Pore size distribution from challenge coreflood testing by colloidal flow

chemical engineering research and design 9 0 ( 2 0 1 2 ) 63–77

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Pore size distribution from challenge coreflood testing by colloidal flow P. Chalk, N. Gooding, S. Hutten, Z. You, P. Bedrikovetsky ∗ Australian School of Petroleum, The University of Adelaide, SA 5005, Australia

a b s t r a c t The transport of colloidal and suspension particles and the resultant particle retention occur in a wide range of porous media. The micro scale pore throat size distribution is an important characteristic of porous media, allowing for evaluation of important transport properties. An effective method based on micro scale modelling for the determination of overall pore throat size distribution (PSD) by injection of colloidal particle suspensions into engineered porous media with monitored inlet and breakthrough particle concentrations is developed. The treatment of inlet and outlet colloidal particle concentrations obtained in coreflooding results in a good agreement between the modelling and experimental data. Yet, some deviation was observed between the obtained PSD and that calculated by the Monte Carlo simulation based on the Descartes’ theorem. © 2011 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. Keywords: Pore size distribution; Porous media; Suspension; Colloid; Size exclusion; Monte Carlo; Challenge testing

1.

Introduction

The pore size distribution (PSD) is an important characteristic of the porous media allowing prediction of transport and volumetric properties (Amix et al., 1964; Dullien, 1992; Bedrikovetsky, 1993; Selyakov and Kadet, 1996). Different methods for determination of PSD have been applied extensively in many industrial sectors: chemical and environmental engineering, petroleum, natural gas, food, medical and pharmaceutical, etc. Presently, there are two commonly used methods in the industry for the determination of PSD: porosimetry and challenge testing. The method of porosimetry suggests injection of non-wetting phase in the core under increasing pressure. During the displacement, the injected phase fills the pores in decreasing order of their radii and the curve “injection pressure versus phase saturation” allows calculating PSD (Amix et al., 1964; Brakel et al., 1981; Dullien, 1992; Yortsos, 1999). Different options of the method use mercury, gas and other non-wetting displacing fluids. Disadvantages of the method include: it underestimates concentration of thin pores because high pressures must be applied in order to force the nonwetting fluid into thin pores; it can be destructive under high pressure hence not applicable for deformable and fragile

∗

porous materials; the last but not the least is environmental unfriendliness of mercury for the case of mercury porosimetry. The challenge testing method, which was recently significantly improved, utilises the injection of particle suspension into porous media (Purchas and Sutherland, 2002; Rideal, 2007, 2009). Since particle cannot pass the pore throat with the size smaller than the particle size, the maximum size of passed-through particle determines the maximum pore throat size. Besides the maximum pore throat size, the inlet and breakthrough concentrations for different size particles allow determining the overall PSD curve, but only for thin filters or membranes (Aimar et al., 1990; Frising et al., 2003; Rideal, 2009). The method is based on calculation of the particle capture probability in a single sieve and cannot be applied for deep bed filtration. Yet, usually the reservoir rock sample sizes vary as 0.5–10 cm; a thin slice can be submitted to flow only for highly consolidated cores. To the best of our knowledge, there is no available in the literature method to determine the overall PSD of core plugs from particle size distribution in the injected and produced suspensions. Determination of PSD from challenge testing data, i.e. using inlet and breakthrough concentrations of different size particles, is an inverse problem for suspension flow in porous media. Since pore and particle sizes affect the passage of

Corresponding author. Tel.: +61 8 83033082. E-mail address: [email protected] (P. Bedrikovetsky). Received 2 February 2011; Received in revised form 7 August 2011; Accepted 18 August 2011 0263-8762/$ – see front matter © 2011 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.cherd.2011.08.018

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chemical engineering research and design 9 0 ( 2 0 1 2 ) 63–77

Inlet particle distribution

Nomenclature C C0 CL c fa fns k1 k1a H h j L PSD p q1 rg rp rs s1 t U V x

suspended particle concentration distribution (m−4 ) inlet (injected) suspended particle concentration distribution (m−4 ) outlet (effluent, breakthrough) suspended particle concentration distribution (m−4 ) total suspended particle concentration (m−3 ) accessible flow fraction inaccessible flow fraction pore conductance (m4 ) accessible pore conductance (m4 ) pore concentration distribution (m−3 ) total pore concentration (m−2 ) jamming ratio length scale (m) pore throat size distribution fraction of pores thinner than particle flow rate in a single pore (m3 s−1 ) grain radius (m) pore radius (m) mean pore radius (m) particle radius (m) cross-sectional area of pore throat (m) time (s) total flux (m s−1 ) column volume (m3 ) coordinate (m)

Greek letters  filtration coefficient (m−1 ) flux reduction factor in a single pore  retained particle concentration distribution ˙ (m−4 ) total retained particle concentration (m−3 )  0 standard deviation for pore size distribution (m)  porosity a accessible porosity area accessibility fraction 

particle through porous media, the pore size is an adequate length scale for modelling in order to determine PSD (Payatakes et al., 1973, 1974). Presently used mathematical models for transport of particulated suspensions in porous materials on micro scale include random walk equations (Cortis et al., 2006; Shapiro, 2007; Yuan and Shapiro, 2010), population balance models (Sharma and Yortsos, 1987a,b,c; Santos and Bedrikovetsky, 2006; Bedrikovetsky, 2008), equations for resident time distributions (Fan et al., 2008; Lin et al., 2009), different versions of Boltzmann model (Shapiro and Wesselingh, 2008; Tang et al., 2010) and direct micro scale modelling (Biggs et al., 2003; Rong et al., 2010). These models are widely used in chemical, environmental and petroleum engineering (Bergendahl and Grasso, 2000; Mays and Hunt, 2005; Gitis et al., 2010). In the present work, the PSD is determined from the challenge testing for “long” engineered porous columns where the deep bed filtration occurs. In order to avoid cumbersome and expensive characterisation of particle–rock interaction, it was proposed to use the liquid and particle materials that promote net repulsion between particles and rock, so size exclusion

Outlet particle distribution Pore size distribution

rs

rpmin

rp

rpmax

Fig. 1 – Schema of determination of pore size distribution from inlet and effluent particle size distributions. becomes the only particle capture mechanism. The population balance model is chosen in this paper because it contains an explicit relationship between particle and pore sizes as a criterion for size exclusion. In the case of particle–rock repulsion, the population balance model becomes simpler and contains the PSD along with the inter-chamber distance as the only input parameters. The inverse problem is solved for the case of short duration injection with low particle concentrations, where the initial PSD remains constant. Five laboratory tests on injection of latex particles into engineered porous media have been performed. The PSD is determined from the laboratory data using solution of the inverse problem. Good match between the experimental and modelling data was observed. Yet, some deviation between PSD as obtained from the Monte Carlo simulation using Descartes’ theorem and that calculated from breakthrough concentrations was observed. The structure of this paper is as follows: Section 2 starts with the general description of the challenge coreflood test for PSD determination. Section 3 presents the derivation of the analytical micro model for suspension flow in porous media with size exclusion particle capture. Subsequently, the laboratory procedure for injection of latex particles into a transparent plastic box with one layer of glass bead grains and into a column packed with these glass beads is presented (Section 4). Determination of PSD from grain size distribution by the Descartes’ model and Monte Carlo simulations is given in Section 5. The data treatment is performed in Section 6. The discussions presented in Section 7 and conclusions in Section 8 complete the paper.

2.

Description of the problem

Let us consider the challenge testing, which includes suspension flow in porous specimen measuring inlet and outlet concentrations of each size particles (Fig. 1). Fig. 2 illustrates the definition of the pore throat size as a radius of the minimum sphere inscribed between three neighbouring grains.

Fig. 2 – Formation of the pore throat by three tangent grains.

chemical engineering research and design 9 0 ( 2 0 1 2 ) 63–77

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Particle passes the pore if the particle size does not exceed the pore size: (1)

rs < rp .

Further in the text, the term “thin pore” means that it is smaller than the passing particle, i.e. inequality (1) is fulfilled. The water composition (salinity, pH) for a given particle and grain material is selected in such a way, that particles repulse from the rock grains, so no particles become attached to the rock and size exclusion mechanism determines particle capture by the rock. The particles larger than the maximum pore do not pass. Therefore, their breakthrough concentration equals zero. It allows calculating the maximum pore size as the minimum size of the particle with zero breakthrough concentration (Fig. 1). The particles smaller than the minimum size pore, pass through the specimen without being captured. It permits calculation of the minimum pore size as the maximum size of particles with coinciding inlet and breakthrough concentrations. So, two end points of the PSD curve can be determined directly from the pair of concentration curves for size exclusion deep bed filtration. Fig. 1 shows that the larger is the particle the higher is the capture probability, i.e. the larger is the difference between inlet and outlet concentration. This phenomenon is described by the population balance model for suspension transport with size exclusion in porous media. The direct problem delivers breakthrough concentration distribution C(rs , x = L, t) for known injected concentration distribution C(rs , x = 0, t) and pore concentration distribution H(rp ). The inverse problem determines the PSD from the inlet and effluent concentration curves. Since the majority of the models uses a cylindrical pore shape, where the pore body and pore throat sizes coincide, the term “pore size distribution” is commonly used instead of the “pore throat size distribution”. As follows from inequality (1), the challenge method determines distribution of pore throats. Keeping it in mind and for the sake of simplicity, we use both terms.

Micro scale model for suspension flow in 3. porous media with particle straining In this section, derivation of the system of governing equations for suspension flow in porous media with size exclusion particle capture mechanism is presented for triangular pore throat form. The parallel tube model intercalated by mixing chambers is used to describe the geometry of the porous medium. The analytical model for low retention filtration and the solution of the inverse problem of PSD determination are provided. The definitions and nomenclature follow Bedrikovetsky (2008).

3.1.

Definition of variables for micro scale model

Let us consider suspension flow in granulated porous media. The triangular shape capillaries (Fig. 2) more precisely describe pore space between spherical grains than the circular capillary (Mason and Morrow, 1991; Patzek and Silin, 2001). The main features of size exclusion deep bed filtration of suspension in three dimensional porous space are as follows:

Fig. 3 – Schematic of suspension flow towards the cross section of porous media represented by the PTM model: (a) flow through capillaries and mixing chambers, (b) size distribution of triangular capillaries. particle flows in a pore with the throat larger than the particle, (1); the particle has choice of entering different pores in the capillary junction; the probability of entering the pore is equal to the fractional flow via this pore; if the particle is larger than the pore, the particle capture occurs upon the entrance into the pore throat. Fig. 3 presents the parallel bundle of different size capillary intercalated by the mixing chambers as a geometric model of porous media, where the size exclusion suspension flow takes place. A suspended particle may only be transported to the next chamber if it is sufficiently small enough to pass through the capillary, (1). Full mixing of the inlet particle fluxes occurs inside the chambers. The particles enter the capillary from the chamber according to the carrier water fluxes via the capillary. Depending on whether the particle is larger or smaller than the entrance capillary, it will be either captured or passed. The particle capture occurs at the chamber exits. So, particles are effectively sieved at the entrance to each capillary bundle according to their sizes. The geometric model of parallel capillary intercalated by the mixing chambers fulfils three above mentioned features. The model is determined by the concentration distribution H(rp , x, t) and the distance between the chambers l. A dimensionless term, the jamming ratio j, is introduced to quantify the ratio between particle size and pore throat size: j=

rs . rp

(2)

The total concentration of all sized pores h(x, t) is defined as the total number of pores per unit surface area at certain distance x and time t:



∞

h(x, t) =

H(rp , x, t)drp ,

(3)

0

where H(rp , x, t)drp is the concentration of pores with sizes between rp and rp + drp . The population of particles in porous media is described in similar manner. The concentration of all suspended particles in porous media, c(x, t), is defined as the total number of particles per unit volume at certain distance x and time t, resulting from the integration of concentration of particles having radii between rs and rs + drs :

 c(x, t) =

∞

C(rs , x, t)drs ,

(4)

0

where C(rs , x, t) is the particle concentration distribution. Similarly, the concentration distribution of retained particles ˙(rs , x, t) is defined as the number of retained particles ˙(rs , x, t)drs

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having radii between rs and rs + drs . The total concentration of retained particles (x, t) is obtained as



∞

(x, t) =

(5)

˙(rs , x, t)drs . 0

The retained particle concentration can also be distributed according to rp since both rp and rs take effect in particle capture. Therefore, we introduce ˙ - (rp , rs , x, t) as the distribution of retained particle concentration over the pore and particle radii:



 (rs , x, t) = o

∞

(rp , rs , x, t)drp .

S1,i (rp ) =

S1,o (rp ) = rp2 . √ S1, (rp ) = 3 3rp2

(7)

Here index i = o corresponds to a circles for cylindrical capillary in PTM; i =  stands for capillary with triangular shape. The accessibility of pore space to a particular sized particle is limited by the particle radius. This leads to the notions of accessible and inaccessible area as shown in Fig. 4. The accessibility factor, (j), describes the ratio between the accessible and total area. As it follows from (7), the accessibility factor for circular pores, where the radius of accessible circle is rp -rs , is 2

(j) = (1 − j) .

(8)

The radius of inscribed circle in the accessible area for triangular pore is equal to rp -rs , which also results in expression (8) for accessibility factor (Ilina et al., 2008; Panfilov et al., 2008; Bedrikovetsky, 2008). The particle capture kinetics requires the definition of the accessible and inaccessible flow fractions fa and fn , respectively. For the PTM, the accessible flow fraction is defined by Ua = fa = U

∞

k1 (rp )(j)H(rp , x, t)drp

∞

rs

o

k1 (rp )H(rp , x, t)drp

.

(9)

where  = (j) is the flux reduction factor, which is the ratio between fluxes via the accessible and the overall pore cross section and k1 = rp4 /8 is the pore conductance determined from the Poiseuille’s flow in a circular tube q1 = −

r4 dp k1 dp =− 8 dx dx

 rs k1 (rp )H(rp , x, t)drp  o∞ .

∞

(6)

The curvilinear pore throat presented in Fig. 2 may be approximated by a circle or a triangle, leading to two definitions of a single pore throat’s cross-sectional area:



fns

Uns = = U

o

(11)

k1 (rp )H(rp , x, t)drp

The flow through inaccessible areas in large pores, fnl , is

 -

For triangular pores, the conductance is calculated from exact solution of the one dimensional viscous steady state flow of Newtonian fluid (see Landau and Lifshitz, 1987): k1 = √ 9 3rp4 /20. Likewise, the inaccessible flow fraction through thin pores (Fig. 4), fns , is defined by

(10)

fnl [H, r] =

rs

k1 (1 − (rs /rp ))H(rp , x, t)drp

∞ o

k1 H(rp , x, t)drp

(12)

.

The accessible conductance formula obtained from k1 replacing rp by rp -rs is described below in Eq. (13) for the two considered pore throat approximations:

⎧  ⎨ k1a,o (rp ) = 8 rp4 (1 − j)2 (1 + 2j − j2 ) √

. k1a,i (rp ) = ⎩ k1a, (rp ) = 9 3 r4 (1 − j)2 1 + 2j − 1 j2 − 4 j3 20

p

3

9

(13)

The flux reduction factors for circular and triangular pore throats follow from (13):

⎧ k ⎪ ⎨ o (rp ) = 1a,o = (1 − j)2 (1 + 2j − j2 ) k1,o i =

. ⎪ ⎩  (rp ) = k1a, = (1 − j)2 1 + 2j − 1 j2 − 4 j3 k1,

3

(14)

9

3.2. Governing equations for suspension flow in porous media In this section, the governing equations of suspension transport in porous media with brief derivations follow Bedrikovetsky (2008). The main assumptions of the model for transport of the colloidal material in porous media include: no particle coagulation or attrition, negligible external filter cake, particle–rock repulsion leading to size exclusion mechanism of the particle capture, no diffusion, one pore is blocked by just one particle (no bridging), homogeneous porous media (Herzig et al., 1970; Sharma and Yortsos, 1987a,b,c; Grenier et al., 2008; Rebai et al., 2010; Daniel et al., 2011). Low retention process is considered, i.e. the amount of retained particles is negligibly smaller that the number of vacant pores available for the particle capture.

Fig. 4 – Accessible and inaccessible pore space.

chemical engineering research and design 9 0 ( 2 0 1 2 ) 63–77

The accessible porosity is the total of accessible pore cross sectional areas per unit of the porous media cross section



∞

a [H, rs ] =

s1,i (rp )(j)H(rp , x, t)drp .

(15)

Substituting the expression for retention rate (19) into mass balance equation (16) yields ∂ ∂ {a [H, rs ]C(rs , x, t)} + U {C(rs , x, t)fa [H, rs ]} ∂t ∂x

rs

where s1,i (rp ) is the cross-sectional area of a single pore throat having radius rp and (j) is the area accessibility fraction. Here round brackets are used for functions of scalar variables and square brackets are used for the functionals of functions. The population balance equation for colloidal particles in porous media for the above assumptions accounts for suspended particles flowing via accessible area and retained particles distributed throughout the overall porous space:

 ∂ ∂ (rs , x, t)} + U {C(rs , x, t)fa [H, rs ]} = 0. {a [H, rs ]C(rs , x, t) + ∂t ∂x (16) Here the flow velocity U is independent of co-ordinate x due to incompressibility of particulate suspension. The derivation of the kinetic equation for particle capture and pore plugging is based on the aforementioned PTM model. Since C(rs , x, t) is particle concentration in suspension, i.e. number of particles per volume of accessible area, the suspension concentration in chambers is fa C(rs , x, t); suspension dissolves in the clean water flowing into the chambers from inaccessible pores. The assumption of perfect mixing between capillary bundles indicates that colloidal particles are randomly presented to a pore throat and there exists no preference for a colloidal particle to a particular throat. Therefore the number of rs particles arriving at pores with size rp from chambers by the flow per unit volume during the time interval t is 1 q1 (rp )H(rp , x, t)C(rs , x, t)fa [H, rs ]t. l

(17)

It is assumed that particles are not captured in inaccessible areas of larger pores (1) under the particle–grain repulsion. The suspended particles are captured by smaller pores only. The rate of retained rs particles being removed by smaller rp pores, rp < rs , is equal to the overall particle flux from chambers into these pores ∂

 -

(rp , rs , x, t)

1 q1 (rp )H(rp , x, t)C(rs , x, t)fa [H, rs ]. l

=

∂

(18)

Integration of (2) in rp for all smaller pores yields the expression for the retention rate of rs particles ∂

 -

(rs , x, t) ∂t

=

1 UC(rs , x, t)fa [H, rs ]fns [H, rs ]. l

(19)

From the assumption that one retained particle plugs one pore, it follows that the rp pore plugging rate due to particle capture is equal to the overall flux of larger particles flowing into rp pores from chambers ∂H(rp , x, t) =− ∂t 1 =− l



 0

∞

∂ ∂t



(rs , x, t)drs

∞

UC(rs , x, t)fa [H, rs ]fns [H, rs ]drs , 0

(20)

67

1 = − fns [H, rs ]C(rs , x, t)fa [H, rs ]U, l

(21)

System of two Eqs. (20) and (21) for two unknowns describes size exclusion deep bed filtration. The initial conditions for the clean bed process include zero suspension concentration and initial uniform PSD for pores: t = 0 : C(rs , x, 0) = 0, H(rp , x, 0) = H0 (rp ).

(22)

The boundary conditions include the model for particle capture and percolation at thin layer with thickness of a few pore lengths. Consider a porous column of the packed glass beads, since this porous media will be used in laboratory tests further described in the paper. It is assumed that particles, which enter thin pores, will remain there after capturing, i.e. the inlet particle flux Cfns U is completely captured in the inlet cross section. The particles entering thick pores via the inaccessible area are not captured due to the particle–grain repulsion, i.e. the entrance flux Cfnl U is not captured. These two assumptions about the particle capture at the inlet x = 0 correspond to the particle retention expression (19) for x > 0. The inlet flux via the accessible area Cfa U is not captured. Mass conservation at the inlet cross section is the equality of the particle flux, entering larger pores with the injected concentration, and the flux of particles, transported by water in accessible pore space with boundary value concentration x = 0 : C0 (rs , t)(fa [H, rs ] + fnl [H, rs ])U = C(rs , 0, t)fa [H, rs ]U,

(23)

Since the injected carrier water flux exceeds the postinlet water flux carrying particles, the post-inlet concentration exceeds the injected particle concentration. Eq. (23) determines the post-inlet concentration and provides the boundary condition for population balance equation (21).

3.3.

The analytical solution

In this section we show that the assumption of low retention filtration results in linearization of the governing system (16), (19) and (20) causing separate independent filtration of particle populations with different sizes. This feature finally enables determining the pore size distribution from the effluent and inlet particle size distributions (Section 6.2). The linearized flow can be described by explicit analytical formulae. Let us consider steady state of suspended particles along the core, where the difference between the number of particles entering and exiting the reference volume is captured by the porous media. Eq. (21) becomes an ordinary differential equation: U

d 1 {C(rs , x)fa [H, rs ]} = − fns [H, rs ]C(rs , x)fa [H, rs ]U, dx l

(24)

Let us discuss the initial stage of size exclusion deep bed filtration, where retained concentration is negligible if compared with the vacant pore concentration. It happens either at small injection times, or at injection of diluted suspensions or at small filtration coefficient (small particle size). Since it

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is assumed that one pore is plugged by one particle and vice versa, the conditions of particle capture do not change during low retention filtration, i.e. the pore concentration remains intact. The same conclusion can be drawn from Eq. (20) for pore plugging rate – the time variation of the PSD tends to zero when either injected concentration tends to zero, the correlation length l (inter chamber distance) tends to infinity or the particle size does not exceed the minimum pore size. The initial PSD for pores is uniform. So, the assumption of low retention filtration leads to constant PSD during the injection, i.e. H(rp , x, t) = H0 (rp ). Therefore, flow fractions become functions of particle size only and will be denoted as fj (rs ), j = a, ns and nl further in the text. In this case, Eq. (24) is solved independently for each particle size, i.e. the particles of the different sizes filter independently from each other and deposit as a bunch of mono disperse suspensions. Boundary condition (23) becomes

The steady state distribution of suspended particles in the overall core is established after arrival of the suspension front to the core outlet (Herzig et al., 1970). As it follows from (21), the velocity of the front is Ufa /a , so the constant value of the effluent concentration (29) is established for t > La /Ufa .

4. Laboratory procedure for forcing suspensions through porous media In this section, the conditions of particle–grain repulsion and the consequent absence of particle retention due to attachment are established by tests in a simplified one-grain-layer engineered porous media. Then the laboratory procedures developed for the flow of colloidal suspensions through glassbead-packed porous media are discussed.

4.1. C0 (rs )(fa (rs ) + fnl (rs )) C(rs , 0) = , fa (rs )

(25)

Eq. (24) becomes 1 dC(rs , x) = − fns (rs )C(rs , x), dx l

(26)

Solving ordinary differential equation (26) subject to initial condition (25) yields the value of the pre-outlet particle concentration C(rs , x) =



C0 (rs )(fa (rs ) + fnl (rs )) x exp −fns (rs ) l fa (rs )

 (27)

,

The outlet upstream particle flux is equal to the downstream particle flux immediately after the core C(rs , x)fa (rs )U = CL (rs )U,

(28)

where CL (rs ) is the particle concentration measured in produced suspension sample. Further in the text CL (rs ) is called the effluent (breakthrough, outlet) concentration. Since the water flux carrying particles upstream the outlet is lower than the downstream flux, the pre-outlet concentration is higher than the breakthrough concentration. The pre-outlet particle flux is dissolved in the overall water flux after passing the core outlet. From (27) and (28) follows that



CL (rs ) = C(rs , x)fa (rs ) = C0 (rs )(fa (rs ) + fnl (rs )) exp −fns (rs )

L l

 , (29)

Formula (29) connects the injected concentration with that measured after the core outlet. It will be used further in the text for determining PSD H0 (rs ) from the values of the breakthrough concentration CL (rs ) as obtained in laboratory tests for different injected particle sizes rs . The formula show that the larger are the injected particles, the higher is the fractional flow through small inaccessible pores fns , the lower is the fractional flow via large pores fa + fnl = 1 − fns and the lower is the exponent term in (29). Finally, the lower is the effluent concentration CL . The effluent concentration vanishes when the particle size reaches the maximum pore size, where the fractional flow via large pores fa + fnl becomes zero.

Preliminary study

An experimental method is developed where particle retention in porous media occurs exclusively due to particle straining. Operational conditions are experimentally determined for the purpose of eliminating particle retention via mechanisms other than straining, in particular retention via attachment. The objective of this preliminary study is to design and construct a micro model unit that provides the visual observation of colloidal suspension flow through porous media. This experiment allows the determination of the solution chemistry that promotes the particle–surface repulsion forces between porous structure and suspended colloids and, therefore, minimises the particle attachment. In the developed experimental method, a colloidal suspension of spherical, fluorescent carboxyl latex microspheres is injected through an engineered porous medium. The yellowgreen (rs = 4.9 ␮m) and blue (rs = 2.3 ␮m) fluorescent latex microspheres (Polysciences Inc., Warrington, PA) are selected as the colloidal particles held in suspension. The surfaces of these colloids are grafted with carboxyl functionalised groups by the manufacturer. This creates a negatively charged hydrophilic colloidal surface possessing a net negative charge when in an alkaline solution. The net charge of the surface prevents coagulation of colloids and reduces electrostatic attraction to the porous media. The colloid concentration is held constant at 20 ppm in all the preliminary tests. A single layer of sieved and cleaned spherical glass beads (Ballotini Bead, Potters Industries Pty. Ltd., Australia) is adopted as a 2D porous medium. The main component of the porous medium, silica (SiO2 ), has a net negative surface charge in alkaline solutions (SiO4 4− ). Silica is also selected because it represents the fundamental material of sandstone reservoirs. According to the manufacturer, the glass beads have a reported chemical composition: 72.0% SiO2 , 15.0% Na2 O, 7.0% CaO, 4.2% MgO, 0.4% Fe2 O3 and 0.3% Al2 O3 . These glass beads are packed homogeneously with a theoretical porosity of 39.6%. Two sizes of beads are chosen for the tests with mean radii 150 ␮m and 300 ␮m, respectively. In order to examine the attachment and straining of colloids moving through a single layer of glass beads, the micro model housings are designed to support the observation under an optical microscope. The housings are milled out of polyvinyl chloride (PVC) plastic and are designed in such a way that two glass slides are held in place to contain the porous media in a single layer. The deionised ultrapure MilliQ water (resisitivity of 18.2 M × cm at 25 ◦ C) after degassing in

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chemical engineering research and design 9 0 ( 2 0 1 2 ) 63–77

Fig. 5 – Images of particles strained in porous media: (a) different salinities and (b) different pH levels. vacuum at pressure ≈ 10−2 Pa is used for the preparation of a colloidal suspension. The salinity, acidity and alkalinity of the prepared suspensions are adjusted with the addition of NaCl(aq), HCl(aq) and NaOH(aq), respectively. The effect of colloidal suspension salinity on the particle retention is investigated under three different NaCl(aq) concentrations (0 mM, 10 mM and 100 mM) at constant acidity/alkalinity. A proportional increase in particle attachment with salinity is observed throughout these experiments, as shown in Fig. 5(a), where the images of colloid particles are small bright “points” while the images of glass beads are semitransparent large circles. Simultaneously, the particle–grain interaction force was calculated for three salinity values using DLVO theory (Landau and Lifshitz, 1980; Khilar and Fogler, 1998). Three plots of the force versus the particle–surface separation distance show that the increase in retention is due to an increased depth of the secondary minimum and a reduced energy barrier to the primary minimum (Tufenkji and Elimelech, 2004, 2005). The same results were obtained by Kuznar and Elimelech (2007). Colloidal particle retention by glass beads is investigated using suspensions with five different pH levels (2.79, 4.26, 7, 8.48 and 10.38) at constant salinity. The monotonic decrease in attachment is found to occur with increased alkalinity. The colloid retention in the unit is comparatively low in all alkaline resident solutions, as shown in Fig. 5(b). The high particle retention in acidic conditions is due to the reduction in the strength of the electrical charge on the surface of the glass beads and the colloidal particles. In order to minimise the colloidal particle attachment, the resident solutions with high pH and pure water must be employed so that the physical capture mechanisms would dominate. The above conclusions indicate that net repulsion exists between the porous media and the colloidal suspensions.

4.2.

Experimental procedure

Using the outcomes of the preliminary study, an experimental setup and procedure simulating the flow of suspended

particles through a 3D pore structure is developed. A plastic column with a packing space of 47 mm in diameter and L = 50 mm in length to simulate the dimensions of core samples is constructed. The column design has incorporated homogenised sections to negate the end effects, allowing for the particle suspension to be injected as a piston-like front into the packing matrix. A 30 ␮m sized stainless steel mesh is utilised to contain the porous media within the column. The application of distribution plates ensures the rigidity of the packed glass beads. An exploded diagram of the resulting column with its components is demonstrated in Fig. 6(a). Fig. 6(b) shows the photo of the column. Monodisperse suspensions of yellow-green fluorescent polystyrene latex microspheres used in the preliminary study are taken as colloidal particles in these experiments. Five sizes of microspheres are used in these experiments. Table 1 summarises the specific physical characteristics of the colloids, where rs denotes the mean radius of the colloid particle,  is the standard deviation, and s is the particle density. Glass beads similar to those used in the preliminary experiments are utilised as the porous media in all experimental procedures. The glass bead sizes, as reported by supplier, are 10–80 ␮m. A thorough sieving process is performed to reduce the size range of the glass beads; the size variation after the sieving becomes 10–66 ␮m. A complete washing procedure is also performed to remove organic impurities using acetone, hexane and hydrochloric acid. Colloids are held in suspension solution made up with 0.1 M sodium hydroxide and degassed ultrapure MilliQ water

Table 1 – Size distributions (mean radius and standard deviation  0 ) and densities of the colloidal particles. Type 1 2 3 4 5

, ␮m

 0 , ␮m

s , g cm−3

1.789 2.297 2.976 4.915 9.844

0.339 0.457 0.342 0.796 0.059

1.055 1.055 1.055 1.055 1.055

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chemical engineering research and design 9 0 ( 2 0 1 2 ) 63–77

through the column to remove any particulate matter from the porous media that may have contaminated the effluent colloid sample. Samples of the 50 mL effluent colloidal solution are collected and passed through the PAMAS S4031 GO portable particle counter to determine the effluent particle concentration. The effluent solution sampling should be stopped after a stable concentration of colloidal particles in the outlet stream is observed. A standard deviation (in %) from the mean value for all colloidal particles ranging from 1.789 to 9.844 ␮m was introduced by repetitive measurement (up to 5 measurements) of the same particle suspension. This standard deviation may be adopted as a measure of experimental uncertainty of PAMAS particle counter. Therefore, if the concentration of effluent colloidal particles for the next sample differs from the previous one by the value less than the above standard deviation, then the sampling process and flow-through test should be stopped, otherwise the experiment should continue.

5. Determination of pore size distribution from grain sizes

Fig. 6 – The schematic of the column used for experiment: (a) exploded diagram, (b) photo of the column.

to create a resident solution with a pH of 10. The effect of pH variation on particle retention is investigated by repeating the same experiment in acidic and neutral pH environment. In order to exclude the possibility of air pockets ingress to the packed glass bead bed, the column is wet packed with glass beads for all experiments. The packed column was placed in the sonic bath for 30 min, allowing achieving the dense and close packing of glass beads. Afterwards, the column is connected to a syringe piston pump (AdelabNE-1000). A steady state bottom-up flow of the colloidal solution with linear velocity of 10−5 m s−1 is established through a vertically placed packed column to minimise sedimentation due to the density difference between the colloids and the resident solution. Five pore volumes of the resident solution are flushed

A predictive method for the PSD from the measured grain size distribution is presented in this section. The validation of Monte Carlo simulation for PSD using Descartes’ theorem is performed by treating the literature data. Afterwards, this method is used for calculation of PSD using known size distribution of glass beads in the porous columns described in Section 4. An idealised porous medium can be represented as a granular packing with distribution of grain sizes. The typical for petroleum reservoirs sandstone rocks are often treated as the packing of spherical grains (Amix et al., 1964). This simplification is commonly used to model and simulate petroleum systems (Dullien, 1992). Assume that the spherical grains are densely and closely packed, and each pore body is formed by 4 grains. This type of packing is the densest for the case of equal radii beads. Four throats exit the same pore body. Therefore, the throats have a form of curvilinear triangle (Fig. 2). Consider a particle approaching the pore throat. The throat is formed by 3 neighbouring grains. The radius of a sphere touching these 3 grains reaches minimum if the sphere centre is located on the plane that crosses centres of 3 grains. Therefore, the radius of a pore throat is effectively defined as the maximum inscribed radius of three mutually tangent grains in the plane formed by the centres of those spherical grains (Fig. 2). Any larger spherical particle does not pass the throat. As a result, analysis of the pore throat size is dissolved to a 2D problem whose solution can be determined by the Descartes’ theorem, which was first published in 1936 by Soddy (Descartes, 1901; Pedoe, 1967). The theorem states that if four circles in a plane touch each other externally, with the radii r1 , r2 , r3 and rp , then the following relationship holds:

2

1 r12

+

1 r22

+

1 r32

+

1 rp2



=

1 1 1 1 + + + r1 r2 r3 rp

2 .

(30)

Because of the random nature of the grain packing, it is beneficial to employ a statistical algorithm for the simulation. In this work, the Monte Carlo method (Binder and Heermann, 1988), which relies on repeated random sampling to calculate

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chemical engineering research and design 9 0 ( 2 0 1 2 ) 63–77

(a)

(b)

7

8 7

6

6

5

f(rp)

f(rp )

5 4 4

3 3 2

2

1

1

0

0 0

0.5

1

1.5

0

0.5

rp (µm)

1

1.5

rp (µm)

Nolan & Kavanagh (close packed c)

Nolan & Kavanagh (close packed d)

MC (Descarte's theorem)

MC (Descarte's theorem)

Fig. 7 – Validation of the Monte Carlo model for pore size distribution using Descartes’ theorem: (a, b) treatment of data by Nolan and Kavanagh (1994) (♦); results of modelling (). Section 6 to validate the method for PSD calculation from the suspension flow test.

6.

Treatment of experimental results

In this Section, it is verified whether the conditions of low retention filtration leading to a simple analytical model (29) have been fulfilled during the laboratory tests. Afterwards, the laboratory data are treated using the analytical model.

6.1.

Validation of the low retention assumption

Let us check whether the assumption of the analytical model (Section 3.3) that the PSD during particle straining in the porous media is invariant have been fulfilled in the laboratory tests. This occurs when the concentration of pore vacancies (thin throats) is significantly higher than the concentration of retained particles, i.e. the particle retention and the consequent consumption of vacant throats do not affect the total number of vacancies: (x, t)  h(x, t).

(31)

This condition can be satisfied if either the injected concentration is very low, or the injection time in the test is short, 0.8

0.06

0.6

f(r )

0.04 f(r )

the results, is applied. The method is suitable for modelling phenomena with significant input uncertainty and studies systems with a large number of coupled degrees of freedom. Applying the Monte Carlo simulation with Latin Hypercube selection criteria utilising Descartes’ theorem, the PSD curve can be determined from the grain size distribution. Let us validate the proposed method for prediction of PSD. Nolan and Kavanagh (1994) determined the PSD from a packed column of spherical particles with a given size distribution using geometric 3D simulation. The pore throat sizes were described by the maximum inscribed radius, geometrically restricted by the confining spheres in a packed system. Fig. 7 presents the results of Monte Carlo simulation for the case when three grains are chosen according to their probability distribution to generate a single pore radius by Descartes’ theorem presented in Section 2. The process is repeated 100,000 times using Latin-hypercube sampling in a Monte Carlo simulator to generate a PSD, and the predicted curves are smooth. On the contrary, the curves obtained by Nolan and Kavanagh and shown in Fig. 7 (a) and (b) exhibit oscillations due to small number of Monte Carlo runs. Despite some deviation between the prediction results by the Descartes’ method and those presented by Nolan and Kavanagh (1994), the quality of the agreement allows for utilisation of the proposed method for the evaluation of PSD from known size distribution of packed glass beads. Several other cases (Rouault and Assouline, 1998; AlRaoush et al., 2003; Reboul et al., 2008) have been investigated as well and show the same order of magnitude in deviation between the Descartes’ prediction and geometric simulation. The proposed method aims at an approximate evaluation and cannot substitute high precision geometric simulation. After validation of the Descartes’ method, the PSD is determined from the grain size distribution for conditions of laboratory experiments. The size distribution of the glass beads adopted as the porous media is determined using a Malvern Mastersizer (2000). The measured grain size distribution and the calculated PSD by Monte Carlo method are given in Fig. 8. The PSD can be approximated by lognormal distribution with the mean of 3.52 ␮m and the standard deviation of 0.53 ␮m. The obtained PSD curve will be used further in

0.4 0.02 0.2

0

0 0

10

20

30

40

50

60

70

r , r (µm) Pore Radius from Monte Carlo Simulaon

Grain Radius

Fig. 8 – Determination of pore size distribution from grain size distribution using Monte Carlo simulation.

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chemical engineering research and design 9 0 ( 2 0 1 2 ) 63–77

or the filtration coefficient is small resulting in low retention concentration even for high suspension concentration. Let us examine whether the inequality Eq. (31) is fulfilled under the conditions of the performed tests. The concentration of vacant (thin) pores is h = pNp /V, where Np is the number of pores, V is the volume of the column and p is the fraction of vacant pores. The number of grains for spherical grain packing is Ng = (1 − )V/(4rg3 /3). For assumed close and dense packing in the column, each pore is formed by four grains and each grain is surrounded by eight pores. Hence, Np = 2Ng : Np =

(1 − )V (2rg3 /3)

(32)

,

The suspended concentration is smaller than the injected one c(x,t) < c0 , the retained particle concentration is limited by (x, t) = c(x, t)Ut ≤ c0 Ut. Let us define the dimensionless time tD = tU/L calculated in pore volumes of injected suspension. Recalculate the injected concentration c0 measured by the number of particles per volume of suspension into volumetric concentration measured in ppm multiplying by the colloid particle volume (4rs3 × 106 /3). Accounting for (32), the condition (31) is transformed to the following form: c0 tD L 2p(1 − )

r 3 g

rs

× 10−6  1,

(33)

The product c0 tD in the numerator of (33) supports the above statement that low retention case corresponds to either small injected concentration, or short injection time, or small filtration coefficient. Substituting the values of parameters, which are typical for the coreflooding tests described in Section 4 of the present work (c0 = 8 ppm,  = 0.4, the measured particle size rs = 5.0 ␮m and the grain size rg = 40 ␮m, the fraction of thinner pores p = 0.3, dimensionless filtration coefficient L = 3.0 as calculated from experimental data and tD = 7 PVI) yields the value of left hand side 0.093, i.e. the condition (33) is fulfilled. Yet, the breakthrough concentrations in 5 tests stabilize after 4–5 PVI. Also, the maximum retention concentration at the inlet

exponentially decreases along the core. It allows concluding that the blocking is negligible under conditions of the conducted tests (Section 4).

6.2.

Results of the coreflood tests

Let us determine PSD for the glass bead packed porous column from breakthrough concentrations as obtained from the injection of different size suspended particles (Section 4.2) using the analytical model (29)



CL (rs ) L = (1 − fns (rs )) exp −fns (rs ) l C0 (rs )

 (34)

,

Here the fractional flux via thick pores fa + fnl is substituted by 1 − fns . Consider n challenge tests using particles with radii rs1 , rs2 , . . ., rsn . The inlet and breakthrough concentrations C0 (rsi ) and CL (rsi ) are known from experimental data for each test. Assume lognormal distribution for pore sizes as determined by mean pore radius and standard deviation  0 . It allows expressing the fractional flow via thin pores fns via and  0 using formula (11). Finally, n tests yield system of n transcendental equations (34) for three unknowns – mean pore radius, standard deviation and dimensionless correlation length l/L. The system is solved by the least square method, i.e. the solution minimises the total quadratic deviation between the experimental data and those predicted by modelling n  

min rp ,0 ,l/L



L (fa (rsi ) + fnl (rsi )) exp −fns (rsi ) l

i=1



CL (r ) − 0 si C (rsi )

2 . (35)

The results of minimisation (35) are shown in Figs. 9–11 and in Table 2: mean pore size is 3.13 ␮m, distance between chambers l = 0.25 mm. The solid piecewise lines 1,2, . . ., 5 in Fig. 9 correspond to breakthrough curves during 5 injections of mono sized particles with different sizes presented in Table 1. The concentrations are constant over the time intervals where the effluent samples were taken. The stabilised relative breakthrough values CL /C0 for type 1 particles almost coincide with

Fig. 9 – Breakthrough concentration curves for five sized particles.

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chemical engineering research and design 9 0 ( 2 0 1 2 ) 63–77

1

Table 2 – Summary of pore size distribution results from the model and Monte Carlo simulation.

model exp MC

1

0.9 0.8

Modelling data

2

0.7

Mean radius (␮m) Standard deviation (␮m) Average relative error

c L/c 0

0.6 0.5

3.13 2.76 1.94e−4

Monte Carlo simulation 3.52 0.53 0.11

3

0.4 0.3 0.2 0.1 0

4 0

2

5

4

6 rs

8

10

12

Fig. 10 – Normalized breakthrough concentration versus jamming ratio for five size particles. line CL /C0 = 0.93, i.e. it is expected that rs1 slightly exceeds minimum pore throat size. The relative breakthrough values CL /C0 for types 4 and 5 particles almost coincide with line CL /C0 = 0, i.e. it is expected that maximum pore size is less than rs4 and rs5 . Fig. 10 presents the normalized breakthrough concentration versus the jamming ratio j = rs /rp0 , where rp0 is a typical pore throat size. In the sake of forthcoming analysis, rp0 for (a) 1

0.25

0.2

0.6

0.15

0.4

0.1

0.2

0.05

fns (rs)

F(rp)

0.8

0

0

2

4

6

8

0 12

10

rp, rs 1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

fns (rs)

F(rp)

(b)

0

0

20

40

60

80

100

0 120

rp, rs

Fig. 11 – Cumulative pore size F(rp ) (continuous curve) and pore flux fns (rs ) (dashed curve) distributions obtained by matching with the model assuming the triangular pores (a) full interval of pore throat sizes; (b) zoom.

calculations presented in Fig. 10 was taken from the results of matching (35) and rp0 = . The stabilised values of the breakthrough concentrations in all tests were adopted as CL values. The solution (,  0 ) of optimisation problem (35) is presented in Table 2 for circular pore shape (first column) and for triangular pore shape (second column). Fig. 11(a) and (b) presents the cumulative PSD curve as calculated from the solution (,  0 ) for triangular pore shapes. Dashed horizontal straight lines in Fig. 9 correspond to stabilised breakthrough concentrations as predicted by analytical model (34) for solution (,  0 ) for five particle sizes for the triangular pore shape using optimisation procedure (35), showing good agreement between the experimental and predicted values of stabilised breakthrough concentrations. Fig. 10 shows that all 5 experimental points match very well with the modelling curve as obtained by formula (34). Dashed curve in Fig. 10 corresponds to the normalized breakthrough curve as obtained by the analytical model (34) using the PSD calculated by the Monte Carlo simulations according to Descartes theorem (30). Some deviation is evident. Despite the Descartes model predicts the averaged pore throat size with a good accuracy, the difference in standard deviations is significant (Table 2). The test would be more complete with the injection of particles smaller than the minimum pore radius resulting in full particle recovery. The minimum pore radius is equal, according to first column of Table 2, to 0.37 ␮m. Yet, such small particles were unavailable. Therefore, the test with the injection of particles smaller than the minimum pore radius was performed for larger particles and glass beads under the same salinity, pH, etc. The glass bead size range was 20–31.5 ␮m corresponding to 2.46–4.58 ␮m range of pore sizes; particle size was 0.886 ␮m. The recovery (normalized breakthrough concentration) was 0.98. Besides the cumulative PSD plot shown in Fig. 11 as a continuous curve, dashed curve shows the cumulative distribution of flow rates via different size pores fns , (11). Since the rate via the pore is proportional to rp4 , dashed curve lays significantly to the right from the continuous curve. It means that even for large pores, where the fraction of smaller pores is large, the flux via the smaller pores is relatively small. For large particles, where fns has order of magnitude of unity, the first term in right hand side of Eq. (34) is small. Since L/l  1, the second term is also small. So, the effluent concentration is negligibly small if compared with the inlet concentration. Therefore, the proposed method allows determining PSD only up to a maximum value rpmax , which is determined by the accuracy of concentration measurements. Let us calculate the maximum pore radius, which the method allows to determine. For porous media under consideration, L/l = 200. Consider three cases of the particle counter accuracy – 0.03, 0.02 and 0.01. The maximum value 0.03 corresponds to the particle

chemical engineering research and design 9 0 ( 2 0 1 2 ) 63–77

0.2

0.08

0.15

0.06

0.1

0.04

0.05

0.02

0

0

1

2

3 rs

4

5

fns

c L/c 0

74

0 6

Fig. 12 – Maximum pore size that can be determined from challenge coreflood test. counter Pamas (see website). The minimum value 0.01 corresponds to Malvern Mastersizer (Mastersizer, 2000). These are the values for left hand side of Eq. (34). The modelling curve CL /C0 (34) allows determining the corresponding particle sizes, which are 4.39, 4.53 and 4.75 ␮m, respectively. Fig. 12 presents the plot of the fractional flow via smaller pores fns (rs ) versus particle radius and the normalized breakthrough concentration CL /C0 versus the particle radius. The graph fns versus rs in Fig. 12 allows calculating the fractional flow values fns for the above the accuracy values, which are 0.0174, 0.0195 and 0.0229, respectively. The medium pore size, as determined above by tuning procedure (35) is 3.13 ␮m (Table 2). It corresponds to 50% of pores smaller than 3.13 ␮m. Formula (11) and Fig. 12 allows calculating that only 0.0048 of the total flux moves via the smaller pores. The product in exponent of (34) is −0.96, the normalized breakthrough concentration is 0.383, i.e. the value which can be measured with high accuracy. For points 1, 2 and 3, the fractional flow fns is equal 0.0004, 0.0013 and 0.0037; despite large value of L/l = 200, the products in exponent of (34) are 0.08, 0.26 and 0.74. The corresponding values of normalized breakthrough concentrations are 0.93, 0.79 and 0.47, i.e. quite “measurable”. The distribution curve for pore throat sizes is located to the left of the distribution curves for fluxes via these pores. So, even for large pore radii rp , the corresponding fns is small enough for the corresponding particle to be detected at the core outlet. It explains why the challenge coreflood test allows determining PSD for pores which are significantly larger than the mean value.

7.

Discussions

The proposed method of PSD determining involves matching the analytical model (29) for deep bed filtration with experimental breakthrough concentrations, as obtained from size exclusion deep bed filtration tests, and uses PSD as calculated from grain sizes to validate the proposed method. The analytical model uses geometrical description of the porous media as a set of parallel capillary intercalated by mixing chambers which is widely used for suspension transport modelling. The main assumption of the model – low retention filtration – can be fulfilled in laboratory challenge tests on colloid transport in porous media. Yet, the model (19)–(21) presents a simplified description of the pore space geometry. The network models better represent the pore and throat

shapes. The percolation and effective media models better represent the topology of the pore network (Bedrikovetsky, 1993; Selyakov and Kadet, 1996; Panfilov et al., 2008). The low retention assumption limits the injection period; the numerical population balance models can simulate long term injection processes. The above shows the ways of the improvement of the proposed method. The key point of the proposed laboratory method is creation of the environment that provides the grain-particle repulsion. It excludes various particle capture mechanisms driven by the particle–grain attraction like attachment, bridging, internal filter cake and avoids cumbersome and expensive physico-chemical characterisation of grain and particle surfaces, interactions, geometry of retained matter, etc. The only particle capture mechanism under the repulsion is size exclusion. The straining mechanism is characterised mainly by pore and particle size distributions, which suits to the goal of this research. Several filtration tests with latex particles in packed glass bead medium with simplified geometry show that low salinity and high pH of water promote the grain-particle repulsion. Therefore, the same glass beads, particle and water have been used further for column experiments in 3D porous media. The preliminary repulsion tests have been performed for latex colloids and glass beads. More research is required to determine the injected water composition providing the colloid–grain repulsion for the natural reservoir rocks (sandstones with clays, dolomites, carbonates, etc.), filters and membranes. The breakthrough concentration curves for five tests, as plotted in Fig. 9, stabilise after injecting 5 pore volumes at the most. The experiments have been conducted with a minimum of 250 mL (7 pore volumes) of injected fluid, i.e. the stabilised values have been reached in all tests. The restriction of the colloid concentration measurement is that the minimum volume required for an accurate test is 50 mL. Therefore, the rig used makes it possible to determine only a stepwise breakthrough curve – the concentration values are constant over the time intervals where a single sample was taken. Yet, this does not affect the quality of the results because only the stabilised outlet concentrations are used for PSD determination, see (34). The Descartes’ model for pore sizes, used for validation of the proposed method for PSD determination, is based on the assumption of dense and close packing, where four grains form one pore body. This assumption effectively defines the radius of a pore throat as the maximum inscribed radius of three mutually tangent grains in the plane formed by the centres of these spherical grains. If the assumption of densely packed media is incorrect, more than three grains may construct a pore throat and hence the calculation of PSD using Descartes’ theorem becomes inaccurate. In the case of close packing not occurring, high permeability streaks may transport the colloidal suspension through the column without exposure to the matrix pore throats. If the assumption of spherical grains is inaccurate, the shape of a pore throat becomes more complex and is less likely to be defined by a curvilinear triangle. In order to ensure that the matrix is closely and densely packed, an extensive sieving process is performed to narrow the size distribution. It is reasoned that similar sized particles will reduce the occurrence of unconstrained matrix particles. Furthermore, the experimental procedure includes the wet packing of the column, an extended settling period and sonification of the glass beads to

chemical engineering research and design 9 0 ( 2 0 1 2 ) 63–77

Fig. 13 – Visual observation of glass bead packing matrix. increase the likelihood that the matrix is closely and densely packed. The assumption of sphericity is confirmed through analysis of the packing matrix under an Olympus BX51 optical microscope. From observation of the glass beads under the microscope, it is evident that any irregularities of the glass beads are within a tolerable range. Despite wet packing have been performed in the tests and the column was immersed into a sonic bath, the ideally dense packing have not been achieved. Fig. 13 shows the cross section of the column under SEM, where the complete contact between grains in some places does not occur. Therefore, despite the results of the Descartes’ modelling are in a good agreement with that of the precise geometrical 3D modelling of the porous space (Fig. 7), the latter is more preferable for validation of the proposed PSD determination method. Another limitation of the Monte Carlo simulation employed in this work is that the grain size distribution is stepwise (Fig. 8). A more accurate size distribution for the glass beads could not be obtained due to the constraints of the particle sizer data (Malvern Mastersizer, 2000). The ratio of total number of injected colloids to the pore vacancies is kept below 0.1 throughout the tests (see (33)) to ensure that there is no competition between colloidal particles for pore throats. It shows that the assumption of low retention filtration is fulfilled during the injection process in the carried out tests. Such a low ratio also prevents the occurrence of concentration decline due to accumulation of particles around the blocked pore throats. The adjustment of PSD using 5 experimental points shows that the experimental points are tuned well by the predicted curve as calculated for pores with triangular shape – five points are located exactly on the dashed curve in Fig. 10. Yet, very limited sizes for particles and glass beads, available on the market, narrowed the amount of experimental data. Just five points have been recovered from the flow tests, where just three points are clearly located between the minimum and maximum pore sizes. More points must be utilised for reliable determination of three model parameters (mean radius and standard deviation of PSD, correlation length). The deviation between the Monte Carlo and the model generated curve, presented in Fig. 10, can be explained by the existence of four-grain patterns in the media. Despite a compact structure of grains in the bed, some inter-grain spaces are

75

formed by four neighbouring grains instead of three. However, Fig. 13 presents the photo of glass beads under SEM microscope; it shows that the concentration of four-grain patterns is significantly lower than that of three-grains. The Monte Carlo simulation based on Descartes’ theorem (Section 5) assumes that the porous space consists of three-grain patterns only, which leads to a deviation of PSD from the real media. Although the validation of the proposed PSD calculation method was successful, more extensive research is required to validate the proposed method and recommend it for practical applications in porous media technologies. First, the proposed Monte Carlo method for calculation of the PSD from the grain size distribution must be improved and further validated. The method assumes the triangular throat shape only. Yet, there may be some caverns which cannot be plugged by particles in hexagonal mode due to particle sizes and quantities, available from the given set of grains. More complex throat shapes, formed by four and more particles, may be considered. Since the data on known grain and PSD for the same porous media are almost not available in the literature, the laboratory studies have to be carried out for validation of the method. Second, the number of “experimental points”, as obtained from our lab tests is very limited. The good match between the PSD curve, as predicted by analytical modelling based on the test data from suspension flow, with that calculated from the grain size distribution, do not validate the method completely – on account of the limited number of points. Since very restricted sizes of the used latex particles are available in the market, it would be a solution to find other sources for particles. The proposed model (Eqs. (20)–(23)) describes long time filtering where the retained particle accumulation affect the filtration coefficient, while in the present paper the short time injections with constant filtration coefficient are discussed. Further research may include treatment of the long term data using optimisation procedure for the general model (Eqs. (20)–(23)). This option is particularly attractive since the time variation of pressure drop across the core can be used as additional information to characterise the filtration system (Bedrikovetsky et al., 2001; Grenier et al., 2008; Rebai et al., 2010; Daniel et al., 2011). The particle pulse injection could be considered for the use instead of the continuous injection. It would shorten the test period and decrease the particle consumption. Yet, the high particle concentration during a short time pulse injection may be restricted by the low retention assumption, besides the higher accuracy of concentration measurement is required if compared with continuous injection. The pulse option of laboratory tests and its combination with already applied continuous injection would also demand an additional research.

8.

Summary and conclusions

The challenge test on colloidal flow with different size particles under the particle–grain repulsion allows for determining the pore throat size distribution in long cores. Creation of the particle–grain repulsion environment allows avoiding the particle capture by attachment and expensive and cumbersome characterisation of particle–grain interaction in order to model the particle attachment. The particle–grain repulsion leaves size exclusion as a single particle retention mechanism. This mechanism is described by pore space geometry and particle sizes only. The fact that size exclusion deep bed filtration is determined by pore space only is in accordance with the goal

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chemical engineering research and design 9 0 ( 2 0 1 2 ) 63–77

of determining the PSD from transport of different size colloids through porous media. The particle–grain repulsion can be achieved by the correct choice of the material and coating of the particles and of water composition (salinity, pH). The geometrical model of a porous space as a set of parallel capillaries intercalated by mixing chambers, which is widely used for colloidal flows in porous materials, is fully determined by PSD and the inter-chamber distance. Therefore, this model was chosen for determining PSD. The colloidal flow test includes injection of different size particles with measurements of the inlet and stabilised breakthrough concentrations. The analytical model for size exclusion deep bed filtration relates the inlet and stabilised breakthrough concentrations with PSD. Finally, PSD is determined from the inlet and stabilised breakthrough concentrations by least square method using the analytical model. The laboratory tests performed in packed glass bead column with 5 types of particles with different sizes show that the determined PSD curve tunes well 5 experimental points. The curve is also in a good agreement with that obtained by the Descartes’ method. The analytical model for triangular shape capillary tunes the experimental points and agrees with the Descartes’ method better than that for circular pore shape. The proposed method needs more substantial validation. More experimental points for particle radii between the maximum and minimum pore sizes must be obtained. It is preferable to use high accuracy geometric simulation of the packed column for validation of the results.

Acknowledgements Many thanks are due to Prof. Y. Yortsos (University of Southern California), Prof. A. Shapiro (Technical University of Denmark) and Prof. P. Currie (Delft University of Technology) for longterm cooperation in suspension flow studies. Prof. A. Shapiro critically revised the text and provided valuable comments. Special thanks go to Dr. A. Badalyan for his helpful suggestions and comments. The authors are grateful to A. Zeinijahromi for calculation of DLVO forces. Fruitful discussions with T. Rodrigues and I. Abbasy (Santos Pty Ltd., Australia) are gratefully acknowledged. Financial supports from the Australian Research Council (ARC) Discovery Project 1094299, Linkage Project 100100613 and Santos Pty Ltd are gratefully acknowledged.

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