Efficient and Accurate simulation of nonlinearly coupled multiphase flow in porous media. 25 August 2015 Meeneesh Kardale

Transcription

1 Efficient and Accurate simulation of nonlinearly coupled multiphase flow in porous media 25 August 2015 Meeneesh Kardale

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3 Title : Efficient and accurate simulation of nonlinearly coupled multiphase flow in porous media Author(s) : Meeneesh Kardale Date : 25 August 2015 Professor(s) : Dr. Hadi Hajibeygi Supervisor(s) : Matteo Cusini Postal Address : Section for Petroleum Engineering Department of Geoscience and Engineering Delft University of Technology P.O. Box 5028 The Netherlands Telephone : (31) (secretary) Telefax : (31) Copyright c 2015 Section of Petroleum Engineering All rights reserved. No parts of this publication may be reproduced, Stored in a retrieval system, or transmitted, In any form or by any means, electronic, Mechanical, photocopying, recording, or otherwise, Without the prior written permission of the Section of Petroleum Engineering

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5 Delft University of Technology Faculty of Civil Engineering and Geosciences Department of Geoscience & Engineering Efficient and Accurate simulation of nonlinearly coupled multiphase flow in porous media Thesis submitted to the Delft University of Technology in partial fulfilment of the requirements for the degree by Meeneesh Kardale Delft, The Netherlands 25 August 2015 Copyright c Meeneesh Kardale. All rights reserved. An electronic version of this dissertation is available at

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7 Author Meeneesh Kardale MSc. Student Track: Petroleum Engineering & Geosciences Specialization: Petroleum Engineering Department of Geoscience & Engineering Delft University of Technology Title Efficient and Accurate simulation of nonlinearly coupled multiphase flow in porous media Committee Members Dr. Hadi Hajibeygi Assistant Professor Department of Geoscience & Engineering Delft University of Technology Prof.dr.ir. J.D. Jansen Professor and Department Head Department of Geoscience & Engineering Delft University of Technology Dr.ir. M.I. Gerritsma Associate Professor Faculty of Aerospace Engineering Delft University of Technology Matteo Cusini PhD. Candidate Department of Geoscience & Engineering Delft University of Technology

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9 Contents List of Figures Abstract Acknowledgement iii v vii 1 Introduction Governing Equations Discretization Spatial Discretization Temporal Discretization Peaceman Well Model Interface properties: upwind and harmonic-average Capillary Treatment Simulation Strategies for Coupled P-S Systems IMPES Formulation Sequential Implicit Formulation (SQ-Imp) Fully Implicit Formulation (FIM) Stability Analysis Non-Linear Stability Upwind Criteria Stability Analysis Viscous forces Viscous and buoyancy forces Viscos forces and Capillarity forces Flux Correction Methods Non-Linear Convergence Trust Region Method Locating Trust Regions Trust Region Chopping Results Numerical Results Viscous Forces Viscos and Buoyancy Forces Enhanced Oil Recovery Foam in Permeable Media i

10 5.2 Numerical Model Results Flux correction strategy for foam flow Conclusion 49 Bibliography III A FIM formulation VI A.1 Derivation of Equations VI A.1.1 Flow Residual w.r.t Pw VI A.1.2 Flow Residual w.r.t Sw VII A.1.3 Transport Residual w.r.t Pw VIII A.1.4 Transport Residual w.r.t Sw VIII ii

11 List of Figures 1.1 Illustration of two neighboring cells with interfaces. The flux (or velocity) is introduced at the interfaces while the pressure unknowns are at the cell centers Illustration of P c (S w ) (a) and C(S w ) functions (b) Flowchart of Sequential Implicit method Flux Functions: (a) Viscous flux (M = 1, N g = 0, P e ); (b) Viscous and Capillary (M = 1,N g = 0,P e = 0.2); (c) Viscous and Buoyancy (M = 1,N g = 5,P e ); (c) Viscous and Buoyancy (M = 1,N g = 5,P e ) (a) Flux function (M = 1, N g = 0, P e, u t = 1); (b) Residual; (c) Derivative of residual w.r.t S upwind Flux Functions: (a) Flux function (M = 1, N g = 0, P e, u t = 1); (b) Derivative of flux function w.r.t S upwind Flux Function Description of Flow Reversal for the Section Flux Functions: (a) Flux function (M = 1, N g = 5, P e ); (b) Flux function (M = 1, N g = 5, P e ) BC: S L = 1 and S R = 0; (a) Residual Function (b) Derivative of Residual; BC: S L = 0.6 and S R = 0.8; (c) Residual Function; (d) Derivative of Residual For the case of M = 1, N g = 5, P e, and u t = 1 (a) Flux Function (b) Derivative of Flux function w.r.t upwind cell; (c) Derivative of Flux function w.r.t downwind cell; Flux Functions: (a) Capillary Pressure; (b) λwλo λ t ; (c) λwλo λ t p c; (d) Flux Function with Phase based upwinding P e = Flux Functions: (a) Viscous; (b) Viscous and Capillary (P e = 0.2); (c) Residual Function; (d) Derivative of Residual Convergence performance of Residual function with an inflection point (a) Standard Newton; (b) Modified Newton Method [14] Modified Newton method proposed by Jenny et al. [14] Trust-Region method, an improved flux-correction strategy [23] The solution update is from X to Y, C is the inflection point and [A, B] is the buffer zone. The chopping ratio can be defined as ζ = XB XY Comparative study of Sequential Implicit (SEQ-Imp) and Fully Implicit (FIM) convergence properties as the time-step sizes increase for Test Case Comparative study of Sequential Implicit (SEQ-Imp) with and without flux correction strategy for Test Case Saturation Solution at 0.25 PVI: (a) t = 10 and (b) t = sec iii

12 4.4 Log Heterogeneous Permeability Comparative study of Sequential Implicit and Fully Implicit convergence properties with increasing time-step sizes for Test Case Comparative study of Sequential Implicit with and without flux correction strategy for Test Case Saturation Solution at 0.5 PVI injected: (a) t = 160 and (b) t = sec. for Test Case Comparative study of Sequential Implicit and Fully Implicit convergence properties with increasing time-step size for Test Case Comparative study of Sequential Implicit with and without flux correction strategy for Test Case Saturation Solution for 0.65 PVI: (a) t = 6.3 and (b) t = 6317 sec. for Test Case Comparative study of Sequential Implicit and Fully Implicit convergence properties as time-step size increases Test Case Comparative study of Sequential Implicit with and without flux correction strategy for Test Case Saturation Solution for 0.65 PVI: (a) t = 142 and (b) t = sec. for Test Case Flux Function for N g Total number of iterations employed to obtain transport solutions with increasing time-step size for Test Case 5. Note that the t is in days and that the sequential implicit approach does not converge for t > Saturation solution at (a) T S = 1 day; (b) T S = 10 days; (c) T S = 100 days; (d) T S = 1000 days Saturation solution for different time-step sizes for Test Case 5, showing the unconditionally stable simulations with flux-correction strategy. Note that the quality of solution needs to be considered for practical purposes Fluid Properties in of gas phase: (a) Relative permeability (b) Total mobility; (c) Fractional flow; (d) Derivative of fractional flow; (e) Second derivative of fractional flow; (f) Residual function for different time-step sizes Saturation solution for (a) gas flooding and (b) foam flooding after 1000 days Comparative study of Sequential Implicit and Fully Implicit convergence properties for different time-step sizes Comparative study of Sequential Implicit with and without flux correction strategy for foam flow. Note that much bigger time step sizes can be taken after employing the flux-correction strategy Saturation solution for foam flow in homogeneous media with (a) t = 100 sec. and (b) t = sec iv

13 Abstract In this work, the non-linear behaviour of saturation transport equations in sequential implicit simulation strategy for multi-phase, immiscible and incompressible displacements are investigated. In reservoir simulation, use of explicit time schemes can lead to severe time-step size restrictions, specially when strong nonlinear terms exist. For highly heterogeneous formations, when a global time step needs to be taken, the CFL numbers can vary by orders of magnitude through the entire domain. In such cases, explicit schemes are not practical, and implicit time schemes are typically followed. The transport equation which describe the fluid flow in porous media in space and time are often highly non-linear and tightly coupled with the flow (pressure) equation. Characterized by S-shaped flux functions, they can also be non-monotone in presence of strong buoyancy (and capillary) forces. Through the phase velocities, which depend on pressure, the transport equation is strongly coupled to pressures equation. In presence of strong capillarity, the pressure solution also depends on the slope of capillary function, which is a function of saturation. An industry standard procedure is to use Newton method to first linearize these coupled governing equations, and iteratively reach a converged solution. To study the nonlinearity within the transport equation, first, a two-dimensional twophase simulator based on all types of Implicit Pressure Explicit Saturation (IMPES), Sequential Implicit, and Fully Implicit strategies are developed. Both capillary and gravitational effects are being considered. As to extend the stability limits of the sequential implicit strategy, flux correction method is investigated and implemented in the transport equation. All cases of viscous, buoyancy and capillary dominated flows are studied, where, in the cases of strong nonlinearities, non-convergence and severe time-step restrictions have been observed. Stability analysis of the flux functions in buoyancy and capillary dominated flows is performed along with a time-step control strategy, which is shown to result in unconditional convergence irrespective of the time-step size selection. Finally, application of flux correction strategies to EOR processes, where foam flows through the rock formation, is presented. Such a study has never been performed in the literature. v

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15 Acknowledgment Fist and foremost, I would like to thank my supervisor Dr. Hadi Hajibeygi for his unconditional support and guidance for the past two years since I became his student. He has been ever inspiring and helped me a lot at good as well as bad times. I would also like to thank Matteo Cusini for his constant help as my daily supervisor. I also thank all the members of DARSim for the constructive criticism throughout the duration of the project. I would also like to thank Francois Pascal Hamon and Dr. Xiaochen Wang for their prompt response for helping me with the code implementation. Last but not the least, I would like to thank my parents for always believing in me and keeping up my spirits. Without them I would not be the person I am today. vii

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17 Chapter 1 Introduction Reservoir simulation is a powerful tool to accurately solve nonlinear coupled equations describing fluid flow through natural formations, with complex fluid and rock physics [8]. It is also an essential tool for several applications, including hydrocarbon production optimization [6]. Fluid flow in porous media is a complex phenomenon. These are described via mathematical formulations, the governing equations are quite nonlinear, span large scales, and entail highly heterogeneous anisotropic coefficients. As such, analytical methods can only be employed if several simplifications in the physics and geological complexities (e.g., heterogeneity) are made. The validity of these assumptions are questionable, specially, in several modern applications involving complex physics and heterogeneous parameters over three-dimensional domains. Computational approaches, i.e., numerical methods, are quite powerful in dealing with several complexities within the equations, thus, offer more accurate solution strategies for realistic scenarios, compared with analytical approaches. However, they come at much higher costs of computational CPU. They first generate discrete elements on the domain, accounting for geometrical complexities and sometimes complex physics, and offer discretization schemes in order to solve the governing equations numerically. There are, generally, three main schemes: finite difference (FD), finite element (FE), and finite volume (FV) approaches. Finite difference methods, in their original form, are applicable for structured Cartesian grid geometries, and are highly sensitive to grid orientation effects. Finite element methods, on the other hand, can handle various complex grid geometries, but are not locally mass conservative in nature. Note that locally conservative velocity fields are crucial for accurate and stable solutions to transport equations. Finite volume methods, by employing control volumes, are locally conservative by construction, and (with the recent developments) are applicable to unstructured grids. Their implication for unstructured grids are typically more challenging than FE methods. And, their monotonicity become negatively affected when employed for complex grid geometries. Also it is important to note that FE methods have extensions (such as Mixed and Discontinuous-Galerkin formulations) which are conservative at additional costs. Due to their mass-conservation and low computational costs, FV methods have been quite attractive for several fluidmechanics applications, including reservoir simulation, and will be the focus of this thesis 1

18 work. As mentioned before, Partial Differential Equations (PDE) describing fluid flow through porous media are typically nonlinear. The transport of phase saturation, e.g., entails several nonlinear complexities for which employing a classical upwind-based strategy would result in many convergence limitations. As such, in presence of strong nonlinear terms (e.g., capillarity and gravity) classical reservoir simulators are typically restricted to very small time step sizes. This sever time-step chopping makes them too expensive to provide reasonable solution run times for realistically challenging applications. Therefore, several studies in the literature have been dedicated to extend the stability of these special types of nonlinear transport equations, arising from multiphase flow in porous media. In this thesis work, the stability of nonlinear transport equations are studied in detail, and a recently developed flux-correction strategy is implemented and assessed for several test cases. This thesis is structured as following. The rest of this chapter is dedicated to governing equations describing multiphase flow in porous media with capillary and gravitational effects. Also, the three different simulation strategies, namely, IMPES, Sequential-Implicit (SQ-Imp), and Fully-Implicit (FIM), are explained. Then, the nonlinear stability studies are presented in Chapter 2, followed by Chapter 3, where the flux-correction strategy is described. Chapter 4 is dedicated to numerical results, obtained by the developed MAT- LAB simulator. This simulator has been developed from scratch, through this limited thesis work period, allowing for investigation of all types of implementation and convergence challenges within each simulation strategy. Specially the published results of the flux-correction strategy could not be reproduced, until several efforts after discussing with the original developers allowed for resolving the challenge by introducing a safe-zone guard. Chapter 5 presents a novel application for the flux-correction strategy, where it has been applied to simulate foam flow. The thesis is concluded in Chapter 6. Nomenclature and an Appendix on derivation of FIM discretization schemes are presented at the end of this manuscript. Next, in Section 1.1, governing equations that describe two-phase flow in heterogeneous porous media are presented. 2

19 1.1 Governing Equations This section explains the set of mass balance equations governing the immiscible, incompressible, two phase flow in porous media. Wetting phase and non-wetting phases are denoted as w for water phase and o for oil phase. The conservation law for a phase can be written as φ S α t + (u α) = q α, (1.1) where S α and u α are phase saturation and velocity, respectively. The accessible void volume in the rock formation is denoted by φ. Moreover, phase injection source term appears in the righ-hand-side (RHS), q α. As the nomenclature used in this thesis is quite common in reservoir simulation literature, the rest of the parameters are selectively introduced, but a complete list is provided before the Appendix section. The phase velocity u α is related to phase-potential gradient by Darcy s law, which reads u α = k k rα µ α ( p α + ρ α g z). (1.2) Here, p α is the phase pressure, k is the absolute permeability, typically highly heterogeneous anisotropic tensorial factor. Also, k rα is the relative permeability and µ α is the phase viscosity. Phase mobilities are defined as λ α = k rα /µ α, where the water and oil relative permeabilities can be experimentally modeled as k rw = k 0 rws 2 eff and k ro = k 0 ro(1 S eff ) 2, respectively, where k 0 rw and k 0 ro are the end point relative permeabilities. The effective saturation can be defined as S eff = S w S wc 1 S wc S or, (1.3) where S wc and S or are connate water and irreducible oil saturations, and are found experimentally. Substituting the phase velocity in the mass balance equations for the two water-oil phases leads to a well-posed system for 4 unknowns (i.e., p w, p o, S w, and S o ), if constraint and S w + S o = 1 (1.4) P c = p o p w (1.5) relationship are also used. 3

20 The capillary term, P c, is a non-linear function of the wetting phase saturation, S w. In this study, the Brooks Corey relation is used where P c = (S w ) 1 λ, with λ being the sorting factor. Substituting Eq. 1.2 into Eq. 1.1 for the two phases and summing up the two equations results in the pressure (flow) equation, i.e., where u t = u w + u o is the total velocity, and can be stated as u t = q t, (1.6) u t = (kλ t p w + kλ o p c + k(ρ w λ w + ρ o λ o )g z). (1.7) Moreover, the total source term reads q t = q w + q o. (1.8) Eq. 1.7 is substituted in 1.6 to obtain the elliptic pressure equation, which in the absence of source/sink terms 1.6 indicates that the total velocity is a divergence free vector, i.e., u t = 0. The pressure (flow) equation can finally be stated as (kλ t p w + kλ o p c + k(ρ w λ w + ρ o λ o )g z) = q t. (1.9) The motivation behind derivation of the pressure equation, i.e., (1.9), is that the pressure is typically a global variable, while phase saturation is local. Thus, for the cases of small capillary effects, the separation of pressure and saturation unknowns by governing pressure-saturation decoupled system of equations would lead to better convergence properties. As such, the pressure equation is obtained by summing up the transport equation and employing the saturation constraint (1.4), which eliminates the explicit representation of saturation in the pressure equation. This is typically the case when capillarity effects are negligible, i.e., when the two equations (i.e., pressure (1.9) and saturation (1.1)) are weakly coupled and that when time-step sizes are small. The phase velocity u α appearing in the saturation transport equation (1.1) can be related to the total velocity u t. Such a re-phrasing is typically followed in order to maintain the consistency between the equations, specially for IMPES or Sequential strategies. Therefore, after some mathematical manipulations (1.1) we be stated as, φ S w t + ( λ w u t + k λ wλ o P c kg λ wλ o (ρ w ρ o ) z) = q w, (1.10) λ t λ t λ t where the convective term explicitly contains capillary and buoyancy terms. Since u t is kept constant in the saturation equation, when a sequential strategy (either IMPES or sequential-implicit) is followed, the phase fractional flow function, considering the buoyancy and capillary terms, can be written as F w = λ w + k λ wλ o P c λ t λ t u t kg λ wλ o (ρ w ρ o ) z. (1.11) λ t u t 4

21 Note that typically u α = F α u t defines the total fractional flow function, and F w + F o = 1. It is also important to note that u t appears in the denominator of capillary and buoyancy terms. Systematic studies of importance of each term is more straightforward if the following non-dimensional numbers are introduced, i.e., the Gravitational number, and the Pèclet number, N g = k(ρ w ρ o )g h µ o u t, (1.12) P e = u tµ o L kp c. (1.13) Here, P c is the characteristic capillary pressure and L is the characteristic cell size. The fractional flow equation can be re-written in terms of these two dimensionless numbers, i.e., F w = λ w λ t + λ wk ro p c λ t P c /LP e k roλ w λ t N g. (1.14) The flow (pressure) and saturation equations, i.e., (1.9) and (1.1), combined, form a twophase system which can be solved to obtain the solution for phase pressure (p w ) and phase saturation (S w ) and the corresponding solution for the other phase can be obtained from Eq. 1.4 and Eq If these equations are solved sequentially, by first solving one and then solving the other, depending on the explicit or implicit treatment of the parameters an IMPES or Sequential-Implicit strategy is obtained. Next, the flow and transport equations are discretized with a FV scheme, which is second order accurate in space for pressure and first-order accurate in time for transport timedependent term. 1.2 Discretization In order to simulate fluid flow in reservoir the equations need to be discretized in time and space, which results in non-linear algebraic equations that need to be solved at every time-step. As mentioned before, there are many schemes available to discretize these equations, but due to the common practice of reservoir simulation community, Finite Volume method is followed in this thesis work. This choice is mainly attributed to the fact that local conservation is needed in every grid cell interface, as for the stable and robust solution of the nonlinear transport equation. The described governing equations in continuous PDF form can be written in discrete forms as explained in the following sub-sections. 5

22 1.2.1 Spatial Discretization Cartesian coordinate system is used to discretize the equations in space. The discretization here employs a two-point flux approximation (TPFA) scheme to discretize the equations where the cell interface values are dependent on the adjacent cells. However, it should be noted that multi-point flux approximation schemes (MPFA) are also used in reservoir simulators when higher accuracies are needed. The spacial discretization of the terms appearing in the flow Eq. 1.9 and trasnport Eq equations is as follows. Note that the capillary term in the transport equation is further expanded as k λ wλ o λ t P c S w, (1.15) while all other terms remain the same. In the following discrete equations, the discretized terms are written for flow (F) and transport equation (T). An illustration of the two neighboring cells with interfaces is provided in Fig , which are used in the following discrete equations. Figure 1.1: Illustration of two neighboring cells with interfaces. The flux (or velocity) is introduced at the interfaces while the pressure unknowns are at the cell centers. Viscous forces: F: (kλ t p w ) and T: ( λw λ t u t ) F : Ki+1/2λ H (p n+1 w,i+1 p n+1 w,i ) t,i+1/2 + K H (p n+1 w,i p n+1 w,i 1) x i 1/2λ 2 t,i 1/2 (1.16) x 2 T : ( λw λt )i+1/2 u t,i+1/2 x ( λw λt )i 1/2 u t,i 1/2 x (1.17) Buoyancy forces: F: (k(ρ w λ w + ρ o λ o )g z) and T: (kg λwλo λ t (ρ w ρ o ) z) F : KH i+1/2 (ρ wλ w,i+1/2 + ρ o λ o,i+1/2 )g(z i+1 z i ) + KH i 1/2 (ρ wλ w,i 1/2 + ρ o λ o,i 1/2)g(z i z i 1) x 2 x 2 (1.18) 6

23 T : KH i+1/2 ψ i+1/2(ρ w ρ o )g(z i+1 z i ) + KH i 1/2 ψ i 1/2(ρ w ρ o )g(z i z i 1), (1.19) x 2 x 2 where ψ = λwλo λ t. Finally, capillary forces in discrete form read as follows. Capillary forces: F: (kλ o P c ) and T: (k λwλo λ t P c ) F : KH i+1/2 λ o,i+1/2p c,i+1/2 (S w,i+1 S w,i ) x 2 + KH i 1/2 λ o,i 1/2Pc,i+1/2 (S w,i S w,i 1 ) (1.20) x 2 T : KH i+1/2 ψ i+1/2p c,i+1/2 (S w,i+1 S w,i ) x 2 KH i 1/2 ψ i 1/2Pc,i 1/2 (S w,i S w,i 1 ) (1.21) x 2 again, ψ = λwλo λ t. For 2D problems, the discrete pressure equation leads to a 5-diagonal system matrix (5- point stencil), and in 3D it becomes 7-point stencil [8]. Next the discretization in time is briefly reviewed. 7

24 1.2.2 Temporal Discretization Time disretization of the governing equations can be done using explicit or implicit schemes. For pressure equation, due to sever time-step restrictions of explicit methods, an implicit formulation is always followed. Such an implicit scheme would calculate the pressure-dependent terms at new time step, (n + 1). As for the transport equation, however, both implicit (Euler backward) and explicit (Euler forward) schemes are used in practice. Therefore, the different treatments of transport equation is explained here. Note that the accumulation term is typically discretized with a first-order scheme, i.e., S/ t (S n+1 S n )/ t, where t is the time-step size. This would lead the discrete form of transport equation be stated as φ (Sn+1 w Sw) n + (Fw ξ u t ) = q α, (1.22) t where, the superscript ξ can be either n (explicit), or (n + 1) (implicit). The explicit treatment of saturation-dependent terms would require the fractional flow function being calculated at the previous time step, i.e., Fw ξ = Fw. n Such a strategy is conditionally stable, therefore, imposes restrictions on the time-step selection which will be addressed later in the solution strategy section. To overcome the stability constraints imposed by the explicit schemes, an implicit method can be employed, where these coefficients are obtained at the new time step Fw ξ = Fw n+1, i.e., F w = ( λw λ t ) n+1 u t + k ( λw λ o λ t ) n+1 P c S w n+1 kg ( ) n+1 λw λ o (ρ w ρ o ) z. (1.23) λ t It is clear that implicit variables need to be first linearized and then iteratively be solved for. For a nonlinear dependent variable β, e.g., standard Newton Lemma can be used in order to linearize it over the independent variable η, i.e., β n+1 β ν + β ν (η ν+1 η ν ), (1.24) η where ν and ν +1 are the iteration index. As for the 2nd order truncation error, the Newton linearization lemma is expected to have second order convergence property. However, it is not always the case in reservoir simulation applications. More precisely, for a problem with only viscous forces existing, the saturation dependent terms in the fractional flow function F w is limited to the mobility ratio λ w /λ t. This ratio leads to S-shaped function with an inflection point. In addition, as more physics is considered, the saturation dependency increases through additional nonlinear terms. And thus, the F w function becomes more complex. In some cases F w can have more than one inflection points, leading to sever nonlinear convergence difficulties. More important is that for cases of dominant nonlinear terms of, e.g., capillarity or buoyancy, the linearized equations would never converge to the nonlinear terms unless extremely small time-step sizes are selected. As such, detailed study of the nonlinear convergence properties in presence of complex physics is crucially important. Of high demand is to find stable and robust linearization and iteration procedures that would guarantee the convergence with an optimum computational costs. 8

25 1.2.3 Peaceman Well Model Production from and injection to natural formations occurs through wells. Wells appear as source terms in the RHS of mass balance equations on the basis of Peaceman model [8] as q α = W Iλ α K(p α p wf ) (1.25) where, p α is the grid cell pressure, p wf is the flowing bottom-hole pressure and W I is the well productivity index. The equation does not add any additional unknown if p wf is provided. For a rate constraint wells, i.e., when q α is known, this equation increases the system of equations to be solved for both fluid phase and well pressures. Note that the total source terms appear in the pressure equation. Thus, once the total rate is known, the individual phase flow rates can be obtained as q α = f α q t. (1.26) For implicit transport solution strategy, all the saturation dependent terms including those appearing in the well equation are treated implicitly Interface properties: upwind and harmonic-average To calculate values at the cell interfaces, various methods can be implemented. Absolute permeability, rock property, at the cell interfaces is calculated using harmonic averaging, i.e., K H = 2K 1K 2 K 1 + K 2. (1.27) This method is used since permeability is synonymous to conductivity and harmonic average preserves mass balance at the interface. Transmissibilities are calculated using upwind scheme. Since the transport equation is hyperbolic in nature the cell downstream the front has little or no information of the speed of the front. The upstream direction is calculated using the pressure gradient. Cells with higher potential determine the upwind cell, i.e., { λα (S λ α,i+1/2 = i ) ( p α,i+1 p α,i ) + ρ x α g z 0 λ α (S i+1 ) ( p α,i+1 p α,i ) + ρ x α g z > 0. (1.28) Typically the upwind direction should be calculated based on characteristic speed. As for the strong nonlinearity in the special functions appearing in reservoir simulation, the phase-based upwind direction has been implemented in most reservoir simulators. In linear situation, both velocities, i.e., characteristic speed and phase velocity, have similar directions. 9

26 1.2.5 Capillary Treatment The capillary term in the transport equation contains the capillary function (an upscaled experimental function of saturation) multiplied by a function of phase mobilities. Typical capillary functions tend to very large values (in the limit, to infinity) as the saturation of wetting phase decreases. Such a behaviour would lead to several stability issues, specially for grid cells with small wetting phase saturations. More precisely, the capillary term in the fractional flow term reads [F w ] P c = λ wk ro P c S w λ t P c /LP e. (1.29) To resolve this difficulty, a new discretization scheme was proposed by Douglas et. al [5], and also used by Cances(2009) [2], which reads C(S) = k u t S 0 λ w λ o λ w + λ o P cds. (1.30) Function C(S) is first obtained in a preprocessing step for 0 S w 1, and replaces with the corresponding terms in the flux function calculation. Note that as S w tends to small values, λ w tends to zero which makes the result of the multiplication (λ w P c) a bounded value. Moreover, since capillary terms are diffusive in conservative form, central discretization scheme is often used to calculate their numerical values at the grid interfaces. Figure 1.2 illustrates a typical P c (S w ) and C(S w ) curves. (a) (b) Figure 1.2: Illustration of P c (S w ) (a) and C(S w ) functions (b) 10

27 1.3 Simulation Strategies for Coupled P-S Systems The main simulation strategies that are in common use are IMPES, Sequential Implicit, and Fully Implicit methods. In this section, a brief description of these strategies are provided IMPES Formulation IMPES (Immplicit Pressure Explicit Saturation) solves flow equation implicitly and the transport equation explicitly. The coupling between flow and transport is also treated in a sequential manner, i.e., flow-dependent terms are kept constant in the transport equation and, visa versa, the transport dependencies are neglected in the pressure equation solution stage. Explicit treatment of transport equation imposes time-step restrictions, and it can severely slow-down the simulations for realistically challenging nonlinear physics. The stability analysis provides the conditional stable criterium, known as CFL condition, which reads t = CF L φ x u t f w S w (1.31) where, 0 < CF L < 1 is required for a stable simulation. Note that the CFL condition is obtained based on linear stability analysis, and that the transport equation is actually nonlinear with strongly nonlinear F w function. This nonlinearity can result in more restricted time-step selection which will be addressed later. For heterogeneous problems, fluid velocity can vary by several orders of magnitude in the whole reservoir. Because the minimum t needs to be taken in a global manner, this can lead to small time-step size selection and long simulation times Sequential Implicit Formulation (SQ-Imp) The solution obtained from the elliptic flow equation is global in nature. For an incompressible system, the flow equation does not depend on the time-step size selection. The transport equation solution is local in nature as the information travels with a finite speed. From the above formulation, Eq. 1.9 and 1.10 can be decoupled to solve the flow equation implicitly, and subsequently solve the transport equation using implicit formulation. Using implicit time scheme ensures unconditional stability for time-step selection. The sequential implicit algorithm can be reviewed as following. First the flow (pressure) equation is solved implicitly, and total velocity is calculated from its solution. Then, the transport equation is solved implicitly, with pressure-dependent terms (total velocity) being treated as constants. After the converged solution of the implicit transport solver is obtained, the solution iterates in an outer loop for global nonlinear convergence of pressure-saturation coupled system. Note that in some cases, one can avoid the outerloop iterations, and simply move to the next time step after implicit solution of transport equation is achieved. What is called as Sequential-Implicit strategy in this thesis always refers to the former case, i.e., where the outer-loop iterations are employed and fully 11

28 converged nonlinear coupled flow-transport solution is obtained. Such a solution is fully consistent with the fully implicit solutions (to be explained in the next sub-section), and thus, is also called as Sequential-Fully-Implicit approach. Due to the presence of strong buoyancy and Capillarity the flux function changes shape and this can lead to serious convergence difficulties. Figure 1.4 illustrats the flux function with strong gravitational and capillary forces. Most solvers as explained above use standard newton method to linearize the implicit system and for higher timesteps standard newton method fails to converge. To overcome this failure, flux-correction strategy is studied in this thesis work. Figure 1.3: Flowchart of Sequential Implicit method 12

29 (a) (b) (c) (d) Figure 1.4: Flux Functions: (a) Viscous flux (M = 1, N g = 0, P e ); (b) Viscous and Capillary (M = 1,N g = 0,P e = 0.2); (c) Viscous and Buoyancy (M = 1,N g = 5,P e ); (c) Viscous and Buoyancy (M = 1,N g = 5,P e ) Fully Implicit Formulation (FIM) Another simulation technique, which is commonly used in most commercial simulators, solves for both P and S simultaneously. In this approach, the P-S coupling is fully accounted for, in a fully-implicit (FIM) system. As a matter of fact, when the flow-transport coupling is strong, this simulation strategy would be expected to be the most stable one among the sequential implicit and IMPES alternatives. Implicit methods provide unconditional stability for linear equations. Nonlinear convergence, though, is quite challenging due to the sensitivity of the standard Newton method to the complexity of flux functions and initialization of the iterations. As such, even fully implicit simulation strategy can suffer from nonlinear convergence when big time step sizes are taken (previous time-step solution does not provide a good initial guess) and when nonlinear terms lead to flux functions with inflection points. To construct the fully implicit formulation, first the residual equation needs to be stated. Equations (1.32) and (1.33) are nonlinear residual equations for p w and S w. These equations are discretized over control volumes, where horizontal (grid index i) and vertical (grid index j) directions are taken in the x and y directions, respectively. Also, gravitational acceleration act in the y-direction. 13

30 The flow and transport residual equation are given as and R p = q t ( λ t K p w λ o K P c K(ρ w λ w + ρ o λ o )g z) (1.32) R s = q w φ S w t + (Kλ w ( p w + ρ w g z), (1.33) respectively, which are nonlinear functions of the main unknowns, p w and S w. As such, the objective is to having residual values at the new time step (n+1) to be both zero, i.e., R n+1 p = R n+1 s = 0. (1.34) As residual equations are nonlinear, they need to be first linearized and iteratively be solved for the main unknowns satisfying Eq. (1.34). The detailed discretized and formulations are provided in Appendix A, at the end of the thesis. Linearization of these equations lead to which can be stated as R n+1 R p R s ν + R p p R p S R s R s } p {{ S } J ν δp ν+1 δs } {{ } δψ = 0, (1.35) J ν δψ ν+1 = R ν. (1.36) In this thesis, to point the different block-entries of the Jacobean matrix, J, J = J pp J sp J ps J ss (1.37) notations have been used. Equation (1.36) is iteratively solved until the convergence is reached. Note that if the off-diagonal blocks in the Jacobean terms are neglected along with their diagonal contributions, the sequential-implicit approach is derived. For strong flow-transport (i.e., P-S) coupling, the off-diagonal terms J ps and J sp are quite important, and play major role in convergence to the solutions p n+1 and Sw n+1. 14

31 Chapter 2 Stability Analysis 2.1 Non-Linear Stability Implicit techniques like FIM and SQ-Imp, once constructed based on proper calculation of interface properties, need to be initialized (initial guess) for the first Newton iteration. For small time-step sizes, previous solution is usually a good initial solution. However, for larger time-steps previous solutions at time n may not be a good initial guess for (n + 1), as for the large changes of the variables during a big time step. In fact, such an initialization can lead to non-convergence, or divergence, of the Newton loop. Jenny et al. [14] proposed a modified Newton method for hyperbolic conservation laws in the absence of gravity and capillarity. The non-linearity of the transport problem can be associated to the shape of the flux function. For the studied viscous problems the flux function has an S-shape with a single inflection point. Inflection point is the value of saturation where the slope of the flux function is maximum. Jenny et al. [14] stated that the convergence difficulties can occur if the initial guess and the final solution lie on different sides of the inflection point. As such, they proposed a correction strategy to bring the saturation back to the inflection point once it crosses this point in a Newton iteration. Such a damping strategy would provide a better initial guess for the next iteration, and convergence can be guaranteed for any big time-step sizes. Later, Wang and Tchelepi [23] extended this approach to the cases with strong capillary and gravitational forces, where flux functions behave more complex with more inflections points. Note that both methods, presented in [14] and [23], were developed for fixed-velocity values, i.e., isolated nonlinear transport equation. In this thesis work, the flux-correction strategy is integrated within the implicit nonlinear outer-loop iterations, where velocity values change in each iteration. Such a study is crucial in correct assessment of the progress achieved with the introduction of flux-correction strategies. And, to further illustrate the challenges within the nonlinear convergence for multiphase time-dependent simulations for strongly nonlinear scenarios. In this section the non-linear behavior of the transport equation is analyzed in detail, which can lead to convergence difficulties due to large time-steps. After this study, the flux correction method will be applied to overcome with the convergence difficulties. Transport equation for incompressible problems (i.e., constant u t ) in 1D domains can be 15

32 written as S n+1 w,i Sw,i n + t u t n+1 (Fi+1/2 x F n+1 i 1/2 ) = 0, (2.1) where, and F n+1 i+1/2 = f w(sw,i n+1 ; Sw,i+1) n+1 + C(Sn+1 w,i+1) C(S n+1 x w,i ) (2.2) f w = λ w λ t λ wk ro N g λ t. (2.3) The viscous and buoyancy terms at the cell interfaces are represented in f w. The residual equation can be written as where S n i = 0 and R = S n+1 w,i + t u t n+1 (FR FL n+1 ) (2.4) x w ) F R = f w (S R ; Sw n+1 ) + C(S w,r) C(S n+1 x (2.5) F L = f w (S L ; Sw n+1 ) + C(Sn+1 w ) C(S w,l ). (2.6) x Upwind Criteria Upwind-based calculation of the phase properties at the interfaces, based on phase velocities as mentioned in Section 1.2.3, can be states as λ α (S i ) λ α,i+1/2 = λ α (S i+1 ) [ pα,i+1 p α,i + ρ x α g z ] 0 [ pα,i+1 p α,i + ρ x α g z ] > 0. (2.7) This upwind scheme stands valid for both phases α = o, w, where co-current flow is observed. For cases with counter-current flow (i.e., oil and water flow in different directions at an interface), buoyancy terms dominate the process. Therefore, upwind scheme needs to be re-defined for saturations where S f=1 is crossed. More precisely, based on the total velocity, one can calculate the counter-current upwind calculation strategy as and { λw (S λ w,i+1/2 = i, ) u t kg ρλ n+1 o > 0 λ w (S i+1 ), otherwise 16 (2.8)

33 { λo (S λ o,i+1/2 = i ), u t + kg ρλ n+1 w > 0 λ o (S i+1 ), otherwise. (2.9) Equations (2.8) and (2.9) do not entail pressure dependency in determination of the upwind direction, but the total velocity and an implicit representation of mobilities, λ n+1 o and λw n+1. Such an implicit representation of mobilities can lead to difficulties, thus, Brenier and Jaffré [24] defined an explicit formulation, i.e., and where, { λw (S λ w,i+1/2 = i ) θ w > 0 λ w (S i+1 ), otherwise { λw (S λ w,i+1/2 = i ) θ o > 0 λ w (S i+1 ), otherwise (2.10) (2.11) θ w = u t kg ρλ o (S i ) (2.12) θ o = u t + kg ρλ w (S i+1 ) (2.13) Kwok and Tchelepi [3] explained this upwind criteria in detail, for both cases with N g > 0 and N g < 0. For more information readers are referred to their paper. Here, only the case of N g < 0 (downdip) with u t < 0 is explained, i.e., θ w is always negative. For such a case, one obtains and λ w,i+1/2 = { λw (S i+1 ) θ w < 0 < θ o λ w (S i+1 ) θ w < θ o < 0, (2.14) λ o,i+1/2 = { λo (S i+1 ) θ w < 0 < θ o λ o (S i ) θ w < θ o < 0 (2.15) for calculation of phase mobilities at the interfaces. Consider a case where oil and water are flowing from left to right. As the saturation in a grid cell crosses S f=1 point, oil phase velocity will change its flow direction and counter-current flow occurs. Note that f w + f o = 1 holds. Therefore, if f w 1 then f o 0 and negative fractional flow will cause negative phase velocities, i.e., counter-current flow. Finally, the upwind scheme for the flux function with viscous and buoyancy terms combined can be stated as f i+1/2 (S i ; S i+1 ) = Mk rw(s i )[1 N gk ro(s i )] Mk rw(s i )+k ros(i) 0 S i S f=1 Mk rw(s i )[1 N gk ro(s i+1 )] Mk rw(s i )+k ro(s i+1 ) S f=1 S i 1. (2.16) 17

34 2.2 Stability Analysis Consider a single cell problem with quadratic relative permeability functions as k rw = S 2 and k ro = (1 S) 2 with mobility ratio of M = 1. The non-linearity within the transport equation can lead to non-convergence in presence of strong buoyancy and capillary effects. The effect of high N g and P e on the non-linearity of the flux-function is studied in this section Viscous forces For a case with only viscous forces, where both fluid viscosities are 1 cp and mobility ratio is 1, the inflection point is at S infl = 0.5. Also, analyses of the residual and derivative of the residual indicate that maximum slope occurs at S infl = 0.5. Dirichlet boundary conditions of S L = 1 and S R = 0 are set at the left and right interfaces, respectively. for the 1D problem with positive velocity direction (left to right), employing upwind direction would lead the residual and its first and second derivatives for cell i be stated as R i = S n+1 w,i + t u t n+1 (FR (S w,i ) FL n+1 (S w,i 1 )), (2.17) x and R i = 1 + t u t x R i = t u t x F R (S w,i ) S w, (2.18) 2 F R (S w,i ). (2.19) Sw 2 Note that since only viscous terms exist, N g = 0, and F w = f w = λ w /λ t. Also note that for a cell i, the values at (i 1) at new time step (n + 1) are all known. For the given case with co-current flow, the maximum value of R gives the inflection point and R = 0 gives the value of the inflection point. Figure 2.1 illustrate the flux function and the residual and its derivative for different time-step sizes and u t = 1. From Fig. 2.1 it can be seen that as the time-step is increased the non-linearity of the problem is increased with inflection point (R = 0) being always at S infl = 0.5. It is important to mention that, clearly, the solution is where the residual is 0, and the point where all the curves with different time-steps converge is the steady-state solution (i.e., t does not impact the residual value). A further step can be taken by analyzing the behavior of discretized flux function and its derivative w.r.t to both upwind and downwind cells, as shown in Fig As no counter-current flow can happen in the absence of gravity terms, i.e., 0 F w 1, the discretized equation only depends on the upwind cell. 18

35 (a) (b) (c) Figure 2.1: (a) Flux function (M = 1, N g Derivative of residual w.r.t S upwind. = 0, P e, u t = 1); (b) Residual; (c) (a) (b) Figure 2.2: Flux Functions: (a) Flux function (M = 1, N g = 0, P e, u t = 1); (b) Derivative of flux function w.r.t S upwind 19

36 2.2.2 Viscous and buoyancy forces For a case with fluid viscosity of 1cp with M = 1 and Ng = 5 (u t < 0), we encounter two inflections points S infl1 = 0.342, S infl2 = and unit flux point S f=1 = where flow reversal occurs. This is shown as case (a) in Figs. 2.3 and 2.4. We consider cases with two boundary conditions (a) S L = 1 and S R = 0, (b) S L = 0.6 and S R = 0.8. Figure 2.3: Flux Function Figure 2.4: Description of Flow Reversal for the Section 2.3 For N g < 0 (case (a), with u t < 0), as the wetting phase saturation of cell i + 1 increases, starting at S f=1 the non-wetting phase velocity switches its direction and the flow becomes counter-current. For N g > 0 (case (d), counter-current with u t > 0), as the wetting phase saturation of the cell i increases, starting at S f=0 the wetting phase velocity switches its direction and the flow becomes co-current. 20

37 (a) (b) Figure 2.5: Flux Functions: (a) Flux function (M = 1, N g function (M = 1, N g = 5, P e ) = 5, P e ); (b) Flux The inflection points in Figs. 2.5 are indicated by red dots, sonic points by green dot, and the point where flow reversal occurs by black dot. The inflection points for N g = 5 are S infl1 = and S infl2 = with S f=1 = And, for N g = 5 the inflection points are S infl1 = and S infl1 = with S f=0 = We can perform the flux function analyses, similar to the previous viscose-flow case, with respect to the upwind and downwind cells. For N g = 5 Fig (a) shows how the flux function behaves for various upwind and downwind cell saturation values. As it can be seen from Fig. 2.6 the derivative of the flux function with respect to upwind cell is continuous until counter current flow is observed. With the same upwind conditions, it can be seen that till counter-current flow is seen, the flux function has no dependence on the downwind cell saturation. 21

38 (a) (b) (c) (d) Figure 2.6: BC: S L = 1 and S R = 0; (a) Residual Function (b) Derivative of Residual; BC: S L = 0.6 and S R = 0.8; (c) Residual Function; (d) Derivative of Residual. 22

39 (a) (b) (c) Figure 2.7: For the case of M = 1, N g = 5, P e, and u t = 1 (a) Flux Function (b) Derivative of Flux function w.r.t upwind cell; (c) Derivative of Flux function w.r.t downwind cell; 23

40 2.2.3 Viscos forces and Capillarity forces Finally, the case where viscose and capillary forces are both present is studied in this section. Fluid viscosity of 1 cp with M = 1 and P e = 0.2 are considered. Figure 2.8 shows the flux function using phase-based upwinding. Here, single point phase-based upwind scheme is employed to discretize the flux function. When saturations tend to zero, P c λ goes to infinity, causing numerical difficulties. As saturation tends to zero, wλ o λ w+λ o also tends to zero. For the term λwλo λ w+λ o p c to be bounded λwλo λ w+λ o should approach to zero faster than p c approaching infinity. This is satisfied when quadratic relative permeabilities are used along with Brooks-Corey model for capillary pressure. The fractional flow function reads F w = λ w λ t + λ wk ro p c λ t P c /LP e, (2.20) where L is the characteristic length of the cell and P e pressure. is the characteristic capillary (a) (b) (c) (d) Figure 2.8: Flux Functions: (a) Capillary Pressure; (b) λwλo λ t ; (c) λwλo λ t p c; (d) Flux Function with Phase based upwinding P e =

41 A two-cell case having interface of i + 1/2 and positive velocity direction u t > 0 is considered. Therefore, P e is also positive. Moreover, the numerical values of N g = 0 and P e = 0.2 are set. With these numbers being fixed, one can calculate the total velocity, from which the and then the fractional flow function F w via the steps presented in the following algorithm. Algorithm 1 Flux function calculation using Phase based Upwinding 1: Calculate u t using the Péclet number. 2: Calculate the sign of u w and u o. 3: Calculate u t using the value of P e 4: Calculate (p w,i+1 p w,i ) x 5: Calculate the new u w and u o 6: if Phase velocity signs are correct then 7: Calculate F using Eq : else if Repeat steps 1 to 5 then 9: end if Note that the fractional flow function F w can be obtained from its definition once the velocities are known, i.e., F w (S i, S i+1 ) = u w. (2.21) u w + u o Moreover, having the phase pressure gradient, one can calculate the phase velocities on the basis of Darcy s law, i.e., and u w = Ki+1/2λ H p w,i+1 p w,i w,i+1/2, (2.22) x u o = Ki+1/2λ H p w,i+1 p w,i + P c (S i+1 ) P c (S i ) o,i+1/2. (2.23) x Also, residual and its derivatives can be calculated as and R i = 1 + t u t x ( F R(S i ) 2 C(S w,i ) ) (2.24) S w x S w R i = t u t F R (S w,i ) x ( 2 2 Sw 2 x 2 C(S w,i ) ), (2.25) Sw 2 respectively. Figure 2.9 shows the flux functions for both the viscose and viscose-capillary cases. Also, the residual and its derivatives are plotted in this figure. Note that for F w > 1 countercurrent flow occurs, where oil phase changes direction due to capillarity effects. With the new discretization method mentioned in Chapter 1, the non-linearity in presence of P c is less severe (due to introducing C(S) function), as infinite slopes are avoided. This approach can also be extended for heterogeneous capillary curves which falls beyond the 25

42 scope of this thesis. (a) (b) (c) (d) Figure 2.9: Flux Functions: (a) Viscous; (b) Viscous and Capillary (P e Residual Function; (d) Derivative of Residual = 0.2); (c) The new discretization scheme (based on C(S) function) gives monotonic fractional flow curve with a single inflection point. The saturation value at which R is maximum and R = 0 is S infl =

43 Chapter 3 Flux Correction Methods 3.1 Non-Linear Convergence The non-convexity and non-monotonicity of the fractional flow equation are major sources of difficulty for non-linear coupled equations. For implicit time-scheme methods large time-steps S-shaped flux functions can lead to non-convergence of Newton method. In 2009, Jenny et al. [14] proposed a modified Newton method for unconditional convergence for non-linear flux functions with viscous forces only. Also, they assumed the total velocity is constant, i.e., isolated saturation equation were considered.the flux correction method splits the flux function into two regions separated by the inflection point. This point corresponds to the point where the flux function is maximum or F = 0. The transport equation can be written in residual form. For standard Newton method, when large time-step sizes are considered, the linearization of the flux function can be a weak assumption and it may lead to non-convergence. Note that for such a case, the initial guess is also typically far from the final solution. The proposed modified Newton method brings the saturation to the inflection point, if it crosses this point in an iteration. The chopping of the saturation provides a better initial guess for the consecutive iterations which can lead to faster convergence. Again, it is re-emphasized that all of these studies were based on constant total velocity and the flux functions which have only 1 inflection points. An illustration of this procedure is depicted in Fig (a) (b) Figure 3.1: Convergence performance of Residual function with an inflection point (a) Standard Newton; (b) Modified Newton Method [14]. 27

44 An overview of the flux-correction algorithm is provided in Fig Figure 3.2: Modified Newton method proposed by Jenny et al. [14]. As mentioned before, Jenny et al. [14] did not consider a case with more than one inflection points. Such a case can happen, e.g., when buoyancy effects are present. This limitation was recently resolved by Wang and Tchelepi [23], who extended this study for cases with buoyancy and capillary forces. Note that they still focused mainly on transport equations, and not much on the nonlinear outer-loop convergence of a multiphase flow simulator, where total velocity changes and the coupling between flow and transport is more pronounced Trust Region Method The behavior of the flux function becomes more complex as more physics is added to the system. Stability analysis suggested that for strong buoyancy cases, there exist two inflection points and a unit flux point. These three saturation points play a vital role in the convergence behavior of the non-linear transport loop. A correction strategy for the unit flux point was considered along with inflection points, as flow reversal occurs and the upwind criteria changes accordingly [23]. This results in change in the Jacobian values as the saturation of cell i depends on the upwind cell (i 1) as well as downwind cell saturation (i + 1), due to flow reversal. An overview of the improved flux-correction method, called as trust-region-based method, is provided in Fig The term "trustregion" referred to the work of [23] since it does not allow the Newton update to cross the trust regions. At every Newton iteration, the non-linear solver ensures that the solution 28

45 update lies in the trust region. If crossed, the correction is damped (modified) such that the crossing does not occur. This chopping then serves as an initial guess for the consecutive Newton updates which allows the non-linear system to converge all the time. Figure 3.3: Trust-Region method, an improved flux-correction strategy [23]. While the mentioned approaches form the base of the studies of this thesis, it is important to note that other approaches have been proposed to guarantee convergence of the Newton linearized system. Examples of these methods include Continuation-Newton [17, 18] and reordering methods [3]. Moreover, it is also important to be mentioned that for very large time-steps, the Newton updates can give unphysical solutions (i.e., saturations below 0 and above 1). In such cases the new solution for saturation are bound as S [0, 1]. 29

46 3.1.2 Locating Trust Regions The inflection points (S infl ) and the critical flux points (S f=1 and S f=0 ) depend on the gravity number (N g ). In sequential implicit strategy, the total velocity (u t ) calculated at interfaces is fixed in the transport solution stage. This velocity can be used to calculate N g. Once the gravity number is known, the trust regions for a given interface can be easily located. Since in a simulation step, the gravity number of the interface can change as the saturation solution is updated, it is computationally inefficient to calculate the trust regions at every iteration. Hence, the trust regions can be pre-calculated and tabulated for various N g values. A simple table look up can be used to locate the trust regions (S f=0, S f=1 and S infl ) during simulation. For homogeneous capillary functions, the inflection point of flux function can be also pre-calculated Trust Region Chopping After the trust regions have been located, a buffer zone for the inflection point S infl ± ɛ, S f=1 ± ɛ needs to be considered. The buffer zone is needed, because the unit flux point S f=1 is a non-differentiable point, while its neighboring points have definite derivatives. For a multi dimensional domains, identification of the cell to chop plays an important role. If the initial guess and the Newton updates cross the buffer zone, the solution is scaled back at the end of buffer zone. Figure 3.4: The solution update is from X to Y, C is the inflection point and [A, B] is the buffer zone. The chopping ratio can be defined as ζ = XB XY. For the solution update from X to Y, the chopping ratio is XB XY (see Fig. 3.4). For cell S ν i,j the chopping ratio in the i direction can be written as ζ i and in the j direction as ζ j. Note that cell interfaces can have different N g values which correspond to different chopping ratios. For a given cell, the chopping ratio is selected as Hence, the new solution update reads ζ i,j = min(ζ i, ζ j ). (3.1) S ν+1 i,j = S ν i,j + ζ i,j δs ν+1 i,j. (3.2) As shown by Li and Tchelepi [1], this global chopping ratio is conservative in nature. 30

47 Chapter 4 Results 4.1 Numerical Results In this section, various numerical results are presented to test the performance of the flux correction strategy. A comparative study between different solution strategies is performed, in order to find the maximum time-step sizes that can be used as the complexity of the process increases. The following list summarizes the test cases which have been considered spot homogeneous test case with viscous forces only spot heterogeneous test case with viscous forces only spot homogeneous test case with viscous and buoyancy forces spot heterogeneous test case with viscous and buoyancy forces. 5. 1D Homogeneous gravity segregation test case Viscous Forces Test Case - 1 A 2D homogeneous case with quarter-five-spot pattern is considered. No-flow condition is applied at the entire boundary of the domain. Also, the rock absolute permeability is 1 Darcy. The reservoir size is m 2, discretized into grid blocks. Injection and production wells are pressure constrained with injection well pressure Pa and production well pressure Pa. The mobility ratio is M = 1. Simulations have been run for the duration of 0.35 days injection. Standard Sequential Implicit (SQ-Imp) and Fully Implicit (FIM) simulations have been compared as the time-step sizes are increased. Figure 4.1 presents the observed performance, where the SQ-Imp iterations are split into outer and inner loops for more precise studied. From Fig. 4.1 it is clear that the standard Newton method does not converge for CFL numbers above 1 (i.e., t 100 sec). To improve the convergence limitations, flux correction strategy is then employed. Results of the SEQ-Imp with and without the flux correction strategy is presented in

48 (a) Figure 4.1: Comparative study of Sequential Implicit (SEQ-Imp) and Fully Implicit (FIM) convergence properties as the time-step sizes increase for Test Case 1. (a) Figure 4.2: Comparative study of Sequential Implicit (SEQ-Imp) with and without flux correction strategy for Test Case 1. Clearly, the flux-correction strategy allows for much more stable simulations when the time-step sizes are increased. This improvement is achieved by avoiding the corrections in each iteration crossing the inflection point. It is important to mention that once the simulator is unconditionally stable, one may or may not consider using it for very large time steps. It is due to the fact that the numerical error increases as the time-step sizes increase. Important is that typically in most of the simulators, including the developed code for this thesis, a first-order upwind scheme is used. This would cause significant 32

49 numerical diffusions once very large time steps are used. Figure 4.3 illustrates this fact, where two simulations are performed for the same amount of injected fluids, i.e., 0.25 pore-volume-injection (PVI). However, their time-step sizes are significantly different, causing quite different estimate of the location of the front. Thus, accuracy-efficiency tradeoff, similar to many other reservoir simulation applications, needs to be considered for realistic cases. The advantage of employing flux-correction strategy is to allow for unconditionally stable simulations. (a) (b) Figure 4.3: Saturation Solution at 0.25 PVI: (a) t = 10 and (b) t = sec. Test case - 2 Test Case 2 is similar to the previous test cases, except that a heterogeneous permeability field as shown in 4.4 is now considered. All fluid properties are similar to those of Test Case 1. The injection well has an injection pressure of Pa and the production well has a pressure of Pa. Final simulation time is 3.5 days. The two different implicit simulation approaches, i.e., sequential implicit and fully implicit, in their classical implementations are compared and presented in Fig The improvement achieved by employing flux-correction strategy is presented in Fig

50 (a) Figure 4.4: Log Heterogeneous Permeability (a) Figure 4.5: Comparative study of Sequential Implicit and Fully Implicit convergence properties with increasing time-step sizes for Test Case Viscos and Buoyancy Forces Test Cases 3 and 4 study the convergence behavior of 2D homogeneous and heterogeneous media, respectively, where both viscose and gravitational forces are present. Absolute permeability of 1 Darcy, similar to Test Case 1, is considered for Test Case 3. Boundary conditions are all the same as in the previous cases. A gravity segregation scenario is considered, where heavy fluid is initially present at the upper half of the reservoir and lighter fluid is at the half bottom. This unstable initial phase saturation imposes severely challenging flow scenario. Density of the heavier fluid is ρ w = 1000kg/m 3, and lighter fluid is ρ o = 500kg/m 3 34

51 (a) Figure 4.6: Comparative study of Sequential Implicit with and without flux correction strategy for Test Case 2. (a) (b) Figure 4.7: Saturation Solution at 0.5 PVI injected: (a) t = 160 and (b) t = sec. for Test Case 2. Test Case - 3 Convergence behavior of the quarter-five-spot settings similar to Test Case 1 is studied. The highest value of N g = 6 is observed, at the corners of the reservoir, where buoyancy forces dominate the viscous forces. The final time for the simulation is 0.5 days. A modified upwind criteria is employed for the transport equation as following. Ignoring capillary terms, the fractional flow equation can be written as F w = ( λ w λ t ) νv u t kg( λ wλ o λ t ) νg (ρ w ρ o ) z (4.1) 35

52 The superscripts of the mobility terms ν v and ν g denote different strategies for viscose and buoyancy terms. The upwind criteria for the viscose terms with superscript ν v is based on phase velocities. The upwind criteria of the buoyancy terms, denoted with superscript ν g, is based on the density difference of the fluids. Heavy fluid has a tendency to flow downwards which suggests that for interface i + 1/2 the upwind cell is i + 1. The lighter fluid has a tendency to flow upwards hence for interface i + 1/2 the upwind cell is i. Mathematically, upwind criteria for viscous terms are { λα (S λ α,i+1/2 = i ) ( p α,i+1 p α,i ) + ρ x α g z 0 λ α (S i+1 ) ( p α,i+1 p α,i ) + ρ x α g z > 0, (4.2) and for buoyancy term are λ α,i+1/2 = { λα (S i ) ρ α < ρ β λ α (S i+1 ) ρ α > ρ β, (4.3) where α and β are the two existing phases [19]. Convergence behavior of sequential implicit and fully implicit methods based on standard Newton method implementation is provided in Fig The flux correction strategy is then employed, the improvement achieved by it is also presented in Fig Note that for this test case, the coupling between flow and transport is much stronger than that of Test Case 1. As such, it is observed that the fully implicit simulations, which accounts for the full coupling terms, performs better than the sequential implicit approach. It is clear that the standard Newton method does not converge for CFL above 1 ( t 63 sec.). For this simple case, flux correction strategy gives unconditional convergence for CFL of 100 ( t = sec.). Simulation results at the same injection time for two different time steps are provided in Fig Note that, similar to other cases, the one with bigger time step sizes is more diffuse. 36

53 (a) Figure 4.8: Comparative study of Sequential Implicit and Fully Implicit convergence properties with increasing time-step size for Test Case 3. (a) Figure 4.9: Comparative study of Sequential Implicit with and without flux correction strategy for Test Case 3. Test case - 4 Similar heterogeneous field as in Test Case 2 is considered. The gravity number in the domain changes between 40 N g 40. The total simulation time is 3 days. For such a challenging test case, total velocity and inflection points change significantly in the domain. Figure 4.11 shows the convergence comparative study, where standard Newton scheme is implemented. The impact of flux correction strategy is studied and presented in Fig And, finally, the solution for two different time step sizes are compared in Fig It is important to re-emphasize that once the simulator is unconditionally 37

54 (a) (b) Figure 4.10: Saturation Solution for 0.65 PVI: (a) t = 6.3 and (b) t = 6317 sec. for Test Case 3. stable, the accuracy of the solutions should be also considered for practical purposes. (a) Figure 4.11: Comparative study of Sequential Implicit and Fully Implicit convergence properties as time-step size increases Test Case 4. 38

55 (a) Figure 4.12: Comparative study of Sequential Implicit with and without flux correction strategy for Test Case 4. (a) (b) Figure 4.13: Saturation Solution for 0.65 PVI: (a) t = 142 and (b) t = sec. for Test Case 4. 39

56 Test Case - 5 A 1D case is considered where gravity forces dominate the flow. Total number of grid cells is 80. Cells 1 : 40 are initially filled with 100% lighter fluid (ρ o = 1 kg/m 3 ) and cells 41 : 80 with heavy fluid (ρ w = 1000 kg/m 3 ). The mobility ratio of the fluids is M = 1. As for no external source terms, total velocity in the reservoir is zero. Therefore, the flux function is only governed by the buoyancy terms and is presented in Fig The final solution time is 1000 days. (a) Figure 4.14: Flux Function for N g The flux function, as shown in Fig. 4.14, has 2 inflection points of S infl1 = 0.28 and S infl2 = 0.72, and a sonic point at S sonic = 0.5. Standard Newton method does not converge for t > 10 days, but the flux-correction strategy provides unconditionally stable simulations. The solutions for 1D gravity segregation case at different time steps are provided in Fig Using flux correction strategy for this test case leads to unconditionally stable simulation. However, the solution quality will decrease as the time-step size increases. Simulation results for different time-step sizes, varying from t = 1 to t = 1000 days, are provided in Fig The solution at t = 1000 days is quite different than t = 1 day, due to time discretization error of the time discretization method. 40

57 (a) Figure 4.15: Total number of iterations employed to obtain transport solutions with increasing time-step size for Test Case 5. Note that the t is in days and that the sequential implicit approach does not converge for t > 10. (a) (b) (c) (d) Figure 4.16: Saturation solution at (a) T S = 1 day; (b) T S = 10 days; (c) T S = 100 days; (d) T S = 1000 days 41

58 (a) Figure 4.17: Saturation solution for different time-step sizes for Test Case 5, showing the unconditionally stable simulations with flux-correction strategy. Note that the quality of solution needs to be considered for practical purposes. 42

59 Chapter 5 Enhanced Oil Recovery In producing oil from a reservoir, the average oil recovery varies between15-40% of the oil originally in place, after primary and secondary recovery methods. Advanced techniques are employed to recover the remaining oil and gas are used namely Enhanced Oil Recovery(EOR) methods. These techniques are used to improve the displacement and volumetric efficiencies by altering the properties of either fluids or their interactions with the rock. One of the most commonly applied EOR methods is gas flooding. Miscible gas drive methods are used wherein the fluid interfaces between the two fluid phases disappears. Since the displacement occurs entirely in a single phase, it can give 100% efficiency at microscopic level. However, unfavorable mobility ratios between the gas and oil and the density difference can lead to viscous fingering and gravity override, resulting in poor oil recovery. Gas sweeps is rapid along the high-permeability regions in the reservoir, and once gas breakthrough happens, subsequent gas injected in the reservoir follows the same path. 5.1 Foam in Permeable Media Foam is a dispersion of gas in water, stabilized by surfactants. Use of foam can significantly improve the oil recovery by mitigating the adverse effects of fingering due to heterogeneities and gravity override problems encountered in gas flooding. Other application of foam include near wellbore stimulation [16], recovering non-aqueous wastes from aquifers [4] and preventing gas coning or water in production wells. There are several models describing the foam flow in porous media. The two main models are Mechanistic "population balance" foam models which attempt to represent the individual process of lamella creation and destruction. The second model is local steady state or Local Equilibrium (LE) model which assumes instantaneous attainment of strong foam state everywhere where water, gas and surfactant are present in sufficient quantities. LE model are based on conservation laws, conserving the phases present in the reservoir where surfactant is considered to be dissolved in either the water or gas phase. Foam is generated as soon the gas comes in contact with the water phase in presence of sufficient amount of surfactant. The generated foam decreases the mobility of the gas phase considerably. The water phase mobility remains unaffected by this model. 43

60 The foam process is described by immiscible two phase model where gas is injected in porous media filled with mixture of water and surfactant. The change from pure gas to foam is incorporated by reduction in gas phase mobility. The generation of foam will cause a rapid increase in the flux function over a small saturation scale. Consequently, the derivative of the flux function can be extremely large which can impose severe stability constraints [22, 15]. 5.2 Numerical Model When the gas comes in contact with sufficient amounts of water and surfactant, foam will be created. This reduces the mobility of the gas phase rapidly, which is modeled using a mobility reduction factor f mr, i.e., k f rg = k0 rg f mr (5.1) where k 0 rg is the gas mobility without foam. The mobility reduction factor can be written as f mr = 1 + fmmob F s F w, (5.2) with F w = arctan(epdry(s w fmdry)). (5.3) π The function F w is from the foam model from STARS [13]. The factor fmmob accounts for the reduction of gas mobility for full strength flow, epdry is the parameter that governs the abruptness of the transition. F s accounts for sensitivity of surfactant concentration to foam. The same parameters as in [21] is used here, i.e., and k rw = (S weff ) , (5.4) k 0 rg = (S weff ) (5.5) The parameters for foam mobility reduction is epdry = 10000, fmmob = and fmdry = Viscosity of water is 1cp (0.01 Pa s) and viscosity of gas is 0.02 cp ( Pa s). The phase mobilities, total mobility and the fractional flow formulation for two cases with and without foam are plotted in Fig

61 (a) (b) (c) (d) (e) (f) Figure 5.1: Fluid Properties in of gas phase: (a) Relative permeability (b) Total mobility; (c) Fractional flow; (d) Derivative of fractional flow; (e) Second derivative of fractional flow; (f) Residual function for different time-step sizes. 45

62 5.3 Results Results for a heterogeneous model using the given fluid properties are presented in Fig Injection takes place at the lower-left corner, and the producer is located at the top-right corner. Very poor sweep efficiency is observed for gas flooding since, while foam flooding has quite an efficient sweep efficiency. (a) (b) Figure 5.2: Saturation solution for (a) gas flooding and (b) foam flooding after 1000 days Flux correction strategy for foam flow As done in the previous chapter, we compare the stability of the fully implicit and sequential implicit formulation with and without flux correction strategy for 2D homogeneous test case. The inflection point for the flux function is S infl = epdry, and introduce a buffer zone of ɛ = Selection of the buffer zone also plays an important role in the convergence of the flux correction strategy. Results are obtained for a five-spot setting, with injector at the lower left corner of the reservoir with the fixed rate of m 3 /s. The producer is located at top right corner with a fixed pressure of Pa. Fully Implicit method outperforms the sequential implicit one, as it is stable and fully convergent for higher time-step size. A comparative study is presented in Fig In this chapter the flux correction strategy is extended for the first time to simulate foam flow in porous media. Due to the abrupt change in the gas phase mobility, which in turn results in abrupt change in fractional flow formulation, severe time-step restrictions exist in foam simulations. Incorporating flux correction strategy for foam fractional flow is studied here, to improve the simulation stability. From Fig. 5.1 (f) the converged and steady-state solution lie on the epdry saturation. As the time-step size is increased, the non-linearity of the solution increases with epdry as the inflection point. Flux correction is implemented keeping the buffer zone small. This is due to the fact that the fractional flow curve is monotonic till and after saturation epdry ± ɛ. For big 46

63 (a) Figure 5.3: Comparative study of Sequential Implicit and Fully Implicit convergence properties for different time-step sizes. buffer zone sizes, the next iteration solution may lie in the other half of the discontinuous curve. This can again lead to non-convergence due to the jump in saturation values as explained in Fig Also a small buffer zone provides a better initial guess to the solver which can lead to faster convergence. The strategy guarantees unconditional convergence for the non-linear transport loop, however, the outer non-linear loop faces convergence issues as the time-step size is increased. The sensitivity of foam to water saturation introduces the challenge that individual grid cells experience mobilities and saturations that are not present in the displacement front. The abrupt changes in the mobilities introduces oscillations in the simulator and poses a challenge to the simulator. A solution to this problem has been proposed by reducing the abrupt change in the gas mobility (f mmob) [22]. This reduction in abrupt changes near epdry saturation reduces oscillation and speeds up the simulation, if these parameters are justified from the core flood data. Note that this MS thesis work was dedicated to flux-correction strategies and the nonlinear loop convergence challenges. Several other challenges, such as the oscillatory solutions, are out of the scope of this work. Using flux correction strategy, the stability of the sequential implicit strategy is improved. Bigger time-step sizes can be taken, as shown in Fig Also, Fig. 5.5 shows the solution obtained with two different time step sizes. It is important to note that the challenge in this aspect is not only due to transport flux functions, but the coupling between flow and transport equations. This coupling plays a major role in the outer loop convergence. Further study needs to be done to extend the non-linear coupling convergence of the flow and transport equation. 47

64 (a) Figure 5.4: Comparative study of Sequential Implicit with and without flux correction strategy for foam flow. Note that much bigger time step sizes can be taken after employing the flux-correction strategy. (a) (b) Figure 5.5: Saturation solution for foam flow in homogeneous media with (a) t = 100 sec. and (b) t = sec. 48