**4.6 Effect of salinity**

 The shape of surfactant micelles (known as aggregates of surfactant molecules) changes with the change in surfactant packing parameter. The packing parameter is expressed as P = v/aolc, where ao is the area of the surfactant headgroup, v is the volume of the surfactant tail, and lc is the tail length of the surfactant molecule [74]. Due to the increase in salinity, the transformation of spherical shape micelles of ionic surfactants into wormlike micelles takes place. These micelles are elongated spherocylindrical having two hemispherical end caps and a cylindrical body. The neutralization of repulsive forces between micelles due to the addition of salt reduces the effective area of surfactant head and alters the packing parameter of surfactant [14, 75]. The wormlike micelles entangle with each other and generate three dimensional networks, which impart viscoelasticity, and therefore, the behavior of surfactants becomes similar to that of viscoelastic polymer solutions [74]. Anionic surfactants have the ability to form such wormlike micelles when sufficient electrolyte is added [76]. For surfactants, such as sodium lauryl ether sulfate (SLES) and sodium dodecyl sulfate (SDS), the addition of electrolyte increases the surfactant packing parameter as well as solution viscosity [74]. It has also been reported that surfactant ability to form a strong foam depends on its hydrophilic/lipophilic balance (HLB) which may vary due to the addition of salinity [77]. Foam of decyltrimethylammonium bromide (DTAB) surfactant has been reported to decrease the foam viscosity with the increase in salinity; however, the cetyl trimethylammonium bromide (CTAB) provided an increasing trend as the salinity in the solution was increased. Another surfactant, Mackam CB-35, by Rhodia provided a decrease in foam viscosity until 3 wt% salinity, and beyond that a prominent increase in foam viscosity was reported [77].

### **4.7 Foam rheological models**

The rheology of foam determines various characteristics of fracture growth, and therefore, it is important to accurately estimate the rheology in order to predict the fracture geometry. Different rheological models have been developed that describe

the foam flow behavior, from the widely used power law model to the Herschel-Bulkley model, which have different degrees of success [8, 32].

The Herschel-Bulkley model incorporates a yield stress for non-Newtonian fluid. In this model, the yield stress becomes negligible at high shear rate and the model becomes similar to the power law model. Mathematically, the Herschel-Bulkley model is presented as Eq. (2) [8].

$$
\pi = \pi\_o + K\gamma^n \tag{2}
$$

 where τ is the shear stress, γ is the shear rate, τ*o* is the yield stress, *K* is the consistency index, and n is flow behavior index.

 Power law or Ostwald-de Waele model is one of the most commonly used models for describing the non-Newtonian behavior of foam [8, 63, 78–80]. The power law model can be mathematically expressed as shown in Eq. (3) below [63, 81].

$$
\mu = K \gamma^{n-1} \tag{3}
$$

where *μ* is the viscosity, *K* is the flow consistency index, γ is the shear rate, and n is the flow behavior index.

 A straight line appears when log μ is plotted versus log γ. By taking the logarithm of both sides of Eq. (5), the parameters of the power law model can be determined as shown below.

$$
\log \mu = \log K + (n - 1)\log \gamma \tag{4}
$$

If the solution viscosity of the solution is plotted against the corresponding shear rate on a log-log paper, a straight line appears with the intercept as *K*  at a shear rate (1/s), and as shown in **Figure 7**, (*n* − 1) will be the slope of the straight line.

 The n value explains the behavior of the solution, i.e., when *n* < 1, the fluid shows shear thinning behavior, whereas for the shear thickening fluids, *n* > 1. For foams, *n* < 1.0 indicates a pseudoplastic behavior. The extent of shear thinning behavior of solutions can be quantified by the value of *n*. The value of n is significantly lower than unity if the solution is highly shear thinning, whereas if the n value is equal to unity which is the case of Newtonian fluid, the *K* value will become the Newtonian viscosity.

Foam behavior indices (*K* and *n*) are the function of foam quality, chemical concentration, temperature, and pressure. The rheological behavior of the foam is somewhat similar to the polymers. Foam system is considered complex and its model parameters are reliant on foam geometry, temperature, pressure, and foam properties [8, 41, 78]. Previous studies performed on foam rheology concluded that it is important to control various parameters such as gas volume fraction (i.e. foam quality), foam texture, pressure, temperature, chemical types, concentrations, etc. while measuring the foam apparent viscosity [82]. Many studies also reported higher performance of CO2 foam fluid with higher recoveries as compared to other fluids [21, 83]. However, it is difficult to understand and model the behavior of such energized fluid [83].

 Ahmed et al. investigated the effect of various process variables such as pressure, temperature, salinity, surfactant concentration, and shear rate on CO2 foam apparent viscosity under high pressure high temperature conditions and presented a set of empirical correlations [39, 84]. In their study, the polymer free foam was generated using a conventional surfactant, i.e., alpha olefin sulfonate (AOS) and a

*CO2 Foam as an Improved Fracturing Fluid System for Unconventional Reservoir DOI: http://dx.doi.org/10.5772/intechopen.84564* 

**Figure 7.**  *Power-law model [8, 78].* 

 foam stabilizer, and it was noticed that all the aforementioned process variables are strongly dependent on foam apparent viscosity (discussed above in Section 4). All the foams exhibited a typical shear thinning behavior within the tested shear rate range (10–500 s<sup>−</sup><sup>1</sup> ) and the power law model was fitted on experimental data. They presented a set of empirical correlation to predict apparent viscosity of CO2 foam. Power law indices were found to be strongly dependent on all process variables. The new equations for K and n were developed as a function of process variables which were then substituted in the power law model. These correlations could cover a wide range of conditions and were found accurate in predicting the viscosity of CO2 foam fracturing fluid. These developed models could be integrated into any fracturing simulator in order to evaluate the efficiency of foam fracturing fluid.

 Gu studied foam fracturing using polymer free foam considering ultralightweight proppants (ULWPs) [8, 32]. They also developed empirical correlations through the modification of the power law model, which were then applied in a fracturing and reservoir model using a commercial simulator CMG IMEX. They used ULWPs to predict the formation productivity with both slickline and polymer free foams. They have been able to present foam-based hydraulic fracturing fluid which has efficiently propped the fractures and utilized less water compared to that of slickline fluid. Furthermore, he evaluated the designed foam fluid and proppant using a combined experimental and computational modeling technique which helped in identifying the optimal proppant amount and gas liquid fraction (or quality) of foam.

### **4.8 Experimental study of foam rheology**

 Numerous experimental studies used pipes with a small diameter to investigate foam rheology. This is a more reliable method of studying foam behavior in wellbores. Foam deteriorates due to its unstable nature which is caused by liquid drainage under the action of gravity [85]. Accumulation of the liquid takes place at the bottom of the samples and foam cannot be taken as a homogeneous system. Foam is made to flow through a steel recirculation loop in which pressure drop over a certain length is measured and apparent viscosity was calculated. Hagen-Poiseulle equation is used to compute the apparent viscosity of foam in pipe or tubing and it is represented in Eq. (5) [41, 53, 57, 74, 77].

*Exploitation of Unconventional Oil and Gas Resources - Hydraulic Fracturing...* 

$$
\mu\_{app} = \frac{D^2 \Delta P}{32LU} \tag{5}
$$

 where *μapp* is the foam apparent viscosity, D is the diameter of tubing, ΔP is the differential pressure between the test sections, L is the tubing length, and U is average velocity determined from the total volumetric flow rate of foam.

 Before carrying out any measurements, a constant shear rate needs to be set to ensure uniformity across the foam. Once the foam is equilibrated in the recirculation loop, the pressure drop is measured rapidly at different flow rates while ensuring that the foam texture does not vary over time. Patton et al. measured foam viscosity as a function of shear rate using a viscometer apparatus [86]. They mixed constituents of the foam and passed it through the foam generator, i.e., packed bed. Flow rates, temperature, and pressure drop were measured after displacing the foam through the small diameter tube [87].

Xue et al. [74], Li et al. [57], and Sun et al. [41] recently used a flow loop system for fracturing foam studies by using CO2 foam. These rheology studies involved foam as fracturing fluid and the investigation was made at downhole condition using a flow loop system. The effects of temperature, pressure, foam quality, and shear rate on fracturing foam were studied. Sudhakar and Shah (2002, 2003), Bonilla and Shah [88], and Sani et al. (2001) used a recirculation loop rheometer to investigate the rheology of polymer foam and the power law behavior was observed [88]. These experiments determined that foam behaves as a non-Newtonian fluid. The foam apparent viscosity decreases as the shear rate increases and such behavior is termed as pseudoplastic.
