Todd Arbogast. Department of Mathematics and Center for Subsurface Modeling, Institute for Computational Engineering and Sciences (ICES)

Transcription

1 CONSERVATIVE CHARACTERISTIC METHODS FOR LINEAR TRANSPORT PROBLEMS Todd Arbogast Department of Mathematics and, (ICES) The University of Texas at Austin Chieh-Sen (Jason) Huang Department of Applied Mathematics and National Center for Theoretical Sciences National Sun Yat-sen University (Taiwan)

2 Outline 1. The PDE s of Transport Problems and Local Conservation Principles 2. Characteristic Methods for Approximating the solution 3. Satisfying the Local Volume Conservation Principle Discretely 4. Numerical Results 5. Conclusions

3 Transport Problems and Local Conservation

4 Suppose Conservative Fluid Flow ξ is a conserved quantity ξ (mass/volume) v is the fluid velocity (length/time) ξv is the flux of ξ (mass/area/time) 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 dt ξ dx = ξv ν da(x) + Q dx } R{{}} R {{}} R {{} Change in R Flow across R Sources/sinks = conservation locally on R R ξ t dx = (ξv) dx } R {{} Divergence Theorem This is true for each region R, so in fact ξ t + (ξv) = Q + R Q dx

5 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 + (ξv) = 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 + (cu) = 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

6 A Transport Problem 2 However, transport is not the only process occurring! Mass flux. v = cu D c (Transported plus Diffusive Flux) where D is the diffusion/dispersion coefficient Chemical reactions. q = q c (c) + R(c) (Wells plus Reactions) where R is the reaction term Tracer Transport. Conservation of tracer mass gives φc t + (cu D c) = q c (c) + R(c)

7 Operator Splitting of Transport Equation 1 φc t + (cu 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 φ cn+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 (Reaction) (Transport) cn φ c t = R(c n ) φc t = R(c) φĉ c t + (cn u) = q c (c n ) φc t + (cu) = q c (c) (Diffusion) φ cn+1 ĉ (D c n+1 ) = 0 φc t (D c) = 0 t with some intermediate c and ĉ.

8 Operator Splitting of Transport Equation 2 Nonlinear Ordinary Differential Equation part (Reaction) φc t = R(c) Linear Hyperbolic part (Transport) φc t + (cu) = q c (c) Linear Parabolic part (Diffusion/dispersion) φc t (D c) = 0 We discuss approximations of the transport step only.

9 Locally Conservative Methods A locally conservative method is one for which the approximate solution satisfies the conservation principle, but only over certain discrete regions. Normally, one would take the grid elements R and require ( φct + (cu) ) dx = q dx R R but we will need to be more general than this. Remark: The reactive and diffusive steps must also be solved by locally conservative methods, or local conservation will break down!

10 Characteristic Methods for Linear Transport

11 Characteristic Tracing of Points The characteristic trace-forward of the point x is denoted ˇx = ˇx(x; t). It satisfies the ordinary differential equation dˇx u(ˇx, t) = dt φ(ˇx), tn < t t n+1 ˇx(t n ) = x In the absence of sources/sinks and diffusion, fluid particles simply travel along the characteristics of the equation. Time t n+1 ˇx t n x Space The concentration is constant along this space-time path, since dc(ˇx, t) dt = c t dˇx + c dt = c t + c u φ = 1 ( φct + u c ) = 0 φ

12 Characteristic Trace-back of Points The characteristic trace-back of the point x is denoted ˆx = ˆx(x; t). It satisfies the (time backward) ordinary differential equation dˆx dt = u(ˆx, t) φ(ˆx), tn t < t n+1 ˆx(t n+1 ) = x 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 ˆx Space Again, the concentration is constant along this space-time path.

13 Modified Method of Characteristics (MMOC) (Douglas and Russell, 1982) Key idea: Use a finite difference approximation of the characteristic derivative dc dt c t(x, t) + u(x, t) This results in the approximation φ c(x, t) c(x, t + t) c(ˆx, t) t c(x, t + t) c(ˆx, t) φ t = (c I c)q + t n+1 c(x) at each grid point t n c(ˆx) Problems: Because the method is based on points, it violates local mass conservation constraints for both the bulk fluid and the tracer.

14 Characteristic Trace-back of Regions To obtain mass conservation,... Key idea: Trace regions rather than points! The particles in a grid element E trace back to a region Ê Ê = {ˆx Ω : ˆx = ˆx(x; t n ) for some x E}. In space and time, we actually trace a region E = E(E) given by E = {(ˆx, t) Ω [t n, t n+1 ] : ˆx = ˆx(x; t) for some x E}. t n+1 E t n Ê E

15 Local Mass Conservation of the Tracer φc t + (cu) = q c (c) Integrate in space-time over E and use the divergence theorem ( ) ( ) cu u x,t dx dt = E φc c ν E φ x,t dσ ( ) u = E φcn+1 dx Ê φcn dx + c ν S φ x,t dσ The( last ) term is integration on the space-time sides S of E, u but is orthogonal to ν φ x,t there! The local mass constraint: E φcn+1 dx = + Ê φcn dx E q c dx dt S B cu ν dσ t n+1 t n S B Ê E E ν x,t

16 Local Mass Conservation of the Bulk Fluid A similar local mass constraint holds for the bulk fluid (c 1) φ t + u = u = q Since we are dealing with incompressible fluids, we call this the local volume constraint. The local volume constraint: E φ dx = Ê φ dx + E q dx dt S B u ν dσ

17 Characteristics Mixed Method (CMM) (A., Chilakapati, and Wheeler, 1992; A. and Wheeler, 1995) Use lowest order Raviart-Thomas mixed finite elements. The scalar test function is a constant on each element in space. E φcn+1 dx = Ê φcn dx + q c dx dt cu ν dσ E S B Remark: Practical implementation requires that Ê be approximated by Ẽ Ê, a polygon. This is equivalent to modifying the velocity field, so tracer mass is still conserved locally by the above equation Ê Ẽ Vol(Ê) Vol(Ẽ)

18 Volume Error Problem: The local volume constraint may be violated for this perturbed velocity! This is because the volume constraint does not enter into the definition of the method. This leads to incorrect densities of the tracer, which leads, e.g., to bad approximation of reaction dynamics Relative volume errors around an injection well.

19 Local Volume Conservation in Characteristic Methods

20 Volume Conservation Key idea: To obtain volume conservation, define Ẽ Ê to be a simple shape that satisfies the volume constraint E φ dx Ẽ φ dx = E q dx dt S B u ν dσ Strategy: Suppose that E is a rectangle. Perturb the vertices and midpoints of Ê only a little so that we get a polygon Ẽ with 8 sides such that the above constraint is satisfied. We call this method the Volume Corrected Characteristics Mixed Method (VCCMM) Ê Ẽ Vol(Ê) = Vol(Ẽ) Problem: It is easy to introduce systematic bias into the flow field and thereby produce unphysical flows. We must do the adjustment very carefully!

21 An Example of Unphysical Flow Volume not conserved Volume conserved 10 years years Biased trace-back adjustment has introduced unphysical flow corresponding to a large, incorrect velocity channel.

22 Forward Trace of Injection Wells Most of the error is near injection wells. Characteristic tracing back in time traces into the well, which is difficult to approximate. Key idea 1: Trace the well forward (out of the well). Adjust region to satisfy the volume constraint (cf. Healy and Russell, 2000). The characteristic trace-forward of the point x is denoted ˇx = ˇx(x; t), and it satisfies the ordinary differential equation dˇx u(ˇx, t) = dt φ(ˇx), tn < t t n+1 ˇx(t n ) = x

23 Conservation Near Injection Wells Well volume constraint: Adjust W so W φ dx = W φ dx + E f q dx dt E Ẽ W W Element volume constraint: Adjust Ẽ so E φ dx = Ẽ φ dx+ Vol(E W) Vol( W) E q dx dt Transport: E φcn+1 dx = Ẽ φcn dx + Vol(E W) Vol( W) E q c dx dt

24 Inflow boundaries Like injection wells, inflow boundaries trace back out of the domain. Idea: Either trace inflow boundaries forward, or fold the time axis down to the xy-plane to create a ghost region. y y t t n t n+1 x t x u ν φ Ẽ Ω Ω Volume constraint: Replace φ by u ν in the ghost region: E Ẽ Ω φ dx φ dx + u ν dσ = S B E Ẽ φ dx φ dx = Mass constraint: Replace φc n by c n Iu ν in the ghost region: E φcn+1 dx Ẽ φcn dx = q c dx dt. E E q dx dt,

25 Trace-back Point Adjustment Key idea 2: Adjust points in the direction of the flow; that is, along the characteristics in time (cf. Douglas, Huang, and Pereira, 1999). To define x, for τ n t n, we solve (backwards) d x dt = u( x, t) φ( x), τn < t t n+1 x(t n+1 ) = x We convert space error into time error: t n+1 x t n τ n ˆx x

26 Trace-back (or Forward) Point Adjustment 1 Proceed away from injection wells and inflow boundaries by layers. For each layer, obtain volume conservation in two steps. 1. Volume conservation of the layer. Adjust the exterior contour of the entire layer along the characteristics until the volume of the layer is correct (within a small tolerance). That is, in the absence of other sources, inflow boundaries, and sinks, E in the layer E φ dx = E in the layer Ẽ φ dx Flow Adjusted point (fixed) Points adjusted simultaneously in the direction of the characteristic (we use a type of bisection algorithm) Points adjusted individually transverse to the flow in Step 2

27 Trace-back (or Forward) Point Adjustment 2 2. Element volume conservation. Within the layer, sequentially adjust the interior midpoint of each element transverse to the flow until the volume of the element is correct (within a small tolerance). Flow > x i,j < > xi,j+1/ x i,j 3 Adjusted point (fixed) Points individually adjusted transverse to the flow Remark: This is an extremely fast direct algorithm.

28 Numerical Results

29 Numerical Results Darcy s Law completes the equations: u = k µ p p is the fluid pressure k is the permeability µ is the fluid viscosity Measure the variability of k by the dimensionless coefficient of variation where the mean is C v = 1 M k ( 1 Vol(Ω) M k = 1 Vol(Ω) Ω (k(x) M k) 2 dx Ω k(x) dx ) 1/2

30 A Nuclear Contamination Problem 1 The permeability is log-normal and fractal. M k = cm 2 (about 20 md) C v = (varies over five orders of magnitude). 256 Inflow Y 64 1E-08 1E-09 1E-10 1E-11 1E-12 1E-13 Outflow Injection well

31 A Nuclear Contamination Problem 2 If we use a small time step of t = 1.5 years, we can trace back into the injection well CMM volume errors VCCMM (volume errors 10 9 )

32 A Nuclear Contamination Problem 3 Concentration at 30 years on a grid, with t = 1.5 yr 2.6CFL E E E E E E E E E E E E E E E E CMM VCCMM CMM overshoots the maximum concentration of 1E-5 by 34% up to 1.34E-5.

33 The Courant-Friedrichs-Lewy (CFL) Condition Explicit methods in 1-dimension have a time step restriction, known as the Courant-Friedrichs-Lewy (CFL) time-step, given by where h is the grid spacing. t t CFL,1-D = max x Ω hφ(x) u(x) In 2-dimensions, we should limit t to half this value, t t CFL,2-D = max x Ω hφ(x) 2 u(x) Godunov s method is a popular method, that is unstable if the CFL condition is violated. In principle, characteristic methods are not subject to this constraint, and large time steps can be used.

34 A Nuclear Contamination Problem 4 Concentration at 30 years on a grid E E E E E E E E E E E E E E Godunov t = yr = 1 CFL VCCMM-TF t = 3 yr = 5.1 CFL We use trace-forwarding near the well. No overshoot for either method. Less numerical diffusion for VCCMM-TF. 51 Godunov steps vs. 10 for VCCMM-TF.

35 A Nuclear Contamination Problem 5 Concentration at 30 years on a grid E E E E E E E E E E E E E E Godunov t = yr = 1 CFL VCCMM-TF t = 1 yr = 6.8 CFL We use trace-forwarding near the well. No overshoot for either method. Less numerical diffusion for VCCMM-TF. 205 Godunov steps vs. 30 for VCCMM-TF.

36 A Nuclear Contamination Problem 6 Concentration at 30 years on a grid E E E E E E E E E E E E E E Godunov t =.0366 yr = 1 CFL VCCMM-TF t = 0.5 yr = 13.7 CFL We use trace-forwarding near the well. No overshoot for either method. Less numerical diffusion for VCCMM-TF. 820 Godunov steps vs. 60 for VCCMM-TF.

37 A Quarter Five-Spot Problem 1 Geostatistically generated permeability. M k = 100 md C v = 2.58 (varies over four orders of magnitude). 5E-12 1E-12 5E-13 1E-13 5E-14 1E-14 5E-15 1E-15

38 A Quarter Five-Spot Problem 2 Concentration at 3.36 years using t = yr = CFL CMM VCCMM-TF CMM shows both overshoot and undershoot. Very large initial volume imbalances throughout the domain. If t = yr = CFL initially creates degenerate trace-back regions, which cannot be used.

39 A Linear Flood Problem 1 A test with an inflow boundary. Permeability field of the quarter five-spot problem Linear pressure drop across the domain in the x-direction. Concentration at 25 years using t = 0.18 year = 3 CFL CMM VCCMM CMM has severe overshoots up to c = Initial volume errors exceed 10% in the interior of the domain. If t = 9 CFL, initial relative volume errors are around 25%.

40 A Linear Flood Problem 2 Concentration at 25 years using trace-forwarding of the inflow boundary VCCMM-TF t = 0.18 yr = 3 CFL VCCMM-TF t = 0.36 yr = 6 CFL

41 A Fluvial Domain Problem 1 Domain feet 2 solved on a grid. Permeability of 3 values, M k = Darcy and C v = φ = 0.2. Wells in opposite corners, injecting 1 pore volume every 3 years. t = years = CFL The permeability, in Darcies

42 A Fluvial Domain Problem 2 VCCMM-TF concentration Time 1.05 years (step 70) Time 1.65 years (step 110) Remark: CMM alone produces negative concentrations on a grid with t = 0.01 year, indicating that the trace-back regions self intersect.

43 A Fluvial Domain Problem 3 CMM with trace-forwarding of wells only Time 1.05 years (step 70) Time 1.65 years (step 110)

44 Comparison with an Analytic Solution Comparison of VCCMM-TF concentration with an analytic solution for radial flow from a well in a horizontally infinite, uniform porous medium Analytic Analytic VCCMM-TF 0.6 VCCMM-TF Longitudinal dispersion 20 cm Longitudinal dispersion 100 cm The analytic solution is derived by a code from P.A. Hsieh (1986). VCCMM-TF uses t = 8 CFL.

45 Conclusions 1. A critical aspect of approximating hyperbolic transport problems is to conserve the mass of the tracer locally. 2. It is just as critical to the conserve locally the mass of the bulk fluid. 3. When adjusting trace-back points, care must be used to avoid introducing systematic bias into the transport computation. The key is to adjust along the characteristics when possible. 4. Trace-forwarding is needed around injection wells. 5. Inflow boundaries can be treated transparently through a space-time fold-down strategy, or with trace-forwarding. 6. In principle, the only restriction on the time step is that the traceback regions not degenerate or self-intersect (this problem can be alleviated by tracing more points of the boundary E). 7. VCCMM-TF allows us to take large time steps (such as 14 times the 2-D CFL limited time step) with no overshoots, which results in less numerical dispersion than Godunov s method.