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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
- Gradient method: It utilizes gradients of objective function. Gradient method takes negative gradient direction as search direction.

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-1E(Ak)

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

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

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