## 1. Introduction

[2] Capturing the influence of physical heterogeneity on flow and transport in geological media is still one of the great challenges facing us today. Even for linear problems, such as single phase flow and transport many questions remain unanswered and while many have been presented with some success, no single clear model has emerged as capable of capturing all effects of heterogeneity [see, e.g., *Dagan*, 1989; *Gelhar*, 1993; *Neuman and Tartakovsky*, 2009]. Similarly, accounting for the influence of buoyancy on single phase flow [e.g., *Henry*, 1964; *Kalejaiye and Cardoso*, 2005; *Huppert and Woods*, 1995; *Dentz et al.*, 2006] and transport [e.g., *Graf and Therrien*, 2008; *Bolster et al.*, 2007] in porous media is a challenging problem that has a rich body of work dedicated to it.

[3] Many interesting and relevant problems in porous media involve the flow and interaction of two immiscible fluids. Relevant examples that receive much attention include CO_{2} sequestration [e.g., *Bachu*, 2008; *Bachu and Adams*, 2003; *Bryant et al.*, 2008; *Riaz and Tchelepi*, 2008] and enhanced oil recovery [e.g., *Lake*, 1989; *Ferguson et al.*, 2009; *Dong et al.*, 2009; *Tokunaga et al.*, 2000]. Accounting for the effects of mobility (viscosity differences between phases) and capillarity introduces significant complexity and results in highly nonlinear and coupled governing equations [e.g., *Binning and Celia*, 1999]. Add to this buoyancy effects when the two phases are of differing density and one has a very interesting and challenging problem (even in the absence of heterogeneity).

[4] In this work we focus on the interaction of buoyancy and heterogeneity effects on multiphase flows. To do so, we consider a displacement problem where an invading phase displaces another one as depicted in Figure 1. We neglect the influence of capillarity by using the commonly used Buckley-Leverett approximation, which we discuss in more detail in section 2. In such a displacement problem there is typically a sharp interface between the invading and displaced phases. Spatial variability in the flow field, induced by heterogeneity, cause this sharp interface to vary in space, which results in spreading of the front. At the same time buoyancy plays its role. In the case of a stable displacement, the spreading ultimately induces lateral pressure gradients that slow down the spreading of the interface. Similarly, an unstable injection will result in greater spreading due to buoyancy. This is illustrated clearly in Figure 1 where the results of three numerical simulations are presented, one with no buoyancy effects (Figure 1, left), one with stabilizing buoyancy (Figure 1, middle) and one with destabilizing buoyancy (Figure 1, right).

[5] To date, in the field of single phase flows, the approaches to capture the effect of heterogeneity that have achieved most success are stochastic methods. The theory of such approaches is described extensively in the literature [e.g., *Dagan*, 1989; *Brenner and Edwards*, 1993; *Gelhar*, 1993; *Rubin*, 2003]. In the context here, if one averages transversely across the transition zones depicted in Figure 1, the resulting transition zone between high and low saturation of the displacing fluid can have the appearance of a dispersive mixing zone. It should of course be noted that this averaged dispersive zone does not represent actual mixing as only spreading occurs. However, for applications where the fluid-fluid interfacial area is important, it is important to have model predictions that quantify the spreading zone.

[6] Dispersive transition zones in solute transport problems have typically been characterized by spatial moments and a wide body of literature exists doing so [e.g., *Aris*, 1956; *Gelhar and Axness*, 1983; *Dagan*, 1989; *Kitanidis*, 1988; *Dentz and Carrera*, 2007; *Bolster et al.*, 2009b]. Similar approaches have been applied to two-phase flow, but most work along these lines has been limited to horizontal displacements that neglect buoyancy effects. *Cvetovic and Dagan* [1996] and *Dagan and Cvetkovic* [1996] applied a Lagrangian perturbation theory approach in order to determine the averaged cumulative recovery of the displacing fluid and the spatial moments of the fluid distribution. They found that the heterogeneities cause a dispersive growth of the second moment. However, they did not quantify it. Similarly, *Zhang and Tchelepi* [1999] found a dispersion effect for the immiscible displacement in the horizontal direction. This dispersion coefficient was calculated semianalytically by numerical means by *Langlo and Espedal* [1995], who also applied a perturbation theory approach. Their approach was extended by *Neuweiler et al.* [2003] to quantify the dispersion coefficient analytically and later by *Bolster et al.* [2009a] to include temporal fluctuations in the flow field. Within the validity of perturbation theory and in direct analogy to single phase flow, they showed that the dispersive growth for neutrally stable displacement was directly proportional to the variance and the correlation length of the permeability field. As such a natural question arises: given the additional influence of buoyancy, can we anticipate the same behavior?

[7] For vertical immiscible displacement in the presence of buoyancy effects we anticipate a similar quasi-dispersive transition zone of the averaged front, which will be augmented or suppressed due to buoyancy. The heterogeneity still leads to fluctuations in the velocity field as illustrated in Figure 1. However, the process will be more complicated and not solely due to the stabilizing and destabilizing processes mentioned above. After all, such stabilization/destabilization effects will occur even for single phase miscible displacement [e.g., *Welty and Gelhar*, 1991; *Kempers and Haas*, 1994], leading to the question what additional role the multiphase nature of this flow plays?

[8] In the absence of buoyancy effects the Buckley-Leverett problem is governed by a single dimensionless parameter, which is the viscosity ratio (or ratio of the viscosities of the two phases). This dimensionless number does not depend on any of the parameters associated with the flow or porous medium. This means that while heterogeneity in the porous medium induces fluctuations in the flow field it does not affect the fundamental fluid properties in an equivalent homogeneous medium. Thus, the (mean) front positions obtained from the solutions of the homogeneous and heterogeneous media are identical.

[9] On the other hand, when one includes buoyancy effects, a second dimensionless number is necessary to describe the system, namely, the gravity number. The gravity number physically reflects the ratio of buoyancy to viscous forces. The buoyancy number (defined formally and discussed further in section 2) is directly proportional to the permeability of the porous medium. Therefore when the permeability field is heterogeneous in space, so too is the buoyancy number. This means that while the viscosity ratio is insensitive to heterogeneity, the gravity number can potentially vary over orders of magnitude depending on how variable the permeability field is. This raises another important and potentially problematic question: as this system is so inherently nonlinear, does the arithmetic mean (or for that matter any other mean) of the gravity number provide a good representative measure of the behavior of the heterogeneous system?

[10] In fact, as the buoyancy number varies in space, in a manner directly proportional to the spatial variations in permeability one might anticipate a local contribution to the dispersion front spreading effect beyond the nonlocal contribution that arises from fluctuations in the velocity field. In this paper we aim to answer the following questions regarding buoyancy influenced multiphase immiscible displacement in a heterogeneous medium.

[11] 1. Can we, using perturbation theory, asses the rate of front spreading that occurs?

[12] 2. What measures of the heterogeneous field (e.g., variance, correlation length) control this spreading? Also, why and how do they?

[13] 3. What influence does the heterogeneity in gravity number have? And does the arithmetic mean of the gravity number represent a mean behavior in the heterogeneous system considering that the problems considered here are highly nonlinear?