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FUNDAMENTALS OFFUNDAMENTALS OF

Reservoir Engineering II (PCB2053)

RESERVOIR FLUID FLOWUnsteady-State

Diffusivity Equation

RESERVOIR FLUID FLOWUnsteady-State

Diffusivity EquationDiffusivity Equation Diffusivity Equation

Ali F. M. Alta’ee

Outcomes

• To Derive Diffusivity Equation.y q• To apply constant terminal pressure solution

(CTP)• To apply constant terminal rate solution

(CTR)

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UNSTEADY-STATE FLOWPi Pi

Const. Flow rate

Pi Pi

r3 r2 r1 r1 r2 r3

Pressure disturbance as a function of time

Const. Pwf

Pi Pi

r3 r2 r1 r1 r2 r3

• Pressure disturbance moves away from wellbore at a rate determined by

• Permeability• Porosity• Viscosity• Rock & Fluid Compressibilities

Transient flow is defined as the time period pboundary has no effectduring which the

on the pressure behavior in the reservoir its infinite and the reservoir will behave as

.in size

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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:– Time, t– Porosity, φ– Total compressibility, ct

Basic Transient Flow Equation

• The flow rate into & out of an element of volume Q1 = Q2• The variables in unsteady-state flow:

Q1

Q2

The variables in unsteady-state flow:– Time, t– Porosity, φ– Total compressibility, ct

Transient Flow equation must have these independent variables + limits

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• The mathematical formulation is based on:

– Continuity Equation MBE

– Transport Equation

C ibilit E ti

Darcy

I th ff– Compressibility Equation

– Initial and Boundary Conditions

Isotherm coeff

• Continuity Equation– A material balance equation that accounts for

every pound mass of fluid produced, injected, i i i th ior remaining in the reservoir.

• Transport EquationThe transport equation is Darcy’s equation in– The transport equation is Darcy s equation in its generalized differential form

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• Compressibility Equation– Expressed in terms of density or volume– Is used in formulating the unsteady-state g y

equation with the objective of describing the changes in the fluid volume as a function of pressure

• Initial and Boundary Conditions– Boundary Conditions:

• The formation produces at a constant rate into the wellbore

• There is no flow across the outer boundary and the reservoir behaves as if it were infinite in size, i.e., re = ∞

– Initial Condition• The reservoir is at a uniform pressure when

production begins, i.e., time = 0

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Diffusivity Equation

where k = permeability mdwhere k = permeability, mdr = radial position, ftp = pressure, psiact = total compressibility, psi−1

t = time, hrsφ = porosity, fractionμ = viscosity, cp

• According to the concept of the material-balance equation

Mass enteringl l t

Mass leavingl l t

CONTINUITY EQUATION

volume element during interval Δt

volume element during interval Δt

rate of mass((5656))accumulation

during interval Δt---------------- ((5656))

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Illustration of radial flow

• Mass entering the volume element during time interval Δt

---------------- ((5757))where ν = velocity of flowing fluid, ft/dayρ = fluid density at (r + dr), lb/ft3ρ fluid density at (r + dr), lb/ft3A = Area at (r + dr)Δt = time interval, days

• The area of element at the entering side is:

---------------- ((5858))

• Combining Equation (58) with (57) gives:

---------------- ((5959))

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• Mass leaving the volume element

---------------- ((6060))

• Total Accumulation of MassThe volume of some element with a radius of r

DifferentiatingDifferentiating with respect to r

OR ---------------- ((6161))

Total mass accumulation during Δt = dV [(φρ)t + Δt − (φρ)t]

Substituting for dV

Total mass accumulation = (2πrh) dr [(φρ)t + Δt − (φρ)t ] ---------------- ((6262))Replacing terms of material balance Equation with calculated relationshipsReplacing terms of material balance Equation with calculated relationships

Dividing by (2πrh) dr

OR ---------------- ((6363)) where φ = porosityρ = density, lb/ft3ν = fluid velocity, ft/day

continuity equation

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• The transport equation– Darcy’s Law is essentially the basic motion

equation• The velocity is proportional to the pressure gradient (∂p/∂r)

((6464))------------ ((6464))

where k = permeability, mdν = velocity, ft/dayCombining Equation (64) with Equation (63) results in:

---------------- ((6565))

Expanding the right-hand side by taking the indicated derivatives eliminates the porosity from the partial derivative term on the right-hand side:

---------------- ((6666))

• porosity is related to the formation compressibility

---------------- ((6767))

Applying the chain rule of differentiation to ∂φ/∂t

COMPRESSIBILITY EQUATION

Applying the chain rule of differentiation to ∂φ/∂t,

Substituting Equation (67) into this equation

substituting the above relation into Equation (66) and the result into Equation (65), gives:

---------------- ((6868))

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Radial Flow of Slightly Compressible Fluids

• the permeability and viscosity are constant over pressure time and distance rangespressure, time, and distance ranges

---------------- ((6969))

Expanding

• Using the chain rule in the above relationship yields:

Dividing by ρ

Recalling that the compressibility of any fluid is related to its density by:

Combining the two equations

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ll d b i dvery small and may be ignored

---------------- ((7070))

---------------- ((7171))

Define total compressibility, ct, as:

• Combining Equations (70) with (701)and rearranging gives:

---------------- ((7272))

diffusivity equation

where the time t is expressed in days

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Diffusivity Equation• It is one of the most important equations in

petroleum engineering• Particularly used in analysis well testing data

where the time t is commonly recorded in hours

---------------- ((7373))

where k = permeability, mdr = radial position, ftp = pressure, psiact = total compressibility, psi−1

t = time, hrsφ = porosity, fractionμ = viscosity, cp

The Diffusivity Constant

• When the reservoir contains more than one fluid, total compressibility should be computed as:

---------------- ((7474))

Co,w,g= compressibility of oil, water and gas

So,w,g = fractional saturation of oil, water and gas

The Diffusivity Constant η

---------------- ((7575))

The diffusivity equation can then be written in a more convenientform as:

---------------- ((7676))

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• The diffusivity equation is essentially designed to determine the pressure as a function of time t and position r

• The assumptions and limitations used inThe assumptions and limitations used in developing the diffusivity equation– Homogeneous and isotropic porous medium– Uniform thickness– Single phase flow– Laminar flow– Rock and fluid properties independent of pressure

• Laplace’s Equation

• for a steady-state flow condition, the pressure at any point in the reservoir is constant and does not change with time

∂p/∂t = 0Substitute in diffusivity equation

---------------- ((7777))((7777))

Laplace’s equation