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Reservoir Engineering II, PDB(3023)

Ch.1: Unsteady State Flow

Derivation of the Diffusivity Equation Dr. Mohammed Abdalla Ayoub U n iv ers iti Tek n ologi P E T RO N A S , Ba n d a r S e r i I s k anda r, 3 1 7 50 Tr o n oh , P e r a k, Ma la ys ia | Te l:+ 6 05 3 6 8 7 0 8 6 | F a x :+ 605 3 6 5 5 6 7 0 E - ma il : ab d alla.ayoub@utp. edu. my

Learning objectives CO1: Apply concepts of fluid flow in porous media (unsteady state) to predict behavior of reservoir fluid flow LO: Derive the Diffusivity Equation To apply constant terminal pressure solution (CTP) To apply constant terminal rate solution (CTR)

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Contents Pressure distribution as a function of time Transient (unsteady-state) flow Derivation of the Diffusivity Equation

Radial Flow of Slightly Compressible Fluids Solutions to the Diffusivity Equation 3

Fluid Flow Equations The fluid flow equations that are used to describe the flow behavior in a reservoir can take many forms depending upon the combination of variables presented previously, (i.e., types of flow, types of fluids, etc.).

By combining the conservation of mass equation with the transport equation (Darcy’s equation) and various equationsof-state, the necessary flow equations can be developed.

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Flow Regimes There are basically three types of flow regimes that must be recognized in order to describe the fluid flow behavior and reservoir pressure distribution as a function of time:

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Steady-state flow 2

Unsteady-state flow 3

Pseudosteady-state flow 5

Steady-state Flow This type of flow occurs when the pressure at every location in the reservoir remains constant (does not change with time)

In reservoirs, the steady-state flow condition can only occur when the reservoir is completely recharged and supported by strong aquifer or pressure maintenance operations. 6

Unsteady / Transient State Flow This type of flow is defined as the fluid flowing condition at which the rate of change of pressure with respect to time at any position in the reservoir is not zero or constant. The rate of travel of the pressure transient is proportional to a property of the reservoir

As long as the leading front of the pressure transient has not reached the outer boundaries of the reservoir, fluid flow toward the well is in the transient state, and the reservoir acts as if it is infinite in size. During the transient flow phase, pressure distribution in the reservoir is not constant and depends on time and distance from the well. 7

Pseudosteady-state Flow The flow is achieved in the reservoir after the pressure transient has reached the outer boundaries of the reservoir and the pressure distribution at every position is changing at a constant rate over time. The time for starting of PSS flow is given by:

It should be pointed out that the pseudosteady-state flow is commonly referred to as semisteady-state flow and quasi-steady-state flow. 8

A schematic comparison of the pressure declines as a function of time of the three flow regimes

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Diffusivity Equation Diffusion is a process by which there is a net flow of matter from a region of high concentration to a region of low concentration

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Pressure distribution as a function of time Pi

Pi

r3

r3

r2

r2

r1

r1

r1

r2

r1

r2

Pi

Pi

dp/dt = f(r,t)

Pi

Constant Flow rate

Pi

Constant Bottom hole Pressure. Pwf

r3

r3

Pressure disturbance as a function of time 11

Pressure distribution as a function of time cont’d The pressure at the wellbore, i.e., Pwf, will drop instantaneously as the well is opened.

The pressure disturbance will move away from the wellbore at a rate that is determined by: ◦ ◦ ◦ ◦

Permeability Porosity Fluid viscosity Rock and fluid compressibilities

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Transient (unsteady-state) flow

Transient flow takes place during the early life of a well, when the reservoir boundaries have not been felt and the boundary has no effect on the pressure behaviour in the reservoir. The reservoir will behave as its infinite in size, and hence said to be infinite-acting.

Both the pressure and pressure derivative, with respect to time, are functions of both position and time, i.e., P = f(r, t). It shows also that the transient flow period occurs during the time interval 0 < t < tt. 13

Basic transient flow equation The flow rate into an element of volume of a porous media may not be the same as the flow rate out of that element. The fluid content of the porous medium changes with time. The variables in unsteady-state flow additional to those already used for steady-state flow, therefore, become: ◦ Time, t ◦ Porosity, φ ◦ Total compressibility, ct

The applications of the unsteady-state flow include:  

Radial flow of slightly compressible fluids Radial flow of compressible fluids

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The mathematical formulation of the transient-flow equation is based on combining three independent equations. These equations are: Continuity Equation

MBE

Transport Equation

Darcy

Compressibility Equation

Isotherm coefficient

Initial and Boundary Conditions

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1.

2. 3. 4.

Continuity Equation  accounts for every pound mass of fluid produced, injected, or remaining in the reservoir. Transport Equation  Darcy’s equation in its generalized differential form Compressibility Equation  describing the changes in the fluid volume as a function of pressure Initial and Boundary Conditions  2 Boundary Conditions: • a constant rate of fluids into the wellbore • No flow across the outer boundary (i.e., re = ∞)  Initial Condition, a uniform pressure prior to production at time = 0 16

Derivation of the Diffusivity Equation Assumptions ◦ Single phase radial flow ◦ Constant formation properties (k, h, ф)

◦ Constant fluid properties (μ, B, Co) ◦ Slightly compressible fluid with constant compressibility ◦ Pore volume compressibility is constant (Cf) ◦ Darcy’s law is applicable (Laminar flow Re < 1) ◦ Effect of gravity is neglected

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Concept of Continuity Equation

m   Av 1   Av 2 mass L   mass L  mass  m   L2  3     L2  3    L T 1  L T 2 T   Av 1

 Av 2

Bulk Volume of Element= Δx. Δy. Δz Pore Volume of Element = ϕ. Δx. Δy. Δz Volume of Fluid = Pore Volume = ϕ. Δx. Δy. Δz Mass of Fluid = ρ. ϕ. Δx. Δy. Δz Change of mass ={ (ρ. ϕ)t +Δt-(ρ. ϕ)t} 18

Mass Balance Mass into the Mass out of the  Rate of change of mass      element at x element at x +  x inside the element       The mass balance for the control element is then written as

Massin  Massout

m  t

…..(1)

 Av x   Av x x Mass changes in y direction =  Av y   Av y  y

Mass changes in x direction =

Mass changes in z direction =

 Av z   Av z  z

Mass originally in the cube m =  * v * 

…..(2) …..(3) …..(4)

  (xyz )

The rate of accumulation of mass in the elementary volume is:

 t  t (x.y.z )   t (x.y.z )

…..(5)

Since there is no generation of mass 19

Mass Balance Combining equation 2,3,4 and 5 in Equation 1 will results in:



 

 Ax v x x  v x x  x   Ay v y y  v y y  y  t  t   t (x.y.z )  t   Az v z z  v z z  z  Dividing by txyz Ax  yz A z  xy Ay







t  t

  t

t

  t  v   v    v   v   

x x  x

x x

xt

y y  y

y y

yt

   xz

  v   v 

z z  z

z z

zt

  

and taking the limits as ∆x -> 0 , ∆y -> 0, ∆z -> 0 ….and ∆t -> 0, we obtain the final form as:

  v x   v y   v z           t y z   x

…..(6)

The above equation is the General Continuity Equation for 3-dimensional Cartesian system. 20

Continuity Equation for 3-dimensional Cartesian system (x, y, z)

    v x   v y   v z     0 t x y z

 What is the final continuity equation for one dimension flow?  What is the final Continuity equation if the flow is under steady-state?  What is the final continuity equation if the fluid is incompressible?

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Derivation of the Diffusivity Equation cont’d Consider the flow element shown below. The element has a width of dr and is located at a distance of r from the centre of the well. According to the concept of the materialbalance equation, the mass rate of accumulation during the time interval is:

Mass entering volume element during interval Δt

Mass leaving volume element during interval Δt

rate of mass accumulation during interval Δt

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Derivation of the Diffusivity Equation cont’d Continuity equation: ◦ is an equation that describes the transport of a conserved quantity, such as mass, energy, electric charge, momentum, number of molecules, etc.

1   rv     r r t ◦ The dimension of flux is "amount of quantity under investigation per unit time, per unit area". For example, the average mass flux j inside the pipe is (1 gram / second) / cm2.

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Derivation of the Diffusivity Equation cont’d Transport equation ◦ Basically, the transport equation is Darcy’s equation in its generalized differential form

k p v  r

Volume compressibility equation ◦ Expressed in terms of density, volume or porosity ◦ Describes the changes in the fluid volume as a function of pressure.

1  cf   p

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Derivation of the Diffusivity Equation cont’d Boundary conditions: ◦ The flow rate at the wellbore is given by Darcy’s law. ◦ q=const. at r=rw for t>0

◦ The reservoir is infinite acting, i.e. No boundary effect. ◦ P=Pi as r∞, there is no flow across the outer boundary

Initial condition: ◦ Uniform pressure distribution throughout the system at t=0 for all values of r ◦ P(r, 0) = Pi

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Derivation of the Diffusivity Equation cont’d The law of conservation of mass (mass balance):

In – Out + Generation = Accumulation (mass flow rate in)r+Δr – (mass flow rate out)r= (mass accumulation) Mass flow rate “mass”= (Volume flow rate) x (Density) x (Time) Volume flow rate= (Area) x (Velocity) Mass accumulation = Change of mass over time ∆t Bulk volume of the control volume = π((r+∆r)2 – r2 )h ≈ 2πr∆rh This is the volume of cylindrical shell

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Derivation of the Diffusivity Equation cont’d

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Derivation of the Diffusivity Equation cont’d Illustration of the radial flow

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Derivation of the Diffusivity Equation cont’d Mass entering the control volume during the time interval ∆t is: ------------------- eq. (1)

The area of an element at the entering side is: ------------------- eq. (2)

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Derivation of the Diffusivity Equation cont’d Combining eq. (1) and (2) ------------------- eq. (3)

Adopting the same procedure above, mass leaving the control volume during the time interval ∆t is: ------------------- eq. (4)

Total mass accumulation during time interval ∆t is:

------------------- eq. (5)

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Derivation of the Diffusivity Equation cont’d The volume of a control volume element with radius r is given by:

Differentiation with respect to r gives:

Therefore, after substituting expression of dV into eq. (5), the total mass accumulation becomes: ------------------- eq. (6)

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Derivation of the Diffusivity Equation cont’d So, substituting the above terms into the mass balance equation gives:

2hr  dr t v r  dr  2hrt v r  2rh dr  t  t   t  Divide both sides of the above eq. by:

(2πrhdr∆t)

After Taking the limit of the above eq. as both: ∆r

and ∆t approach zero ------------------eq. (7)

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Derivation of the Diffusivity Equation cont’d Eq. (7) is the continuity equation in radial coordinate .

The transport equation: ◦ relates the fluid velocity to the pressure gradient ◦ Darcy’s Law: the velocity ‘v’ is proportional to the pressure gradient (∂p/∂r).

------------------eq. (8)

Where is the –ve sign?

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Derivation of the Diffusivity Equation cont’d Substituting eq. (8) into eq. (7) gives: ------------------eq. (9)

Expanding the RHS: ------------------eq. (10)

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Derivation of the Diffusivity Equation cont’d Porosity is related to formation compressibility by the following: ------------------eq. (11)

Applying the chain rule of differentiation to ∂φ/∂t, ------------------eq. (12)

Substituting eq. (11) into eq. (12) gives:

----------------eq. (13)

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Derivation of the Diffusivity Equation cont’d Finally, substituting eq. (13) into eq. (10) and the result into eq. (9) gives:

---------eq. (14)

Equation 14 is the general partial differential equation used to describe the flow of any fluid flowing in a radial direction in porous media. 36

Derivation of the Diffusivity Equation cont’d In addition to the initial assumptions, Darcy’s equation has been added, which implies that the flow is laminar. Otherwise, the equation is not

restricted to any type of fluid and is equally valid for gases or liquids.

Compressible and slightly compressible fluids, however, must be treated separately in order to develop practical equations that can be used to

describe the flow behaviour of these two fluids.

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Radial Flow of Slightly Compressible Fluids The definition of a slightly compressible fluid is one whose density is a linear function of pressure:

This approximation is valid only if c = constant and c∆P