## 1. Introduction

[2] The movement of solutes within two-phase systems is important in many environmental and engineering applications because in almost all cases each fluid phase consists of different components rather than just one. Consequently, both the unsteady flow of the two phases and miscible displacement within each phase occur at the same time. Displacement processes of this kind take place, for example, if water is pumped into a geological formation or aquifer contaminated with nonaqueous phase liquids (NAPL) and the ionic composition of the connate water is different from that of the injected water. In this scenario, both for the purpose of bioremediation and the cleanup of NAPL and enhanced oil recovery, surfactants and polymers are dissolved in the injected aqueous phase to mobilize the NAPL [*West and Harwell*, 1992; *Khan et al.*, 1996; *Sorbie*, 1991]. Here the appropriate design of an efficient chemical flood crucially depends upon the brine composition since the interfacial activity, phase behavior, and mobility control of the chemical flood depend as much on the concentration of the chemicals as they depend on the composition and mixing behavior of the ionic environment itself [*Lake and Helfferich*, 1977]. Similarly, for the design of aquifer remediation schemes, a vital step is to identify the location and distribution of the NAPL. To this end, tracer tests can be performed where a range of both partitioning and nonpartitioning solutes are injected into the subsurface and recovered down gradient at the extraction wells [*Datta-Gupta et al.*, 2002]. Another example is carbon sequestration. In recent years, a growing awareness of the hazardous consequences of anthropogenic greenhouse gases has been seen, and one helpful mitigation method seems to be the sequestration of carbon dioxide in the subsurface, i.e., the reaction of carbon dioxide molecules (CO_{2}) with mineral grains [*Javadpour*, 2009; *Xu et al.*, 2006]. In this case, CO_{2} is dissolved in the water phase, and the mixing with the brine triggers a number of aqueous reactions that lead to the CO_{2} being trapped by the minerals. If water is pumped into a hydrocarbon reservoir in order to produce oil, the two aqueous solutions mix while replacing the oil, and the otherwise inert brine components react. In many reservoirs, this leads to the precipitation of minerals, such as barium sulfate (BaSO_{4}), and formation of scale [*Sorbie and Mackay*, 2000; *Mackay*, 2003] that can hinder production. In other scenarios, different ionic compositions have been shown to enhance oil recovery if the injected brine has a salinity different from that of the connate brine [*Zhang and Morrow*, 2007; *Austad and Standnes*, 2003; *Hiorth et al.*, 2010], and a good understanding of the transport of the different compositions within the two phases due to the interplay of dispersion and viscous and capillary forces builds the fundament for appropriate upscaling of transport and for determining its field efficiency [*Stoll et al.*, 2008; *Stephen et al.*, 2001]. In all of these cases, a proper understanding of miscible displacement and dispersive mixing is fundamental to properly assess the amount of reactive solutes involved in chemical reactions [see, e.g., *Emmanuel and Berkowitz*, 2005; *De Simoni et al.*, 2005, 2007; *Cirpka*, 2002; *Dentz et al.*, 2011]. Although the effects of dispersion and even spatial heterogeneity on miscible displacement and mixing for single-phase flow are increasingly well understood and a significant body of literature exists [see, e.g., *Werth et al.*, 2006; *Rahman et al.*, 2005; *Bolster et al.*, 2010; *Paster and Dagan*, 2008; *Willingham et al.*, 2008], investigations for two-phase systems so far only focus on the spreading of the phases themselves [*Neuweiler et al.*, 2003; *Cvetkovic and Dagan*, 1996; *Langlo and Espedal*, 1994; *Panfilow and Floriat*, 2004]. This is surprising given the practical importance of simultaneous flow and transport in two-phase systems, but it can be explained with the complexity of the governing equations where capillary, viscous, and dispersive terms are coupled in a highly nonlinear way.

[3] Clearly, for gaining a full understanding of all the mechanisms and effects involved, numerical simulations are important, and in recent years, there has been substantial progress in the development of numerical methods [see, e.g., *Blunt et al.*, 1996; *Geiger et al.*, 2004; *Huber and Helmig*, 1999; *Reichenberger et al.*, 2006; *Hoteit and Firoozabadi*, 2005; *Lunati and Jenny*, 2006]. However, the development of numerical models requires verification and validation as to which new analytical solutions contribute an important part. Also, analytical solutions allow a deeper insight into the structure of a problem and, thus, as to which parameters control processes and often act as a building block for numerical methods [e.g., *Lie and Juanes*, 2005; *Blunt et al.*, 1996]. Finally, constitutive relationships, such as relative permeabilities, are normally gained from core flood experiments where numerical simulations are matched with flow rates. However, interpretation of experimental data would be faster and more reliable if the forward problem was solved analytically [*Juanes*, 2003]. Our solutions provide the framework for the common situation of two-phase core floods, where both flow rates and breakthrough curves of tracers are available and where all physical mechanisms, i.e., viscous and capillary forces and hydrodynamic dispersion, are considered.

[4] The outline of this paper is as follows: In section 2, we introduce the mathematical model and basic notation and give a short overview of existing solutions for immiscible two-phase flow without transport. In section 3, we solve the advection problem exactly by two different methods: On the basis of a known integral solution for two-phase flow, we first combine a variable transformation with the physical notion for which in the dispersion-free limit, the solutes can be written as functions of their carrying fluid only; second, we use the method of characteristics. We show that if the boundary and initial conditions of the flow problem satisfy the McWhorter and Sunada problem [*McWhorter and Sunada*, 1990], the solution to the transport equation can be represented by a modified Welge tangent [*Welge*, 1952]. In section 4, we use a perturbation expansion to derive analytical expressions for hydrodynamic dispersion for the case where the dispersion coefficient is small compared to the characteristic length of the system. On the basis of these equations, we are able to obtain an analytical expression for the growth rate of the dispersive mixing zone. These solutions are the first known analytical expressions for hydrodynamic dispersions in two-phase flow. In section 4, we compare the obtained solutions against the numerical reference solution for the cases of cocurrent and countercurrent imbibition and for the capillary-free limit, the Buckley-Leverett problem [*Buckley and Leverett*, 1942], and then finish with some conclusions.