JutulDarcy.jl

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Buckley-Leverett two-phase problem ​

The Buckley-Leverett test problem is a classical reservoir simulation benchmark that demonstrates the nonlinear displacement process of a viscous fluid being displaced by a less viscous fluid, typically taken to be water displacing oil.

Problem definition ​

This is a simple model without wells, where the flow is driven by a simple source term and a simple constant pressure boundary condition at the outlet. We define a function that sets up a two-phase system, a simple 1D domain and replaces the default relative permeability functions with quadratic functions:

krα(S)=min(S−Sr1−Sr,1)n,Sr=0.2,n=2

In addition, the phase viscosities are treated as constant parameters of 1 and 5 centipoise for the displacing and resident fluids, respectively.

The function is parametrized on the number of cells and the number of time-steps used to solve the model. This function, since it uses a relatively simple setup without wells, uses the Jutul functions directly.

julia
using JutulDarcy, Jutul
function solve_bl(;nc = 100, time = 1.0, nstep = nc)
    T = time
    tstep = repeat([T/nstep], nstep)
    domain = get_1d_reservoir(nc)
    nc = number_of_cells(domain)
    timesteps = tstep*3600*24
    bar = 1e5
    p0 = 100*bar
    sys = ImmiscibleSystem((LiquidPhase(), VaporPhase()))
    model = SimulationModel(domain, sys)
    kr = BrooksCoreyRelativePermeabilities(sys, [2.0, 2.0], [0.2, 0.2])
    replace_variables!(model, RelativePermeabilities = kr)
    tot_time = sum(timesteps)
    pv = pore_volume(domain)
    irate = 500*sum(pv)/tot_time
    src  = SourceTerm(1, irate, fractional_flow = [1.0, 0.0])
    bc = FlowBoundaryCondition(nc, p0/2)
    forces = setup_forces(model, sources = src, bc = bc)
    parameters = setup_parameters(model, PhaseViscosities = [1e-3, 5e-3]) # 1 and 5 cP
    state0 = setup_state(model, Pressure = p0, Saturations = [0.0, 1.0])
    states, report = simulate(state0, model, timesteps,
        forces = forces, parameters = parameters)
    return states, model, report
end
solve_bl (generic function with 1 method)

Run the base case ​

We solve a small model with 100 cells and 100 steps to serve as the baseline.

julia
n, n_f = 100, 1000
states, model, report = solve_bl(nc = n)
print_stats(report)
Jutul: Simulating 1 day as 100 report steps
╭────────────────┬───────────┬───────────────┬──────────╮
│ Iteration type │  Avg/step │  Avg/ministep │    Total │
│                │ 100 steps │ 100 ministeps │ (wasted) │
├────────────────┼───────────┼───────────────┼──────────┤
│ Newton         │      3.31 │          3.31 │  331 (0) │
│ Linearization  │      4.31 │          4.31 │  431 (0) │
│ Linear solver  │      3.31 │          3.31 │  331 (0) │
│ Precond apply  │       0.0 │           0.0 │    0 (0) │
╰────────────────┴───────────┴───────────────┴──────────╯
╭───────────────┬────────┬────────────┬────────╮
│ Timing type   │   Each │   Relative │  Total │
│               │     ms │ Percentage │      s │
├───────────────┼────────┼────────────┼────────┤
│ Properties    │ 0.0145 │     0.16 % │ 0.0048 │
│ Equations     │ 0.3180 │     4.53 % │ 0.1371 │
│ Assembly      │ 0.0069 │     0.10 % │ 0.0030 │
│ Linear solve  │ 3.3754 │    36.90 % │ 1.1173 │
│ Linear setup  │ 0.0000 │     0.00 % │ 0.0000 │
│ Precond apply │ 0.0000 │     0.00 % │ 0.0000 │
│ Update        │ 0.2724 │     2.98 % │ 0.0902 │
│ Convergence   │ 0.9084 │    12.93 % │ 0.3915 │
│ Input/Output  │ 0.3121 │     1.03 % │ 0.0312 │
│ Other         │ 3.7859 │    41.38 % │ 1.2531 │
├───────────────┼────────┼────────────┼────────┤
│ Total         │ 9.1484 │   100.00 % │ 3.0281 │
╰───────────────┴────────┴────────────┴────────╯
╭────────────────┬───────────┬───────────────┬──────────╮
│ Iteration type │  Avg/step │  Avg/ministep │    Total │
│                │ 100 steps │ 100 ministeps │ (wasted) │
├────────────────┼───────────┼───────────────┼──────────┤
│ Newton         │      3.31 │          3.31 │  331 (0) │
│ Linearization  │      4.31 │          4.31 │  431 (0) │
│ Linear solver  │      3.31 │          3.31 │  331 (0) │
│ Precond apply  │       0.0 │           0.0 │    0 (0) │
╰────────────────┴───────────┴───────────────┴──────────╯
╭───────────────┬────────┬────────────┬────────╮
│ Timing type   │   Each │   Relative │  Total │
│               │     ms │ Percentage │      s │
├───────────────┼────────┼────────────┼────────┤
│ Properties    │ 0.0145 │     0.16 % │ 0.0048 │
│ Equations     │ 0.3180 │     4.53 % │ 0.1371 │
│ Assembly      │ 0.0069 │     0.10 % │ 0.0030 │
│ Linear solve  │ 3.3754 │    36.90 % │ 1.1173 │
│ Linear setup  │ 0.0000 │     0.00 % │ 0.0000 │
│ Precond apply │ 0.0000 │     0.00 % │ 0.0000 │
│ Update        │ 0.2724 │     2.98 % │ 0.0902 │
│ Convergence   │ 0.9084 │    12.93 % │ 0.3915 │
│ Input/Output  │ 0.3121 │     1.03 % │ 0.0312 │
│ Other         │ 3.7859 │    41.38 % │ 1.2531 │
├───────────────┼────────┼────────────┼────────┤
│ Total         │ 9.1484 │   100.00 % │ 3.0281 │
╰───────────────┴────────┴────────────┴────────╯

Run refined version (1000 cells, 1000 steps) ​

Using a grid with 100 cells will not yield a fully converged solution. We can increase the number of cells at the cost of increasing the runtime a bit. Note that most of the time is spent in the linear solver, which uses a direct sparse LU factorization by default. For larger problems it is recommended to use an iterative solver. The high-level interface used in later examples automatically sets up an iterative solver with the appropriate preconditioner.

julia
states_refined, _, report_refined = solve_bl(nc = n_f);
print_stats(report_refined)
Jutul: Simulating 23 hours, 60 minutes as 1000 report steps
╭────────────────┬────────────┬────────────────┬──────────╮
│ Iteration type │   Avg/step │   Avg/ministep │    Total │
│                │ 1000 steps │ 1000 ministeps │ (wasted) │
├────────────────┼────────────┼────────────────┼──────────┤
│ Newton         │      3.265 │          3.265 │ 3265 (0) │
│ Linearization  │      4.265 │          4.265 │ 4265 (0) │
│ Linear solver  │      3.265 │          3.265 │ 3265 (0) │
│ Precond apply  │        0.0 │            0.0 │    0 (0) │
╰────────────────┴────────────┴────────────────┴──────────╯
╭───────────────┬────────┬────────────┬────────╮
│ Timing type   │   Each │   Relative │  Total │
│               │     ms │ Percentage │      s │
├───────────────┼────────┼────────────┼────────┤
│ Properties    │ 0.0832 │     3.46 % │ 0.2717 │
│ Equations     │ 0.0660 │     3.59 % │ 0.2814 │
│ Assembly      │ 0.0431 │     2.34 % │ 0.1839 │
│ Linear solve  │ 2.0755 │    86.36 % │ 6.7766 │
│ Linear setup  │ 0.0000 │     0.00 % │ 0.0000 │
│ Precond apply │ 0.0000 │     0.00 % │ 0.0000 │
│ Update        │ 0.0256 │     1.06 % │ 0.0835 │
│ Convergence   │ 0.0182 │     0.99 % │ 0.0776 │
│ Input/Output  │ 0.0344 │     0.44 % │ 0.0344 │
│ Other         │ 0.0422 │     1.76 % │ 0.1377 │
├───────────────┼────────┼────────────┼────────┤
│ Total         │ 2.4033 │   100.00 % │ 7.8468 │
╰───────────────┴────────┴────────────┴────────╯
╭────────────────┬────────────┬────────────────┬──────────╮
│ Iteration type │   Avg/step │   Avg/ministep │    Total │
│                │ 1000 steps │ 1000 ministeps │ (wasted) │
├────────────────┼────────────┼────────────────┼──────────┤
│ Newton         │      3.265 │          3.265 │ 3265 (0) │
│ Linearization  │      4.265 │          4.265 │ 4265 (0) │
│ Linear solver  │      3.265 │          3.265 │ 3265 (0) │
│ Precond apply  │        0.0 │            0.0 │    0 (0) │
╰────────────────┴────────────┴────────────────┴──────────╯
╭───────────────┬────────┬────────────┬────────╮
│ Timing type   │   Each │   Relative │  Total │
│               │     ms │ Percentage │      s │
├───────────────┼────────┼────────────┼────────┤
│ Properties    │ 0.0832 │     3.46 % │ 0.2717 │
│ Equations     │ 0.0660 │     3.59 % │ 0.2814 │
│ Assembly      │ 0.0431 │     2.34 % │ 0.1839 │
│ Linear solve  │ 2.0755 │    86.36 % │ 6.7766 │
│ Linear setup  │ 0.0000 │     0.00 % │ 0.0000 │
│ Precond apply │ 0.0000 │     0.00 % │ 0.0000 │
│ Update        │ 0.0256 │     1.06 % │ 0.0835 │
│ Convergence   │ 0.0182 │     0.99 % │ 0.0776 │
│ Input/Output  │ 0.0344 │     0.44 % │ 0.0344 │
│ Other         │ 0.0422 │     1.76 % │ 0.1377 │
├───────────────┼────────┼────────────┼────────┤
│ Total         │ 2.4033 │   100.00 % │ 7.8468 │
╰───────────────┴────────┴────────────┴────────╯

Plot results ​

We plot the saturation front for the base case at different times together with the final solution for the refined model. In this case, refining the grid by a factor 10 gave us significantly less smearing of the trailing front.

julia
using GLMakie
x = range(0, stop = 1, length = n)
x_f = range(0, stop = 1, length = n_f)
f = Figure()
ax = Axis(f[1, 1], ylabel = "Saturation", title = "Buckley-Leverett displacement")
for i in 1:6:length(states)
    lines!(ax, x, states[i][:Saturations][1, :], color = :darkgray)
end
lines!(ax, x_f, states_refined[end][:Saturations][1, :], color = :red)
f

Example on GitHub ​

If you would like to run this example yourself, it can be downloaded from the JutulDarcy.jl GitHub repository as a script, or as a Jupyter Notebook

This example took 15.970120069 seconds to complete.

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