SPE 134906

Can Formation Relative Permeabilities Rule Out a Foam EOR Process? E. Ashoori and W.R. Rossen, Delft University of Technology

Copyright 2010, Society of Petroleum Engineers This paper was prepared for presentation at the SPE Annual Technical Conference and Exhibition held in Florence, Italy, 19–22 September 2010. This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.

Abstract Foam is a promising means of increasing sweep in miscible- and immiscible-gas enhanced oil recovery. SAG (surfactant-alternating-gas) is a preferred method of injection. Numerous studies verify that the water relative-permeability function krw(Sw) is unaffected by foam. Studies of foam have used a variety of krw functions.

This paper shows a connection between the krw(Sw) function and SAG foam effectiveness that is independent of the details of how foam reduces gas mobility. For simplicity we analyze SAG processes in the absence of mobile oil; success without oil is a precondition to success with oil, and our analysis also applies to a miscible-gas process with oil in 1D in the absence of dispersion.

Fractional-flow methods have proved useful and accurate for modeling foam EOR processes. The success of SAG depends on total mobility at a point of tangency to the fractional-flow curve, which defines the shock front at the leading edge of the foam bank. One can determine total mobility directly from the coordinates of this point (Sw, fw) if the function krw(Sw) is known.

Geometric constraints limit the region in the fractional-flow diagram in which this point of tangency can occur. For a given krw(Sw) function, this limits the mobility reduction achievable for any possible SAG process. We examine the implications of this limitation for different krw functions.

These implications include the following: Increasing nonlinearity of the krw function is advantageous for SAG processes, regardless of how foam reduces gas mobility. SAG is inappropriate for naturally fractured reservoirs if straight-line relative permeabilities apply, even if extremely strong foam can be stabilized in fractures. It is important to measure krw(Sw) separately for any formation for which a SAG process is envisioned. Introduction Injected gas (CO2, steam, light hydrocarbons, or N2) can recover oil effectively on the pore scale, but suffers from poor sweep efficiency (Lake, 1989). The poor sweep efficiency arises from permeability variations, segregation of injected gas under gravity, and viscous instability between gas and oil. Foam can rectify these problems (Kovscek and Radke, 1994; Schramm, 1994; Rossen, 1996). To work, foam must reduce gas mobility sufficiently to overcome or modify permeability differences between layers, and to reduce gravity override (Shan and Rossen, 2004; Rossen et al., 2010). Depending on the design of the process, foam must reduce gas mobility sufficiently in the presence of oil to divert gas flow, or reduce gas mobility in the absence of oil sufficiently to divert gas into unswept, oil-rich layers.

Surfactant-alternating-gas (SAG) is a promising method of injection, for both operational reasons (Matthews, 1989; Heller, 1994) and optimal sweep efficiency. SAG processes are especially beneficial for field application where injection pressure is constraining, because in a SAG process foam dries out and collapses in the immediate vicinity of an injection well and allows high injectivity (Shan and Rossen, 2004).

Method of Characteristics for SAG Foam Processes

The method of characteristics (MOC), also known as fractional-flow theory, is a useful tool in understanding and analysis of foam processes (Zhou and Rossen, 1995; Rossen et al., 1999; Shan and Rossen, 2004; Ashoori et al., 2010a) as it is for other EOR processes (Lake, 1989). Insights from the MOC have guided the development of improved designs to overcome gravity override (Shan and Rossen, 2004), highlighted the key mechanisms in complex foam displacements (Rossen et al., 1999), indicated the best conditions under which experiments should be conducted for scale-up to the field (Rossen et al. 1999), exposed numerical artifacts

2 SPE 134906

in simulations (Shan and Rossen, 2004), and identified mechanisms in foam models that introduce serious errors into simulations of foam processes (Namdar Zanganeh et al., 2009). If entrance effects, the width of the shock front, or time for foam to reach steady state is significant on the laboratory scale, then fractional-flow analysis based on steady-state data is a more-reliable way to scale-up a SAG flood than conducting a dynamic SAG coreflood (Xu and Rossen, 2004). Fractional-flow methods proved accurate and provided key insights into a SAG foam field trial in the Snorre field (Martinsen and Vassenden, 1999).

In a region swept of mobile oil, or in a miscible displacement of oil (Ashoori et al., 2010a), a foam process can be represented as a two-phase displacement. An implicit assumption of this work is that a foam process that cannot succeed without mobile oil present also cannot succeed with mobile oil. Three-phase MOC analysis of foam with mobile oil is complex, and to date studies have been limited to relatively simple mobility functions and oils that have no mutual solubility with injected gas (Mayberry et al., 2008; Namdar Zanganeh et al., 2009). As a starting point, with one exception these studies assumed linear relative-permeability functions in the absence of foam, including the water relative-permeability function krw(Sw). Only one of the three-phase MOC studies of foam (Namdar Zanganeh et al., 2009) modeled SAG foam processes, and in that study the SAG foam processes modeled with oil would have failed the test for foam processes without oil discussed below; that is, it would not have produced a low-mobility foam bank even in the absence of oil.

The two-phase MOC or fractional-flow analysis of foam processes proceeds briefly as follows. Details can be found elsewhere (Zhou and Rossen, 1995; Ashoori et al., 2010a). One plots the fractional-flow function for water fw(Sw) in the presence of foam, based on steady-state behavior determined in the laboratory. Though foam processes include complex dynamics (Kovscek and Radke, 1994; Rossen, 1996; Kam et al., 2007), it is now recognized that steady-state behavior controls foam processes on the field, or even the laboratory, scale (Rossen et al., 1999; Chen et al., 2010; Ashoori et al., 2010b). The fractional-flow function is defined as

( ) /( )

( ) / ( ) /rw w w

w w f frw w w rg w g

k Sf S

k S k Sμ

μ μ=

+ ......................................................................................................................... (1)

where μi is viscosity of phase i and superscript f indicates effective gas properties in the presence of foam. Strictly, it is impossible to distinguish the components of gas mobility, i.e. effective gas relative permeability and effective gas viscosity, in foam flow (Rossen, 1992), but most models make this separation either for convenience or theoretical reasons. If immobile, insoluble oil is present in the region of interest, then its saturation is present everywhere; it does not change during the displacement, but its presence alters the mobilities of gas and water and therefore must be taken into account in fw(Sw) (cf. e.g. Kloet et al., 2009).

If there is a change of surfactant concentration in the displacement, e.g. if water with surfactant solution is injected into a formation with no surfactant initially present, there is a second fractional-flow curve representing the formation ahead of the surfactant front (Zhou and Rossen, 1995; Ashoori et al. 2010a). In this paper we focus on gas injection in SAG processes with a surfactant preflush injected ahead of the gas; therefore the entire region of interest has the surfactant at the injected concentration in the water and only one fractional-flow curve applies.

Next one locates the initial condition of the formation I (a specified water saturation) and the injection condition (a specified fractional-flow of water) J on the fractional-flow curve. For gas injection in a SAG process, I is either at zero gas saturation (if no gas has previously been injected) or at residual gas saturation Sgr. J, representing gas injection, is at fw = 0. One seeks a path along the fractional-flow curve fw(Sw) from J to I with monotonically increasing (or, more precisely, non-decreasing) slope dfw/dSw. In such a displacement, each saturation in the path advances through the formation with a dimensionless velocity dxD/dtD equal to the slope of the function dfw/dSw at that saturation. (Dimensionless position is the fraction of the pore volume in the formation between the injection point and the given position. Dimensionless time is normalized by the volume of fluids injected up to that time, divided by the pore volume of the formation.) Each saturation implies a mobility and other properties corresponding to that saturation. Therefore, though MOC represents a one-dimensional (1D) displacement, one can determine from the sequence of saturations whether viscous instability would likely lead to fingering or channeling in 2D or 3D.

If the path from J to I along fw(Sw) does not have monotonically increasing slope, then there is a jump from one point along the curve to another, representing a discontinuity or shock front in saturation in the displacement. The dimensionless velocity of the shock is Δfw/ΔSw across the jump, a condition that derives from material balances on water and gas at the shock. The shock must fit into the sequence of monotonically increasing slopes (velocities) from J to I.

Figure 1 shows a typical fractional-flow function for foam (red curve) and no-foam (blue curve) along with J and I for gas injection into a medium with gas initially present at residual saturation Sgr. (If one allows for multiple steady states, the curve can become much more complex (Kam and Rossen, 2003; Kam, 2008; Ashoori et al., 2010b).) The large reduction in gas mobility caused by foam shifts the gas-water fractional-flow functions upward and to the left on such a plot. At some water saturation, corresponding to the "limiting capillary pressure" (Khatib et al., 1988), foam abruptly weakens or collapses: gas mobility rises, and fw(Sw) plunges nearly vertically to a small value fw near zero (Zhou and Rossen, 1995; Dong and Rossen, 2007).

As illustrated in Figure 1, for gas injection in a SAG process, dfw/dSw is not monotonically increasing from J to I. Instead, there is a shock from I to a point of tangency to the fractional-flow curve at a small value of fw. The properties of foam for all values of

SPE 134906 3

fw larger than the value at the point of tangency are irrelevant to the displacement except within the narrow traveling wave at the shock front (Rossen and Bruining, 2007; Ashoori et al., 2010b). The total mobility at the saturation corresponding to the point of tangency is critical to the success of the process, because mobility only increases from this point back point to J. For any foam process foam must collapse completely at connate water saturation Swc, where capillary pressure becomes extremely large (Khatib et al., 1988). Therefore, if there is insufficient mobility reduction at the point of tangency, then the foam process would fail (Zhou and Rossen, 1995). This failure can be masked in computer simulations (Shan and Rossen, 2004).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

Sw

fw

I

J Sw

*

foam fw foam-free fw shock line

Figure 1. Foam fractional-flow curve, with construction of shock front formed when injecting gas following surfactant preflush. The fractional-flow curve is nearly vertical at Sw*.

The total relative mobility λrt at this point of tangency, or at any point in the fractional-flow plot, is a simple function of the coordinates of the point and the relative-permeability function krw(Sw):

( ) /( , ) rw w w

rt w ww

k Sf S

fμλ = ........................................................................................................................................... (2)

which derives from the definition of fw(Sw). Equivalently, one could define the effective viscosity of foam at this location μ f, in effect representing foam as a single-phase fluid, as

1( )

f w w

rt rw w

fk S

μμλ

≡ = . ................................................................................................................................................. (3)

krw(Sw) Function It is widely reported that the krw(Sw) function is unaffected by the presence or strength of foam (Bernard et al., 1965; Sanchez

and Schechter, 1989; de Vries and Wit, 1990; Friedman et al., 1991). In other words, it is an inherent property of the formation, and not altered by a foam process (as long as the formation is initially water wet - if the formation is initially oil-wet, surfactant itself may reverse the wettability to water-wet (Sanchez and Hazlett, 1992)).

Different studies of foam have used different krw(Sw) functions, e.g. chierichi, Burdine and Corey functions (Vassenden and Holt, 2000). Most use Corey-type functions (Corey, 1954) for krw(Sw):

( )1

B

w wcrw w

wc gr or

S Sk S A

S S S⎛ ⎞−

= ⎜ ⎟⎜ ⎟− − −⎝ ⎠ ............................................................................................................................... (4)

where Equation (4) allows for a uniform saturation of inert, residual oil, Sor, within the region of interest. For the remainder of this paper we assume for simplicity Sor = 0 within the foam-swept zone. The Corey-type relative-permeability function is widely accepted for its simplicity. It requires limited input data and it is fairly accurate for consolidated porous media with intergranular porosity (Honarpour et al., 1986).

Jones (1966) proposes a Corey-type krw function with exponent 3 for water-gas systems. Johnson (1968) reports an approximate exponent B of 4 for water relative permeability for consolidated rocks. He reports that the exponent has a value of 3 for rocks with perfectly uniform pore-size distribution. Several other authors propose similar water relative-permeability equations with different exponents for other types of porous media. Values of 3 (Irmay, 1954) and 3.5 (Averganov, 1950) are proposed for unconsolidated sands with a single grain size, which may not be absolutely uniform in pore size but should have a narrow range of pore sizes.

4 SPE 134906

Frick’s Petroleum Production Handbook suggests exponent B=3 for well-sorted unconsolidated sand, 3.5 for poorly-sorted unconsolidated sand and 4 for cemented sandstone (oolitic limestone) for water relative permeability in drainage. Honarpour (personal communication) recommends values of B from 5 to 8 for water-wet, relatively clean sandstones.

For foam application, earlier studies in our group (e.g., Zhou and Rossen, 1995; Rossen et al., 1999; Kloet et al., 2009) use a Corey exponent B of 4.2, based on a fit to relative-permeability data of Persoff et al. (1991). Kovscek et al. (1995) use a cubic function (B=3). To date, most three-phase MOC studies of oil with foam use linear relative permeabilities (B = 1) for all phases in the absence of foam (Mayberry et al., 2008) though Namdar Zanganeh et al. (2009) extend their study to some nonlinear functions. In naturally fractured reservoirs, it is common to assume linear relative-permeability functions within the fracture network (Romm, 1996; see also, e.g,, van Heel et al., 2008; Zhang et al., 2010). Kibodeaux and Rossen (1997) and Xu and Rossen (2004) measured krw(Sw) in foam corefloods, but over too narrow range in Sw to allow reliable fitting to Corey-type functions. Xu and Rossen (2004) fit a Corey-type krw to their limited data for two experiments, in effect assuming negligible Swc and Sor. Their fit results in large exponents (A=170, B=10, and A=20 and B=7.2) not intended to apply outside the narrow range of their data.

In this paper, we show that the krw(Sw) function for a formation, which is independent of the foam process, can place limits on the mobility reduction achievable during gas injection in SAG foam processes for any conceivable foam formulation. In some cases, then, one might consider co-injection of gas and surfactant solution instead of SAG, or SAG with slugs so small that they mix nearby well and give nearly constant mobility some distance away (Faisal et al., 2009). Simulation studies show a strong advantage of SAG with extremely large slugs that give low mobility at the displacement front during gas injection (Shan and Rossen, 2004; Kloet et al., 2009), so a limit on this mobility is a serious constraint on the process. This limit derives from Equation (2) and the geometrical constraints on possible positions of a point of tangency in gas injection in a SAG process. This shows that the choice of krw(Sw) function for a formation is not simply an arbitrary choice, but an important constraint in design of a foam process for a field application. It is important to measure this function directly in the laboratory for each formation for which a SAG process is envisioned.

Results In this study, we examine three cases with increasingly restrictive but more-realistic assumptions. In the first case, it is assumed that foam can survive in any water saturation. In the second case we restrict foam existence to water saturations greater than a "limiting" water saturation Sw* that corresponds to the "limiting capillary pressure" Pc* at which foam collapses (Khatib et al., 1988; Zhou and Rossen, 1995). In the third case we assume a smooth, continuous foam-fractional-flow function near the limiting water saturation. We consider five different Corey-type relative-permeability functions (Equation (4)):

• Linear (A=B=1). Linear relative-permeability functions are often assumed to apply to fractured reservoirs (Romm, 1996). Moreover, initial MOC studies of foam with oil included linear relative-permeability functions (Mayberry et al., 2008, Namdar-Zanganeh et al., 2009).

• Quadratic (A=0.8, B=2). This is similar to the fit Kam and Rossen (2003) make to gas-water flow in unconsolidated sandpack in Collins (1961).

• Cubic (A=0.7, B=3). This function is taken from that in Kovscek et al. (1995). • Quartic (A=0.2, B=4.2): This is taken from curve-fitting of gas-water relative-permeability data in sandstone (Persoff et

al., 1991) and used in a number foam studies (e.g. Zhou and Rossen, 1995; Rossen et al., 1999, Kloet et al., 2009)). • Quintic (A=0.5, B=5): Taken from the studies of Zitha (2006) and Farshbaf Zinati et al. (2008).

We present results for two porous media with distinctly different properties. The first is an unconsolidated sand, for which we assume Swc~0, Sw*=0.015 and Sgr=0.2 (Vassenden and Holt, 2000; see also Cheng et al., 2000). For consolidated sandstone, we assume Swc~0.2, Sw*=0.37 and Sgr=0.2 (Zhou and Rossen, 1995; see also Persoff et al., 1991). Honarpour et al. (1986) report Swc higher than 20% to 25% and water relative permeability at the residual gas or oil saturation less than 0.3 for water-wet consolidated sandstones.

a) No foam collapse

We start with the assumption that foam does not collapse at any water saturation; the only restriction on the foam fractional-flow curve is that, obviously, water fractional flow must be zero at Swc, where krw = 0. The point of tangency for gas injection in SAG must lie within the triangles (below the red lines) shown in Figure 2 for the sandpack and the sandstone, respectively. This requirement derives ultimately from the material balance at the shock and the condition of monotonically increasing velocities from J to I in the displacement. Strictly, the region below the foam-free gas-water fractional-flow curve should be excluded from triangles. In the region of greatest interest, at low water saturation, however, the foam-free water fractional-flow is nearly zero and this curve is indistinguishable from the horizontal axis of the diagram (see, e.g., Zhou and Rossen, 1995; Rossen et al., 1999).

According to Equation (2), the greater the water fractional flow the lower the total mobility; hence, the most promising SAG process is one with point of tangency on the red line in Figure 2:

SPE 134906 5

1w wc

wwc gr

S Sf

S S−

=− −

. ................................................................................................................................................... (5)

0.2 0.4 0.6 0.80

0.5

1

Sw

fw

Swc

Sw*

1-Sgr

0 0.80

0.5

1

Sw

fw

Swc

Sw*

1-Sgr

(a) (b) Figure 2. Yellow triangles representing the possible locations of the point of tangency for gas injection in SAG in (a) sandpack and (b) sandstone, without presuming foam collapse at any water saturation. Therefore, the lowest possible foam mobility for any conceivable foam could be determined by inserting fw from Equation (5) and a Corey-type krw into Equation (2) or (3):

1 1

; 1 1

B B

fw wc w w wcrt

w wc gr wc gr

S S S SAS S A S S

μλ μμ

− −⎛ ⎞ ⎛ ⎞− −

≥ ≤⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟− − − −⎝ ⎠ ⎝ ⎠ . ..................................................................................... (6)

Equation (6) reveals a limit on the lowest possible total mobility of a foam bank, or greatest effective viscosity, in gas injection in SAG process for any conceivable foam, for given Corey relative-permeability parameters. Figure 3 and Figure 4 show the lowest possible foam-bank mobilities (according to Equation (6)) as a function of the water saturation at the point of tangency for the sandpack and sandstone, respectively. The more concave the water relative-permeability function (the larger the exponent B) and the smaller the pre-exponential factor A, the lower the total mobility at the leading edge of the foam bank. A larger exponent correlates with more-strongly water-wet rock, which also favors the stability of foam (Sanchez and Hazlett, 1992; Rossen, 1996). This finding also implies that foam-simulation results with a high Corey exponent could be optimistic if applied to formations with a smaller exponent, and vice versa.

With linear krw(Sw), the lowest conceivable total mobility of the foam bank in SAG is A/µw. If A = 1, the foam bank cannot have an effective viscosity greater than that of water. Thus studies with a linear water relative-permeability function are pessimistic for SAG for a formation with a more-convex krw(Sw) function.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

200

400

600

800

1000

Sw

λt (

pa.

s)-1

A=1, B=1 A=0.8, B=2 A=0.7, B=3 A=0.2, B=4.2 A=0.5, B=5

Figure 3. The lowest possible foam-bank relative mobility in gas injection in SAG, as a function of water saturation at the leading edge of the foam bank, for the five krw(Sw) functions, in sandpack; foam not assumed to collapse at any saturation. Foam effective viscosity μf in (Pa s)-1 is (1/λrt); in cp, it is (1000/λrt).

6 SPE 134906

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

200

400

600

800

1000

Sw

λt (

pa.

s)-1

A=1, B=1 A=0.8, B=2 A=0.7, B=3 A=0.2, B=4.2 A=0.5, B=5

Figure 4. The lowest possible foam-bank relative mobility in gas injection in SAG, as a function of water saturation at the leading edge of the foam bank, for the five krw(Sw) functions, in sandstone; foam not assumed to collapse at any saturation. Foam effective viscosity μf in cp is (1000/λrt). b) Foam collapses at Sw* > Swc

Khatib et al. (1988) showed the abrupt collapse of foam at a limiting capillary pressure Pc*. This finding is in agreement with a large body of evidence showing that pressure gradient is independent of gas superficial velocity in the "high-quality" or "coalescence" regime (Persoff et al., 1991; Ettinger and Radke, 1992; Osterloh and Jante, 1992; Rossen and Zhou, 1995; Vassenden and Holt, 2000; Alvarez et al., 2001; Mamun et al., 2002). The limiting capillary pressure corresponds to some water saturation Sw* below which foam is broken. In our second case we assume that there is some Sw*>Swc below which foam does not exist. It is not yet confirmed experimentally whether foam collapses completely for Sw<Sw*, (cf. Kibodeaux and Rossen, 1997; Xu and Rossen, 2004) but here we assume this is the case.

Some foam models represent a collapse of foam in the vicinity of Sw*; not at Sw* (cf. Rossen et al., 1999; Dong and Rossen, 2007); in some models foam does not collapse completely at any value of Sw (Cheng et al. 2000; Computer Modeling Group, 2006). Other models (cf. Kovscek et al., 1994; Bertin et al., 1998; Vassenden and Holt, 2000) do assume foam collapses completely at Pc*, or, equivalently, at Sw(Pc*). Here, we use Sw* to mean the water saturation at which foam collapses completely.

Since foam collapses at Sw*, fw(Sw*) ≅ 0, and the point of tangency for gas injection in SAG must be located in triangles depicted in Figure 5. Therefore, the line

*

*1w

w

ww

gr

S Sf

S S−

=− −

....................................................................................................................................................... (7)

gives the lowest possible total mobility of foam if foam collapses at Sw*>Swc. Accordingly, the lowest total mobility at the tangent point is given by substituting fw from Equation (7) and a Corey-type krw into Equation (2), as we did in previous case:

*

*

11

w

w

B

grw wcrt

w wc gr w

S SS SAS S S S

λμ

⎛ ⎞⎛ ⎞ − −−≥ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟− − −⎝ ⎠ ⎝ ⎠

. .................................................................................................................. (8)

In some foam models with an abrupt collapse of foam at Sw*, there can be more than one shock with gas injection in SAG (Dong and Rossen, 2007; cf. also Renkema and Rossen, 2007): there is a shock to a modest value of fw, a narrow spreading wave, and a second shock to a smaller value of fw representing collapsed (or nearly collapsed) foam. For geometric reasons (which derive from the material balance at the shock), however, the entire displacement must still take place within the triangles in Figure 5, and Equation (8) still applies to all foam banks.

Figure 6 and Figure 7 show the minimum possible foam mobility at the tangent point for the case of foam collapse at Sw*>Swc, applying Equation (8), for the sandpack and the sandstone, respectively. Clearly, total mobility at the point of tangency can in principle be lower for the sandpack because Sw* is closer to Swc. The limit Sw* Swc is given in the previous section.

SPE 134906 7

0.2 0.4 0.6 0.80

0.5

1

Sw

fw

Swc

Sw*

1-Sgr

0 0.80

0.5

1

Sw

fw

Swc

Sw*

1-Sgr

(a) (b)

Figure 5. Yellow triangles representing the possible locations of the point of tangency for gas injection in SAG in (a) sandpack and (b) sandstone, for foam that collapses at Sw*>Swc.

0 0.5 10

500

1000

1500

2000

Sw

λt (

pa.

s)-1

A=1, B=1

0 0.5 10

200

400

600

800

1000

Sw

λt (

pa.

s)-1

A=0.8, B=2

(a) (b)

0 0.5 10

200

400

600

800

1000

Sw

λt (

pa.

s)-1

A=0.7, B=3

0 0.5 10

200

400

600

800

1000

Sw

λt (

pa.

s)-1

A=0.2, B=4.2

(c) (d)

0 0.5 10

200

400

600

800

1000

Sw

λt (

pa.

s)-1

A=0.5, B=5

(e) Figure 6. The lowest possible foam-bank relative mobility in gas injection in SAG, as a function of water saturation at the leading edge of the foam bank, for the five krw(Sw) functions, in sandpack; foam collapses at Sw*>Swc. Table 1 gives the minimum value of mobility in each case.

8 SPE 134906

Table 1. Lowest mobility for each case in Figure 6 (sandpack). A B wS wf rtλ (Pa s)-1

1 1 0.8 1 1000 0.8 2 0.0300 0.0191 58.8750 0.7 3 0.0225 0.0095 1.6299 0.2 4.2 0.0197 0.0060 0.0058 0.5 5 0.0187 0.0048 0.0007

0 0.5 10

500

1000

1500

2000

Sw

λt (

pa.

s)-1

A=1, B=1

0 0.5 10

200

400

600

800

1000

Sw

λt (

pa.

s)-1

A=0.8, B=2

(a) (b)

0 0.5 10

200

400

600

800

1000

Sw

λt (

pa.

s)-1

A=0.7, B=3

0 0.5 10

200

400

600

800

1000

Sw

λt (

pa.

s)-1

A=0.2, B=4.2

(c) (d)

0 0.5 10

200

400

600

800

1000

Sw

λt (

pa.

s)-1

A=0.5, B=5

(e) Figure 7. The lowest possible foam-bank relative mobility in gas injection in SAG, as a function of water saturation at the leading edge of the foam bank, for the five krw(Sw) functions, in sandstone; foam collapses at Sw*>Swc. Table 2 gives the minimum value of mobility in each case. Table 2. Lowest relative mobility for each case in Figure 6 (sandstone).

A B wS wf rtλ (Pa s)-1

1 1 0.8 1 1000 0.8 2 0.5400 0.3953 649.7778 0.7 3 0.4550 0.1977 271.8406 0.2 4.2 0.4231 0.1236 25.4019 0.5 5 0.4125 0.0988 28.1895

SPE 134906 9

In every case the minimum possible mobility for a given value of Sw is at the maximum value of fw, i.e. on the red line in Figure 4. Except for the case of linear krw(Sw), there is a minimum possible value of λrt for some Sw > Sw*. The reason is that at Sw*, fw = 0 and mobility is very large (Equations (7) and (8)). For some increase in Sw beyond this, the possible increase in fw more than compensates for the increase in krw in Equation (8). For the case of linear krw(Sw), the minimum mobility is, ironically, at fw = 1: a "foam bank" with no gas injected, for which total relative mobility is A/µw. For all smaller values of Sw, λrt is larger, and much larger for foam banks with water saturation close to Sw*. One can conclude that it is difficult, if not impossible, to achieve a successful SAG process for medium with a linear krw(Sw) function if foam collapses at Sw*>Swc.

For nonlinear krw(Sw), Table 1 and Table 2 give the global minimum mobility for the cases in Figure 6 and Figure 7. Differences in the krw function can give substantially different SAG performance: more nonlinearity in the krw function allows lower total mobility in the foam bank during gas injection in SAG.

c) Smooth, continuous fw(Sw) function for Sw near Sw*

It is difficult to construct a fractional-flow function that would give a point of tangency close to the red line in Figure 5. Therefore next we consider a range of possible fw(Sw) functions and their points of tangency. Specifically, we assume that foam collapses at Sw*, as in the previous case, but the fractional-flow function decreases smoothly and continuously as water saturation approaches Sw*; fw and dfw/dSw are almost zero at Sw* because there is no foam for Sw≤ Sw*. So, one could approximate fw for Sw≤ Sw* as

*( )Dw w wf C S S= − ......................................................................................................................................................... (9)

where C and D are two parameters. This function is intended to approximate fw(Sw) for small values of fw, where one expects the point of tangency for the shock to occur. Experimental data can be used to estimate the approximate values of these parameters.

Since measuring the fractional-flow function in the range of low fw is difficult, data for fw(Sw) are scarce. For instance, Persoff et al. (1991) report steady-state foam mobility and water saturation for fw down to 0.004, but still not low enough to predict the mobility of the gas bank in a SAG process (Zhou and Rossen, 1995). Kibodeaux and Rossen (1997) measured the fractional-flow curve for one foam formulation in Berea sandstone. They weighed their core continuously to determine average water saturation during the coreflood. Foam gradually weakened as fw decreased, up to a distinct point where water saturation rose and total mobility jumped by a factor of between 10 and 20. Wassmuth et al. (2001), using NMR imaging to measure water saturation, also found fw to be multi-valued with respect to Sw. Xu and Rossen (2004) conducted experiments to measure fractional-flow curves for two surfactant formulations in Berea sandstone core, with no oil present, at extremely high foam qualities. One experiment resulted in a single-valued fw with respect to Sw while the other one repeated the findings of some previous studies: multi-valued fractional-flow function.

We fit the function introduced in Equation (9) to the experimental data of Xu and Rossen (2004) because it is the only published single-valued fractional-flow function. The power-law function fits the data reasonably well (see Figure 8) with C= 63800 and D=5.7. Since Sw* is not specified in that study, we also fit Sw*= 0.2633. The saturation behind the shock in a SAG process assuming zero residual gas (as assumed in Xu and Rossen (2004)) is found by drawing a shock line from (fw, Sw)=(1,1) to the fitted curve, which results in Sw=0.3347 and fw=0.01846.

0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.360

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Sw

fw

experimental data fitted curve shock line

Figure 8. fw, Sw at low fw: black dots are experimental data from Xu and Rossen (2004); the blue solid line is the fitted curve and the green dashed line is the SAG shock line.

For gas injection in a SAG process, the point of tangency must obey

10 SPE 134906

11

w w

w gr w

df fdS S S

−=

− − . ............................................................................................................................................... (10)

Combining Equations (9) and (10), one can find an implicit relation for Sw at point of tangency:

* * 1

1( ) ( ) (1 )D D

w w w w gr w

CS S D S S S S−=

− + − − − . .................................................................................................... (11)

The total mobility at the tangent point is given by substituting fw from Equation (9) and C from Equation (11) into Equation (2): * * 1

*

( ) ( ) (1 )( )( , )

( )

D Dw w w w gr wrw w

rt w w Dw w w

S S D S S S Sk Sf S

S Sλ

μ

−⎡ ⎤− + − − −= ⎢ ⎥

−⎢ ⎥⎣ ⎦ ............................................................................. (12)

where the value in bracket is 1/ fw at the tangent point. We choose three values of D surrounding that fit to the data of Xu and Rossen: 2, 4 and 7.

Figure 9 and Figure 10 show the foam relative mobility vs. water saturation at the tangent point for the case of foam collapse at Sw*>Swc, following the fw function in Equation (9), for the sandpack and the sandstone, respectively. We apply Equation (12) to calculate the relative foam mobility vs. water saturation at the tangent point assuming Sw

*<Sw≤1- Sgr. To save room in these two figures we show only the results for linear, quadratic and quartic krw functions.

Differentiating Equation (12) with respect to Sw, one can find the lowest total mobility at the tangent point for a given D and water relative-permeability function. Table 3 and Table 4 give the global minimum mobility for the cases in Figure 9 and Figure 10 for all five krw functions.

Requiring a smooth fw(Sw) function, rather than a point of tangency on the diagonal in Figure 5, leads to optimal foam-bank mobilities significantly greater in the sandpack, especially for less nonlinear krw (smaller B) and larger D (cf. Table 1 and Table 3). For the sandstone (Table 2 and Table 4), the greatest difference with having the point of tangency on the diagonal is for more-nonlinear krw (smaller B). The curvature of the fw(Sw) function (exponent D) has relatively little effect on optimal mobility for a given krw function (Table 3 and Table 4), though mobility is somewhat smaller for less-nonlinear functions (smaller D). This suggests that the somewhat arbitrary choice of our fw(Sw) functions (Equation (9)) has relatively small effect on our conclusions.

For given D, the krw function has a major impact on mobility in the foam bank: surprisingly, little effect on Sw at the optimal point of tangency, but a large impact on fw and mobility at this point of tangency.

The nearness of Sw* to Swc has a large impact on foam-bank mobility: in all cases but linear krw, mobility is much lower in the sandpack than in the sandstone. The difference between sandpack and sandstone is even greater for more-nonlinear krw.

Differences in the krw function can give significantly different SAG performance: more nonlinearity in the krw function allows lower total mobility in the foam bank during gas injection in SAG. As in the previous cases, a linear relative-permeability function gives poor results: the lowest total mobility of the foam bank is at the highest water saturation, for which total relative mobility is A/µw. The more nonlinear the relative-permeability curve, the lower the total mobility can be.

According to Figure 10 and Table 4, less-nonlinear krw functions (linear and quadratic for D=2 and linear, quadratic and cubic for D=4 and 7) in sandstone behave poorly for SAG and the minimum total mobility occurs at the maximum water saturation, (1-Sgr): a "foam" process without gas injection. In reality, the point of tangency would occur are lower Sw, with much greater mobility (Figure 9 and Figure 10). Even in these cases, however, the behavior for nonlinear krw is still better than the case with linear relative-permeability function. This means that an fw function that is highly concave around Sw

* can lead to a successful SAG process only for highly nonlinear krw functions.

Discussion These results show that the more nonlinear is the krw(Sw) function, the lower the potential mobility of the foam bank in a SAG process. Greater nonlinearity in this function is associated with increasing water-wetness of the formation (Honarpour, 1986). Water-wet conditions in the formation are already known to favor, or even be essential to, foam (Sanchez and Hazlett, 1989; Rossen, 1996); this work provides another advantage to strongly water-wet formations for foam, based on the water relative-permeability function and its effect on mobility in a SAG process.

The closer Sw* is to Swc, the greater the potential reduction in gas mobility in a SAG process, as illustrated here in the sandpack and sandstone. Reduced Sw* reflects increasing foam stability (Khatib et al., 1988), which likewise has long been recognized as desirable for foam. This work provides another advantage for a formulation with Sw* close to Swc based on the material balance at a shock front and constraints this puts on saturations and mobility in the foam bank.

Comparing foam in a sandpack and sandstone, Sw* can be closer to Swc in the sandpack (see, e.g., Cheng et al., 2001), probably because most sandpacks have lower capillary pressure than most sandstones, allowing foam to be stable closer to Swc. However, krw is more nonlinear for most sandstones than sandpacks; see discussion above, with B ranging in the literature from 4 to 8 for water-

SPE 134906 11

wet sandstones, and from 3 to 3.5 for sandpacks. In our examples, the nearness of Sw* to Swc in sandpacks outweighs the effect of the krw function. Mobilities with the largest values of B in Table 4 are consistently above below those for B = 3 in Table 3.

0 0.2 0.4 0.6 0.81000

35000

Sw

λt (

pa.s

)-1

A=1 ,B=1, D=2

0 0.2 0.4 0.6 0.8

0

200

400

600

800

1000

Sw

λt (

pa.s

)-1

A=0.8 ,B=2, D=2

0 0.2 0.4 0.6 0.80

50

100

150

200

Sw

λt (

pa.s

)-1

A=0.2 ,B=4.2, D=2

0.020.01

0.015

(a) (b) (c)

0 0.2 0.4 0.6 0.81000

70000

Sw

λt (

pa.s

)-1

A=1 ,B=1, D=4

0 0.2 0.4 0.6 0.8

200

400

600

800

1000

1200

Sw

λt (

pa.s

)-1

A=0.8 ,B=2, D=4

0 0.2 0.4 0.6 0.80

100

200

Sw

λt (

pa.s

)-1

A=0.2 ,B=4.2, D=4

0.016 0.03

0.02

0.04

(d) (e) (f)

0 0.2 0.4 0.6 0.81000

120000

Sw

λt (

pa.s

)-1

A=1 ,B=1, D=7

0 0.2 0.4 0.6 0.8

0

500

1000

1500

2000

Sw

λt (

pa.s

)-1

A=0.8 ,B=2, D=7

0 0.2 0.4 0.6 0.80

100

200

300

Sw

λt (

pa.s

)-1

A=0.2 ,B=4.2, D=7

0.02

0.04

0.05

(g) (h) (i) Figure 9. Foam-bank total relative mobility in gas injection in SAG, as a function of water saturation at the leading edge of the foam bank, for the three krw(Sw) functions, in sandpack; foam collapses at Sw*>Swc following the fw function in Equation (9). A and B are Corey exponents and D is a parameter in Equation (9). Table 3 gives the minimum value of mobility in each case. Table 3. Lowest relative mobility for each case in Figure 9 (sandpack) for five krw(Sw) functions.

A B C wS wf rtλ (Pa s)-1

D=2

1 1 1.623 0.8 1 1000 0.8 2 42.045 0.0303 0.0098 116.6139 0.7 3 84.720 0.0226 0.0048 3.2443 0.2 4.2 135.753 0.0197 0.0030 0.0117 0.5 5 169.749 0.0188 0.0024 0.0015

D=4

1 1 2.6334 0.8 1 1000 0.8 2 87538.24 0.0304 0.0050 232.0700 0.7 3 735883.3 0.0226 0.0024 6.4730 0.2 4.2 3051270 0.0197 0.0015 0.0233 0.5 5 5979541 0.0188 0.0012 0.0029

D=7

1 1 5.4439 0.8 1 1000 0.8 2 1.32E+10 0.0305 0.0029 405.2600 0.7 3 9.56E+11 0.0226 0.0014 11.3160 0.2 4.2 1.66E+13 0.0197 0.0009 0.0408 0.5 5 6.37E+13 0.0188 0.0007 0.0052

12 SPE 134906

0.4 0.6 0.81000

250000

Sw

λt (

pa.s

)-1

A=1 ,B=1, D=2

0.4 0.6 0.8

800

20000

40000

60000

Sw

λt (

pa.s

)-1

A=0.8 ,B=2, D=2

0.4 0.6 0.8

0

200

400

600

800

1000

Sw

λt (

pa.s

)-1

A=0.2 ,B=4.2, D=2

(a) (b) (c)

0.4 0.6 0.81000

600000

Sw

λt (

pa.s

)-1

A=1 ,B=1, D=4

0.4 0.6 0.8

800

150000

Sw

λt (

pa.s

)-1

A=0.8 ,B=2, D=4

0.4 0.6 0.850

1000

1800

Sw

λt (

pa.s

)-1

A=0.2 ,B=4.2, D=4

0.4490

100

(d) (e) (f)

0.4 0.6 0.81000

900000

Sw

λt (

pa.s

)-1

A=1 ,B=1, D=7

0.4 0.6 0.8

800

200000

Sw

λt (

pa.s

)-1

A=0.8 ,B=2, D=7

0.4 0.6 0.8

0

1000

2000

3000

4000

Sw

λt (

pa.s

)-1

A=0.2 ,B=4.2, D=7

0.4 0.45

160

170

(g) (h) (i)

Figure 10. Foam-bank relative mobility in gas injection in SAG, as a function of water saturation at the leading edge of the foam bank, for the three krw(Sw) functions, in sandstone; foam collapses at Sw*>Swc following the fw function in Equation (9). A and B are Corey exponents and D is a parameter in Equation (9). Table 3 gives the minimum value of mobility in each case. Table 4. Lowest relative mobility for each case in Figure 10 (sandstone) for five krw(Sw) functions.

A B C wS wf rtλ (Pa s)-1

D=2

1 1 5.4083 0.8 1 1000 0.8 2 5.4083 0.8 1 800 0.7 3 12.7411 0.4738 0.1372 484.6006 0.2 4.2 21.3903 0.4283 0.0727 47.5185 0.5 5 26.9776 0.4155 0.0559 53.4970

D=4

1 1 29.2500 0.8 1 1000 0.8 2 29.2500 0.8 1 800 0.7 3 29.2500 0.8 1 700 0.2 4.2 2748.633 0.4319 0.0403 91.4700 0.5 5 5952.027 0.4174 0.0300 103.9400

D=7

1 1 367.8924 0.8 1 1000 0.8 2 367.8924 0.8 1 800 0.7 3 367.8924 0.8 1 700 0.2 4.2 5663880 0.4338 0.0243 157.2600 0.5 5 28894812 0.4183 0.0178 179.5300

SPE 134906 13

Conclusions We derive geometric constraints that limit the region in the fractional-flow diagram in which the point of tangency defining the foam bank for a SAG process can occur. For a given krw(Sw) function, this limits the mobility reduction achievable for any possible SAG process. We examine the implications of this limitation for various krw functions assuming three different foam behavior.

• Increasing nonlinearity of the krw function is advantageous for SAG processes, regardless of how foam reduces gas mobility.

• SAG is inappropriate for naturally fractured reservoirs if straight-line relative permeabilities apply, even if extremely strong foam can be stabilized in fractures: If foam does not collapse at any water saturation, the lowest conceivable total mobility of the foam bank in SAG is A/µw with linear krw(Sw). If foam collapses at Sw*>Swc, it is difficult, if not impossible, to achieve a successful SAG process for medium with a linear krw(Sw) function.

• It is important to measure krw(Sw) separately for any formation for which a SAG process is envisioned. Acknowledgment We thank M. M. Honarpour for useful discussions of krw(Sw) functions in petroleum reservoirs. Nomenclature For equations in this text, it is only important that a consistent set of units be employed. IS units are specified below. fw = water fractional flow function krw = water relative permeability f

rgk = gas relative permeability in presence of foam Pc = gas-water capillary pressure, Pa tD = dimensionless time xD = dimensionless position in 1-D Sw = water saturation Swc = connate water saturation in a water-gas system Sgr = residual gas saturation in a water-gas system Sor = residual oil saturation in a water-gas-oil system *

wS = critical water saturation A = Corey parameter in Equation (4) B = Corey parameter in Equation (4) C = parameter in Equation (9) D = parameter in Equation (9) I = initial state J = injection state Greek Symbols λrt = total relative mobility of water and gas, (Pa.s)-1 fμ =effective viscosity of foam defined in Equation (3), Pa.s f

gμ = gas viscosity in presence of foam, Pa.s μw = viscosities of water, Pa.s

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