Chapter 10 – Simultaneous Laminar Flow of Miscible Fluids 10.1 10.1 Introduction In the previous chapter the fluids displaced each other without mixing, resulting in a distinct fluid-fluid interface within the pores. In this chapter, we will discuss the simultaneous flow of two fluids that are soluble; subsequently a distinct interface does not exist. This process is termed miscible displacement. Applications in the petroleum industry exist in areas of improved oil recovery, contamination plumes, and tracer tests. In remaining sections of this chapter we will discuss the mathematical foundation for miscible displacement, and then provide applications. 10.2 Transport Processes Consider a homogeneous, isotropic porous media, saturated, with Darcy’s law applies. Under the Darcy assumption flow occurs by convection, described by the volumetric flux. If this was the only transport mechanism, then solutes in the fluid would move as a plug. In reality, mixing occurs caused by variations in the microscopic velocity within the pores; i.e., hydrodynamic dispersion or just dispersion. Subsequently, to account for this microscopic mixing on a macroscopic scale, it is necessary to introduce a second mode of transport. The conservation of mass for the transport of solutes in porous media over a fixed elemental volume can be given as: reactions toduemass solute ofgainor losselement theinto soluteofflux element theofout soluteofflux element within thesolute of mass ofchange of ratenet (10.1) The physical processes that control the flux into and out of the elemental volume are convection and dispersion. Loss or gain of the solute mass in the elemental volume can occur as a result of chemical reactions. Convection is the component of solute movement attributed to fluid flow. The rate of transport is a function of the average linear velocity, /vv . In the x-direction, CdAvx termconvection (10.2) where C is the concentration of the solute defined as the mass of solute per unit volume of solution. The process of dispersion occurs as a result of mechanical mixing and molecular diffusion. In the x-direction,Chapter 10 – Simultaneous Laminar Flow of Miscible Fluids 10.2 dAxCDx termdispersion (10.2) where Dx is the dispersion coefficient in the x-direction. This coefficient can be expressed in terms of two components, *DvDxx (10.3) where x is a characteristic property of the porous media known as dispersivity [L] and D* is the coefficient of molecular diffusion for the solute in the porous media [L2/T]. If we define Fx as the total mass of solute per unit cross-sectional area transported in the x direction per unit time, then xCDCvFxxx (10.4) The difference in the amount of solute entering and leaving in the x-direction is: VxFAdxxFFAFxxxxxx (10.5) Assuming the dissolved substance is nonreactive, then the accumulation in the element is given by, VtC (10.6) Combining Eqs. 10.4 through 10.6 results in a linear conservation of mass expression,  tCCvxxCDxxx (10.7) This equation represents the convection – dispersion equation for solute transport in saturated porous media. The solution will provide the solute concentration as a function of space and time. The physical picture of the convection – dispersion equation is illustrated in Figure 10.1. In the experiment (shown on the left), a non-reactive tracer of concentration Co is continuously injected through a homogeneous porous media. For convenience, define a relative concentration as C/Co, where C is the concentration in the column or at the output. Thus in (b) the tracer input can be represented as a step function. The concentration profile as a function of time is shown in (c). This figure represents outflow concentration and thus reflects the breakthrough of tracer at the outflow face. Notice theChapter 10 – Simultaneous Laminar Flow of Miscible Fluids 10.3 effect of dispersion is to smear the front, subsequently tracer first appears in the outflow at time denoted as t1, before the arrival of the water (t2) traveling at the average velocity. If no dispersion or diffusion exists, then the front will be sharp and plug flow will occur through the sample, as shown by the dashed line in (c). As the front moves through the sample over time, the tracer will increase in spreading. This is illustrated in (d), where the points (1) and (2) represent t1 and t2, respectively. Figure 10.1 Experimental tracer setup for uni-directional dispersion and the associated tracer profiles. The mathematical solution for Eq. (10.7) requires appropriate initial and boundary conditions; 0),(),0(0)0,(tCCtCxCo (10.8) C/Co01distancet1t2C/Co01timet1t2Effect ofdispersionC/Co01timet0inflowoutflowbcdC/Co01distancet1t2C/Co01timet1t2Effect ofdispersionC/Co01timet0inflowoutflow C/Co01distancet1t2C/Co01timet1t2Effect ofdispersionC/Co01timet0C/Co01distancet1t2C/Co01C/Co01distancet1t2C/Co01C/Co01timet1t2Effect ofdispersionC/Co01C/Co01timet0inflowoutflowbcdChapter 10 – Simultaneous Laminar Flow of Miscible Fluids 10.4 The solution to this problem (assuming steady state and homogeneous and constant dispersion coefficient, D) is: tDtvxerfcDvtDtvxerfcCCxxxxo2exp221 (10.9) where erfc is the complementary error function. The magnitude of the spread of the concentration profile is a function of both mechanical dispersion and molecular diffusion. Figure 10.2 is a schematic for the experimental conditions from Figure 10.1. The contribution of diffusion only is shown as a dotted line in the figure. Figure 10.2 Schematic of spread of the tracer front for step function input At low velocity, diffusion dominates and therefore the dispersion coefficient is given by *DDx. As velocity increases the mechanical mixing becomes dominant in the dispersion, in which case, vDxx. To