B aquifer constant, bbl/psi

Cr rock compressibility, psi – 1

Cw water compressibility, psi -I

Cwr effective

compressibility of water and rock in aquifer

f fraction of perimeter of circle that original

oil/water boundary constitutes

rO radius

to perimeter of reservoir, ft

t = time, days

tD = dimensionless time

tj = cumulative elapsed time at end of the interval, days

We = cumulative water influx, bbl

=

aquifer porosity

?p

= pressure drop at OWC

re = radius to perimeter of aquifer

PD = dimensionless pressure

PD’

= first derivative of

dimensionless pressure

tj

=

cumulative elapsed time at end of jth interval, days

Water Influx

Modelling using The van Everdingen-Hurst unsteady-state Model

In 1949, one of the most significant solution for the water

influx problem was established by van Everdingen and Hurst.

Klins

et al, (1988), they have developed mathematically the solutions to the radial

diffusivity equation for radial, unsteady state and single phase flow with

respect to the pressure dispersed in water aquifer.

In addition, Utilizing van Everdingen-Hurst

solution to the diffusivity equation, as mentioned below, this is in case of

water encroachment into the reservoir for radial aquifers.

We(tDj)=BqD (tDj- tDk)

Where:

B=1.119hCwr ro2

f

?Pk=

tDJ=

The equations for hydrocarbon flow

system into a wellbore which had expressed mathematically is the same as

expressed for those equations to define the flow of water from an aquifer into

a cylindrical reservoir.

For instance, if a specific well is

producing at a constant flow rate (q) after a shut-in period, the pressure

behavior is primarily controlled by the unsteady state flowing behavior. This

flowing behavior expressed as the time period during which the boundary has no

effect on the pressure behavior.

Van Everdingen and Hurst’s constant-terminal pressure

solution, that has observed a significant value problems related to the water-encroachment. If some average pressure is specified at the

interface over a given time, flow rate and hence water influx into the

reservoir can be estimated. However, in case if pressure continues to drop at

the oil/water contact (OWC) over time, a number of constant-pressure steps can

replace this declining pressure and superposition can be used.

In addition, the diffusivity

equation which is considered as the dimensionless form is essentially the

general equation that is designed to model the unsteady state flow behavior in

reservoirs.

The

van Everdingen-Hurst style and Carter-Tracy alteration gives the rigorous

solutions to the radiaI-diffusivity equation. In the other hand, the application

of these solutions depends on the correct values of either dimensionless

pressure function which is the (PD) or the dimensionless rate influence

function which is mentioned as (qD)

Anyway, those Values of (PD) and (qD) are in general presents

and derived from tables provided in the official results and study of van

Everdingen and Hurst.

·

Fetcovish,

M. (1971). A simplified approach to water influx calculations. Journal of Petroleum Technology.

·

Klins, M.

and Bouchard, A. (1988). A Polynomial Approach to the van Everdingen-Hurst

Dimensionless Variables for Water Encroachment. Society of Petroleum Engineering.

Figure 1. Water influx into a

cylindrical reservoir.

However, in case of the constant terminal

rate boundary, the rate of water encroachment is considered as a constant for that

given period and the pressure drop is calculated at the reservoir-aquifer

boundary and then the water influx rate is determined. In the expression and explanation

of the water encroachment from the water aquifer to the reservoir, there is significant

chance that we can determine and calculate the encroachment rate rather than

the pressure.

Hurst and later Carter tried to

utilize van

Everdingen Hurst constant-terminal-rate solution to enhance a new method and develop

it so that to approach to analyze water encroachment to the reservoir that

eliminated the superposition. Eventually which is estimated by this equation below:

In Addition, A whole and exact set to

exchange the van Everdingen and Hurst tables adequately for both the terminal-pressure

and terminal-rate, in result the radial flow

cases finite and infinite aquifers is then applied.

In case of the infinite aquifers, the value of (qD)

as a function of dimensionless time is determined and calculated by the

integral which is mentioned below:

However,

in case of finite aquifers acting infinitely. It is obvious that all aquifers will

present as like they are infinite for tiny values of dimensionless time. Then for

the next time and during the times, the boundary affects will be observed

gradually in time and finite aquifer actions strays consequently after that.

The benefit

of this section is to estimate for a given aquifer ratio the time at which the boundary

affects are observed and known. Also once

this crossover value of tD is predicted and known then after that the engineer

then can decide whether the case will be finite or infinite.

Conclusion:

These

simple equations mentioned values of PD and qD as exact as the

original van Everdingen and Hurst tables. However, for water encroachment process,

these equations means and will represent the controllable replacement to

tabular guides for the van Everdingen and Hurst dimensionless functions.

Anyway, the van Everdingen-Hurst style and Carter-Tracy alteration gives the rigorous

solutions to the radiaI-diffusivity equation. In the other hand, the application

of these solutions depends on the correct values of either dimensionless

pressure function which is the (PD) or the dimensionless rate influence

function which is mentioned as (qD).