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

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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
• 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.
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
• Benson-Anli
• Brooks-Coery
• Thomeer
• O\'Meara Unimodel
• O’Meara Bimodel
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
• 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
• 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
• The search sequence has the following general form: Ak = Ak +λk Dk
• Search method: it only utilizes values of objective function

Dk = -E(Ak)

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)

• 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, , 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
• 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)
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
• 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
• 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