ESTIMATION OF OIL SATURATION

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ESTIMATION OF OIL SATURATION

• Hong Li
• Computer System Technology
• NY City College of Technology CUNY
• Ali Setoodehnia Kamal Shahrabi
• Technology Department
• Kean University

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introduction

• Estimation of oil saturation has been an
  important issue for petroleum engineer
• Collectable data includes pressure, rock type,
  depth and etc.
• Permeability and saturation are not easy to
  measure during their study of the oil fields
• Engineers attempt to determine parameters that
  produce the best match with observation.

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• Using Leverett J function to estimate initial oil
  saturation has become the problem of parameter
  estimation by applying different fitting
  functions

Fitting function
saturation
J value
Leverett J function
pressure
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Fitting fuctions

• Benson-Anli
• Brooks-Coery
• Thomeer
• O'Meara Unimodel
• OMeara Bimodel

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Assumptions

• Suppose that saturation S is function of Leverett
  J function with unknown parameters a ( a1, a2,
  , an), i.e. S S(J, a), where J function value
  is determined by capillary pressure.
• (Ji, Smi ) is a set of measured data, J function
  value and saturation

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Problem statement

• Determine parameters (ak) in fitting functions
  that produce the best match with observation, in
  the sense that minimizes an objective function
  depended on parameters (ak).
• objective function is defined as

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Numerical method

• A numerical method of optimization generally
  consists of three steps
• Choose a starting point, i.e. given initial value
  of parameters.
• Designate a way to generate a search sequence,
  A1, An, such that
• E(Ak) lt E(Ak-1)
• 3. Stipulate a convergence criterion

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Search algorithm

• The search sequence has the following general
  form Ak Ak ?k Dk
• Search method it only utilizes values of
  objective function
• Gradient method It utilizes gradients of
  objective function. Gradient method takes
  negative gradient direction as search direction.
• Dk -?E(Ak)

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Newton Method

• Newton Method It utilizes the gradient of
  objection function and Hessian matrix (second
  order derivatives of objection function with
  respect to parameters), denoted by G and set the
  search direction
• Dk -G-1 ??E(Ak)

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Advantage and disadvantage

• rapidly converge and be more robust when number
  of parameters is small
• When is not close to the minimum, is not
  necessarily positive definite

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• Given initial guess of parameters, ?, suppose
  that the first derivative of E(?) with respect
  to parameters is denoted by ?E(?) and the second
  derivative of E(?) with respect to parameters is
  called Hessian matrix, denoted by
• G ?2 E(?) / ??i ??j

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Modified Newton Method

• A descent algorithm using the Newton (or near
  Newton) direction.
• E(?) E(?0) (?- ? 0 )??E(?- ? 0 )
• (?- ? 0 )G (?- ? 0 )
• so, ?E(?) ?E(?0) G (?- ? 0 )
• Set ?E(?)0 to determined the next iteration
  point
• ? ? 0 G-1 ?E(?0)

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• For the Newton direction to be a descent
  direction, we must have that the Hessian matrix G
  be positive definite
• ? chosen to assure that G?I is invertible and
  satisfies

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Summary

• The modified Newton method
• applied the second order derivatives of the
  objective function with respect to the parameters
  
• promised convergence in computer simulation.
• Numerical analysis was driven to prove the
  problem solvability and the convergence.
• Computer simulation with collected data from oil
  field has shown improvement in convergence speed
  and estimation accuracy.  

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Future Research

• Neural Network has been widely applied in
  different fields to solve problem with parameter
  estimation
• Preliminary research was done to estimate the
  oils saturation in simplified situation.
• Prospect of neural network applied in saturation
  estimation