Fully Conservative Characteristic Methods for Flow and Transport: Part II, Theoretical Considerations Todd Arbogast Department of Mathematics and Center for Subsurface Modeling, Institute for Computational Engineering and Sciences (ICES) The University of Texas at Austin 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 1 The University of Texas at Austin, USA
Outline 1. The Volume Corrected Characteristics-Mixed Method 2. Monotonicity and Stability of VCCMM 3. Convergence of VCCMM 4. Numerical Convergence Tests 5. Some Numerical Examples 6. Extension to Compressible Flows 7. Summary and Conclusions Center for Subsurface Modeling Institute for Computational Engineering and Sciences 2 The University of Texas at Austin, USA
The Volume Corrected Characteristics-Mixed Method Center for Subsurface Modeling Institute for Computational Engineering and Sciences 3 The University of Texas at Austin, USA
Characteristic Trace-backs Points. The characteristic trace-back of x is ˇ x = ˇ x ( x ; t ): Time d ˇ dt = u (ˇ x , t ) x t n ≤ t < t n +1 ✻ , t n +1 x φ (ˇ x ) x ( t n +1 ) = x ˇ Assume u · ν = 0 so no particles trace to t n ✲ Space ˇ x the domain boundary. Elements. Let T h be the collection of elements. ✻ t n +1 E Particles in E ∈ T h trace back in space-time to x , t ) ∈ Ω × [ t n , t n +1 ] : ˇ E E E E = { (ˇ x = ˇ x ( x ; t ) , x ∈ E } . t n The “bottom” is ✲ ˇ E x ( x ; t n ) , x ∈ E } . ˇ E = { ˇ x ∈ Ω : ˇ x = ˇ � Let the pore volume of R be | R | φ = R φ dx . �� | E | φ = | ˇ The local volume constraint: E | φ + q dx dt E E Center for Subsurface Modeling Institute for Computational Engineering and Sciences 4 The University of Texas at Austin, USA
The VCCMM ② ② Approximate local volume constraint: ② ② ② ② Perturb The polygonal approximation of ˇ E is perturbed to ˜ E so that ② ② E ≈ ˜ ˇ E �� ② ② Vol( ˇ E ) = Vol( ˜ | E | φ = | ˜ E ) E | φ + q dx dt ˜ E E ② ② ② ② ② ② Volume Corrected Characteristics Mixed Method (VCCMM): c ( x , t n ) ≈ c n h ( x ) = c n E ∈ R when x ∈ E For each E ∈ T h , define c n +1 by E � �� c n +1 E φc n | E | φ = h dx + q c n h dx dt E ˜ ˜ E E Center for Subsurface Modeling Institute for Computational Engineering and Sciences 5 The University of Texas at Austin, USA
A Note on Updating Concentrations V 1 V 2 V 1 V 3 V 2 V 8 It is convenient to store the trace-back elements ˜ V 6 V 5 E as a V 4 Polygon V 3 polyline, i.e., a doubly linked V 7 list of vertices. V 4 V 6 V 7 V 9 =V 1 V 8 V 5 Polyline class For each E ∈ T h , in the absence of sources, E ? � c n +1 E φc n | E | φ = h dx E ˜ � � E ∩ F φc n = F dx ˜ ˜ E F ∈T h � c n F | ˜ = E ∩ F | φ F ∈T h F � ˜ ˜ E ∩ F is calculated efficiently by a 2-D E F Sutherland-Hodgman clipping algorithm (Foley, Van Dam, Feiner, & Hughes, 1995) Center for Subsurface Modeling Institute for Computational Engineering and Sciences 6 The University of Texas at Austin, USA
An Algebraic View of VCCMM—1 VCCMM reduces to � � � �� c n +1 = | E | − 1 E n c n h dx + q c h dx dt E φ E n ˜ ˜ E � � � � � �� �� E n ∩ F | φ + q − dx dt c I q + dx dt = | E | − 1 c n | ˜ + F φ E n ˜ E ∩ I n E n ˜ F ∈T h F E � A n E,F c n F + b n = E , F where I n F is the space-time cylinder F × ( t n , t n +1 ) over F . Consider the piecewise constant function c n h as a vector indexed by T h c n = { c n E } E ∈T h Thus �� c n +1 = A n c n + b n c I q + dx dt. E = | E | − 1 b n where φ E n ˜ E Center for Subsurface Modeling Institute for Computational Engineering and Sciences 7 The University of Texas at Austin, USA
An Algebraic View of VCCMM—2 ∈ T h,P ( not at any production well, q − = 0 in ˜ E n Case 1: E / E ) E n ∩ F | φ A n E,F = | E | − 1 φ | ˜ Case 2: E ∈ T h,P (a production well) ✻ t n +1 E Note that production wells expand during I E E , and q − = 0 for F trace-back, so I n E n E ⊂ ˜ q = 0 q < 0 q = 0 near E , F � = E . Thus ˜ t n E E ✲ �� E n ∩ F | φ + q − dx dt | E | φ A n E,F = | ˜ F E E n ˜ E ∩ I n F � � �� E n ∩ F | φ + δ E,F q − dx dt = (1 − δ E,F ) | ˜ | E | φ + I n E � �� � V n E where δ E,F = 1 if E = F and 0 if E � = F . Note: V n E ≥ 0 is the volume of fluid remaining in E after production. Problem: However, V n E might be negative if production is large! Center for Subsurface Modeling Institute for Computational Engineering and Sciences 8 The University of Texas at Austin, USA
An Algebraic View of VCCMM—3 Modification when V n E < 0 : 1. Set V n E = 0 in above formula. 2. Reduce the volume of fluid in each nearby F proportionately by E n ∩ F | | ˜ E n \ E | V n E | ˜ Case 2 (Modified): E ∈ T h,P (a production well) E n ∩ F | φ � � E n ∩ F | φ + | ˜ E ) + | E | φ A n ( V n E ) − + δ E,F ( V n | ˜ E,F =(1 − δ E,F ) E n \ E | φ | ˜ � �� � � �� � if negative if positive Remarks: • This is consistent with the unmodified case when V n E ≥ 0. • Note that the local volume constraint still holds. • In practice, we might ignore the production wells, since there fluid leaves the domain (and we know how much by mass conservation). Center for Subsurface Modeling Institute for Computational Engineering and Sciences 9 The University of Texas at Austin, USA
An Algebraic View of VCCMM—4 VCCMM (Vector Form/Algebraic View): c n +1 = A n c n + b n Matrix A n = ( A n E,F ) E,F ∈T h and vector b n = ( b n E ) E ∈T h are E n ∩ F | φ | ˜ , E / ∈ T h,P , | E | φ E n ∩ F | φ | ˜ ( V n E ) − � � A n 1 + E ∈ T h,P , F � = E, E,F = , E n \ E | φ | ˜ | E | φ ( V n E ) + , E = F ∈ T h,P , | E | φ �� 1 c I q + dx dt, b n E = E n | E | φ ˜ E �� q − dx dt. V n E = | E | φ + I n E Center for Subsurface Modeling Institute for Computational Engineering and Sciences 10 The University of Texas at Austin, USA
Monotonicity and Stability of VCCMM Center for Subsurface Modeling Institute for Computational Engineering and Sciences 11 The University of Texas at Austin, USA
Properties of Matrix A n —1 Lemma 1: Each A n E,F ≥ 0. Proof: We need only consider E ∈ T h,P and F � = E . Then �� E n \ E | φ + ( V n E ) − ≥ | ˜ E n \ E | φ + q − dx dt A n E,F ∝ | ˜ I n E �� E n | φ − | E | φ + = | ˜ q dx dt = 0 . � E n ˜ E � �� � local volume constraint Center for Subsurface Modeling Institute for Computational Engineering and Sciences 12 The University of Texas at Austin, USA
Properties of Matrix A n —2 � A n Lemma 2: Each row sum E,F ≤ 1. F ∈T h Proof: We need only consider E ∈ T h,P and F � = E . Then, since production wells expand, when E / ∈ T h,P , E n | φ E,F = | ˜ �� 1 � � � A n = ≤ 1 , | E | φ − q dx dt | E | φ | E | φ E n ˜ F ∈T h E � �� � ≥ 0 again using the local volume constraint. When E ∈ T h,P , E n \ E | φ � � E ) + E,F = | ˜ ( V n E ) − + ( V n � A n 1 + E n \ E | φ | ˜ | E | φ | E | φ F ∈T h � � �� 1 1 E n \ E | φ + V n q − dx dt E n | φ + ( | ˜ | ˜ = E ) = I n | E | φ | E | φ E � � �� 1 E n | φ + | ˜ = q dx dt = 1 . � E n | E | φ ˜ E � �� � local volume constraint Center for Subsurface Modeling Institute for Computational Engineering and Sciences 13 The University of Texas at Austin, USA
Properties of Matrix A n —3 Corollary: Matrix A n does not increase the l ∞ -norm of a vector, i.e., | A n c | ∞ ≤ | c | ∞ , where | · | ∞ is the maximum component in absolute value: | x | ∞ = max | x E | . E ∈T h Proof: For E ∈ T h , � � � � � � � � ( A n c ) E � A n � A n A n � � � = E,F c F � ≤ E,F | c F | ≤ E,F | c | ∞ ≤ | c | ∞ . � � � � F ∈T h F ∈T h F ∈T h Center for Subsurface Modeling Institute for Computational Engineering and Sciences 14 The University of Texas at Austin, USA
Monotonicity of VCCMM VCCMM satisfies a maximum and minimum principle. Theorem: Up to the effect of injection wells, VCCMM can produce neither overshoots nor undershoots. Proof: Without b n (i.e., injection wells), c n +1 = A n c n , and so each c n +1 is a positive average of previous values c n F . � E Center for Subsurface Modeling Institute for Computational Engineering and Sciences 15 The University of Texas at Austin, USA
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