Asymptotic Approach to the Generalized Brinkman s Equation with Pressure-Dependent Viscosity and Drag Coefficient

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1 Journal of Alied Fluid Mechanics, Vol. 9, No. 6,. 3-37, 6. Available online at ISSN , EISSN Asymtotic Aroach to the Generalized Brinkman s Equation with Pressure-Deendent Viscosity and Drag Coefficient I. Pažanin, M. C. Pereira and F. J. Suárez-Grau 3 Deartment of Mathematics, Faculty of Science, University of Zagreb, Bijenička 3, Zagreb, Croatia Deartamento de Matemática Alicada, Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão,, CEP 558-9, São Paulo, Brazil 3 Deartamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Avenida Reina Mercedes S/N, 4 Sevilla, Sain Corresonding Author azanin@math.hr (Received October 6, 5; acceted May 4, 6) ABSTRACT In this aer we investigate the fluid flow through a thin (or long) channel filled with a fluid saturated orous medium. We are motivated by some imortant alications of the orous medium flow in which the viscosity of fluids can change significantly with ressure. In view of that, we consider the generalized Brinkman s equation which takes into account the exonential deendence of the viscosity and the drag coefficient on the ressure. We roose an aroach using the concet of the transformed ressure combined with the asymtotic analysis with resect to the thickness of the channel. As a result, we derive the asymtotic solution in the exlicit form and comare it with the solution of the standard Brinkman s model with constant viscosity. To our knowledge, such analysis cannot be found in the existing literature and, thus, we believe that the rovided result could imrove the known engineering ractice. Keywords: Brinkman s equation; Pressure-deendent viscosity; Pressure-deendent drag coefficient; Transformed ressure; Asymtotic analysis. NOMENCLATURE F dimensionless flux H thickness of the channel l length of the channel ˆk ressure-viscosity coefficient k dimensionless ressure-viscosity coefficient M ε Characteristic number ˆP ressure P dimensionless ressure dp dimensionless ressure dro P referent ressure q û u V ˆβ β ε ˆ σ transformed ressure velocity dimensionless velocity referent velocity drag coefficient dimensionless drag coefficient small arameter viscosity dimensionless viscosity auxiliary arameter. INTRODUCTION The steady-state flow of an incomressible, viscous fluid through a orous media is given by the conservation of mass and conservation of linear momentum rinciles. While the conservation of mass is described by the standard continuity equation div u ˆ, () different laws have been used to describe the conservation of the linear momentum. Without doubt, Darcy law (Darcy 856) is the most commonly used model stating that the filtration

2 I. Pažanin et al. / JAFM, Vol. 9, No. 6,. 3-37, 6. velocity is roortional to the alied ressure gradient. However, it is based on several (restrictive) assumtions and, thus, its range of alicability is limited. In articular, exressed as a first order PDE for the velocity, Darcy law cannot sustain the no-sli boundary condition for the fluid velocity imosed on an imermeable wall. That insired H. Brinkmann (Brinkman 947) to roose the correction of the Darcy law in order to be able to imose such condition on an obstacle submerged in orous medium. In the absence of the external force, Brinkman equation can be written as ˆ uˆ ˆ βˆ u ˆ, () Note that the second-order Eq. () can handle the resence of a boundary on which (hysically relevant) no-sli condition is imosed. Nevertheless, in many geohysical roblems the variations of the viscosity with ressure cannot be ignored if the flow is subjected to very high ressure dros. Such situation naturally occurs in etroleum engineering, namely in roblems such as enhanced oil recovery and CO sequestration. In view of that, the Brinkman equation needs to be generalized in order to be able to cature the effects of the ressure-deendent viscosity. The notion that the fluid viscosity can deend on the ressure goes back to the celebrated work by Stokes (845). Since then, numerous exerimental investigations (see e.g. Binding et al. (998), Goubert et al. (), Del Gaudio and Behrens (9)) confirmed that, as the ressure is increased by several orders of magnitude, the variations of the viscosity with ressure should be taken into account while the flow is still incomressible. The viscosityressure relation is most commonly described by the Barus law (Barus (893)) stating that the viscosity increases exonentially with ressure: k ˆˆ ˆ ( ) ˆ e, ˆ, k ˆ const.. (3) In the context of Eq. (), the ressure-deendent viscosity imlies that the orous medium arameter ˆβ (drag coefficient) also varies with the ressure (see e.g. Srinivasan et al. (3)). More recisely, the exonential deendence (3) leads to a drag coefficient of the form ˆ ˆ k ˆˆ ( ) ˆ e, ˆ, k const.. (4) As a consequence, we arrive at the following generalized version of the Brinkman s equation: ˆ div ˆ ( ˆ) Duˆ ˆ β ˆ u ˆ, (5) where is Du T ˆ ˆ ˆ u u the symmetric art of the velocity gradient tensor. The aim of this aer is to study the flow governed by Eqs. () and (5) from the analytical oint of view. Note that when ˆ and ˆβ is constant, Eq. (5) reduces to simle Darcy equation, while for ˆβ and ˆ constant, we get the Stokes equation. Finally ˆ and ˆβ are assumed to be constants (i.e. k = ), we have the Brinkman's equation (). Introducing the exonential deendence (3), (4) into the Brinkman s equation makes the roblem highly nonlinear and, thus, we cannot exect to derive the exact solution of the full system (), (5) even in the case of the simle two-dimensional channel (i.e. fracture with lane-arallel walls). Therefore, we introduce the small arameter ε (denoting the ratio between thickness of the channel and its length) and roose the asymtotic aroach as ε. After rewriting the Eqs. (), (5) in the non-dimensional form, we aly the concet of the transformed ressure originally roosed by Marušić-Paloka and Pažanin (3). By doing that, we transform the momentum Eq. (5) into the equation with small nonlinear erturbation that we can control. In such transformed system, we comare the characteristic (non-dimensional) number M ε describing the flow with small arameter ε trying to identify the critical case in which all the effects we seek for are balanced at the main order. We comute the asymtotic solution of the transformed roblem in the most interesting (critical) case and then recover the solution of the governing roblem by alying the inverse transformation. As a result, we obtain the effective model described by the exlicit exressions for the velocity and ressure distribution. It enables us to clearly observe the influence of the viscosityressure deendance and orous structure on the effective flow. In articular, we can easily comare our asymtotic solution with the solution of the standard Brinkman s model with constant viscosity. To conclude the Introduction, we rovide some bibliograhic remarks on the subject. For the flow through orous media, as governed by the Brinkman equation and its various generalizations, analytical treatments can be found only in the constant viscosity case ( k ˆ ). We refer the reader to Durlofsky and Brady (987), Kuznetsov (998), Malashetty et al. (), Merabet et al. (8), Marušić-Paloka et al. (), Khan et al. (4)). In the variable viscosity case ( k ˆ ), the numerical aroach has been develoed and we refer the reader to Naskhatrala and Rajagoal (9), Srinivasan et al. (3). In those aers, the results based on numerical simulations have been rovided clearly suggesting that the corresonding solutions exhibit significantly different characteristics than the solutions of the classical Darcy or Brinkman s equation. We confirm those findings in the resent aer using analytical (asymtotic) aroach and without making any simlifying assumtions on the governing system of PDEs and its solution. If we assume that û= û (ˆx) i (in fact, û (ˆx) = const., due to ()), then the Eq. (5) is reduced to a simle ODE for the ressure which can be easily solved by the searation of variables (see Subramanian and Rajagoal (7)). As far as we know, this is the first attemt to carry out such an 3

3 I. Pažanin et al. / JAFM, Vol. 9, No. 6,. 3-37, 6. analysis and, thus, we believe that it could imrove the known engineering ractice. If we assume that ˆu = ˆu (xˆ)i (in fact, ˆu(xˆ) = const., due to ()), then the Eq. (5) is reduced to a simle ODE for the ressure which can be easily solved by the searation of variables (see Subramanian and Rajagoal 7).. FORMULATION OF THE PROBLEM We consider the flow in a simle two-dimensional domain ˆ xˆ, yˆ R : xˆ l, yˆ h. (6) The ratio ε = h/l is assumed to be small meaning that the channel under consideration is either very thin or very long. Such situation aears naturally in the alications we aim to address. As exlained in the Introduction, the channel is filled with a fluid saturated orous medium and the flow is modeled by the generalized Brinkman s equation with ressure-deendent viscosity and drag coefficient. Thus, we have the following system satisfied by the unknown fluid velocity û and ressure ˆ : ˆ div ˆ ( ˆ) Duˆ ˆ β ˆ u ˆ, (7) div u ˆ, (8) where the deendence of the functions ˆ and ˆβ uon the ressure is given by Eqs. (3) and (4) resectively. To comlete the roblem, suitable boundary conditions have to be added. We imose a standard no-sli boundary condition for the velocity on the channel walls and we assume that the flow is governed by the rescribed ressure dro dp ˆ ˆl. In view of that, the boundary conditions read: u ˆ for yˆ, h, (9) uˆ i for xˆ, l, () ˆ ˆi for xˆ i, i, l. () Our goal is to investigate the asymtotic behavior of the flow described by (7) (), as ε. 3. ANALYSIS 3. Equations in Non-Dimensional Form The first ste is to write the governing roblem in the non-dimensional form reresenting the aroriate framework for our analysis. We do it in a standard way by introducing xˆ yˆ uˆ ˆ x, y,,, l l V P u () kˆ ˆ βˆ k,, β= (3) ˆ β ˆ P where V, P denote referent values of the velocity and ressure. Setting P V (4) β l we obtain k u M e M u kd ( ), ε ε u (5) divεu. (6) Note that the above equations are osed in the ε- indeendent domain Ω = (,). The nondimensional characteristic number Mε aearing in the Eq. (5) reresents the ratio between the frictional effect due to drag and the frictional effect due to viscosity, namely ˆ l M (7) ˆ Finally, hereinafter we use the following notation for the artial differential oerators: v v v v div εv, v, x y x y φ φ T ε i j, D ( ) ε εv ε v x y v v x y εv, v v +v i j v v x y 3. Transformed Pressure Following the idea from Marušić-Paloka and Pažanin (3), we introduce a new function qε, called the transformed ressure, such that k e ε ε q. (8) From (8) we deduce k k q e e, k R. (9) For the sake of the further analysis, it is imortant to observe that arameter σ in (9) can be chosen in an arbitrary way. Since k ε e εq εq, k e kq () from (5)-(6) we get the following system satisfied by the velocity u and transformed ressure q: k εu Mε εq Mεu Dε( u ) q, k e kq () 33

4 I. Pažanin et al. / JAFM, Vol. 9, No. 6,. 3-37, 6. divεu. () The transformed Eq. () is still nonlinear but we are able control the nonlinearity aearing on the right-hand side. Indeed, the liberty in choice of arameter σ (see (9)) enables us to choose it small enough so that k lim. k e kq (3) Such choice of the arameter σ will be justified in the sequel by the fact that the effective ressure does not deend on σ at all (see (35)). That means that, throughout the analysis, σ lays the role of just an auxiliary arameter, i.e. by choosing σ such that (3) holds, we do not imose any additional constraints in the rocess. 3.3 Asymtotic Solution Now we have to construct the asymtotic solution of the transformed system ()-(). Before roceeding, it is imortant to make the following observation: if we ket characteristic number M ε constant (i.e. indeendent of ε), then a simly calculation would yield the model with no contribution of the orous structure at the main order. Thus, we need to comare M ε with small arameter ε trying to identify the critical case in which those effects remain in the macroscoic model. It can be easily verified that the critical case takes lace when Mε O(ε ). Indeed, if we take Mε O(ε ) we would obtain a simle Darcy law not accounting the effects of the Brinkman (viscous) term. On the other hand, the assumtion Mε O(ε ) would yield the situation already described above. In view of that, we set M Mε ε, M O () (4) and erform a careful analysis under this assumtion. Note that the assumtion (4) suggest that the frictional effects between the fluid and the ores in the solid are dominant over the frictional effects within the fluid due to viscosity. Exanding the solution as uu ε u... q q ε q... and taking into account (3), we first conclude that q q ( x). Next, we obtain u dq dq : M im jmu ε y dx dy u : in, ε y u for y,. (5) We deduce u u( x, y) i, q q( x), leading to u dq Mu = M in. y dx (6) We can treat x (,) as a arameter and solve the above equation as a linear second-order ODE with resect to y. We get My My dq u ( x, y) C( x) e C( x) e. (7) dx The functions C ( x), C ( x) can be comuted by taking into account that u for y,. We obtain M M e dq e dq C ( x), C( x). M M dx M M e e e e dx (8) It remains to determine the transformed ressure q. From the divergence equation we deduce u : u in. (9) x y Integrating the above equation with resect to y gives udy udy D const. x Taking into account (7), we obtain (3) M M M e e M q( x) D xd. M M e e (3) In view of (9), the boundary conditions for q are given by k k q() i e e i, i,, (3) k ˆ with ˆ, l. Consequently, we arrive at k k k k q( x) e e x e e. k k (33) In order to reconstruct the effective ressure eff related to the governing (dimensionless) roblem, we aly the inverse transformation (see (9)) to obtain eff ( x) ln. k k e kq( x) In view of (33) we deduce ( x) ln. k e e e x eff k k k (34) (35) The effective velocity is rovided as 34

5 I. Pažanin et al. / JAFM, Vol. 9, No. 6,. 3-37, 6. ueff ( y) u ( y) i, where u is given by (7).Since, it follows k k e e imlying that eff is well-defined. Moreover, observe that the effective ressure does not deend on the arameter σ at all. This fact justifies the above transformation rocedure, namely the choice of the arameter σ such that (3) holds. 4. DISCUSSION Let us write our (dimensionless) asymtotic solution. In view of (7), (8), (33), and (35), it has the following form: Fig.. Plots of ub( k ) and u for various values of k >. ueff ( y) u ( y) i, k k e e M M y u ( y) [( e ) e M M ke ( e ) M M y M M +( e ) e ( e e )], (36) ln, k e ( e e ) x eff k k k (37) for xy, (,) On the other hand, it can be easily verified that the solution corresonding to the standard Brinkman model ()-() with constant viscosity reads: ub( y) ub( y) i, ( ) M M y ub y [( e ) e M M ( e e ) M M y M M +( e ) e ( e e )], Fig.. Plots of B( k ) and eff for various values of k >. (38) b ( x ) ( ) x. (39) The above velocity and ressure rofiles have been lotted in Figs. - for different values of the ressure-viscosity coefficient k. According to the exerimental results that one can find in the literature (see Srinivasan et al. (3) for details), it is reasonable to choose values k.,.,.35 in the case of the referent 6 ressure P. In the following we also take M = and ut 3 and ensuring the setting with high (dimensional) ressure dro. From Fig. we conclude that the velocity rofiles exhibit different characteristics in constant (k = ) and variable viscosity case (k > ). Indeed, though the symmetry (around y =.5) is reserved in both settings, the values of the velocity for k > are more than ercent lower than those for k =. Furthermore, as the ressure-viscosity coefficient k increases, the velocity rofile becomes more flattened indicating that the velocity, for higher values of k, does not vary too much with resect to y. Fig. 3. Plots of FB (k = ) and F for various values of k >. The difference between constant and variable viscosity case is even more aarent when we comare the ressures (see Fig. ). While for k = we have a simle linear function describing the ressure distribution, for k > the outcome is comletely different. We observe that the ressure exhibits a huge decrease within a short distance near the left end of the channel. After that it does not vary too much with resect to x. That suggests the aearance of the ressure boundary layer in the vicinity of x = in case of high ressure dros between the channel s ends. Note that the decrease near x = becomes more significant as the ressureviscosity coefficient k increases. Last but not least, it is interesting to investigate how the corresonding flow rate (flux) changes with resect to the ressure dro dp (,3) By a simle integration, for Brinkman s model (k = ), we obtain that the flux increase linearly with the 35

6 I. Pažanin et al. / JAFM, Vol. 9, No. 6,. 3-37, 6. change of the ressure difference across the channel, namely B ( ) ( ) dp ( 3 F dp u y dy e e e e 4). (4) On the other hand, in the variable viscosity case, from (36), we deduce FdP ( ) u eff. idy u( ydy ), 3 k k(3 dp) e e = ( e3e 4). ke ( e ) (4) As we can observe from Fig. 3., the values of the flux are significantly lower for k > than those for k = so we may conclude that the standard Brinkman s model in some sense overredicts the flux. Moreover, the flux in variable viscosity case increase very slowly, with the increase of dp. In fact, the increase in the flux is negligible u to some (critical) value of the ressure dro dp (being rather high even for moderate values of k). Such henomena becomes more dominant, as the value of the ressure-viscosity coefficient k goes u. For instance, for k =.35, the critical value of the ressure dro is almost 5, i.e. the variations of the flux are negligible for ressure dros dp < CONCLUDING REMARKS In the resent aer we address one imortant alication of the orous medium flow: the fluid flow governed by the high ressure dro through a thin channel (i.e. thin fracture with lane-arallel walls) filled with fluid saturated orous medium. Such flows aear naturally in the industrial alications, in articular in etroleum engineering. The framework with high values of ressure induce the significant variations of the viscosity with resect to ressure making standard Brinkman s model with constant viscosity inaroriate for describing such flows. Thus, we consider the generalized Brinkman s model with ressuredeendent viscosity and drag coefficient. The viscosity and drag-ressure relation is assumed to be exonential (Barus law) and the effective flow is found using asymtotic aroach with resect to the thickness of the channel. The key idea is to introduce the notion of the transformed ressure enabling us to control the nonlinearity in the momentum equation. The obtained result clearly indicates that the asymtotic solution exhibits utterly different characteristics than the solution corresonding to the classical constant viscosity case. To our best knowledge, this is the first attemt to solve this roblem using analytical aroach and without making any simlifications on the starting roblem. Finally, one of the benefits of the analysis resented here lies in the fact that it can be straightforwardly generalized in two directions. Instead of addressing simle fracture with lanearallel walls, we can consider more comlex (and realistic) domain, namely the fracture with constrictions (see e.g. Giouloux and Marušić- Paloka ()). Besides the effects of the orous structure and ressure-deendent viscosity, in such setting we are articularly interested to detect the effects of the shae functions describing the constrictions. Another ossible generalization would be to consider general viscosity-ressure deendence ˆ ˆ ( ˆ) satisfied by Barus law and other emiric laws (see e.g. Marušić -Paloka and Pažanin (3)). Instead of (9), we would simly introduce the dξ transformed ressure q as B( ) and (ξ) continue with the same rocedure as above. ACKNOWLEDGMENTS The first author of this work has been suorted by the Croatian Science Foundation (scientific roject 3955: Mathematical modeling and numerical simulations of rocesses in thin or orous domains. The second author has been artially suorted by FAPESP 3/75-, CNPq 396/4-7 and 47/3-7, Brazil. The third author has been artially suorted by the rojects MTM P of the Ministerio de Economı a y Cometitividad and FQM39 of the Junta de Andalucı a. The authors would like to thank the referees for their helful comments and suggestions that heled to imrove the aer. REFERENCES Barus, C. (893). Isothermals, isoiestics and isometrics relative to viscosity. AM. J. Sci. 45, Binding, D. M., M. A. Chouch and K. Walters (998). The ressure deendence of the shear and elongational roerties of olymer melts. J. Non-Newtonian Fluid Mech. 79, Brinkman, H. (947). A calculation of the viscous force exerted by a flowing fluid on a dense swarm of articles. Al. Sci. Res. A, Darcy, H. (856). Les fontaines ubliques de la ville de Dijon, Victor Darmon, Paris. Del Gaudio, P. and H. Behrens (9). An exerimental study on the ressure deendence of viscosity in silicate melts. J. Chem. Phys. 3. Durlofsky, L. and J. F. Brady (987). Analysis of the Brinkman equation as a model for flow in orous media. Phys. Fluids 3, Giouloux, O. and E. Marušić-Paloka (). Asymtotic behavior of the incomressible Newtonian flow through thin constricted fracture. Multiscale Problems in Science and Technology, Sringer 89. Goubert, A., Vermant, J., P. Moldenaers, A. Gotffert and B. Ernst (). Comarison of measurement techniques for evaluating the ressure deendence of viscosity. Al. Rheol.,

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