Within the time scale of the experiment, the theoretical predictions provide good approxima-tions except in the region close to the propagating front. The suppression of the Saffman-Taylorinstability is likely due to the inherent buoyant segregation in the system.34

ACKNOWLEDGMENTS

We thank the Princeton Carbon Mitigation Initiative for support of this research. The work wasalso supported, in part, by the Department of Energy under Grant No. DE-FE0009563. We alsothank I. C. Christov, H. E. Huppert, Q. X. Li, J. M. Nordbotten, A. J. Smits, and R. H. Socolow forhelpful discussions.

APPENDIX A: NUMERICAL SIMULATIONS

We solve nonlinear advection-diffusion equation (11) numerically on a fixed domain (0,L),instead of simulating the moving boundary problem on

�0,RN1(T)

�.13,35 The appropriate boundary

conditions are provided at R = 0 and R = L. Since fluid injection begins at T = 0, and we as-sume that there is only displaced fluid in the porous medium before injection; therefore, the initialcondition is given by

H(R,0) = 0. (A1)


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Note that H(R,T) = 0 holds ahead of the front, i.e., R ≥ RN1(T); thus, the boundary condition atR = L is given by

H(L,T) = 0. (A2)

We multiply R to both sides of Equation (11), integrate it from R = 0 to R = L, and we obtain

ddT

L

0RHdR +

ΓMH(1 − H)R1 + (M − 1)H

∂H∂R+

1 − H1 + (M − 1)H

L

0= 0. (A3)

We consider constant fluid injection in this paper, which gives L

0 RHdR = T . Then, employingEquation (A2) and lim

R→ LH ∂H

∂R= 0, i.e., no fluid entrainment at the front, Equation (A3) can be

rewritten to provide a boundary condition at R = 0,ΓMH(1 − H)R1 + (M − 1)H

∂H∂R+

1 − H1 + (M − 1)H

��R=0= 1. (A4)

Thus, we have obtained the appropriate boundary and initial conditions for the numerical study,i.e., Equations (A1), (A2), and (A4). A central-difference scheme is employed to provide thenumerical solutions.38

APPENDIX B: DERIVATION OF BOUNDARY CONDITION (16b)

We first note that Equation (15) can be rearranged asη − 2M

(1 + (M − 1)H)2 +4ΓM(1 − 2H)η1 + (M − 1)H

dHdη+

4ΓMH(1 − H)1 + (M − 1)H

− 4ΓM(M − 1)H(1 − H)η(1 + (M − 1)H)2

dHdη

dHdη+

4ΓMH(1 − H)η1 + (M − 1)H

d2Hdη2 = 0. (B1)

Using boundary condition (16a), i.e., H(ηN1) = 0, we evaluate each term in Equation (B1) asη → η−N1. We further assume lim

η→η−N1

H dHdη = 0 and lim

η→η−N1

H d2Hdη2 = 0 and obtain

(η − 2M + 4ΓMη

dHdη

)dHdη

��η→η−N1

= 0. (B2)

Then, we obtain two values for the slope of the interface dH/dη as η → η−N1,

dHdη

��η→η−N1

= 0 (B3a)

and

dHdη

��η→η−N1

=1

2ηN1Γ− 1

4ΓM. (B3b)

Nontrivial slope (B3b) is used in the shooting procedure, together with other boundary condition(16a), to solve Equation (15).

APPENDIX C: NEGLIGIBLE INFLUENCE OF THE SLOW FRONT

When solving for the interface shape f using Equation (44) subject to boundary conditions(45a) and (45b), we have neglected the influence of the slow front at the bottom boundary. Here, weinvestigate under which conditions this is a good approximation.

From Equation (44), we obtain that f |y→0+ → −∞; thus, the interface f always intersects withthe bottom boundary where H = 1, or equivalently, f = ΓM/ηN1. We now define yN2 as

yN2 ≡ ηN2/ηN1, (C1)


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FIG. 10. The location of the slow front yN2, defined in (C1), and the error for the global mass Em, defined in (C2), bothappear to decay exponentially versus MΓ from our numerical simulation. In addition, when MΓ ≫ 1, both yN2 and Em

are negligibly small for the analysis in Section IV (regime IV).

which represents the location of the slow front. In addition, we define the error for the global massEm introduced by not considering the slow front yN2 as

Em ≡��yN2

ΓMηN1+

1

yN2

f dy − 1

0f dy

��

1

0f dy. (C2)

Both the location of the slow front yN2 and the error Em are functions of MΓ, which can becomputed from the numerical solution to Equation (44) subject to boundary conditions (45a) and(45b), as shown in Figure 10. From numerical simulation, both yN2 and Em appear to decay expo-nentially versus MΓ, and this is consistent with the asymptotic behavior as y → 0+, see Equation(2.16a) in Lyle et al.1 In particular, when MΓ ≈ 5, we obtain yN2 ≈ 10−3 and Em ≈ 10−2. We notethat in regime IV, MΓ ≫ 1 always holds; thus, the influence of the slow front yN2 is negligiblysmall in computing the location of the fast front yN1 and the shape of the fluid-fluid interface awayfrom the narrow region near the injection site.

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