Q913 re1 w4 lec 15

Reservoir Engineering 1 Course (1st Ed.)

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http://www.about.me/AlamiNia

1. Future IPR Approximation

2. Generating IPR for Oil WellsA. Wiggins’ Method

B. Standing’s Method

C. Fetkovich’s Method

3. Horizontal Oil Well Performance

4. Horizontal Well Productivity

1. Vertical Gas Well Performance

2. Pressure Application Regions

3. Turbulent Flow in Gas WellsA. Simplified Treatment Approach

B. Laminar-Inertial-Turbulent (LIT) Approach (Cases A. & B.)

IPR for Gas Wells

Determination of the flow capacity of a gas well requires a relationship between the inflow gas rate and the sand-face pressure or flowing bottom-hole pressure.This inflow performance relationship may be established

by the proper solution of Darcy’s equation. Solution of Darcy’s Law depends on the conditions of the flow

existing in the reservoir or the flow regime.

2013 H. AlamiNia Reservoir Engineering 1 Course: Gas Well Performance 5

Gas Reservoir Flow Regimes

When a gas well is first produced after being shut-in for a period of time, the gas flow in the reservoir follows an unsteady-state behavior until the pressure drops at the drainage boundary of the well. Then the flow behavior passes through a short transition

period, after which it attains a steady state or semisteady (pseudosteady)-state condition. The objective of this lecture is to describe the empirical as well

as analytical expressions that can be used to establish the inflow performance relationships under the pseudosteady-state flow condition.


Exact Solution of Darcy’s Equation for Compressible Fluids under PSSThe exact solution to

the differential form of Darcy’s equation for compressible fluids under the pseudosteady-state flow condition was given previously by:

Where Qg = gas flow rate,

Mscf/dayk = permeability, mdψ–r = average reservoir

real gas pseudo-pressure, psi2/cp

T = temperature, °Rs = skin factorh = thicknessre = drainage radiusrw = wellbore radius


Productivity Index for a Gas Well

The productivity index J for a gas well can be written analogous to that for oil wells as:

With the absolute open flow potential (AOF), i.e., maximum gas flow rate (Qg)max, as calculated by:

Where J = productivity index, Mscf/day/psi2/cp


Steady-State Gas Well Flow

In a linear relationship as:

Above Equation indicates that a plot of ψwf vs. Qg would produce a straight line with a slope of (1/J) and intercept of ψ–r, as shown in next slide.

If two different stabilized flow rates are available, the line can be extrapolated and the slope is determined to estimate AOF, J, and ψ–r.


Steady-State Gas Well Flow (Cont.)


Darcy’s Equation for Compressible Fluids under PSS RegimeDarcy’s equation for compressible fluids under the

PSS regime can be alternatively written in the following integral form:

Note that (p/μg z) is directly proportional to (1/μg Bg) where Bg is the gas formation volume factor and defined as:


Typical Plot of the Gas Pressure Functions vs. PFigure shows a

typical plot of the gas pressure functions (2p/μgz) and (1/μg Bg) versus pressure.

The integral in previous equations represents the area under the curve between p–r and pwf.

Gas PVT data2013 H. AlamiNia Reservoir Engineering 1 Course: Gas Well Performance 13

Pressure Regions

As illustrated in Figure, the pressure function exhibits the following three distinct pressure application regions:Region III. High-Pressure Region

Region II. Intermediate-Pressure Region

Region I. Low-Pressure Region


Region III. High-Pressure Region

When both pwf and p–r are higher than 3000 psi, the pressure functions (2p/μgz) and (1/μg Bg) are nearly constants.

This observation suggests that the pressure term (1/μg Bg) in Equation can be treated as a constant and removed outside the integral, to give the following approximation to Equation:

Where Qg = gas flow rate, Mscf/dayBg = gas formation volume factor, bbl/scfk = permeability, md


Region III. High-Pressure Region, P MethodThe gas viscosity μg and formation volume factor Bg should

be evaluated at the average pressure pavg as given by:

The method of determining the gas flow rate by using below Equation commonly called the pressure-approximation method.

It should be pointed out the concept of the productivity index J cannot be introduced into above Equation since it is only valid for applications when both pwf and p–r are above 3000 psi.


Region II. Intermediate-Pressure RegionBetween 2000 and 3000 psi, the pressure function

shows distinct curvature.

When the bottom-hole flowing pressure and average reservoir pressure are both between 2000 and 3000 psi, the pseudopressure gas pressure approach (i.e., below Equation) should be used to calculate the gas flow rate.


Region I. Low -Pressure Region, P2 MethodAt low pressures, usually less than 2000 psi, the

pressure functions (2p/μgz) and (1/μg Bg) exhibit a linear relationship with pressure. Golan and Whitson (1986) indicated that the product

(μgz) is essentially constant when evaluating any pressure below 2000 psi.


Region I. Low -Pressure Region,P2 Method (Cont.)

Implementing above observation gives (pressure-squared approximation method):

Where Qg = gas flow rate, Mscf/dayk = permeability, mdT = temperature, °Rz = gas compressibility factorμg = gas viscosity, cp

It is recommended that the z-factor and gas viscosity be evaluated at the average pressure pavg as defined by:


Region I. J Calculation

If both p–r and pwf are lower than 2000 psi, the equation can be expressed in terms of the productivity index J as:

With

Where


Laminar Vs. Turbulent Flow

All of the mathematical formulations presented thus far in this lecture are based on the assumption that laminar (viscous) flow conditions are observed during the gas flow. During radial flow, the flow velocity increases as the

wellbore is approached. This increase of the gas velocity might cause the development

of a turbulent flow around the wellbore.

If turbulent flow does exist, it causes an additional pressure drop similar to that caused by the mechanical skin effect.


PSS Equations Modification (Turbulent Flow)As presented earlier, the semisteady-state flow

equation for compressible fluids can be modified to account for the additional pressure drop due the turbulent flow by including the rate-dependent skin factor DQg.

The resulting pseudosteady-state equations are given in the following three forms:


PSS Equations Modification (Turbulent Flow) (Cont.)

First Form: Pressure-Squared Approximation Form

Second Form: Pressure-Approximation Form

Third Form: Real Gas Potential (Pseudopressure) Form


Empirical Treatments to Represent the Turbulent Flow in Gas WellsThe PSS equations, which were given previously in

three forms, are essentially quadratic relationships in Qg and, thus, they do not represent explicit expressions for calculating the gas flow rate. Two separate empirical treatments can be used to

represent the turbulent flow problem in gas wells.


Empirical Treatments to Represent the Turbulent Flow in Gas Wells (Cont.)

Both treatments, with varying degrees of approximation, are directly derived and formulated from the three forms of the pseudosteady-state equations. These two treatments are called:Simplified treatment approach

Laminar-inertial-turbulent (LIT) treatment


The Simplified Treatment Approach

Based on the analysis for flow data obtained from a large member of gas wells, Rawlins and Schellhardt (1936) postulated that the relationship between the gas flow rate and pressure can be expressed as:

Where Qg = gas flow rate, Mscf/day

p –r = average reservoir pressure, psi

n = exponent

C = performance coefficient, Mscf/day/psi2

The exponent n is intended to account for the additional pressure drop caused by the high-velocity gas flow, i.e., turbulence. Depending on the flowing conditions, the exponent n may vary from

1.0 for completely laminar flow to 0.5 for fully turbulent flow.


Deliverability or Back-Pressure Equation

The performance coefficient C in the equation is included to account for:Reservoir rock propertiesFluid propertiesReservoir flow geometry

The Equation is commonly called the deliverability or back-pressure equation. If the coefficients of the equation (i.e., n and C) can be

determined, the gas flow rate Qg at any bottom-hole flow pressure pwf can be calculated and the IPR curve constructed.


Deliverability or Back-Pressure Equation (Logarithmic Form)Taking the logarithm of both sides of the Equation

gives:

This equation suggests that a plot of Qg versus (p–r2 − p2wf) on log-log scales should yield a straight line having a slope of n.

In the natural gas industry the plot is traditionally reversed by plotting (p–r2 − p2wf) versus Qg on the logarithmic scales to produce a straight line with a slope of (1/n).


Well Deliverability Graph or the Back-Pressure PlotThis plot as

shown schematically in Figure is commonly referred to asthe

deliverability graph or

the back-pressure plot.


Calculation of N & C

The deliverability exponent n can be determined from any two points on the straight line, i.e., (Qg1, Δp12) and (Qg2, Δp22), according to the flowing expression:

Given n, any point on the straight line can be used to compute the performance coefficient C from:


Gas Well Testing

The coefficients of the back-pressure equation or any of the other empirical equations are traditionally determined from analyzing gas well testing data.

Deliverability testing has been used for more than sixty years by the petroleum industry to characterize and determine the flow potential of gas wells. There are essentially three types of deliverability tests

and these are:Conventional deliverability (back-pressure) testIsochronal testModified isochronal test


Deliverability Testing

These tests basically consist of flowing wells at multiple rates and measuring the bottom-hole flowing pressure as a function of time. When the recorded data are properly analyzed, it is

possible to determine the flow potential and establish the inflow performance relationships of the gas well.

The deliverability test is out of scope of this course and would be discussed later in well test course.


The Laminar-Inertial-Turbulent (LIT) ApproachThe three forms of the semisteady-state equation

as presented earlier in this lecture can be rearranged in quadratic forms for separating the laminar and

inertial-turbulent terms composing these equations as follows:


Case A. Pressure-Squared Quadratic Forma. Pressure-Squared

Quadratic Form

With

Where a = laminar flow coefficientb = inertial-turbulent flow

coefficientQg = gas flow rate, Mscf/dayz = gas deviation factork = permeability, mdμg = gas viscosity, cp

The term (a Qg) in represents the pressure-squared drop due to laminar flow while the term (b Q2g) accounts for the pressure squared drop due to inertial-turbulent flow effects.


Case A. Graph of the Pressure-Squared DataAbove equation can be linearized by dividing both

sides of the equation by Qg to yield:

The coefficients a and b can be determined by plotting ((p–r^2-pwf^2)/Qg) versus Qg on a Cartesian scale and should yield a straight line with a slope of b and intercept of a.

Data from deliverability tests can be used to construct the linear relationship.


Graph of the pressure-squared data


Case A. Current IPR of the Gas Well

Given the values of a and b, the quadratic flow equation, can be solved for Qg at any pwf from:

Furthermore, by assuming various values of pwf and calculating the corresponding Qg from above Equation, The current IPR of the gas well at the current reservoir

pressure p–r can be generated.


Case A. Pressure-Squared Quadratic Form Assumptions It should be pointed out the following assumptions

were made in developing following Equation:

Single phase flow in the reservoir

Homogeneous and isotropic reservoir system

Permeability is independent of pressure

The product of the gas viscosity and compressibility factor, i.e., (μg z) is constant.

This method is recommended for applications at pressures below 2000 psi.


Case B. Pressure-Quadratic Form

The pressure-approximation equation, i.e., can be rearranged and expressed in the following quadratic form.

The term (a1 Qg) represents the pressure drop due to laminar flow, while the term (b1 Q2 g) accounts for the additional pressure drop due to the turbulent flow condition. In a linear form, the equation can be expressed as:


Case B. Graph of the Pressure-Method DataThe laminar

flow coefficient a1 and inertial-turbulent flow coefficient b1 can be determined from the linear plot of the equation as shown in Figure.


Case B. Gas Flow Rate Determination

Having determined the coefficient a1 and b1, the gas flow rate can be determined at any pressure from:

The application of following Equation is also restricted by the assumptions listed for the pressure-squared approach.

However, the pressure method is applicable at pressures higher than 3000 psi.


1. Ahmed, T. (2006). Reservoir engineering handbook (Gulf Professional Publishing). Ch8

1. Turbulent Flow in Gas Wells: LIT Approach (Case C)

2. Comparison of Different IPR Calculation Methods

3. Future IPR for Gas Wells

4. Horizontal Gas Well Performance

5. Primary Recovery Mechanisms

6. Basic Driving Mechanisms

Q913 re1 w4 lec 15

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