Bingham’s model in the oil and gas industry

The classic Bingham relationship provides simplified characterization of drilling fluid rheology. Established American Petroleum Institute (API) standards for assessing drilling fluid rheology stipulate torque data acquisition at 300 and 600 rpm using a Fann ^Ⓡ Model 35 viscometer, providing fitted values of the Bingham viscosity and yield stress. Quiescent wait times of 10 s or 10 min, followed by slow shear deformation, provide 10-s or 10-min gel strengths, respectively (American Petroleum Institute 1980). In addition to the common API protocols, yield points are also accessible using stress sweep or oscillatory amplitude sweep protocols.

Most service companies use additional viscometer readings in fluid design and laboratory characterization (typically 6 or 12, on a logarithmic scale), so that other rheological parameters may be fitted. Thus, the Herschel-Bulkley model has become a common standard, replacing and incorporating the earlier 2-parameter Bingham and power law models. The enduring popularity of these models operationally stems from the availability of analytical and semi-empirical closure expressions and approximations for hydraulic design calculations, dating from the late 1950s and onwards. These approaches are summarized in, e.g., Bourgoyne et al. (1986) and Govier and Aziz (1977), but are continually evolving. Many companies include internally researched results and/or geometry-specific approximations (e.g., the eccentric annulus) that make their predictions distinct, and such calculations are generally embedded within proprietary engineering design software that also calculate many other features relevant to drilling operations, e.g., torque and drag, hole cleaning parameters, swab/surge.

At the rig site, the drilling fluid is the responsibility of the mud engineer. The job here involves constant monitoring and adjustment. Mud weight (density) is the most important property controlled, followed by yield point and solid (cuttings) content. Drilling fluids in circulation are constantly changing, due to the incorporation of fine particles from the drill cuttings and due to mechanical degradation. Thus, initial designs of rheology in the lab are different from those that evolve on the rig. The mud engineer adjusts the drilling fluid rheology in response to monitoring and measurement. In the high-pressure operational environment, standardization of protocols and ease of application are the key. Continual rheological measurement is conducted using the Marsh funnel (Marsh 1931). Although the basic funnel design is still from the 1930s, efforts have been made to improve interpretation of the readings (Balhoff et al. 2011; Guria et al. 2013), and we should note that what is monitored with this apparatus is rheological change. Drilling rigs are mostly equipped with standard 6- or 12-speed viscometers, which are typically used daily to quantitatively characterize the mud shear rheology.

Conceptual simplicity and the above outline of design and operational procedures helps understand why models such as the Bingham fluid, traditionally, have played and will continue to play an important role in oil and gas well drilling. More complex rheological features (reviewed below) are certainly of importance and are incorporated in fluid design. Indeed, this industry is remarkably innovative rheologically. However, pragmatism and inertia, together with absence of clearly defined and widely accepted new standards, maintain Bingham’s name. Another reason for the adoption of these models concerns the study of more complex fluid flows, beyond hydraulics, e.g., solid transport, conditioning and displacement flows, fluids loss. Where the flow itself is complex and there is a high degree of process uncertainty (geometry, in-situ rheology, etc), the first aim industrially is to understand the leading order effects of intuitively understood and accepted parameters, e.g., yield stress, shear-thinning, and viscosity.

Thixotropy is natural in many drilling fluids due to their composition. Although it might be thought that rapid aging and development of a large static gel strength would be ideal for solid suspension in static conditions, this also makes re-establishing mud circulation and pipe movement difficult, so that in practice, a compromise is sought and the net benefits of thixotropy to drilling are under constant review.

In short and medium distance wells, thixotropy generally benefits drilling operations. During static conditions, which occur during breaks in fluid circulation, thixotropic structural buildup prevents barite sag and provides suspension of the rock cuttings. In conventional water-base drilling fluids formulated with bentonite clay, attractive forces arise between opposing electric charges located on the basal and edge surfaces of the bentonite platelets, driving assembly of a colloidal gel structure at quiescent and low-shear-rate conditions (similarly with sepiolite, laponite, or montmorillonite particles). The colloidal structure imparts a yield stress to the fluid. Thixotropic structural buildup allows a strong gel to form with a relatively low clay content. Upon resumption of shearing, the colloidal gel structure undergoes fragmentation, driving a thixotropic reduction in viscosity. During continuous circulation, low viscosity facilitates efficient removal and transport of cuttings as well as efficient energy transfer to the mud motor. Thixotropic viscosity reduction thereby facilitates high drilling penetration rates by reducing energy losses associated with the drilling fluid in contact with the drill string and bit. Thixotropic viscosity reduction also facilitates efficient separation of rock cuttings and entrained gas in surface separation units where fluid agitation is maintained. In sum, thixotropic structural buildup and viscosity reduction facilitate efficient drilling operations in conventional wells.

However, in extended reach and deepwater wells, the balance shifts. Thixotropy contributes to detrimental pressure swings (surge/swab pressures) arising in the borehole during operations such as casing insertion, drill string positioning, cementing, and the initiation of circulation. Thermal disparities along the flow path of the drilling fluid exacerbate the pressure swings. Deepwater wellbores are particularly vulnerable to pressure fluctuations. In deepwater reservoirs, the envelope between local pore pressure and local fracture pressure is often narrow, as it is with extended reach horizontal wells. In order to prevent formation fracturing and intrusion of formation fluids into the wellbore, downhole pressure conditions must be maintained within the pore-frac pressure envelope. Pressure variations exceeding these limits may ultimately compromise the integrity of the well or formation. Formation fractures usually lead to substantial loss of drilling fluid to the formation. Conversely, large pressure reductions may lead to hole collapse or invasion of reservoir fluids (loss of primary well control). In deepwater drilling, conventional drilling fluid formulations are typically unable to maintain borehole pressures within the respective limits, due to a combination of thixotropic and temperature-dependent fluid rheology.

Most cementing operations involve a pipe flow from surface down the well. Cement slurries are usually denser than drilling fluids, so that this displacement process is frequently mechanically unstable. Efforts are made to separate fluids physically with rubberized plugs, but operational constraints mean that these are frequently missing or only separate one or two interfaces. In plug cementing and remedial operations, smaller diameter tubing is common and separating plugs are not common. Consequently, it is of interest to study density unstable displacement flows of miscible fluids in long inclined pipes.

Miscible Newtonian displacement flows in pipes have been studied for many years. High Péclet number flows at low-moderate Reynolds numbers have been studied computationally (Chen and Meiburg 1996) and experimentally (Petitjeans and Maxworthy 1996), for limited ranges of pipe inclination and density differences. Effects of flow rate and viscosity ratio were studied in vertical displacement flows by Scoffoni et al. (2001), identifying stable finger, axisymmetric and corkscrew modes. Other experimental studies of vertical displacement flows include (Kuang et al. 2004; Balasubramaniam et al. 2005) investigating instabilities due to viscosity and density effects. All these flows are more structured than those found in cementing, which although laminar are significantly inertial, buoyant and include non-Newtonian effects.

A systematic extension of these studies towards cementing displacements is ongoing, focusing initially on Newtonian fluids, buoyancy, viscosity differences, effects of pipe inclination, and flow rate. The effects of increasing the mean flow velocity (\(\hat {V}_{0}\)) on near-horizontal displacement flows are studied in Taghavi et al. (2010), identifying three main regimes as \(\hat {V}_{0}\) was increased from zero. At low \(\hat {V}_{0}\), the flow resembles the exchange flows of Seon et al. (2005). As \(\hat {V}_{0}\) is increased, the front velocity \(\hat {V}_{f}\) was found to vary linearly with \(\hat {V}_{0}\). The first two of these regimes may be either viscous or inertial-dominated. When the mean speed is further increased, we enter the turbulent regime where \(\hat {V}_{f}=\hat {V}_{0}\). The behavior of the trailing displacement front was studied in Taghavi et al. (2011). A synthesis of the results on iso-viscous nearly horizontal displacement flows is presented in Taghavi et al. (2012c), based on a mix of experimental, numerical, and analytical results. These studies have been extended to the full range of pipe inclinations (Alba et al. 2013a), partly also to density stable displacements (Alba et al. 2012). Ongoing work is focused on studying viscosity ratio effects and shear-thinning behavior, where a variety of interesting instabilities are found.

Regarding yield stress effects, the field is less well explored. When the displaced fluid has a yield stress, it is possible for the flow to leave behind residual fluid layers stuck to the wall, which remain permanently. These are illustrated in the elegant study of Gabard-Cuoq (2001) and Gabard-Cuoq and Hulin (2003) in which vertical displacement of Carbopol solutions by glycerin results in beautifully uniform stationary residual layers. More recent work has focused on the case of a dominant yield stress (e.g., a drilling mud that is hard to displace) and displacing with density unstable Newtonian fluids; see Taghavi et al. (2012b), Alba et al. (2013), and Alba and Frigaard (2016). These flows result in two primary flow types: central displacement and slump displacements, distinguished parametrically by an Archimedes number. The slump displacements show a wonderful range of complex flow patterns, including those that rupture the displaced fluid and spiral patterns; see, e.g., Fig. 1. The stratified viscous regimes of Taghavi et al. (2010) and Taghavi et al. (2012c) have been modelled for two Herschel-Bulkley fluids; see Moyers-Gonzalez et al. (2013), but experimental reality in cementing regimes rarely conforms to the strict model assumptions. Ongoing research has studied the central regime extensively (in the absence of any density difference; Moises 2016) and studied vertical pipes with a range of positive and negative density differences.

The second and most critical displacement geometry is the annular space formed by the outside of the steel casing and the inside of the borehole. Typically, the mean annular gap is in the range 1–3 cm, but even when wells are vertical, the annulus is eccentric. Modern wells typically start with a vertical section (surface casing) and end up aligning directionally with the reservoir (production casing). Cemented sections are typically many hundreds of meters long, and the diameters of the steel casings decrease with depth. The annulus is initially filled with drilling mud which should be pre-circulated for conditioning prior to the displacement. Displacing fluids enter the annulus at the bottom and move upwards to surface: the detrimental unstable density difference inside the casing is now stabilizing. Whatever mixing has occurred inside the casing between fluids is now transferred to the annular displacement.

The majority of fluid mechanic studies have focused on laminar displacement flows. A popular approach has been to average the velocity field across the narrow annular gap, thus reducing the flow to a 2D problem for the gap-averaged velocity field. The earliest developments were by Martin et al. (1978). A further-simplified pseudo-2D approach was developed and validated against a series of experiments in Tehrani et al. (1992, 1993), and this style of model was also derived and solved computationally in Bittleston et al. (2002). Fully 2D computations, a rigorous analysis of the model and comparisons with some of the rule-based systems can be found in the series of papers (Pelipenko and Frigaard 2004a, b, c), targeted at near-vertical displacements. For example, in Pelipenko and Frigaard (2004c), it is shown that rule-based systems such as the earlier (Couturier et al. 1990), although physically sensible, can be extremely conservative in the requirements needed for an effective displacement. Near-vertical experiments and model comparisons were made in Malekmohammadi et al. (2010). Strongly inclined and horizontal wells have been studied in Carrasco-Teja et al. (2008a, b) and more recently, the effects of casing rotation have been studied in Carrasco-Teja and Frigaard (2009, 2010) and Tardy and Bittleston (2015). Qualitatively, this level of modelling is adaptable to rather complex wellbore geometries and has been shown to identify bulk features of the flow, such as mud channels remaining stuck on the narrow side of the wellbore, see Fig. 2, for an example. Such models are appropriate for process design and predict well the dominant effects of wellbore eccentricity, rheology, density differences, and inclination. Variants of this approach are increasingly widely used in industry, e.g., Tardy and Bittleston (2015), Guillot et al. (2007), Chen et al. (2014), and Bogaerts et al. (2015). It is interesting to reflect that the above approach is mathematically analogous to the LPG reservoir flows outlined in “Reservoir flows of visco-plastic heavy oils,” with varying annular gap width corresponding varying permeability.

Aside from Tehrani et al. (1992, 1993) and Malekmohammadi et al. (2010), other experimental studies include that of Jakobsen et al. (1991) that investigated a subset of density and rheology differences, eccentricity, inclination, and Reynolds number. A number of authors have studied the annular flows in 3D computationally. For example, Szabo and Hassager (1995, 1997) studied Newtonian displacements in eccentric annular geometries. Comparisons between the 3D computational fluid dynamics (CFD) results of Vefring et al. (1997) and earlier experiments of Jakobsen et al. (1991) are generally favorable. In a modern era of massively parallel computation, one might ask why 3D CFD has not had more impact? The first point here is that advantages over the 2D models come from resolving the scale of the annular gap (cm scale). 3D meshes at that resolution become unmanageable over circumferential distances of ∼0.5 m and wellbore lengths of many hundreds of meters (e.g., \(\gtrsim 10^{9}\) mesh nodes). Secondly, many of the critical features of mud removal displacements concern the yield stress and the residual fluid left behind in the annulus. Reliable implementation of yield stress models into CFD codes, in a way that resolves the unyielded regions properly, results in considerable additional computational iteration compared to a Newtonian fluid flow. Thirdly, there is a question of resolution, data processing, and analysis: the coarse-graining of an averaged approach leads to fairly simple interpretations of displacement results, in much of the annulus nothing much is happening, etc.. Most critical however is certainly the large dimensionless parameter space discussed earlier (10–12 parameters). This rules out systematic study on the scale of the wellbore. Experiments also have issues of scale. In lab scale displacements, the annular lengths used are limited (typically <10 m), which makes interpretation of these studies harder in comparison to 2D models on the wellbore scale.

A different way of resolving the through-gap distribution of fluids is to consider longitudinal sections of the narrow annulus, i.e., as a plane channel displacement. Firstly, lubrication/thin-film approaches have been used, giving a simplified pseudo-2D prediction. This approach dates back to Beirute and Flumerfelt (1977), but with errors in the derivation. Symmetric displacements were considered by Allouche et al. (2000) and inclined channels by Taghavi et al. (2009). The latter work has been extended to include weak inertial effects (Alba et al. 2013b) and more recently to converging-diverging 2D sections by Mollaabbasi and Taghavi (2016). These models allow one to predict the maximal layer of drilling mud that may remain stationary on the wall of the channel (=annulus) and to predict qualitative behaviors of the displacement fronts. Fully 2D simulations and analysis can be found in Taghavi et al. (2012a, c), Allouche et al. (2000), Wielage-Burchard and Frigaard (2011), and Alba et al. (2014). These simulations cover a limited subspace of parameters, which is being currently extended. Figure 3 shows an example of a displacement of a Bingham (drilling) fluid by a Newtonian (spacer) fluid at along a uniform channel of width \(\hat {D}\) at mean imposed velocity \(\hat {V}_{0}\). Two different viscosity ratios are considered: thicker static layers are evident for the more viscous displaced fluid. The focus of these studies is to predict the so-called micro-annuli, i.e., annular wall layers of undisplaced mud extending along the wellbore. As the cement eventually hydrates, these layers dry into porous longitudinal conduits, compromising the annular seal integrity.

Many boreholes are drilled into unconsolidated formations. The combination of drill string vibration, jetting through the drill bit and geological weakness, often results in washout sections, i.e., where the annular geometry has a local expansion into the rock formation. These features are largely unpredictable geometrically although they are increasingly measured using caliper logs prior to cementing. It is of interest that some of the earliest experimental studies considered the effects of sudden expansions on the annular geometry, e.g., Clark and Carter (1973) and Zuiderwijk (1974), but this approach was then abandoned experimentally until quite recently, e.g., Kimura et al. (1999). However, although studied experimentally, these works are largely in the form of yard tests: using limited ranges of realistic fluids but not allowing one to draw more general fluid mechanic understanding.

The main issue with irregular washout shapes is that fluids with yield stress (e.g., drilling muds) are known to have regions of zero strain (plugs) and irregular geometries can promote regions of low shear stress close to wall, which result in static zones. In primary cementing, it is common to pre-circulate drilling mud prior to pumping cement, to condition the mud. Thus, it becomes operationally important to estimate the flowing volume of the annulus, particularly washouts. Although single phase, the requirement now is to determine the yield surface bounding immobile mud. Static wall regions also occur in regular uniform ducts, e.g., with cross-sections having corners, (Mosolov and Miasnikov 1965, 1966). Mitsoulis and co-workers (e.g., Mitsoulis and Huilgol 2004) studied both planar and axisymmetric expansion flows, showing significant regions of static fluid in the corner after the expansion. Flow of yield stress fluids through an expansion-contraction has been studied both experimentally and computationally by de Souza Mendes et al. (2007), Naccache and Barbosa (2007), and Nassar et al. (2011). In de Souza Mendes et al. (2007), Carbopol solutions were pumped through a sudden expansion/contraction, i.e., narrow pipe–wide pipe–narrow pipe, with yield surfaces visualized by particle seeding. Stagnant regions first appear in the corners of the expansion, grow with increasing yield stress, and become asymmetric with increasing Reynolds number. In Roustaei and Frigaard (2013), large amplitude wavy-walled channel flows were studied numerically, predicting the onset of stationary fluid regions, which occur initially at the walls in the widest part of the channel. A more comprehensive study of geometrical variation was carried out in Roustaei et al. (2015). Yield stress fluid becomes trapped in sharp corners and small-scale features of the washout walls and fills the deepest parts of the washout as the depth (\(\hat {H}\)) is increased. For sufficiently large yield stress (\(\hat {\tau }_{Y}\)) and sufficiently deep washouts (\(\hat {H}\)), the actual washout geometry has little effect on the amount of fluid that is mobilized: for a deep washout, the flowing fluid “self-selects” its geometry. Figure 4 shows an example of this flowing area invariance. Having established stationary regions within the depths of the washout, further increasing \(\hat {H}\) does not significantly affect the position of the yield surface. In Roustaei and Frigaard (2015), inertial effects were considered, for similar flows as in Roustaei et al. (2015). Surprisingly, moderate Reynolds numbers (but laminar) can in fact result in a reduction in flowing area, contrary to industrial intuition that pumping faster is better.

However, the pipe/tubing used to place the plugs is generally smaller than the casing in primary cementing. Thus, the annular placement geometry is no longer narrow. Indeed, some jurisdictions require the existing casing to be milled out into the surrounding rock formation. The fluids within the well may then be either old production fluids, possibly weighted brines, or drilling muds from the milling operation. Undoubtedly, this all makes the annular displacement problem harder. As a further complication, while the cement is pumped, the tubing is often slowly withdrawn from the hole, which leads to buoyancy-driven motion re-balancing of the static pressures between tubing and annulus.

A comprehensive introduction to cement chemistry, oilfield additives, and slurry rheology may be found in Nelson and Guillot (2006).

Fresh cement slurries are essentially concentrated suspensions that possess yield stress, thixotropy, and sometimes elasticity.

Cement is composed of calcium silicate and calcium aluminate phases. At the moment cement particles and water come into contact during mixing to form the slurry, chemical reactions begin. These reactions are collectively called hydration. The hydration products of silicate phases are CHS (calcium hydrosilicate) and Ca(OH) ₂ (calcium hydroxide). The calcium aluminate phases react rapidly with water causing rapid hardening, and hence, the addition of calcium sulfate is needed to avoid early setting (Taylor 1997).

In the early stages, the reactions go through a dormant period (the induction stage) of typically a few hours, after which setting initiates and the slurry progressively hardens. During the dormant period, the slurry is said to be fresh. A fresh slurry can be pumped and flow to the region where it is supposed to harden later on. Therefore, a reliable design of cementing operations requires a thorough understanding of the mechanical behavior of the fresh cement slurries (Banfill 1997). In well cementing, retarders are used to control the length of the induction stage, allowing a safety margin for pumping operations to complete.

The rheology of fresh cement slurries is a strong function of the mixing method (Yang and Jennings 1995), because hydration kinetics will depend on the mixing efficiency. At the moment mixing is started, a suspension of aggregates of cement particles forms. The particles are held together in the aggregate by action of an enveloping membrane of hydrated minerals that forms instantaneously. The strength of this membrane is quite high, approaching that of a typical chemical bond between atoms, whereas links between particles—due to van der Waals attraction force—are one order of magnitude weaker (Banfill 1997). Therefore, hydration efficiency will depend directly to what extent the mixing process is successful in rupturing the membranes and thus breaking the initially formed aggregates.

Other factors also have important effect on the rheology of fresh cement slurries, namely the water/cement ratio, temperature, cement fineness, cement type, and the content of admixtures, polymer latexes, flyash, slag, limestone, microsilica, and so on Banfill (1997).

Rheological measurements with cement slurries are rather difficult, due to many potential sources of measurement error. Therefore, good laboratory data requires sophisticated rheometers operated by experienced rheologists. In practical applications of the oil and gas industry, however, it is seldom possible to employ advanced laboratory rheometers, and the usual consequence is lack of reproducibility. The main experimental difficulties and suggested cures are now briefly discussed. A thorough discussion about this topic is found elsewhere (Roussel 2012). Sedimentation is one of the great challenges found in the rheometry of cement slurries. The large density difference between the dispersed phase and water often leads to sedimentation, especially in the high end of the range of water/cement ratio. To reduce and control sedimentation, chemical additives are often included in the slurry composition (Al-Yami 2015). The additives are selected to perform satisfactorily for application purposes. However, even for a slurry that does not exhibit significant settling problems when pumped downhole, sedimentation may still undermine the quality of rheological data. For example, for the parallel-plate geometry, a depleted layer is formed adjacent to the upper plate, leading to grossly underestimated viscosities.

For the Couette geometry, sedimentation causes a stratified viscosity distribution, and the measured value again does not correspond to the viscosity of the homogeneous sample. When it is not possible to obtain reliable data before appreciable settling occurs, one remedy to circumvent sedimentation includes the usage of a modified bob in the Couette geometry that possesses helical grooves which help maintaining homogeneity. The grooves cause a significant departure from the purely tangential flow assumed in the rheometer theory, and therefore, an error is introduced. It is important to estimate the effect of the grooves and re-calibrate, e.g., by running preliminary tests with standard oils.

An interesting alternative to reduce sedimentation is to increase the viscosity of the continuous phase with the aid of some additive and then present the data in the form of relative viscosity, namely the viscosity of the slurry divided by the viscosity of the thickened continuous phase. Therefore, to obtain the viscosity of the original slurry (without the additive), it suffices to multiply the measured relative viscosity by the viscosity of water. Of course, this method is not free of artifacts and should be used cautiously. The viscosity thus obtained will to some extent deviate from the correct one due to possible qualitative changes of the interactions between the continuous and dispersed phase.

Rheological measurements are also useful to characterize the evolution of viscosity due to setting. The performance of chemicals used to control the setting time can be evaluated nicely with the aid of rotational rheometry. In the industry, a consistometer is used for this purpose.

Waxy crude oils are characterized by a large amount of n-paraffins in their composition. At high enough temperatures, the paraffins are fully dissolved, and the oils behave as Newtonian fluids. When the temperature falls below the so-called wax appearance temperature (WAT), wax crystals nucleate in the bulk and can lead to a sol-gel transition when the mass of wax solids exceeds 1–2% (Vignati et al. 2005).

Clathrate hydrates are ice-like inclusions that—under appropriate thermodynamic conditions—form at the interface between water and hydrocarbons or low molecular weight gases, by trapping the guest species within cages of hydrogen-bonded water (Sloan and Koh 2008; Leopércio et al. 2016).

Specifically, they are typically formed at high pressures and low temperatures. Consequently, they often represent a severe flow assurance problem in deep and ultradeep water oil production, when the content of dispersed water in the produced oil is high enough. The hydrates form as the gas (e.g., methane) dissolved in the oil migrates to the water-oil interface. In steady-state production, hydrates can be avoided by pipe insulation, which prevents the temperature to fall within the so-called hydrate thermodynamic envelope.

The insulation solution, however, is not effective for the case of a long enough unexpected production shutdown, because in this case, the temperature may reach the one of water at the sea floor, attaining the necessary conditions for hydrate formation and maybe pipeline blockage. An alternative is to inject additives such as ethanol, which move the thermodynamic envelope so that the current state becomes outside it.

Interestingly, it has been observed that hydrates may form in some oil emulsions without plugging the pipeline. That is, in some cases, the formed hydrates do not agglomerate to form a percolated structure. Rather, they remain disperse forming a slurry. This phenomenon may be due to the presence of indigenous anti-agglomerants in the oil (de Oliveira et al. 2012; de Oliveira and Goncalves 2012). Or else, it may be due to a small water content such that water depletion occurs and the hydrate formation is halted before the hydrate crystals have a chance to touch each other and agglomerate.

For pipeline and operation design purposes, it is important to know the rheological properties of the hydrate slurries. The rheological characterization of this kind of material requires elaborate rheometrical experiments. A rheometer equipped with a pressure cell is needed, in order to provide the appropriate thermodynamic conditions during the measurements. These cells are not easy to operate. To allow pressurization and eliminate leaks, the cell connects to the shaft of the rheometer motor by means of a magnetic coupling. This coupling has a non-negligible residual torque that hinders measurements at low stresses, thus severely limiting the range of accurate rheological measurements.

The rheology of hydrate slurries formed from a water-in-oil emulsion has been investigated (Peixinho et al. 2010; Webb et al. 2012a, b). Viscosity and pressure of the slurry were measured during hydrate formation, growth, aggregation, and dissociation, using an interesting experimental procedure. In addition to the dependence on shear rate, water content, and temperature, the slurry viscosity was shown to increase rapidly with time when hydrates form and then decay after going through a maximum as hydrate aggregates breakup or rearrange. The yield stress initially increases with time and then levels off. The authors also observed that the slurries are shear thinning and that both viscosity and yield stress increase with the water content.

It is not uncommon the presence of solid particles in the oil, such as rock fines, clays, and especially asphaltenes. When the oil becomes in contact with formation water, the asphaltenes and other particles work as indigenous surfactants that adsorb onto the water-oil interface to help forming highly viscous and very stable water-in-oil emulsions (Sjöblom et al. 2003). Wax and hydrate crystals, and naphthenic acids can also stabilize oil-in-water emulsions, in the same fashion as the asphaltenes and other particles. Water-in-oil emulsions are present in different stages of oil production and constitute a major flow assurance problem (de Oliveira et al. 2012; de Oliveira and Goncalves 2012).

In order to devise strategies to circumvent the problems associated with stable water-in-oil emulsions, one of the key aspects is to understand the mechanisms that control the rheological properties of the water/oil interface, because the interfacial rheology controls the emulsion stability and affects the bulk rheological properties. Therefore, a thorough knowledge of the properties and behavior of the different indigenous surfactants and understanding how they affect the interfacial rheology is necessary (Sjöblom et al. 2003). For example, the interface activity of asphaltenes is a strong function of their solubility in the bulk of the oil. As the solubility decreases, the concentration of asphaltenes at the interface is increased, leading to higher interface activity. However, a too low solubility tends to cause a too large amount of asphaltenes at the interface, leading to aggregation and ultimately to a low surface activity.

Water-in-crude oil emulsions generally undergo a sol-gel transition as temperature is decreased, especially for waxy oils (Hemmingsen et al. 2005; Visintin et al. 2008; Paso et al. 2009; Maia Filho et al. 2012; Haj-shafiei et al. 2013; Barbato et al. 2014; Sun et al. 2014). The yield stress of these gelled emulsions may be very high (of the order of 1000 Pa), and an accurate and reproducible method for its measurement still lacks (Barbato et al. 2014). To the large number of difficulties already found in the determination of the rheological properties of waxy crude oils (Marchesini et al. 2012), several other arise, due to many additional factors that influence the gel structure and hence the bulk rheology (Visintin et al. 2008; Paso et al. 2009; Maia Filho et al. 2012; Sun et al. 2014). Among these, we can mention the oil composition and molecular weight, drop size distribution, interfacial tension and rheology, salinity, and pH, just to name but a few. Many of these emulsion characteristics are not easily accessible, which severely undermines the controllability and reproducibility of the rheological measurements.

After crystallization and structural buildup of a wax-oil gel, the colloidal solid wax crystals exist in a volume-spanning physical network that entrains the remaining liquid oil among the crystals. Van der Waals interactions between the individual wax crystals collectively impart a yield stress to a wax-oil gel. The yield stress consists of a degradable portion and a non-degradable portion. The degradable portion of the yield stress maps linearly to the structural parameter λ and the non-degradable portion of the yield stress is constant. Similarly, the overall Bingham viscosity term consists of a degradable portion and a non-degradable portion. The degradable portion of the Bingham viscosity maps linearly to the structural parameter λ, and the non-degradable portion of the Bingham viscosity is constant. The structural parameter λ follows a modified thixotropic dynamic relation in which an exponent n is placed on the structural parameter in the breakdown term, as shown in the following equation In a well-justified first approximation, the buildup term is negligible after initial structural buildup of the gel has been completed, due to an irreversible nature of the gel rupture process that is consistent with irreversible crystal-crystal bond rupture. Post-rupture buildup effects such as Ostwald ripening occur on a timescale substantially larger than the relevant rheological timescales, justifying negligence of the thixotropic buildup term. The thixotropic rate equation is simplified as follows and on implementing the chain rule This indicates that the structural state of the wax-oil gel, λ, follows a point function of the imposed shear deformation γ and is otherwise time-independent. This revealing transformation effectively removes the \(\dot {\gamma }\) parameter (shear rate) from informing the structural breakdown dynamics (which is otherwise viewed as problematic in many applications). The exponent n indicates the kinetic order of the breakdown rate with respect to shear deformation γ.

It is important to note that Eq. 16 applies only to the irreversible rheological breakage phenomena of a wax-oil gel. When shear stress is initially imposed upon an unperturbed wax-oil gel, the first rheological response is a conventional viscoelastic response derived from straining the crystal network, which involves bending of crystal-crystal contact junctions and possibly subtle structural rearrangement of crystals in the colloidal network. Subsequent to network straining and/or network rearrangement up to the yield strain, rheological breakage ensues. Therefore, rheological breakage is the second sequential rheological response, which necessarily occurs at a timescale larger than the initial low-deformation viscoelastic response, in accordance with the rigorous definition of non-ideal thixotropy provided by Larson (2015). Furthermore, when (16), with an absence of a buildup term, is applied analytically to inform the trajectories of the degradable terms within the extended Bingham formalism, the overall rheology description rigorously meets the definition of irreversible non-ideal thixotropy. This categorization of the governing rheology of wax-oil gels is applicable to gelled pipe restart applications and has been rigorously confirmed for model wax-oil gels in which crystal-crystal contact junctions are of a crystallographic nature (Paso et al. 2009). Adsorption of heavy polar colloidal components onto the wax crystal interfaces may alter the nature of the junction sites to a partially amorphous nature. However, even for waxy crude oil gels containing resins and asphaltenes, the primary rheological response remains deformation-dependent, lending credence to the dynamic thixotropic relation in Eq. 16.

The irreversible thixotropic nature of wax-oil gels carries several implications for gelled pipeline restart operations. Deformation-driven rupture of waxy crude oil gels results in localization of applied stress in the axial direction when flow is restarted in gelled oil pipelines. Rheological yielding occurs primarily at the pressure wave front, while flow rate variations govern the evolving pressure profile behind the pressure front. Another implication of deformation-driven rheology for pipeline restart applications is that gel breakage occurs sequentially along the axial length of the pipeline, allowing flow to restart in substantially longer pipe segments as compared to the incompressible case where gel breakage must occur simultaneously at all axial locations. A significant time delay is required in order for sustained flow to restart in pipe sections longer than the conventional critical pipe length established by the linear force balance methodology. This time delay allows the gel to break sequentially along the axial length of the pipeline commensurate with the pressure propagation process. Finally, because the gel breaks with deformation instead of time, repetitive pressure swinging remains a highly viable option to restart flow in difficult plugging cases. Pressure swings may be applied at one or both ends of the pipeline, and internal pressure monitoring may inform the appropriate pressure swing intervals.

Rheometric geometries used for assessment of waxy crude oil gels include vane and vane-like geometries, cone and plate geometries, and concentric cylinders. Vane and vane-like measuring geometries ensure fixation of the gel to the measuring surface, precluding artifacts related to adhesive breakage (slippage of the gel past the measuring surface). Disadvantages of the vane and vane-like measuring geometries are that deformation fields and flow fields are non-uniform in the geometry volume during measurement, resulting in artificial broadening of the observed gel breakage peak accompanied by artificial reduction in the peak height. Uniform deformation fields and flow fields are provided by cone and plate geometries. However, the disadvantage of cone and plate geometries is that adhesive rupture effects are prevalent. Gel fixation may be promoted by sandblasting the contact surfaces. Cone and plate geometries are also vulnerable to micro-sedimentation of wax crystals at the upper cone surface during gel formation, resulting in a locally reduced wax content and reduced strength of the gel in contact with the upper cone surface, affecting non-uniform gel breakage with shearing localized at the upper surface. The micro-sedimentation phenomenon is consistent with a noticeably detached gel at the upper cone surface after an experiment. Upright concentric cylinder geometries ensure representative wax content of the fluid in contact with the measuring surface, precluding micro-sedimentation artifacts while providing relatively uniform flow fields. However, upright concentric cylinder geometries are vulnerable to conventional adhesive breakage effects.

Occurrence of adhesive breakage must be properly accounted for during selection of measuring geometry. In general, for low-wax-content fluids in which cohesive breakage inherently occurs, vane-like geometries underestimate the yield stress; therefore, cone/plate or concentric cylinder geometries should be selected in order to obtain representative yield stress values. Conversely, for high-wax-content fluids where adhesive rupture is prevalent, vane or vane-like geometries should be used, cognizant of the fact that yield stress peaks are somewhat underestimated. In a recent development in Brazil, a grooved coaxial geometry was introduced for waxy crudes that combines the benefits of vane and concentric cylinder geometries (Barbato et al. 2014), confirming occurrence of slippage in comparative concentric cylinders. Implementation of grooved coaxial geometries likely provides the most representative values of true yield stress for the high waxy crude oil gels that otherwise exhibit adhesive breakage.

Various rheometric protocols are useful to assess waxy gel rheology. Optimally, a waxy crude sample should be loaded into the geometry at a temperature above the wax solubility limit, and a representative thermal profile should be imposed prior to measurement. The yield stress value may be accurately and rapidly attained using a linear imposed stress ramp with time. However, rapid acceleration of shearing after rupture precludes meaningful rheological analysis. Linear imposed shear rate ramps are used for obtaining equilibrium flow curves of waxy oil. In order to properly assess the entire deformation-dependent rheology of wax-oil gels, constant shear rate protocols should be used, which allow proper spacing of acquired data points with respect to the deformational state. Within a constant shear rate protocol, prescribed step changes in shear rate allow inference of the full deformation-dependent rheology of a wax-oil gel sample, including trajectories of yield and plastic viscosity values with respect to imposed deformation.