# wormlike micelles in oilfields

The rich phase behaviour, rheological diversity, and responsiveness to a variety of chemical and physical triggers exhibited by colloidal liquids make them ideal building blocks for a wide variety of commercial fluids. To meet the specific demands of each application, such commercial liquids are usually multicomponent, multiphase and complex in composition, structure and dynamical behaviour. Their design involves an equally complex formulation process, in which the number of parameters is so large that the approach is often empirical and far from optimised. This is of particular importance for designing industrial responsive oilfield fluids.

In this project we worked together with scientists at Schlumberger Cambridge Research and focused on the design of new generation fracturing fluids. The production of oilwells is sometimes limited by low natural reservoir permeability. The creation of hydraulic fractures is one major stimulation commonly used to overcome these limitations. This is usually done by injecting polymer gels, which transmit hydraulic pressure to the rock to induce fractures, into which proppants are transported to keep the fractures open on removal of the fluid pressure. Once the proppant is in place, delayed oxidative or enzymatic breakers are used to degrade the gel retained within the fracture. A disadvantage of many of these polymeric fluids is that, despite the gel-breaking step, small cross-linked fragments comparable in size to the proppant pore throats reduce the hydraulic conductivity significantly. An important new generation of fracturing fluids based upon wormlike micelles, forming viscoelastic surfactants (VES), has recently been introduced which attempts to overcome this problem. Similarly to polymer gels, entanglements between these micelles impart viscoelastic properties to the solution, i.e. the wormlike micelles function as thickening and rheology-control agents. However, once the wormlike micelles come into contact with hydrocarbon produced from the fracture through the proppant pack, the VES revert to small spherical micelles or micro-emulsions, the fluid viscosity/elasticity falls by orders of magnitude, and the fluid residues flow easily out of the pack (see Fig. 1).

**Fig. 1. A schematic diagram of the production of wormlike micelles, their organisation into entangled networks, and their destruction by oil.**

Much progress in the design and development of VES fracturing fluids could be made if a fundamental understanding of the relationship between interactions at the molecular and colloidal (wormlike micelle) levels on the one hand, and the thermodynamic and transport properties on the other hand, is achieved. Also, for optimal performance, it is important to understand the strong coupling between shear/extensional flow and structure and phase behaviour (shear banding, shear induced structure, thixotropy) of such fluids. Such understanding and the associated methodology will also be applicable to a range of industrial fluids based on structured surfactants, across a wide range of applications and industries.

# Results from a bead-spring model

Surfactant molecules in solution have a strong tendency to reversibly assemble into extended structures. Depending on the molecular geometry, the mesoscopic packing units can range from small spheres (conventional micelle), through long, sometimes flexible cylinders (wormlike micelles), on to bilayers. Wormlike micelles are subject to reversible scission and recombination and can be viewed as living meso-polymers. The diameter of such a micellar polymer is typically of the order of 3 nm and typical persistence lengths are 10 - 30 nm. The contour length is variable, governed by thermodynamic equilibrium, and in some systems extremely large (up to about 1 mm).

It is impossible to reach the sufficiently long length and time-scales to study the most interesting rheological phenomena by direct atomistic molecular dynamics simulations. Coarse-graining techniques are therefore critical to the advancement of understanding of VES fluids, and have been extensively studied in the literature.

We started by exploring the properties of a previously known simulation model for wormlike micelles, the Finitely-Extensible Nonlinear Elastic spring, Cut-bond version (i.e. the bond breaks if the tension is beyond a certain predefined value): abbreviated to FENE-C. We found evidence that, at high concentrations, the recombination kinetics in this model cannot be described by a mean-field approach, but is diffusion controlled and dominated by self-recombination events. We then studied the influence of shear flow on the formation of rings in the FENE-C model. We found that under equilibrium conditions, rings are dominating in dilute solutions, while linear chains are dominating in strongly overlapping and concentrated solutions. Importantly, we found that shear flow induces a net shift of micellar mass from linear chains to rings. At the same time, the average aggregation size of linear chains is decreasing, while the average aggregation size of rings is increasing.

We hypothesize that the increased abundance and size of rings is caused by a decreased entropy gain associated with ring opening under shear flow. Linear chains and rings are elongated in the flow direction and contracted in the gradient direction. This leaves an essentially two-dimensional free volume, which two newly created chain ends can explore after being disconnected. We studied the ratio of ring and linear chain distribution functions to substantiate this hypothesis. Finally, we studied the rheology and discuss how the observed increase of ring abundance can provide a positive feedback between strain and ring connectivity. Such a positive feedback can contribute to shear thickening behavior, observed in micellar solutions near the overlap concentration.

- J.T. Padding and E.S. Boek,
*Evidence for diffusion-controlled recombination kinetics in model wormlike micelles*, Europhys. Lett.**66**, 756 (2004). - J.T. Padding and E.S. Boek,
*Influence of shear flow on the formation of rings in wormlike micelles: A nonequilibrium molecular dynamics study*, Phys. Rev. E**70**, 031502 (2004).

# Constitutive equations

We made a small digression into solving a long-standing problem with a particular constitutive equation for wormlike micelles. A constitutive equation is the relation between the deformation (rate) of a fluid and the resulting stress in that fluid. There are many different possible constitutive equations, and the "art" is to pick (or construct) the right one for a particular fluid and then to fit its parameters to experimental results. The particular constitutive equation we looked at had convergence problems in flow-solving programs, particularly when relatively large extensional flow rates were present. We traced the cause of the problem, which in fact is related to the anomalous divergence of extensional stress in other, similar, constitutive equations, and offered a possible solution.

- E.S. Boek, J.T. Padding,
*et al*.,*Constitutive equations for extensional flow of wormlike micelles: stability analysis of the Bautista-Manero model*, J. Non-Newtonian Fluid Mech.**126**, 39 (2005).

A problem with constitutive equations, as well as the bead-spring model discussed above, is that the link with the chemistry of the micelles is missing. The parameters of constitutive equations need to be found by comparing the results with experiments. Similarly, in case of bead-spring models, the parameters as well as the relation between simulation time and length scales can only be found by mapping onto experimental results, if at all possible.

# Mesoscopic model

A more fundamental understanding of the relationship between interactions at the molecular and micellar levels on the one hand, and the thermodynamic and transport properties on the other hand, can be achieved by a multi-scale modeling approach. In our research we simplified the extended hierarchy to two levels of coarse-graining:

- From microscopic atomistic models to particle based mesoscopic models. Here the emphasis is on how microscopic chemical details of the individual surfactant molecules lead to specific properties, like the bending rigidity and persistence lengths of amphiphilic bilayers and worms.
- From mesoscopic models to rheological properties. At this level, properties like the persistence length, bending rigidities, and mesoscale objects such as micellar entanglements are linked to rheological behaviour.

A good understanding of both levels is necessary for industrial applications. Questions at level 1 help formulation chemists design the right kind of surfactant molecules, while understanding at level 2 helps engineers design the desired physical properties for a particular application.

We have investigated both level 1 and level 2 type coarse-graining schemes. In the case of polymer melts, a direct coarse-graining from an atomic level to a mesoscopic "soft blob" model was possible. For these surfactant systems this was too large a step to begin with. The smallest length-scale of a worm-like micelle, which is of relevance for its rheological behaviour, is the persistence length. In a first instance we have described long worms by means of a rod-spring model, dubbed MESOWORM. The rods in this model have a length of one persistence length, but for simplicity have no excluded volume. In order to detect and process possible entanglements the twentanglement algorithm, which was previously applied to the case of coarse-grained polymer melts, has been adapted to these new systems. A mechanism was added by which bonds can break and reform. Brownian Dynamics (BD) simulations have been performed with the above model under various conditions of flow. Because of the high worm concentration, hydrodynamic interactions are largely screened, so that they can be ignored in a first instance.

We have investigated the equilibrium properties, linear rheology and non-linear rheology of this model and generally find good (often quantitative) agreement with experiments. Recently, we have also used the model to study the flow of wormlike micelles though contraction-expansion geometries.

- J.T. Padding, E.S. Boek, and W.J. Briels,
*Rheology of wormlike micellar fluids from Brownian and Molecular Dynamics simulations*, J. Phys.: Condens. Matter**17**S3347 (2005). - E.S. Boek, J.T. Padding, W.K. den Otter, and W.J. Briels,
*Mechanical properties of surfactant bilayer membranes from atomistic and coarse-grained molecular dynamics simulations*, J. Phys. Chem. B**109**, 19851 (2005). - E.S. Boek, J.T. Padding, V.J. Anderson, W.J. Briels, and J.P. Crawshaw,
*Flow of entangled wormlike micellar fluids: mesoscopic simulations, rheology and micro-PIV experiments*, J. Non-Newtonian Fluid Mech.**146**, 11 (2007). - M.R. Stukan, E.S. Boek, J.T. Padding, W.J. Briels and J.P. Crawshaw,
*Flow of wormlike micelles in an expansion-contraction geometry*, Soft Matter**4**, 870 (2008). - M.R. Stukan, E.S. Boek, J.T. Padding and J.P. Crawshaw,
*Influence of system size and solvent flow on the distribution of wormlike micelles in a contraction-expansion geometry*, European Physical Journal E**26**, 63 (2008). - J.T. Padding, E.S. Boek and W.J. Briels,
*Dynamics and rheology of wormlike micelles emerging from particulate computer simulations*, J. Chem. Phys.**129**, 074903 (2008). - J.T. Padding, W.J. Briels, M.R. Stukan, and E.S. Boek,
*Review of multi-scale particulate simulation of the rheology of wormlike micellar fluids*, Soft Matter**5**, 4367 (2009).

# Collaboration

During my stay at the University of Cambridge from 2003 until 2006 I was financially supported by the EPSRC through the Impact Faraday programme. Through this grant I was able to collaborate with scientists at Schlumberger Cambridge Research, particularly with Dr Edo S. Boek. From 2006 until 2008 a new postdoc, Dr Mikhail R. Stukan, has continued the research and implemented and studied wormlike micelles in contraction-expansion geometries.