Direct Inverse Modeling with Stream Functions

1
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Direct Inverse Modeling with Stream
Functions Brouwer, G.K., July 16, 2007 Poster
Brouwer, G.K., W. Zijl, P. Fokker, TNO, the
Netherlands
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Contents Intro well measurements, processes,
inversion Revival of stream function Direct
inversion with the Double Constraint Constraints
(P,Q), (water cut) Example assimilating water
cut measurements Conclusions Future work
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• Well data in an oil reservoir with water
  injection
• Pressure (t), or flow potential f (t)
• Flow oil (t), or Qo (t) (flux), qo(t) (flux
  density)
• Flow water (t)
• Two dynamical processes with (very) different
  time scale
• Pressure propagation
• displacement of the phases
• Direct inversion model (Darcy)
• kvtot/(flow potential gradient) or

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l1
l2
(qydy/dx0) qx-dy/dyconst df/dx/const
k1
k2
w
Finding k in flow direction k is hidden in
f Solve by requiring S(li/ki)constant at all
times (M/1)
l
Finding k perpendicular to flow An arrival time
(y) problem !
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Solving y
Implemented in MoReS prescribed boundary
conditions
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Double constraint (Wexler, 1985, medical
application)
pressure BC (wells) gt guessed pressure field
f(x,y)
(MoReS pot. run)
posterior perm field k (x,y)
iterate
prior permeability field k (x,y)
flux BC (wells) gt guessed flow field vx, vy or
y(x,y)
(MoReS flow run)
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Constraining on P, Q
For each stream tube
j porosity DSSbf-Swc Vvolume
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Correcting on wells first arrival stream tube
travel time by changing dy/ds
Apply DC
iterate
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LINE truth Blue circles assim. of fluxes and
pressures Blue squares restored watercut
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• conclusions
• double constraint honours hard data
• Water cut ( saturation fronts from 4-D
  seismics) can be assimilated
• methodology can be applied with any forward
  simulator
• (pressures and velocities have to be available)

• recommendations
• reformulation for 3-D
• software development for 3D, multi- phase flow
• (prototype 3-D available)
• integrate with (extended) Kalman filter
• explore DC for upscaling