Fully Conservative Characteristic Methods for Flow and Transport: Part I, Linear Transport Part II, Theoretical Considerations Part III, Nonlinear Two-Phase Flow Todd Arbogast Department of Mathematics and Center for Subsurface Modeling, Institute for Computational Engineering and Sciences (ICES) The University of Texas at Austin Chieh-Sen (Jason) Huang, National Sun Yat-sen University (Taiwan) Thomas F. Russell, U.S. National Science Foundation Wenhao Wang, The University of Texas at Austin This work was supported by • U.S. National Science Foundation • KAUST through the Academic Excellence Alliance Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA
Fully Conservative Characteristic Methods for Flow and Transport: Part I, Linear Transport Todd Arbogast Department of Mathematics and Center for Subsurface Modeling, Institute for Computational Engineering and Sciences (ICES) The University of Texas at Austin Chieh-Sen (Jason) Huang, National Sun Yat-sen University (Taiwan) This work was supported by • U.S. National Science Foundation • KAUST through the Academic Excellence Alliance Center for Subsurface Modeling Institute for Computational Engineering and Sciences 1 The University of Texas at Austin, USA
Outline 1. Transport Problems and Local Conservation Principles 2. Characteristic Methods for Linear Transport 3. Local Volume Conservation in Characteristic Methods 4. Some Numerical Results 5. Summary and Conclusions Center for Subsurface Modeling Institute for Computational Engineering and Sciences 2 The University of Texas at Austin, USA
Transport Problems and Local Conservation Principles Center for Subsurface Modeling Institute for Computational Engineering and Sciences 3 The University of Texas at Austin, USA
Conservative Fluid Flow Suppose ξ is a conserved quantity ξ (mass/volume) ❩❩❩❩ v ✓ ✼ v is the fluid velocity (length/time) ✓ ✓ ❩❩❩❩ ξ v is the flux of ξ (mass/area/time) A Q is an external source or sink of fluid (mass/volume/time) Within a region of space R , the total amount of ξ changes in time by � � � d R ξ dx = − ∂R ξ v · ν da ( x ) + R Q dx dt � �� � � �� � � �� � v Change in R Flow across ∂R Sources/sinks ❩ ❇ ▼ ❩ ❇ ❩❩❩❩ v · ν ❇ ❩ ✼ ✓ ❇ ✓ ❩ = ⇒ conservation locally on R ❇ ✓ ❩❩❩❩ ∂R � � � R ξ t dx = − R ∇ · ( ξ v ) dx + R Q dx � �� � Divergence Theorem This is true for each region R , so in fact ξ t + ∇ · ( ξ v ) = Q Center for Subsurface Modeling Institute for Computational Engineering and Sciences 4 The University of Texas at Austin, USA
A Transport Problem–1 One incompressible fluid (tracer) flowing miscibly in another incompressible fluid, within an incompressible medium. Velocity of the bulk fluid. Conservation of bulk fluid mass ( ξ = φρ ) gives ξ t + ∇ · ( ξ v ) = Q = ⇒ ∇ · u = q u is the (unknown) bulk fluid velocity ( v = u /φ ) φ is the porosity (constant in time) ρ is the (constant) density q is the source/sink (wells, Q = ρq ) Simple Tracer Transport. Conservation of tracer mass ( ξ = φc ) gives φc t + ∇ · ( c u ) = c I q + + cq − ≡ q c ( c ) c is the (unknown) tracer concentration c I is the given concentration of injected fluid q + /q − is q when positive/negative Center for Subsurface Modeling Institute for Computational Engineering and Sciences 5 The University of Texas at Austin, USA
A Transport Problem–2 However, transport is not the only process occurring! Mass flux. v = c u − D ∇ c (Transported plus Diffusive Flux) D is the diffusion/dispersion coefficient Chemical reactions. q = q c ( c ) + R ( c ) (Wells plus Reactions) R is the reaction term Tracer Transport. Conservation of tracer mass gives φc t + ∇ · ( c u − D ∇ c ) = q c ( c ) + R ( c ) Center for Subsurface Modeling Institute for Computational Engineering and Sciences 6 The University of Texas at Austin, USA
Operator Splitting of Transport Equation—1 φc t + ∇ · ( c u − D ∇ c ) = q c ( c ) + R ( c ) Discretization in time: ∆ t > 0 and t n = n ∆ t . We want to solve the transport and reactive part of the equation explicitly and the diffusive part implicitly. Thus, we want φc n +1 − c n + ∇ · ( c n u ) − ∇ · ( D ∇ c n +1 ) = q c ( c n ) + R ( c n ) ∆ t This is equivalent to the three steps φc r − c n = R ( c n ) (Reaction) φc t = R ( c ) � ∆ t φc r t − c r + ∇ · ( c n u ) = q c ( c n ) (Transport) φc t + ∇ · ( c u ) = q c ( c ) � ∆ t φc n +1 − c r t − ∇ · ( D ∇ c n +1 ) = 0 (Diffusion) φc t − ∇ · ( D ∇ c ) = 0 � ∆ t with some intermediate ˜ c and ˆ c . Center for Subsurface Modeling Institute for Computational Engineering and Sciences 7 The University of Texas at Austin, USA
Operator Splitting of Transport Equation—2 Nonlinear Ordinary Differential Equation part (Reaction) φc t = R ( c ) Linear Hyperbolic part (Transport) φc t + ∇ · ( c u ) = q c ( c ) Linear Parabolic part (Diffusion/dispersion) φc t − ∇ · ( D ∇ c ) = 0 We discuss approximations of the transport step only. Center for Subsurface Modeling Institute for Computational Engineering and Sciences 8 The University of Texas at Austin, USA
Locally Conservative Methods A locally conservative numerical method is one for which the approximate solution satisfies the conservation principle, but only over certain discrete regions. Bulk volume conservation: � � E ∇ · u dx = E q dx We solve for the flow u using a locally conservative mixed finite element method, conserving it over the grid elements E . Tracer mass conservation: A compatible method would require � � � � φc t + ∇ · ( c u ) dx = E q c dx E Characteristic methods do not use the grid elements. They use more general space-time regions E . Question: Is there a numerical incompatibility in the conservation of the two fluids? Center for Subsurface Modeling Institute for Computational Engineering and Sciences 9 The University of Texas at Austin, USA
Compatibility Tracer mass conservation: �� �� � � φc t + ∇ · ( c u ) dx dt = E q c dx dt E Ambient fluid mass conservation: dx dt ? �� �� � � φ (1 − c ) t + ∇ · ((1 − c ) u ) = E ( q − q c ) dx dt E ⇐ ⇒ E ∇ · u dx dt ? �� �� = E q dx dt For mixed finite element methods, ∇ · u = P q pointwise ( P is projection onto the elements E ). Thus there is no incompatibility (up to treatment of the wells). Remark: This may not be true for nonconservative Galerkin methods. Remark: The reactive and diffusive steps of the operator splitting must also be solved by locally conservative methods (e.g., mixed finite elements or cell-centered finite differences)! Center for Subsurface Modeling Institute for Computational Engineering and Sciences 10 The University of Texas at Austin, USA
Characteristic Methods for Linear Transport Center for Subsurface Modeling Institute for Computational Engineering and Sciences 11 The University of Texas at Austin, USA
Characteristic Tracing of Points The characteristic trace-forward of the point x is denoted ˆ x = ˆ x ( x ; t ). It satisfies the ordinary differential equation d ˆ dt = u (ˆ x , t ) x t n < t ≤ t n +1 , φ (ˆ x ) x ( t n ) = x ˆ In the absence of sources/sinks and diffusion, fluid particles simply travel along the characteristics of the equation. Time ✻ t n +1 ˆ x t n ✲ Space x The concentration is constant along this space-time path, since dc (ˆ x , t ) = ∂c ∂t + ∇ c · d ˆ φ = 1 � � dt = c t + ∇ c · u x φc t + u · ∇ c = 0 dt φ Center for Subsurface Modeling Institute for Computational Engineering and Sciences 12 The University of Texas at Austin, USA
Characteristic Trace-back of Points The characteristic trace-back of the point x is denoted ˇ x = ˇ x ( x ; t ). It satisfies the (time backward) ordinary differential equation d ˇ dt = u (ˇ x x, t ) t n ≤ t < t n +1 , φ (ˇ x ) x ( t n +1 ) = x ˇ In the absence of sources/sinks and diffusion, fluid particles simply travel along the characteristics of the equation. Time ✻ t n +1 x t n ✲ Space x ˇ Again, the concentration is constant along this space-time path. Center for Subsurface Modeling Institute for Computational Engineering and Sciences 13 The University of Texas at Austin, USA
Modified Method of Characteristics (MMOC) (Douglas and Russell, 1982) Key idea: Use a finite difference approximation of the characteristic derivative dt ≡ c t ( x , t ) + u ( x , t ) · ∇ c ( x , t ) ≈ c ( x , t + ∆ t ) − c (ˇ x, t ) dc φ ∆ t This results in the approximation ✻ t n +1 c ( x ) φc ( x , t + ∆ t ) − c (ˇ x, t ) = ( c I − c ) q + ∆ t t n at each grid point ✲ c (ˇ x ) Problems: Because the method is based on points, it violates local mass conservation constraints for both the bulk fluid and the tracer. Center for Subsurface Modeling Institute for Computational Engineering and Sciences 14 The University of Texas at Austin, USA
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