Elsevier

Chemical Engineering Science

Volume 198, 28 April 2019, Pages 16-32
Chemical Engineering Science

Suspension-colloidal flow accompanied by detachment of oversaturated and undersaturated fines in porous media

https://doi.org/10.1016/j.ces.2018.12.033Get rights and content

Highlights

  • Oversaturated and undersaturated states of fine particles in porous media.

  • Basic equations for mobilisation and straining of oversaturated and undersaturated fines.

  • Exact solutions for 1D suspended-colloidal transport of oversaturated and undersaturated fines.

  • Fingerprints for transport of oversaturated and undersaturated fines.

  • High agreement between the laboratory and modelling data.

Abstract

Mechanical equilibrium of particles attached to rock during flow is defined as torque balance of attaching and detaching forces. Oversaturated state of fines means that detaching torque exceeds the attaching torque, and fines detach as the flow starts. In undersaturation, the attaching torque dominates the detaching torque, and fines mobilisation can occur only if there is either additional detaching torque or a weakening of the attaching torque. Preliminary laboratory tests determine the mechanical equilibrium velocity and salinity, to ensure that oversaturation or undersaturation can be readily reproduced in further corefloods. We formulate the basic governing equations and derive exact solutions for over and undersaturated cases. The exact solutions allow formulating the fingerprints for transport of oversaturated and undersaturated fines in porous media. The coreflood results reveal the characteristic features of the oversaturated case of fines migration. Moreover, the analytical modelling results are found to be in high agreement with the experimental data, under the typical values of the fitted model coefficients.

Introduction

Suspension-colloidal transport in porous media with detachment of fines is encountered in environmental, chemical, civil, and petroleum engineering. Applications include disposal of industrial wastes in aquifers, cold water injection into geothermal reservoirs in enhanced energy projects, storage of freshwater in aquifers, contamination of aquifers by viruses and bacteria, and low-salinity waterflooding in oilfields (Bradford et al., 2013, Khilar and Fogler, 1998). Electrostatic attachment and capture of the mobilised particles reduce their suspended concentration, and straining and size exclusion of migrating fines reduces permeability. This detachment and capture affect disposal of produced water in aquifers and propagation of contaminants, where the suspended concentration must not exceed its environmentally safe threshold (Khilar and Fogler, 1998). Size exclusion and straining undermine the productivity of artesian and oil-gas wells, and the induced permeability decline reduces well injectivity (Chequer et al., 2018, Marquez et al., 2014).

Most of the migrating fines are clay-particles that have left the pore walls. Fig. 1a shows kaolinite platelets that are aggregated into leaflets in highly saline brines and are attached to grain surfaces. These platelets, which can dissociate from leaflets in low-salinity water, are thin and have faces comparable in size to pore throats. Fig. 1b shows plugging of a large pore throat by a kaolinite platelet, thereby significantly decreasing pore connectivity and network conductivity. Fig. 2 shows that kaolinite leaflets are located mostly in thin crevices near grain intersections. Kaolinite particles also coat the grain surface. Therefore, kaolinite detachment does not significantly increase permeability. However, plugging of thin pore throats during fines migration can significantly increase hydraulic resistance and consequently reduce permeability. Fig. 3 shows an SEM image of the outlet core section, where a pore has been plugged by migrating kaolinite fines. Other sources of movable fines are illite clays, chlorite clays, and silica silts (Khilar and Fogler, 1998, Kia et al., 1987). The detached chlorite flakes and illite bars also plug thin pore throats, yielding a significant permeability reduction.

Fig. 4, Fig. 5 illustrate an attached particle subject to drag, electrostatic, lift and gravity forces (Fd, Fe, Fl, and Fg, respectively) (Bergendahl and Grasso, 2000, Bradford et al., 2013). The detachment of particles that coat the rock surface does not significantly increase permeability, but their straining in thin pore throats makes the particle trajectory more tortuous and reduces permeability. In porous media, gravity and lift have been estimated as between 10−14 and 10−13 N, and drag and electrostatic forces as between 10−11 and 10−8 (Bedrikovetsky et al., 2011, Brady et al., 2015, Elimelech et al., 2013, You et al., 2015). Therefore, in determining the conditions for particle detachment, we disregard gravity and lift and consider only drag and electrostatic force.

It is assumed that at the instant of detachment, a particle rotates around the touching point where the particle is still attached. Fig. 5a shows this rotation around the rock-particle contact, which is due to deformation by the electrostatic attaching force. Fig. 5b shows the particle's rotation around the surface asperity. Therefore, mechanical equilibrium is equality of the torques of the applied forces (Bradford et al., 2013, You et al., 2015):τ=Fd(U,rs)l-Fe(γ,rs)=0,l=ld/lnwhere τ is the resultant torque; ld and ln are the lever arms for drag and electrostatic force, respectively; l is the ratio between drag and normal lever arms; U is the velocity of carrier water; and γ is the salinity. Different expressions for drag and electrostatic force can be found in Bradford et al., 2011, Brady et al., 2015, Chrysikopoulos and Syngouna, 2014, Derjaguin and Landau, 1941, Elimelech et al., 2013, Khilar and Fogler, 1998, and Xie et al. (2017). The formulae for forces used in the present paper are given in Appendix A.

Fig. 6 shows electrostatic energy potential of fines attached to the rock surface. The details of the calculations are presented in Appendix A and Table 1. The electrostatic constants were measured for the Berea sandstone core containing kaolinite clay that was used for coreflooding in the present work (Section 5). Salinities 0.6 M, 0.035 M, and 0.018 M, considered in Fig. 6, correspond to initial brine salinity, and two salinities of the injected water, respectively. The higher the salinity, the stronger the particle-rock attraction. The electrostatic force in Eq. (1) strongly depends on salinity γ. Fig. 6b corresponds to intermediate values of separation distance h and indicates the existence of a secondary energy minimum for injected salinities. The plot of the energy potential shows that electrostatic attraction takes place for all three salinities, i.e., the electrostatic force keeps the fine particles attached to the rock surface. Eq. (A.1) shows proportionality between drag and velocity U, determining U-dependency of drag in Eq. (1).

The torque balance (Eq. (1)) allows determining whether for a given velocity and salinity, each particle on the pore wall remains attached or detaches. So, values of U and γ define the attached particle concentration, which determines the so-called maximum (critical) retention function (Bedrikovetsky et al., 2011)σa=σcrγ,Uwhere σa is the attached particle concentration, and σcr is the maximum (critical) retention concentration.

Fig. 7 shows the plot of the maximum retention function versus salinity. The higher the salinity, the higher the electrostatic particle-rock attachment, the more particles secured by electrostatic force on the rock surface, and the higher the maximum retention function. The region above the maximum retention curve is where total torque τ as expressed in Eq. (1) is positive, i.e., the detaching torque exceeds the attaching torque. This means that the porous media is oversaturated, i.e., flow at velocity U detaches particles. The lowest velocity that yields fines detachment under initial reservoir salinity is given byσcrγI,Ucr=σaI

Eq. (3) defines so-called critical velocity Ucr; the fines are lifted at higher velocity (U > Ucr). The notion of critical velocity was introduced by Miranda and Underdown (1993), and since was widely used to determine maximum well rate where fines are not produced (Bedrikovetsky et al., 2011, Khilar and Fogler, 1998). Eq. (4) below defines so-called critical salinity γcr by fixing injection velocity in Eq. (3). Water injection with salinity that is lower than the critical salinity yields fines mobilisation. The notion of critical salinity was introduced to forecast well index decline during production or injection of low-salinity water (Khilar and Fogler, 1984, Kia et al., 1987).

Oversaturation corresponds to reservoir velocity below the critical velocity Ucr. Injection of low-salinity water corresponds to point J in Fig. 7. The trajectory on the phase diagram corresponds to vertical drop Io → A and then to continuous movement A → J along the maximum retention curve. Here point A is located on the maximum retention curve; this corresponds to instant release of fines with concentration Δσ = σaI − σcr(γI) with further release according to torque balance, corresponding to equilibrium (saturated) conditions (Eq. (2)).

For undersaturation resultant torque τ in Eq. (1) below the curve is negative, i.e., the detaching torque is lower than the attaching torque. The corresponding path on the phase diagram is Iu → B → J, which corresponds to decrease of salinity until its critical value B, and then particle release along the maximum retention curve. The critical salinity γcr for a given attached concentration σaI corresponds to the low salinity, where the first fine particle is detached during continuous salinity decrease from γI to γJ (Fig. 7):σcrγcr,U=σaI

Fig. 8 shows the plot of σcr versus U and γ. At a given salinity, the higher the velocity, the higher the drag and the lower the fines concentration that can be held on the rock surface by electrostatic force. Therefore, the curve σcr versus U decreases at constant salinity. The resulting surface shows σcr to monotonically increase with salinity and monotonically decrease with velocity. Here green, red, and pink curves correspond to constant salinity, and salinity for the pink curve is lower than that for the green curve; thus, the pink curve is located below the green curve, and the red one is in the middle. Black, blue and yellow curves correspond to constant velocity. The velocity increases from the black to blue and yellow curves, so the black curve is located above the blue curve, and the blue one is above the yellow one.

So, over and undersaturation are salinity- and velocity-dependent conditions. The oversaturated rock-fines systems are located above the critical surface (Fig. 8), while the undersaturated systems are below the surface.

Eq. (2) models fines release during velocity increase and salinity decrease (Fig. 7). Eq. (2) closes system of mass balance of suspended and attached particles, and the kinetics equation for size exclusion rate, comprising the mathematical model for fines mobilisation, migration and straining (Bedrikovetsky et al., 2011). Exact solutions for 1D linear and axisymmetric particle transport problems have been found for steady-state flows of oil and gas (Zeinijahromi et al., 2012), for fines migration with nanoparticles (Yuan and Moghanloo, 2017), for fines mobilisation by low-salinity water (Chequer et al., 2018) and for fines-assisted displacement of oil by water (Borazjani and Bedrikovetsky, 2017). The above modelling has been conducted under saturation conditions, where the initial concentration of attached fines, velocity, and salinity are related by Eq. (2). This corresponds to injection or production velocity by which sedimentary reservoir rock was saturated by fine clay particles during geological times. However, the transport velocity is determined by well injection rate, causing the former to vary widely and sometimes be significantly higher or lower than critical velocity.

We are aware of no published mathematical model of suspension-colloidal flow in porous media where, relative to injected salinity or velocity, the rock is oversaturated or undersaturated with fines.

The current paper fills this gap. The main objective of this work is to investigate 1D suspension-colloidal transport for over and undersaturated fines using mathematical and laboratory modelling.

We develop governing equations for transport of fines in oversaturated and undersaturated rock and derive exact solutions for 1D transport problems. Critical velocity and salinity are determined in laboratory tests, allowing controlling oversaturation and undersaturation during the corefloods. The fingerprints for oversaturated transport, indicated by the exact solution, are observed during laboratory coreflood. The analytical model is found to accurately match the laboratory data, and the tuned model coefficients belong to common intervals.

The structure of the text is as follows. Section 1 introduces maximum retention function as a mathematical model for particle detachment, and it defines oversaturation and undersaturation of rock surface having attached fines. Section 2 formulates the basic equations. Section 3 derives exact solutions for oversaturated and undersaturated flows. Section 4 compares the exact and numerical solutions. Section 5 presents the methodology for the laboratory study and the test results. Section 6 presents the accuracy of the analytical model in matching the laboratory data. Section 7 examines the model's validity. Section 8 concludes the paper.

Section snippets

Governing equations for 1D suspension-colloidal flows

This section presents a mathematical model for 1D suspension-colloidal transport that accounts for oversaturation and undersaturation of the natural reservoir fines. Section 2.1 gives the model assumptions, Section 2.2 presents the governing system and its non-dimensional form, and Section 2.3 formulates initial and boundary conditions for oversaturated and undersaturated fine particles.

Analytical models for unsaturated suspension-colloidal flows

This section derives exact solutions for 1D flow with fines migration due to salinity alteration for oversaturated and undersaturated fines. We consider the two specific conditions of initial attached particle concentration in the porous media discussed in Section 2.3. The solution strategy to solve both problems is the method of characteristics after applying the mass balance condition to the salinity shock front (Landau and Lifshitz, 1987, Logan, 2008).

Comparison of exact solution and numerical solution

In order to validate the discontinuous solutions of the governing system (12), (13), (14), (15), (16) subject to initial and boundary conditions (17), (18) and (18), (20), we solved the problem numerically and then compared the results with the exact solutions. We used the Matlab computer code given by Shampine (2005a), which implements a two-step Lax–Friedrichs finite-difference method (Shampine, 2005b). That code with annotations is available from http://faculty.smu.edu/shampine/current.html.

Laboratory study

This section provides a detailed description of the core and brine properties (Section 5.1), the laboratory setup (Section 5.2), the methodology adopted in the laboratory study (Section 5.3), and the test results (Section 5.4).

Matching the experimental data by the analytical model

The section develops the methodology of tuning the model coefficients from the corefloods with oversaturated fines (Section 6.1), presents the values obtained by matching the laboratory data (Section 6.2), and provides calculations of movable fines concentration (Section 6.3).

Discussion

Three states of fines in porous media were distinguished: (1) oversaturation, where the initial fines concentration exceeds maximum retained concentration for flow velocity and initial salinity; (2) undersaturation where the initial fines concentration is below the initial maximum value but exceeds the maximum retained concentration for flow velocity and injected salinity; (3) undersaturation where the initial fines concentration is below the maximum retained concentration for flow velocity and

Conclusions

Laboratory coreflooding by low-salinity water with fines migration, analytical modelling of coreflooding under oversaturated and undersaturated conditions, and matching the laboratory data with the proposed model allow drawing the following conclusions:

  • 1.

    One-dimensional flow problems for low-salinity water injection with migration of oversaturated and undersaturated fines in linear geometry allow for exact solutions.

  • 2.

    The analytical model provides explicit formulae for suspended, strained, and

Conflict of interest

The authors declare that there is no conflict of interest.

Acknowledgements

The authors are grateful to Thomas Russell (The University of Adelaide) for fruitful discussions and close reading of the manuscript. L. Chequer acknowledges CNPq for financial support. Many thanks are due to David H. Levin (Murphy, NC, USA), who provided professional English-language editing of this article.

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