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{  BLACK_OIL.PDE  

 

 This example considers the transport of oil and water in soil.

 The model is given in Gelinas, et al, "Adaptive Forward-Inverse Modeling

 of Reservoir Fluids Away from Wellbores",  (Lawrence Livermore National

 Laboratory report UCRL-ID-126377) and in Saad & Zhang, " Adaptive Mesh for

 Two-Phase Flow in Porous Media" (in Recent Advances in Problems of Flow and

 Transport in Porous Media, Crolet and El Hatri, eds., Kluwer Academic Publishers,

 Boston, 1998).

 The saturation of water is represented by S, with the saturation of oil defined

 as 1-S.  The relative permeabilities of water and oil are assumed to be S^2 and

 (1-S)^2, respecitvely.  The total mobility M is defined as

       M = S^2/muw + (1-S)^2/muo,

 where muw and muo are the viscosities of water and oil.

 The total velocity, V, and the fractional flux, f, are defined as

       V = - K*M grad(P)

       f = [S^2/muw]/M,

 where K represents the saturation-independent permeability coefficient, and

 P is the pressure, assuming capillary to be zero and oil and water pressures

 equal.

 If the porosity Phi is taken as constant and gravity effects are negligible, the

 PDE's governing the system reduce to

       Phi*dt(S) + div(V*f) = 0

       div(V) = 0.

 Here we study the flow through a 30-meter box with an inlet pipe in the upper

 left and   an outlet pipe in the lower right.  The box is initially filled with oil,

 and water is pumped into the inlet pipe at a constant pressure.  Time is measured

 in seconds.

        --  Submitted by Said Doss, Lawrence Livermore National Laboratory.

}

 

TITLE 'Black Oil Model'

 

SELECT

     smoothinit   { Smooth the initial conditions a little, to minimize

                       the time wasted tracking the initial discontinuity }

 

VARIABLES

     s(1), p         { Saturation and Pressure }

 

DEFINITIONS

     muo = 4.e-3               { oil viscosity }

     muw = 1.e-3               { water viscosity }

     K = 1.e-12               { Saturation-independent permeability coefficient }

     Pin = 1.5e6               { Inlet pressure }

     Pout = 1.e6               { Outlet pressure }

     M = S^2/muw + (1-S)^2/muo { Total mobility }

     f = S^2/muw/M             { Fractional flux }

     krw = S^2/muw             { Relative permeability of water }

     phi =.206                 { porosity }

 

     xmax = 30                 { Box dimensions }

     ymax = xmax

     out_ctr = 8

     tfrac = 2*out_ctr

     diam = 2

     in_ctr = ymax-out_ctr

     rad = diam/5

     pipe = 2*rad         {an extended inlet and outlet pipe}

 

     epsvisc = 1.e-6       { A little artificial diffusion helps smooth the solution }

 

     sint = integral(s)   { the total extraction integral }

 

     hour = 60*60

     day = hour*24         { seconds per day }

 

INITIAL VALUES

      s = max(0,-x/pipe )         { start with all oil , but ramp the value in the inlet pipe to speed startup }

      p = Pin + (Pout-Pin)*x/xmax { start with a rough approximation to the pressure }

 

EQUATIONS

      s:  phi*dt(s) - div(K*krw*grad(p)) - epsvisc*div(grad(s)) = 0

      p:  div(K*M*grad(p)) = 0

 

MONITORS

    for cycle=5

      contour(s) as "Saturation" range(0,1)

      contour(s) zoom(xmax-tfrac+pipe,0, tfrac,tfrac) as "Outflow Saturation"

          range(0,1)

      contour(p) as "Pressure"

      vector(-K*M*grad(p)) norm as "Flow Velocity"

 

PLOTS

  for t = day by day to 20*day

              by 10*day to 120*day

      grid(x,y)

      contour(s) as "Saturation" range(0,1) painted

      surface(s) as "Saturation" range(0,1) painted viewpoint(60,-120,30)

      contour(s) zoom(xmax-tfrac+pipe,0,  tfrac,tfrac) as "Outflow Saturation"

          range(0,1)  painted

      contour(p) as "Pressure" painted

      vector(-K*M*grad(p)) norm as "Flow Velocity"

      contour(K*M*magnitude(grad(p))) norm as "Flow Speed" painted

 

HISTORIES

      history(sint) at (0,0) as "Extraction"

 

END