modeling multiphase flow in porous mediaide/data/teaching/amsc663/14... · 2017-06-28 ·...

lamed-fixed

BackgroundMethods

ImplementationProject Outcome

Modeling Multiphase Flow in Porous Media

Quan Bui

Applied Math, Statistics, and Scientific Computation,University of Maryland - College Park

December 9th, 2014

AdvisorsDr. Amir Riaz, Department of Mechanical Engineering, UMCP

Dr. David Moulton, Applied Math Group (T-5), Los Alamos National Lab

http://amsc.umd.edu/

http://www.umd.edu/

lamed-fixed

BackgroundMethods


Applications of Multiphase FlowGoalsContinuum ModelIncompressible Two-phase Flow

Applications of Multiphase Flow

Carbon sequestrationGroundwater managementContaminant transportOil and gas recovery

Source: Groundwater/Vadose Zone Integration ProjectSpecification. DOE/RL-98-48, Draft C, U.S. Department of Energy,Richland, WA, 1988.

Source: Wikipedia

1 / 28

lamed-fixed

BackgroundMethods



Overall Goals

Implement the nonlinear complimentarity constraints approach to solvea system of PDEs and constraint equations modeling multiphase flowin Amanzi.Provide a capability to model fully coupled 2-phase, 2-component,non-isothermal, miscible flow with phase transitions.Develop unit tests for verification.Pursue application to realistic problem such as desiccation.

2 / 28

lamed-fixed

BackgroundMethods



Phase I

Develop confidence working with Amanzi.Formulate a coupled system of pressure and saturation equations.Implement a fully implicit approach to solve this system of equations.Develop unit tests for pressure and saturation equations.

3 / 28

lamed-fixed

BackgroundMethods



System of Equations

∂

∂t

(φ∑α=w,n

ρmol,αXKαSα

)+∇ ·

(−∑α=w,n

(ρmol,αXK

αvα + DKpm,αρmol,α∇XK

α

))= f K

∂

∂t

(φ∑α=l,g

ρmass,αuαSα + (1− φ)ρscsT)

+

∇ ·(−

M∑α=1

(ρmass,αhK

αvα)

+

N∑K=1

∑α=w,n

(DKpm,αρmol,αhK

αMK∇XKα)− λpm∇T

)= f h

vα =κrα

µαK(∇pα − ρmass,αg)

pc(Sl) = pn − pw,

Sw + Sn = 1, 0 ≤ Sα ≤ 1,N∑

K=1

XKα = 1, 0 ≤ XK

α ≤ 1

4 / 28

lamed-fixed

BackgroundMethods



Simplied Model

We made the following simplification

Immiscibility, no phase transition.No molar fractions.No additional constraints for local thermal dynamic equilibrium.

IsothermalNo energy equation.Densities, porosity, and viscosities independent of temperature.

IncompressibilityThe fluid and solid structure (rock matrix) is incompressible.Densities and porosity constant in space and time.

5 / 28

lamed-fixed

BackgroundMethods



Mass Balance

For phase α, we have

∂(φραSα)

∂t+∇ ·

(ραvα

)= qα, α = w, n (1)

in whichφ is the porosityρα is the density of phase αSα is the saturation of phase αvα is the Darcy’s velocity of phase αqα is the source term of phase α.

In addition, we have the constraint∑α

Sα = 1 (2)

6 / 28

lamed-fixed

BackgroundMethods



Constitutive Relations

We use extended Darcy’s law for multiphase flow

vα = −krα

µαK(∇Pα − ραg), α = w, n (3)

whereK is the absolute permeabilitykrα is the relative permeability of phase αµα is the viscosity of phase αPα is the pressure of phase αg is gravity

The phase pressures are related through capillary pressure Pc

Pc = Pn − Pw (4)

7 / 28

lamed-fixed

BackgroundMethods



Relative Permeability

Common models for relative permeability krα

Power law (Brooks-Corey type)

Seα =Sα − Srα

1−∑β Srβ

(5)

krα =(

Seα

)n(6)

Van Genuchten [Genuchten, 1980]

krw(Sw) =√

Sew

(1−

(1− S1/m

ew

)m)2(7)

krn(Sw) =√

1− Sew

(1− S1/m

ew

)2m(8)

8 / 28

lamed-fixed

BackgroundMethods



Capillary Pressure

Models for capillary pressure Pc

Brooks-Corey [Brooks and Corey, 1964]

Pc(Sw) = PdS−1/λw (9)

Van Genuchten [Genuchten, 1980]

Pc(Sw) = Pr

(S−1/m

ew − 1)1/n

(10)

m = 1− 1/n (11)

9 / 28

lamed-fixed

BackgroundMethods



Pressure Equation

Sum the mass balance equations, with∑α Sα = 1,

∇ ·(∑

α

krα

µαραvα)

)=∑α

qαρα

(12)

In short form

∇ · v = q (13)

where v = vw + vn is the total velocity and q is the total source term.

v = −(

Kλw(∇Pw − ρwg) + Kλn(∇Pn − ρng))

(14)

q =qw

ρw+

qn

ρn, λw =

krw

µw, λn =

krn

µn(15)

10 / 28

lamed-fixed

BackgroundMethods



Fractional Flow Formulation

Introducing fractional flow of the wetting phase fw, with total mobilityλ = λw + λn

fw =λw

λ(16)

Total velocity becomes

v = λK(∇Pn − fw∇Pc −G) (17)

G =λwρw + λnρn

λg (18)

Define global pressure such that

∇P = ∇Pn − fw∇Pc (19)

11 / 28

lamed-fixed

BackgroundMethods



Fractional Flow (cont.)

One common choice for global pressure P [Chavent and Jaffre, 1978] is

P = Pn − π(Sw) (20)

π(Sw) =

∫ Sw

S0

fw(ξ)∂Pc

∂Sw(ξ)dξ + π0 (21)

Then the total velocity is reduced to

v = λK(∇P−G) (22)

The pressure equation only depends on the global pressure P

−∇ ·(λK(∇P−G)

)= q (23)

⇒ Diffusion equation

12 / 28

lamed-fixed

BackgroundMethods



Saturation Equation

Using the mass balance equation for the wetting phase,

φ∂Sw

∂t+∇ · vw =

qw

ρw(24)

Rewrite the Darcy’s velocity of the wetting phase vw using total velocity vand fractional flow fw

vw = fwv + λnfwK(∇Pc + (ρw − ρn)g) (25)

⇒ Nonlinear advection equation

13 / 28

lamed-fixed

BackgroundMethods


ApproachDiscretization

Approach

We have a coupled system ofPressure equation: linear elliptic PDE.Saturation equation: nonlinear scalar hyperbolic PDE.

Common approachImplicit pressure explicit saturation (IMPES) in which the saturationequation is solved explicitly.Semi-implicit schemes.Fully implicit approach in which the saturation is solved implicitly. (Ourproject)

For simulation, we solve pressure and saturation equations sequentially inthat order, with some initial condition for saturation.

14 / 28

lamed-fixed

BackgroundMethods



Pressure Equation

We use finite volume method in space

−∑j∈ηi

∫γij

λK∇P · ndS = Viq (26)

On each face, the pressure gradient is approximated with two point fluxapproximation (TPFA)

−∫γij

λK∇P · ndS = −|γij|2(Pi − Pj)

∆xi + ∆xj(λK)ij (27)

We compute the transmissibility using harmonic average.

(λK)ij = (∆xi + ∆xj)( (λK)i(λK)j

∆xi(λK)j + ∆xj(λK)i

)(28)

15 / 28

lamed-fixed

BackgroundMethods



Saturation Equation

We use finite volume in space and backward Euler in time. For each controlvolume,

VφSn+1

w − Snw

∆t+

∫∂V

fw(Smw)v · n = V

qmw

ρw

In IMPES approach, m = n, and for fully implicit approach, m = n + 1.

The fractional flow is upwinded

fw(Sw)ij =

{fw(Sw)i if vij · n ≥ 0fw(Sw)j if vij · n < 0

The total velocity v is calculated from the solution of the pressure equation.

16 / 28

lamed-fixed

BackgroundMethods


HPC FrameworkMilestonesUnit Tests for Verification

Amanzi: The ASCEM Multi-Process HPC Simulator

Wide Range of Complexity

Wide Range of Platforms

Modular HPC simulationcapability for waste formdegradation, multiphaseflow and reactive transport.

Efficient, robust simulationfrom supercomputers tolaptops.

Design and build foremerging multi-core andaccelerator-based systems.

Open-source project withstrong communityengagement.

17 / 28

lamed-fixed

BackgroundMethods



Amanzi: Approach and Features

Structured / Unstructured mesh capability:Leverages mesh frameworks (MSTK, STKmesh)for general unstructured meshes.Leverages structured AMR techniques andlibraries (BoxLib)

Leverages the Trilinos framework (Epetra,Thyra) and supporting tools/solvers.

Leverages advances in Mimetic FiniteDifference (MFD) methods to enableaccurate solutions.

mixed-hybrid (or local) formulationarbitrary polyhedra (layered media, pinch-outs)distorted meshes (capture topography andhydrostratigraphy)discontinous and highly variable anisotropiccoefficients (permeability)

18 / 28

lamed-fixed

BackgroundMethods



Amanzi Design

driver

Visualization

Checkpoint

Observations

process tree

MPC

strong coupler

PK:: Flow PK:: Energy

State

Mesh

FieldMap

• ...

• ...

dependency graph

Components of Amanzi. [Coonet al., 2014]

Process-Kernel TreeHierarchical view of themathematical modelSelect PKs at run-timePKs use high-level operatorclasses to encapsulatediscretizations, solvers

Directed Acyclic Graph (DAG)Dynamic plug-and-play PKs viaself-registering factories.Automated Multi-Process Couplers(MPC) for both “weakly” and“strongly” coupled systems.Customization of strong couplers isusually required.

Goal: Ensure domain scientists can easily understand,extend, and develop PK implementations.

19 / 28

lamed-fixed

BackgroundMethods



Amanzi Design (cont.)

MPC

strong or weak coupler

PK:: Flow

Richards

PK:: Energy

advection-diffusion

PK:: Surface Flow

diffusion wave eqn

PK:: Surface Energy

advection-diffusion

PK:: Surface Balance

precipitation,radiation, evaporation

MPC

customstrongcoupler

An example of a process kernel tree for amodel of thermal hydrology in the arctic.[Coon et al., 2014]

Arctic thermal-hydrology model:Thermal hydrology withsurface and suburface flow isstrongly coupled.This MPC is weakly coupledto the surface energybalance.The hierarchical use of MPCsto makes these couplingscenarios easy to expresss.

20 / 28

lamed-fixed

BackgroundMethods



What have we accomplished?

For phase I, we have implemented in AmanziA pressure PK to solve the pressure equation.A saturation PK to solve the saturation equation.Coupled pressure PK and saturation PK (the equations are solvedsequentially).Unit tests and convergence test for pressure equation.

21 / 28

lamed-fixed

BackgroundMethods



Example Code for Pressure PK

// New interface for a PKvirtual void Initialize();virtual bool AdvanceStep(double t_old, double t_new);virtual void CommitStep(double t_old, double t_new){};virtual void CalculateDiagnostics(){};

// Main methods of this PKvoid InitializeFields();void InitializePressure();void InitTimeInterval(Teuchos::ParameterList& ti_list);void CommitState(const Teuchos::Ptr<State>& S);

// Time integration interface new_mpc, implemented in Pressure_PK_TI.cc// computes the non-linear functional f = f(t,u,udot)virtual void Functional(double t_old, double t_new,

Teuchos::RCP<TreeVector> u_old,Teuchos::RCP<TreeVector> u_new,Teuchos::RCP<TreeVector> f);

// applies preconditioner to u and returns the result in Puvirtual void ApplyPreconditioner(Teuchos::RCP<const TreeVector> u,

Teuchos::RCP<TreeVector> Pu);// updates the preconditionervirtual void UpdatePreconditioner(double t,

Teuchos::RCP<const TreeVector> up, double h);

22 / 28

lamed-fixed

BackgroundMethods



Pressure Equation

Unit test for pressure equation.

−∇ · (λ∇P) = f

Example problem

λ = 1

f = 5π2 sin (πx) sin (2πy)

P = sin (πx) sin (2πy)

23 / 28

lamed-fixed

BackgroundMethods



Presure Equation (cont.)

Analytic solution (left) and numeric solution (right) for test problem

24 / 28

lamed-fixed

BackgroundMethods



Presure Equation (cont.)

Convergence rate for test problem, with mesh size h = 8, 16, 32, 64, 128.

25 / 28

lamed-fixed

BackgroundMethods



Unit test for saturation equation

Unit test for 1D advection.

ut + f (u)x = 0

Buckley-Leverett problem for one-dimensional 2-phase, 2-component,immiscible, incompressible flow in 1D

St = U(S)Sx

U(S) =QφA

dfdS

where S is the saturation, Q is the flux, A is the surface area, φ is theporosity, and f is the fractional flow.This is basically a nonlinear advection equation with non-convex fluxf (S).

26 / 28

lamed-fixed

BackgroundMethods


Timeline

Timeline

Phase IOctober

Gain confidence working with Amanzi coding standard, build tools,interfacing with other libraries, etc. 3

Get this new PK to interface correctly with both the multi-processcoordinator (MPC) and input files. 3

Start with building a simple solver for the pressure equation. 3

NovemberImplement a simulator for incompressible 2-phase flow. 3

Fully coupled approach for incompressible 2-phase flow. 7

Unit tests for pressure equation. 3

Unit tests for saturation equation. 7

DecemberAdd nonlinear complementarity constraints for phase transitions. (Inprogress)Prepare mid-year report and presentation. 3

27 / 28

lamed-fixed

BackgroundMethods


Timeline

Timeline (cont.)

Phase IIJanuary: Develop active sets and semi-smooth Newton method.February: Add miscibility effect.March: Incorporate energy equation for thermal effect.April

Collect the unit tests and make a test suite.Benchmark with existing codes or pursue realistic problems, such asdesiccation if time permits.

May: Prepare final report and presentation.

28 / 28

lamed-fixed

References

R. Brooks and A. Corey. Hydraulic properties of porous media. 1964.G. Chavent and J. Jaffre. Mathematical Models and Finite Elements for

Reservoir Simulation. 1978.E. Coon, J. D. Moulton, and S. Painter. Managing complexity in simulations

of land surface and near-surface processes. Technical Report LA-UR14-25386, Applied Mathematics and Plasma Physics Group, Los AlamosNational Laboratory, 2014. submitted to Environmental Modelling &Software.

M. Van Genuchten. A closed form equation for predicting the hydraulicconductivity of unsaturated soils. 25:551–564, 1980.

28 / 28

modeling multiphase flow in porous mediaide/data/teaching/amsc663/14... · 2017-06-28 ·...

Documents