Mathematicals Models

MATH

EMATI

CAL

MODEL

S

Mathematical models are differential equations that represent changes in time variables such as distance, speed, strength, temperature and gravity.

Through mathematical models we can analyze, identify and represent the dynamics or movement which has an element, a mechanism or process.

MATHEMATICS

MODELS

Definition of the

Problem

Numerical or

Graphical Results

Implementation

The Model

A natural phenomenon is described by a mathematical model, usually by a differential equation. In some (simple) cases, these differential equations have exact solutions which we may use directly. In other cases, we must also solve these differential equations. In either case, we do this using algorithms implemented on a computer.

An engineer must use the most appropriate model, not the best model. For example, Einstein's general theory of relatively is the best model available to describe motion, but for some cases, Newton's laws of motion are more than sufficient, and for such cases, no employer will pay for a solution which has unnecessary precision. However, in others (for example, real-time conversion of optical Gbit/s signals to electric signals), Newton's laws no longer apply and it is necessary to use better models.

Software Implementation

The mathematical models are then implemented in a program on a computer using various algorithms. This class covers many of the introductory algorithms used at this stage.

Simulation

The executions of our simulations take measurements from the real world and produce an approximation of the real world. If we make judicious choices of the model, the algorithms used, and appropriate measurements, the approximation of the real world will be sufficiently good to satisfy the stated requirements. Errors in any of these steps can easily produce an invalid or poor representation of the real world which does not satisfy the given requirements.

SINGLE

-PHASE

TWO-

DIMEN

SIONAL

SIMULA

TOR

Imple

men

tatio

n of a

Mat

hemat

ical

Mod

el

SINGLE-PHASE TWO-DIMENSIONAL SIMULATOR

In petroleum engineering simulators are used to relate the data flow observed in the field with reservoir properties such as permeability, porosity and pressure distribution.

In addition, the primary function of a simulator is to help engineers understand the behavior of pressure and production, to predict the rates at each well in function of time.

All simulators are the reservoir as a large set of cells. Each cell corresponds to a volume of the reservoir and contains information of rock properties and fluid characteristics of the reservoir. The simulator solves the equations for each of these cells taking into account the values of permeability, porosity, viscosity, compressibility and others.

However, current simulators represent the site as a series of interconnected blocks and flow through these is solved by numerical methods.

The equations are written in the form of finite differences, as the field is seen as a succession of blocks and the production is divided into time slots. In mathematical terms, it attempts to place the problem in time and space.

SINGLE-PHASE TWO-DIMENSIONAL SIMULATOR

SIMULATION MODEL

1(3,1)

2(3,2)

3(3,3)

4(2,1)

5(2,2)

6(2,3)

7(1,1)

8(1,2)

9(1,3)

i

j

1 2 2 3

4 5 5 6

7 8 8 9i = Nxj = 1

i = Nxj > 1

j < Ny

i < Nxi > 1

j = Ny

i = 1j = Ny

i = 1j = 1

i = Nxj = Ny

i < Nxi > 1j < Nyj >1

i < Nxi > 1j = 1

i = 1j > 1

j < Ny

i < Nxi > 1

j = Ny

i < Nxi > 1j = 1

BIBLIOGRAPHY

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