# Network Model of Flow, Transport and Biofilm Effects in Porous Media

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<ul><li><p>Transport in Porous Media 30: 123, 1998. 1 1998 Kluwer Academic Publishers. Printed in the Netherlands.</p><p>Network Model of Flow, Transport and BiofilmEffects in Porous Media</p><p>BRIAN J. SUCHOMEL1, BENITO M. CHEN2;? and MYRON B. ALLEN III21Institute for Mathematics and its Applications, University of Minnesota, Minneapolis,Minnesota, U.S.A.2Department of Mathematics, University of Wyoming, Laramie, WY 82071, U.S.A.</p><p>(Received: 22 November 1996: in final form: 29 July 1997)</p><p>Abstract. In this paper, we develop a network model to determine porosity and permeability changesin a porous medium as a result of changes in the amount of biomass. The biomass is in the formof biofilms. Biofilms form when certain types of bacteria reproduce, bond to surfaces, and produceextracellular polymer (EPS) filaments that link together the bacteria. The pore spaces are modeledas a system of interconnected pipes in two and three dimensions. The radii of the pipes are given bya lognormal probability distribution. Volumetric flow rates through each of the pipes, and throughthe medium, are determined by solving a linear system of equations, with a symmetric and positivedefinite matrix. Transport through the medium is modeled by upwind, explicit finite differenceapproximations in the individual pipes. Methods for handling the boundary conditions betweenpipes and for visualizing the results of numerical simulations are developed. Increases in biomass,as a result of transport and reaction, decrease the pipe radii, which decreases the permeability ofthe medium. Relationships between biomass accumulation and permeability and porosity reductionare presented.</p><p>Key words: biofilm, network model, permeability, transport, numerical diffusion.</p><p>1. Introduction</p><p>Biofilms, complexes of bacterial biomass and extracellular polymer, have a varietyof interesting effects when they grow in porous media. Typically, the presence ofbiofilms in the voids of a porous medium reduces its permeability and porosity. Thisfact suggests a wide range of applications in the control of groundwater contaminanttransport and production from oil reservoirs. This paper presents a network modelof fluid flow and the transport and growth of biofilms in a porous medium.</p><p>There is a fundamental difficulty in modeling the effects of biofilm growth andtransport in porous media: the microscopic dynamics, best understood at the scale ofindividual pores, affect macroscopic flow properties such as permeability and poros-ity, which are meaningful on much coarser scales. This is the scale-up problem. Thepurpose of our model is to help scale up from a simplified but reasonable representa-tion of the microscopic physics to the scale described by macroscopic flow properties.The macroscale we model is in the order of a soil column in a lab. At this scale, soilsamples are fairly homogenous but have heterogeneous microstructure typical of the</p></li><li><p>2 BRIAN J. SUCHOMEL ET AL.</p><p>pore scale. The scaling up to the level of field tests is a different task, which we donot attempt.</p><p>Several researchers have studied biofilms and their effects on porous media. Van-deviver and Baveye (1992) experimentally investigated a relationship between trans-port and clogging for biofilms. Other researchers, including Tan et al. (1994) andLindqvist et al. (1994), developed models of porous media based on the continuumhypothesis, including transport and sorption of bacteria, and used physical experi-ments to test their models. Taylor and Jaffe (1990a,b,c) experimentally showed thatpermeability reductions up to a factor of 5 104 can be obtained and investigatedthe relationship between biomass and permeability reduction. They also investigateda biofilms effect on dispersivity.</p><p>There also exists a large literature on network models of porous media. Amongthe applications most pertinent to our work are those that investigated particle entrap-ment (Rege and Fogler, 1986; Sahimi and Imdakm, 1991). Koplik (1982) comparedflow properties derived from network models to that predicted by effective-mediumtheory. Koplik and Lasseter (1982) investigated immiscible displacement on a two-dimensional network model. Simon and Kelsey (1971, 1972) modeled miscible dis-placement on a network model. Sugita et al. (1995) used particle tracking to modelsolute transport on a network model.</p><p>In this paper, we construct a network flow model, and show that the resulting sys-tem of equations may be solved using an iterative SOR technique. We develop a newmethod for modeling transport in a network model and consider some of the methodsnumerical properties such as stability and numerical diffusion. We describe how thegrowth of a biofilm may be incorporated into the model and how to account forresulting changes in the permeability and porosity of a porous medium. We presentnumerical results showing that the network obeys macroscopically Darcys law. Wealso show that our method of modeling transport behaves similarly to, but not exact-ly the same as, classical continuum solutions to the advection-diffusion equation.Finally, we include results of some numerical simulations of flow and transport ofnutrients accompanied by biofilm growth. A sequel to this paper (Suchomel et al.,1998) describes scale-up results in more detail.</p><p>There are several main contributions of this paper. One is a proof that a networkmodel yields a symmetric, positive-definite linear system. This is curiously lackingfrom the literature, but it implies that the linear systems are easy to solve. A secondis a new method for modeling transport in a network model. This numerical methodmakes it easy to add terms that model other physical phenomena and couple equationsrepresenting a number of solutes. These physical phenomena may include, but arenot limited to, reaction kinetics, adsorption, and erosion. The extra terms may belinear or nonlinear.</p></li><li><p>NETWORK MODEL OF FLOW, TRANSPORT AND BIOFILM EFFECTS 3</p><p>2. Development of the Flow Model</p><p>Scientists have used network models to describe flow through porous media for over40 years. One of the earliest examples is the work of Fatt (1956). The idea is to avoida detailed geometric description of the pore spaces, which is essentially impossiblein natural porous media (Scheidegger, 1957), in favor of idealized descriptions thatpreserve macroscopic properties of the media. In our setting, network models canhelp quantify how bacterial transport and biofilm growth affect permeability andporosity. In this section, we develop a model of porous media based on networksof pipes and junctions. We derive a system of equations that describes flow throughthis network and examine properties of this system that are relevant to its numericalsolution.</p><p>2.1. network geometry</p><p>We represent the pore spaces by a network of interconnected, cylindrical pipes withrandom radii, assigned according to a lognormal probability distribution. We assumeno spatial correlation to simulate the heterogeneities at the pore scale. We refer to thephysics of individual pipes and the junctions that connect them as the microscale.We use lowercase letters to denote microscale quantities. For example, the pressurehead at junction i is hi ; the flow through the pipe connecting junctions i and jis qi;j . (Sometimes it is useful to index pipes according to the junctions that theyconnect; sometimes it is more useful to index pipes with a single subscript.) Our goalis to understand the macroscale, i.e. the overall physics of the network, in terms ofstatistical properties of the microscale. We use uppercase letters to denote macroscalequantities. For instance, Q is the total flow rate through the network.</p><p>The issue arises whether the networks topology should be random or regular, inthe sense that all interior junctions have the same number of intersecting pipes. Sinceregular grids are much easier to work computationally, we use them in this study.By examining Voronoi tesselations as archetypical random networks, Jerauld et al.(1983a,b) argued that properties derived from the analysis of regular networks arevirtually the same as those derived from networks with random topologies.</p><p>Many possible regular configurations exist. Figure 1 shows rectangular and hex-agonal networks in two space dimensions. In three dimensions, we use cubic net-works. The coordination number of a regular network is the number of pipes thatintersect at an interior junction. Thus, a rectangular network has coordination number4; a cubic network has coordination number 6.</p><p>There also arises the issue whether pipe lengths should be random or constant.It is possible to assign pipe lengths according to some probability distribution, toaccount for tortuosity. However, the choice between fixed and random pipe lengthsseems to make little difference to the macroscale flow properties of a network (vanBrakel, 1975). On the strength of this observation, we assume that all pipes have thesame length, .</p></li><li><p>4 BRIAN J. SUCHOMEL ET AL.</p><p>Figure 1. Square and hexagonal networks used to model two-dimensional porous media.Both networks have size 7 5.</p><p>We designate left-to-right as the overall (horizontal) flow direction, determinedby the macroscopic head gradient. Denote by M , the number of junctions in the flowdirection; let N be the number of junctions in the vertical direction and, in threedimensions, let P be the number of junctions in horizontal direction orthogonal tothe overall flow direction. Thus, the size of a two-dimensional rectangular grid isM N , and the size of a three-dimensional cubic grid is M N P .</p><p>2.2. flow physics</p><p>We specify the flow physics at the microscale, using these physics to compute quan-tities needed to characterize the flow at the macroscale. At the microscale, all flowoccurs through the interconnected pipes, and no fluid accumulates at the junctions. Inan individual pipe having radius ai;j , the flow rate qi;j obeys the HagenPoiseuilleequation</p><p>qi;j D Okj;i(hi hj</p><p>; (1)</p><p>where qi;j is the flow into junction j from junction i through the pipe, hj and hi arethe values of head at junctions j and i, respectively, and Okj;i D ga4i;j =.8L/. Here is the density of the fluid, g is the acceleration of gravity, and is the viscosity ofthe fluid.</p><p>Mass balance implies that, at each junction j ,Np;jXiD1</p><p>qi;j DNp;jXiD1</p><p>Okj;i.hi hj / D 0; for all junctions j :</p><p>Here Np;j is the number of pipes connected to junction j . Thus, the flow equationis a linear system of the form Ah D b, where h is a vector of unknown head valuesat the interior junctions. The prescribed head values at the junctions on the inlet andoutlet sides of the network appear in the vector b.</p></li><li><p>NETWORK MODEL OF FLOW, TRANSPORT AND BIOFILM EFFECTS 5</p><p>The macroscale volumetric flow rate Q through the grid is the sum of themicroscale flow rates through the outlet side of the grid. Under the hypothesis thatthe macroscale flow obeys Darcys law, the permeability of the network is</p><p>K D QLA 1H</p><p>;</p><p>where A is the cross-sectional area of the network, L is its length, and 1H is the nethead drop between its left and right boundaries. We discuss the Darcy hypothesis inSection 7.</p><p>2.3. solving the linear system</p><p>We now show that the coefficient matrix, A, for the flow equation is sparse, symmetric,weakly diagonally dominant, irreducible, and positive definite.</p><p>First, consider the zero structure of A. In two dimensions, a square grid has acoordination number of 4, while a hexagonal grid has a coordination number of 3.In three dimensions, a cubic grid has a coordination number of 6. Therefore, thematrix A for a square grid has at most four nonzero off-diagonal elements in a row.The matrix for a hexagonal grid has at most three nonzero off-diagonal elements ina row, and a cubic grid has at most six. Similarly, A has at most four, three, and sixoff-diagonal entries, respectively, in each column.</p><p>Next, we establish that A is symmetric. The mass balance furnishes one equationfor each node. Alternatively, we may view each pipe as directly influencing twoequations. In a two-dimensional square grid, consider the pipe connecting junctions and , with flow coefficient Ok; , as shown in Figure 2. Associated with junction is a row in A representing the equation</p><p>Oka1;.hha1/ C Oka2;.h ha2/ C Oka3;.h ha3/ C Ok;.h h/ D 0:We can rearrange this equation as follows:</p><p>. Oka2; C Ok; C Oka1; C Oka3;/h Ok;h Oka1;ha1 Oka2;ha2 Oka1;ha1 D 0: (2)</p><p>Therefore, the diagonal entries are positive. Also the entry a; in the matrix A isOk; . Associated with junction we have the equation</p><p>. Ok; C Okb1; C Okb2; C Okb3;/h Ok;h Okb1;hb1 Okb2;hb2 Okb1;hb1 D 0: (3)</p><p>Thus, a; D Ok; D a; when and are indices of adjacent junctions. Fornonadjacent junctions and , a; D 0 D a; . It follows that the matrix A is realand symmetric and, therefore, all eigenvalues of A are real. Similar reasoning appliesto hexagonal and cubic networks.</p></li><li><p>6 BRIAN J. SUCHOMEL ET AL.</p><p>Figure 2. Subset of a rectangular network. Junctions are labeled in boldface, and pipes arelabeled in larger letters.</p><p>Equation (2) shows that A is weakly diagonally dominant:</p><p>jaii j >nX</p><p>jD1;j 6Dijaij j: (4)</p><p>Moreover, for rows corresponding to junctions on the boundary of the grid where thepressure head is set to 0,</p><p>jaii j >nX</p><p>jD1;j 6Dijaij j: (5)</p><p>By Gershgorins Theorem (Kincaid and Cheney, 1991, p. 240), all eigenvalues ofA lie in disks located to the right of the imaginary axis, with possibly some but notall of the disks touching the imaginary axis. Therefore, all the eigenvalues are greaterthan or equal to zero. The matrix is irreducible because of the physical system fromwhich it is constructed. The grid is connected. Zero is a boundary point of the unionof the closed disks, but it is not an eigenvalue of A, since it is not a boundary pointof each of the closed disks (Brualdi and Ryser, 1991, 3.6). Hence, the matrix A ispositive definite.</p><p>The symmetry and positive definiteness of A guarantee that the successive over-relaxation (SOR) method converges for the linear system Ah D b for values of therelaxation parameter between 0 and 2 (Young, 1971, Chapter 4). Such an iterativescheme has important advantages for our model, beyond the usual advantages asso-ciated with minimizing storage. When we include bacterial transport and growth inthe model, the radii of the pipes change as biofilm accumulates. In this case, the flowproperties of the network change, and we must solve the flow equation repeatedlyduring a single simulation. By using SOR, we may take the most recent values ofhead as initial guesses for the succeeding time step, thereby speeding convergence.</p></li><li><p>NETWORK MODEL OF FLOW, TRANSPORT AND BIOFILM EFFECTS 7</p><p>We use a residual criteria for convergence. Define the vector r by</p><p>ri D bi nX</p><p>jD1ai;jhj :</p><p>We iterate until maxfrig < ", where " is some pre-set tolerance. We compute r everytenth iteration. The residual at node i amounts to the sum of volumetric flow ratesout of that node.</p><p>3. Development of the Transport Model</p><p>Modeling biofilm growth in the network requires solving for the transport of biofilm...</p></li></ul>

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