Using multiphase flow theory to choose f CO 2

Steady-state core-flooding is a laborious and time consuming experimental procedure. At the outset of a programme of core-floods, the fractional flow-saturation relationship for the core is usually unknown, thus the selection of appropriate fractional flows is not straightforward. It may take many hours or even days to achieve a single relative permeability measurement at a particular fractional flow. Consequently, many workers set a somewhat arbitrary minimum number of pore volumes of fluid that should flow through the core before relative permeability measurements are made. While this is not unreasonable, much time and effort may be wasted either by leaving a particular fractional flow step running longer than is required, or by making measurements before steady-state is reached.

In order to make the task more approachable, multiphase flow theory may be used to estimate the saturation achievable at a particular fractional flow and the time required for steady-state to be reached. Thus the efficiency of experiments can be maximised by eliminating the ‘wait-and- see’ aspect of core-flooding and by selecting fractional flows to measure relative permeability over an appropriate spread of saturations.

In the following section the derivation of the Buckley-Leverett solution for 1D forced displace- ment is reviewed and the application to planning core-floods is outlined.

3.1.7.1 Application of the Buckley-Leverett solution to core-floods

For the CO2-brine system, assuming flow occurs in one dimension and that fluids are both im-

miscible and incompressible, a mass balance may be written for each phase (i = CO2, w):

φ∂Si ∂t + qT

∂fi

where Sw + SCO2 = 1. Substituting the multiphase extension to Darcy’s law (Equation 2.6)

into the mass balance for the case i = CO2 gives

fCO2 = qCO2 qT = λCO2 λCO2 + λw  1 −kλw qT  ∂Pc ∂x + ∆ρgx  , (3.9)

where the mobility is given by λi = kr,i/µi, the capillary pressure is defined by Pc(Sw) =

PCO2 − Pw and gx is the component of gravitational acceleration in the x direction. The

contribution of viscous, capillary and gravity forces to the fractional flow of CO2 is described

by Equation 3.9, and is applicable to a steady-state core-flood between two immiscible fluids, where the flow approximates to a one-dimensional forced displacement.

For a horizontal core-flood with sufficiently high total volumetric flow rate, the contribution from gravity and capillary terms is negligible when compared with the viscous flow. Thus fCO2

is reduced to

fCO2 =

λCO2

λT

, (3.10)

where the total mobility is given by λT = λw+ λCO2.

The dimensionless variables xD = x/L and tD =

Rt

0qT/Vpdt are introduced, where x is the

distance travelled along the core, L is the length of the core and Vp is the pore volume φAL.

Equation 3.8 may now be reformulated for CO2 as a quasi-linear first order partial differential

equation, dependent only on SCO2:

∂SCO2 ∂tD + dfCO2 dSCO2 ∂SCO2 ∂xD = 0. (3.11)

Equation 3.11 can be solved following Buckley and Leverett (1942) [180] to give dxD dtD = dfCO2 dSCO2 (3.12) and dSCO2 dtD = 0. (3.13)

This is the Buckley-Leverett solution for forced displacement [180]. Graphically, Equations 3.11 and 3.12 are represented by lines of constant SCO2 in xD−tD space which have a slope of gradient

dfCO2/dSCO2. In a core-flood experiment, each fCO2 has an associated steady-state SCO2. When

a particular fractional flow is set, the saturation at the inlet of core immediately increases to the steady-state level. This saturation then proceeds along the length of the core at a velocity given by the slope of the fractional flow curve. Thus the time taken to reach steady-state can be calculated from the velocity by converting back from dimensionless parameter space (Figure 3.8). Multiphase flow theory may be used in such a way to estimate the saturation that should be achieved for each fractional flow and the time taken to reach steady-state at

each step. However, this requires some a priori knowledge as to the shape of the fractional flow curve.

Figure 3.8: Top: Relative permeability curves and corresponding fractional flow curve. Dotted line indicates ‘first guess’ relative permeability used to predict time to reach steady-state. Bottom: Gradient in fractional flow curve and corresponding time to reach steady-state. Cumulative time plotted for a set of six fractional flows chosen to create a wide range of saturations to measure relative permeability.

A Brooks-Corey type relative permeability relationship [117, 181] can be used to create a ‘first guess’ relative permeability curve (Figure 3.8, dotted lines). The Brooks-Corey curves are given by kr,nw = kr,nw(Sw,irr)(1 − S ∗ w) 2 (1 − S_w∗n) (3.14) and kr,w = Sw∗ m , (3.15)

where the normalised saturation is given by S_w∗ = (Sw− Sw,irr)/(1 − Sw,irr).

If other measurements of relative permeability or some knowledge as to the wetting proper- ties of the rock are known, first guess relative permeability curves can be created by choosing appropriate values for n, m, Sw,irr and kr,nw(Sw,irr), and a fractional flow curve plotted using

saturation. As the core-flood progresses, the parameters used to generate the relative perme- ability curves may be updated so as to improve the prediction (Figure 3.8, solid and dashed lines).

There are two main benefits to carrying out a core-flood in this way. First, fractional flows may be selected to as to control the range of saturations over which relative permeability is measured. This is of particular importance for the CO2-brine system where the extreme viscosity ratio

causes the fractional flow curve to be very steep. Second, predictions can be made as to the time to reach steady-state for each step. When trying to achieve a high CO2 saturation in

the core, when the choice between an fCO2 of 0.9990 or 0.9998 can make difference between

achieving an SCO2 of 0.65 in a few hours, or 0.75 in a few days.