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Matrices in Petroleum EngineeringUsing Tridiagonal Matrices to ModelReservoirs

Juan F. Pea

Oct. 10, 2003

Juan Pena

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Abstract:

Linear algebra is used in engineering to solve problems that

involve multiple complex equations. In the petroleum

industry, engineers must model reservoirs to find theconditions needed to maximize recovery of hydrocarbons.

Reservoir simulation involves many complex partial

differential equations, which are solved using a computer

program that models the reservoir being studied. To thepetroleum engineer it is great that the computer program

does all the simulating for him as long as all the correct

parameters are put into the program. We will look at howreservoir simulators use linear algebra to model reservoirs.

A hydrocarbon reservoir is a porous media through which hydrocarbons flow to a

well or many wells. It is simple to measure a pipes flow capacity as a function of

pressure; however, porous media flow is different because there are no certain paths

through which the fluid will travel. Reservoir models look at fluid flow through the

reservoirs by determining how saturations and pressures change as functions of time.

Nonlinear partial differential equations make up the fluid flow equations and since they

are not linear they have to be transformed into linear forms using the Taylors series. The

Taylors series makes the derivatives finite differences, which are changes in x and

changes in t. Since the reservoir model is divided into blocks or cells, there is an

equation for each cell. In a model with N cells, we get N equations with N unknowns,

which is a lot of equations and unknowns to be solved by hand. The equations can be

written in a matrix form called a tridiagonal matrix, which has three diagonal elements

and zeros in all the off-diagonal elements.

Cell1 a1Po b1P1+c1P2 =d1

2 a2P1 b2P2+c2P3 =d2

3 a3P2 b3P3+c3P4 =d3N anPn-1 bnPn+CnPn+1 =dn

Juan Pena

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AP=d

[ a1 \ \ ] [ ] [ ][ \ b1 \ ] *[p1] = [d1]

[ \ \ c1 ] [ ] [ ]

Then by using the Thomas algorithm, a modified form of Gaussian elimination,

we solve the equations for the unknown pressures P. Once the pressures are known the

results are given as functions of time. The idea behind reservoir modeling is to examine

how a reservoir will do at some future time. The engineer identifies which cells contain

wells, whether they are producers or injector. By running the simulator, the engineer gets

a table of results, which includes saturations, pressures, and amount of fluids left by the

time they are occurring. This allows an engineer to determine when in the future is a

good time to stop producing. Modeling is very important to the petroleum engineer

because there is no other way of determining how a reservoir will perform under certain

conditions. Since reservoirs last long times, it is impossible to wait to see what a

reservoir will do, so the engineers just model the reservoirs and can determine many

important conditions. Some of the conditions to a reservoir are: where to place the wells,

how many wells to have, what spacing between wells, at what rates the wells should be

produced, and how long the wells should be produced. Modeling simply makes a

petroleum engineers job easier, and without linear algebra, modeling would be very

difficult to perform.

References:Crichlow, Henry B. Modern Reservoir Engineering-A Simulation Approach. Prentice

Hall. New Jersey 1977

Fanchi, John R. Principles of Applied Reservoir Simulation. Gulf Publishing Co.

Houston, TX. 1997

Juan Pena