Reservoir Engineering 2 Course (1st Ed.)

1. Introduction

2. Classification Of Aquifers

3. Recognition Of Natural Water Influx

1. Water Influx ModelsA. The Pot Aquifer Model

B. Schilthuis’ SS Model

C. Hurst’s Modified SS Model

Aquifer uncertainties

It should be appreciated that in reservoir engineering there are more uncertainties attached to this subject than to any other. This is simply because one seldom drills wells into

an aquifer to gain the necessary information about the porosity,

permeability, thickness, and fluid properties.

Instead, these properties frequently have to be inferred from what has been observed in the reservoir.

Even more uncertain, however, is the geometry and areal continuity of the aquifer itself.

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Water influx model requirements

Several models have been developed for estimating water influx that are based on assumptions that describe

the characteristics of the aquifer.

Due to the inherent uncertainties in the aquifer characteristics, all of the proposed models require historical reservoir performance data

to evaluate constants representing aquifer property parameters since these are rarely known from exploration-development

drilling with sufficient accuracy for direct application.

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evaluation of the constants

The material balance equation can be used to determine historical water influx provided original oil in place is known from pore volume estimates.

This permits evaluation of the constants in the influx equations so that future water influx rate can be forecasted.

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mathematical water influx models

The mathematical water influx models that are commonly used in the petroleum industry include:Pot aquiferSchilthuis’ steady-stateHurst’s modified steady-stateThe van Everdingen-Hurst unsteady-stateEdge-water driveBottom-water driveThe Carter-Tracy unsteady-stateFetkovich’s methodRadial aquiferLinear aquifer

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The Pot Aquifer Model; Basis

The simplest model that can be used to estimate the water influx into a gas or oil reservoir is based on the basic definition of compressibility.

A drop in the reservoir pressure, due to the production of fluids, causes the aquifer water to expand and flow into the reservoir.

The compressibility is defined mathematically as:

ΔV = c V Δ p

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The Pot Aquifer Model; all directions water encroachmentApplying the above basic compressibility definition

to the aquifer gives:

Water influx = (aquifer compressibility) *(initial volume of water) * (pressure drop) or

We = cumulative water influx, bblcw = aquifer water compressibility, psi−1cf = aquifer rock compressibility, psi−1Wi = initial volume of water in the aquifer, bblpi = initial reservoir pressure, psip = current reservoir pressure

(pressure at oil-water contact), psi

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The Pot Aquifer Model; initial volume of waterCalculating the initial volume of water

in the aquifer requires the knowledge of aquifer dimension and properties. These, however,

are seldom measured since wells are not deliberately drilled into the aquifer to obtain such information.

For instance, if the aquifer shape is radial, then:ra = radius of the aquifer, ft

re = radius of the reservoir, ft

h = thickness of the aquifer, ft

φ = porosity of the aquifer

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The Pot Aquifer Model; fractional water encroachmentQuite often,

water does not encroach on all sides of the reservoir, or the reservoir is not circular in nature. To account for these cases, a modification must be made

in order to properly describe the flow mechanism. One of the simplest modifications is to include

the fractional encroachment angle f in the equation, to give:

the fractional encroachment angle f is defined by:

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Radial aquifer geometries

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Dake consideration

The Pot Aquifer Model is only applicable to a small aquifer, i.e., pot aquifer, whose dimensions

are of the same order of magnitude as the reservoir itself.

Dake (1978) points out that because the aquifer is considered relatively small, a pressure drop in the reservoir is instantaneously transmitted

throughout the entire reservoir-aquifer system.

Dake suggests that for large aquifers, a mathematical model is required which includes

time dependence to account for the fact that it takes a finite time for the aquifer to respond to a pressure change in the reservoir.

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Schilthuis’ Steady-State Model

Schilthuis (1936) proposed that for an aquifer that is flowing under the steady-state flow regime, the flow behavior could be described by Darcy’s equation.

The rate of water influx ew can then be determined by applying Darcy’s equation:

The above relationship can be more conveniently expressed as:

ew = rate of water influx, bbl/day

k = permeability of the aquifer, md

h = thickness of the aquifer, ft

ra = radius of the aquifer, ftre = radius of the reservoirt = time, days

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Schilthuis’ Steady-State Model; Influx constantThe parameter C is called the water influx constant

and is expressed in bbl/day/psi. This water influx constant C may be calculated

from the reservoir historical production data over a number of selected time intervals, provided that the rate of water influx ew has been determined independently from a different expression. For instance, the parameter C may be estimated by combining

the Equation with ew = Qo Bo + Qg Bg + Qw Bw.

Although the influx constant can only be obtained in this manner when the reservoir pressure stabilizes, once it has been found, it may be applied to both stabilized and changing reservoir pressures.

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Notes

If the steady-state approximation adequately describes the aquifer flow regime, the calculated water influx constant C values

will be constant over the historical period.

Note that the pressure drops contributing to influx are the cumulative pressure drops from the initial pressure.

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the cumulative water influx

In terms of the cumulative water influx We, the Equation is integrated to give the common Schilthuis expression for water influx as:

We = cumulative water influx, bbl

C = water influx constant, bbl/day/psi

t = time, days

pi = initial reservoir pressure, psi

p = pressure at the oil-water contact at time t, psi

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Calculating the area under the curve

When the pressure drop (pi − p) is plotted versus the time t, as shown in the Figure, the area under the curve represents the integral

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Calculating the cumulative water influxThis area at time t can be determined numerically

by using the trapezoidal rule (or any other numerical integration method), as:

The Equation can then be written as:

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the “apparent” aquifer radius ra

One of the problems associated with the Schilthuis’ steady-state model is that as the water is drained from the aquifer,

the aquifer drainage radius ra will increase as the time increases.

Hurst (1943) proposed that the “apparent” aquifer radius ra would increase with time and, therefore the dimensionless radius ra/re

may be replaced with a time dependent function, as:

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Hurst’s Modified Steady-State Model

Substituting the dimensionless radius into Schilthuis’ Steady-State Model gives:

The Hurst modified steady-state equation can be written in a more simplified form as:

and in terms of the cumulative water influx

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Determination of the Hurst modified steady-state constants; a and CThe Hurst modified steady-state equation contains

two unknown constants, i.e., a and C, that must be determined from

the reservoir-aquifer pressure and water influx historical data.

The procedure of determining the constants a and C is based on expressing the Equation as a linear relationship.

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Graphical determination of C and a

The Equation indicates that a plot of (pi − p)/ew versus ln(t) will be a straight line with a slope of 1/C and intercept of (1/C)ln(a), as shown schematically in the Figure.

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1. Ahmed, T. (2010). Reservoir engineering handbook (Gulf Professional Publishing). Chapter 10

1. mathematical Water Influx models;A. The van Everdingen-Hurst Unsteady-State Model

a. Edge-Water DriveI. computational steps for We at successive intervals