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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. introduction. Estimation of oil saturation has been an important issue for petroleum engineer

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Estimation of oil saturation
ESTIMATION OF OIL SATURATION

Hong Li

Computer System Technology

NY City College of Technology – CUNY

Ali Setoodehnia Kamal Shahrabi

Technology Department

Kean University


Introduction
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.


J value

Fitting

function

saturation

pressure

Leverett J

function


Fitting fuctions
Fitting fuctions has become the problem of parameter estimation by applying different fitting functions

  • Benson-Anli

  • Brooks-Coery

  • Thomeer

  • O'Meara Unimodel

  • O’Meara Bimodel


Assumptions
Assumptions has become the problem of parameter estimation by applying different fitting functions

  • 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


Problem statement
Problem statement has become the problem of parameter estimation by applying different fitting functions

  • 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


Numerical method
Numerical method has become the problem of parameter estimation by applying different fitting functions

  • 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) < E(Ak-1)

      3. Stipulate a convergence criterion


Search algorithm
Search algorithm has become the problem of parameter estimation by applying different fitting functions

  • The search sequence has the following general form: Ak = Ak +λk 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)


Newton method
Newton Method has become the problem of parameter estimation by applying different fitting functions

  • 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)


Advantage and disadvantage
Advantage and disadvantage has become the problem of parameter estimation by applying different fitting functions

  • rapidly converge and be more robust when number of parameters is small

  • When is not close to the minimum, is not necessarily positive definite


  • Given initial guess of parameters, has become the problem of parameter estimation by applying different fitting functions, 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


Modified newton method
Modified Newton Method has become the problem of parameter estimation by applying different fitting functions

  • 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)



Summary
Summary have that the Hessian matrix G be positive definite

  • 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.  


Future research
Future Research have that the Hessian matrix G be positive definite

  • 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


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