B dimensionless pressure tj = cumulative elapsed time

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

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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).

 

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