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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.
slide3
Using Leverett J function to estimate initial oil saturation has become the problem of parameter estimation by applying different fitting functions

J value

Fitting

function

saturation

pressure

Leverett J

function

fitting fuctions
Fitting fuctions
  • Benson-Anli
  • Brooks-Coery
  • Thomeer
  • O\'Meara Unimodel
  • O’Meara Bimodel
assumptions
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
problem statement
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
numerical method
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) < E(Ak-1)

3. Stipulate a convergence criterion

search algorithm
Search algorithm
  • 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
  • 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
  • rapidly converge and be more robust when number of parameters is small
  • When is not close to the minimum, is not necessarily positive definite
slide11
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
modified newton method
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)
slide13
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
summary
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.  
future research
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
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