Slide 1

Fluid Flow in Porous Media

Fluid Flow in Porous MediaLESSON Objectives:Understand forces responsible for driving fluid through reservoir

Be aware of models available to represent reservoir and wells

Assess flow properties of reservoir

Introduce concepts used in welltesting

CONTENTSIntroductionDarcys LawLinear FlowRadial FlowSkin EffectDiffusivity EquationsDimensionless variables

INTRODUCTIONDarcys Law is fundamental to reservoir engineeringQuantifies the rate of flow of fluids through porous mediaObjective to understand mechanism of fluid migration in order to understand and improve recovery from the reservoir.Similar in concept to flow in pipesDimensions give rise to scaling problem capillary forces become relatively important (over viscous forces)

INTRODUCTIONFluid flow depends on:reservoir geometryreservoir fluidsreservoir properties

Two methods to represent fluid flow:analytical solutions to diffusivity equationapproximation methods to diffusivity equation using finite difference/ finite element simulations

HENRY DARCY19th century French engineerWhile designing a filter to process his towns water demandVertical flow of water through packed sandIntroduce the concept of permeability (unit: mD)

DARCYS LAWWhat are the parameters that affect fluid flow?

DARCYS LAW

Flow rate (cm3/s)Cross sectional area (cm2)Viscosity of flowing fluid (cp)Permeability (Darcy)Pressure gradient (atm/cm)

qLAdxTransport equation implying velocity is proportional to pressure gradient and reciprocal to viscosityWhy the sign ?

Assumptions of Darcys LawThe fluids exhibit laminar flowMost oil reservoirsGas fields, high gas/oil rate give turbulent flowOnly one phase presentUncommon in petroleum reservoirsExtend Darcys Law for multiphase flowThere are no chemical reactions between fluid and the rockDo not account microscopic effectChanges in structure of porous medium violate Darcys Law (e.g clay swelling)

Darcys Law for Multiphase FlowExtended to multiphase - concept of rel perm

From Darcys Law, basic linear and radial flow equations under steady condition can be derived.

Linear Flow of Incompressible Fluids11

Steady State Linear Flow (Fluids)In field units

Flowrate (rb/d)Permeability (mD)Area (ft2)Viscosity (cp)Pressure Gradient (psi/ft)Specific GravityTilt angle

QP1P2

Steady State Linear Flow (Gases)In field units

Flowrate (MMscf/d)Permeability (mD)Area (ft2)Viscosity (cp)Distance (ft)

Pressure at outer boundary (psi)Pressure at inner boundary (psi)

Darcys Law for Radial Flow

For fluid flow to occur, a pressure gradient must be established between the inner and outer boundary of the reservoir.

Curved surface open to flow

hrwrePressure gradient dp/dr

Drainage

Steady State Radial FlowIn field units

Flowrate (rb/d)Permeability (mD)Height (ft)Viscosity (cp)Outer boundary radius(ft)

Pressure at outer boundary (psi)Pressure at inner boundary (psi)Inner boundary radius(ft)

Radial Flow Pressure Profile

rwreqPwPe

P

Initial steady state, no flow

Flow initiated at constant rate

Pressure profile develops

Pressure disturbance reaches outer boundarySealed therefore the pressure in system drops

Well shut in and pressure builds in well

Steady state, no flow

DIFFUSIVITY EQUATIONDiffusion is a process by which there is a net flow of matter from a region of high concentration to a region of low concentration.

INTRODUCTIONWhy is it important to understand reservoir fluid and flow characteristics?Determine productivity of a reservoirDetermine optimum strategy to maximise recoveryOne major aspect that affect fluid flow is the rock-fluid interactionThis factor will in turn affect the time taken by the pressure to change in the reservoir or for fluid to migrateSurface chemistry, surface tension, capillary pressure, permeabilityWhy is it important to know the pressure change?

Illustrate: Reservoir well flowConsider a water saturated reservoir rockResponse not instantaneousPressure changes may take days, even years to manifest themselvesIn this case, the flow regime would not be steady-state while pressure is finding its equilibriumThus Darcys Law cannot be appliedDiffusion process need to be examined time dependent scenarios must be assessed

Fluid Flow RegimeIn order to produce at constant rate, we need to impose a pressure in the wellbore, lets say 3500 psi.But, the wellbore pressure, Pw will not instantaneously changed to 3500 psi.It takes time for its initial pressure, 4500 psi to become 3500 psi.The time taken is the duration of a non-steady state flow regimeWith time, reservoir pressure will decline, unlessHigh drive mechanism (gascap)Pressure maintenance scheme (injection)In this case, the pressure will equilibrate (becomes steady at 3500 psi)Steady state flow regime

Pres=4500 psiPwf=3500 psi

Pressure Profile with Radial Distance

rwreqPwfPe

P4500t=0

t1t2t3t4 and later3500

Pressure Profile with time

Pwf4500t=0t1t2t3t4 and later3500

TransientSemi steadySteady State

Idealised Flow PatternsThere are a number of idealised flow patternsLINEARRADIALHEMISPHERICALSPHERICAL

Most importantBoth: describe water encroachment from an aquifer into reservoirRadial: describe flow of fluid around wellboreIn order to derive diffusion eqn, it is assumed that cP1 (if not, pressure dependence of compressibility c must be taken into account

Idealised Flow PatternsLinear, radial, spherical, hemispherical

Linear and radial of most use

Assume oil system with cp qin

(which is possible in a pressurised system since the pressure of the fluid in the element may reduce causing it to expand and produce a higher flow rate out of the element)

Fluid Flow in Porous MediaTherefore, there is a relationship between the change in mass, m, along the cuboid and the change in density, r, over time as the mass accumulates or depletes from any element. In terms of mass flowrate, Mass flow rate through the area, A = qr ((m3/s)*(kg/m3) = kg/s)Mass flow rate through the area, A at position x= (qr)xMass flow rate through the area, A at position x+dx = (qr)x+dxMass flowrate into a volume element at x minus mass flowrate out of element at x + dx =(qr)x- (qr)x+ dx

The mass flow rate out of the element is also equal to the rate of change of mass flow in the element, i.e. dqr()dxqr()x+dx=qr()x+

*dxChange in mass flow rate

-dqr()dx

*dx(if change is +ve, element accumulating mass, if ve depleting mass) This must equal rate of change of mass in element with volume A*dx*fRate of change of mass equal to drdt

Afdxhence -dqr()dx

t=Addrf=

flow velocity U = q/A, therefore -dUr()dx

tddr=fordUr()dx

tddr=-fSubstitution of parameters gives

txPkxddrfddmrdd=

Equation shows areal change in pressure linked to temporal change in density. Measure pressure easier than density, therefore use isothermal compressibility to convert to pressurec = - 1V

(dVdP

) T The density equals mass per unit volumer=mV

Hence: c = - rm

d(m/r)dP

= 1r

drdP

; (Quotient Rule, constant mass system) Since drdt

= drdP

dPdt

= crdPdt

(from above) Then ddx

krm

dPdx

= fcrdPdt

Partial differential equation for linear flow of any single phase fluid in porous medium relates spatial and temporal variations in pressure

In core relates pressure distribution along core during flooding, during all time, i.e. from start of flood to staedy state conditions

In linear reservoir where aquifer flows into reservoir as production proceeds

But, non-linear because of pressure dependence of density, compressibility and viscosity.Simple linearisation follows

ddx

krm

dPdx = fcrdPdt

Diffusivity EquationLINEAR FLOW

RADIAL FLOW

Non linear p.d.e since , and c depend on pressure the equations must be linearised.

These equations need to solved in order to be applicable to real reservoirs. It can be solved according to the flow regime transient, semi steady or steady state.Diffusivity equation is fundamental to well testing

RECAP LAST CLASS

Types of fluids

Incompressible fluids

Slightly compressible fluids

Compressible fluids

50

Incompressible fluids

51

Slightly compressible fluidsThe slightly compressible fluids exhibit small changes in volume, or density, with changes in pressure.

Crude oil and water systems fit into this category of fluids.

52

Compressible fluids

53

Different fluid types

Pressure- Volume relationship Pressure-Density relationshipReservoir engineering II. Amin Azdarpour54

Steady-state flow55

The applications of the steady-state flow

The applications of the steady-state flow to describe the flow behavior of several types of fluid in different reservoir geometries are presented below. These include:

Linear flow of incompressible fluidsLinear flow of slightly compressible fluidsLinear flow of compressible fluidsRadial flow of incompressible fluidsRadial flow of slightly compressible fluidsRadial flow of compressible fluids56

Example 1 An incompressible fluid flows in a linear porous media with the following properties. Calculate:Flow rate in bbl/dayApparent fluid velocity in ft/dayActual fluid velocity in ft/day

57PropertiesLkP1P2hwidthvalue2000 ft100 md2000 psi1990 psi15%20 ft300 ft2 cp

Solution Example 158

Linear flow with gravitational force

Pressure difference is not the only driving force and gravitational is the other important driving force.The fluid gradient force (gravitational force) is always directed vertically downward while the force that results from an applied pressure drop may be in any direction.The fluid potential () at any point in the reservoir is defined as the pressure at that point less the pressure that would be exerted by a fluid head extending to an arbitrarily assigned datum level.zi is the vertical distance from a point i in the reservoir to this datum level.

59

Cont60

Example 2 Assume that the porous media with the properties as given in the previous example is tilted with a dip angle of 5 as shown in below figure. The incompressible fluid has a density of 42 lb/ft3. Resolve previous example using this additional information.61

Solution Example 262

Linear flow of slightly compressible fluids

63

Cont64

Example 3Consider the linear system given in Example1 and assuming a slightly compressible liquid, calculate the flow rate at both ends of the linear system. The liquid has an average compressibility of 21105 psi1.65

q1q2P1P2

Solution Example 3

The above calculations show that q1 and q2 are not largely different,which is due to the fact that the liquid is slightly incompressible and itsvolume is not a strong function of pressure.

Linear Flow of compressible fluids (gases) 67

Linear Flow of compressible fluids (gases) 68

Linear Flow of compressible fluids (gases) 69

Example 4A linear porous media is flowing a 0.72 specific gravity gas at 120F. The upstream and downstream pressures are 2100 psi and 1894.73 psi, respectively. The cross-sectional area is constant at 4500 ft2. The total length is 2500 feet with an absolute permeability of 60 md. Calculate the gas flow rate in scf/day (Psc=14.7psia,Tsc= 520R).

70

Solution Example 4

Solution Example 4

REMEMBER :It is essential to notice that those gas properties z and g are a very strong function of pressure, but they have been removed from the integralto simplify the final form of the gas flow equation. The above equationis valid for applications when the pressure < 2000 psi. Temp = 120 F + 460 = 580 R1270731

Radial flow of incompressible fluidsThe formation is considered to a uniform thickness h and a constant permeability k. Because the fluid is incompressible, the flow rate q must be constant at all radii. Due to the steady-state flowing condition, the pressure profile around the wellbore is maintained constant with time.

qr = volumetric flow rate at radius rAr = cross-sectional area to flow at radius r(p/r)r = pressure gradient at radius rV = apparent velocity at radius r

The minus sign is no longer required for the radial system as the radius increases in the same direction as the pressure. In other words, as the radius increases going away from the wellbore the pressure also increases.

73

ContAt any point in the reservoir the cross-sectional area across which flow occurs will be the surface area of a cylinder, which is 2rh:

The flow rate for a crude oil system is expressed in surface units (Stock tank barrels, STB) rather than reservoir units. Using symbol Qo to represent the oil flow rate as expressed in STB/day, then: q= Bo.Qo

Bo is the oil formation volume factor (bbl/STB). Then darcy equation can be expressed in STB/day to give:

Integrating above equation between two radii, r1 and r2, when pressures are p1 and p2 gives:

74

ContFor incompressible system in a uniform formation, above equation can be simplified to:

Performing the integration gives:

The two radii of interest are the wellbore radius rw and the external or drainage radius re, then:

Qo = oil, flow rate, STB/daype = external pressure, psipwf = bottom-hole flowing pressure, psik = permeability, mdo = oil viscosity, cpBo = oil formation volume factor, bbl/STBh = thickness, ftre = external or drainage radius, ftrw = wellbore radius, ft

75

Example 5An oil well in the nameless field is producing at a stabilized rate of 600 STB/day at a stabilized bottom-hole flowing pressure of 1800 psi. Analysis of the pressure buildup test data indicates that the pay zone is characterized by a permeability of 120 md and a uniform thickness of 25 ft. The well drains an area of approximately 40 acres. The following additional data is available as well. Calculate the pressure profile (distribution) and list the pressure drop across 1 ft intervals from rw to 1.25 ft, 4 to 5 ft, 19 to 20 ft, 99 to 100 ft, and 744 to 745 ft.

76rw0.25 ftBo1.25 bbl/STBA40 acreso2.5 cp

Solution Example 5

Solution Example 5

Radial flow of slightly compressible fluidsFlow rate is dependent on pressure:

Applying the above equation into darcy law for radial flow gives:

Separating the variables in the above equation and integrating over the length of the porous medium gives:

OR

Where qref is oil flow rate at a reference pressure pref. Choosing the bottom hole flow pressure pwf as the reference pressure and expressing the flow rate in STB/day gives:

79

co = isothermal compressibility coefficient, psi1Qo = oil flow rate, STB/dayk = permeability, md

Example 6The following data are available on a well in the red river field. Assuming a slightly compressible fluid, calculate the oil flow rate. Compare the result with that of incompressible fluid.

80rw0.25 ftBo1.25 bbl/STBo2.5 cpPe 2506 psi Pwf1800 psire745 ftk0.12 Dh25 ftCo2510-6 psi-1

Solution Example 6

Radial flow of compressible gasesFor a radial gas flow, the Darcys equation can be expressed as:

qgr = gas flow rate at radius r, bbl/dayr = radial distance, fth = zone thickness, ftg = gas viscosity, cpp = pressure, psi0.001127 = conversion constant from Darcy units to field units

The gas flow rate is usually expressed in scf/day. Referring to the gas flow rate at standard condition as Qg, the gas flow rate qgr under pressure and temperature can be converted to that of standard condition by applying the real gas equation-of-state to both conditions, or:

OR82

(1)(2)

ContWhere: psc = standard pressure, psiaTsc = standard temperature, RQg = gas flow rate, scf/dayqgr = gas flow rate at radius r, bbl/dayp = pressure at radius r, psiaT = reservoir temperature, Rz = gas compressibility factor at p and Tzsc = gas compressibility factor at standard condition 1.0

Combination of equation 1 and 2 gives:

Assuming that Tsc = 520 R and psc = 14.7 psia:83

ContIntegrating previous equation from the wellbore conditions (rw and pwf) to any point in the reservoir (r and p) gives:

Imposing Darcys Law conditions on above equation, i.e.:Steady-state flow which requires that Qg is constant at all radiiHomogeneous formation which implies that k and h are constant

And expanding the integration gives:

84

ContCombination gives:

The is called the real gas potential or real gas pseudopressure and it is usually represented by m(p) or (psy). Thus:

Therefore previous equation can be written in terms of the real gas potential to give: 85

OR

ContThe flow rate can be given by:

In the particular case when r = re, then:

Where:e = real gas potential as evaluated from 0 to pe, psi2/cpw = real gas potential as evaluated from 0 to Pwf, psi2/cpk = permeability, mdh = thickness, ftre = drainage radius, ftrw = wellbore radius, ftQg = gas flow rate, scf/day

The gas flow rate is commonly expressed in Mscf/day:

The above equation can be expressed in terms of the average reservoir pressure pr instead of the initial reservoir pressure pe as:

86

Example 7The following PVT data from a gas well in the Anaconda Gas Field is given below. The well is producing at a stabilized bottom-hole flowing pressure of 3600 psi. The wellbore radius is 0.3 ft. The following additional data is available. Calculate the gas flow rate in Mscf/day.

87

k65 mdPe4400 psih15 ftre1000 ftT600 R

87

Approximation of the gas flow rateThe exact gas flow rate as expressed by the different forms of Darcys Law, i.e., can be approximated by removing the term outside the integral as a constant. It should be pointed out that the zg is considered constant only under a pressure range of < 2000 psi. Therefore:

Removing the term and integrating gives:

The term (g. z)avg is evaluated at an average pressure that is defined by the following expression: 88

Example 8Using the data given in previous example, re-solve for the gas flow rate by using the pressure-squared method. Compare with the exact method (i.e., real gas potential solution).89

The applications of the unsteady-state flowBasic transient flow equation:

In unsteady-state flow condition, 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. Accordingly, 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, porosity and total compressibility.

a. Continuity Equation: The continuity equation is essentially a material balance equation that accounts for every pound mass of fluid produced, injected, or remaining in the reservoir.

b. Transport Equation: Basically, the transport equation is Darcys equation in its generalized differential form. The continuity equation is combined with the equation for fluid motion (transport equation) to describe the fluid flow rate in and out of the reservoir.

90

Cont

c. Compressibility Equation: The fluid compressibility equation (expressed in terms of density or volume) is used in formulating the unsteady-state equation with the objective of describing the changes in the fluid volume as a function of pressure.

d. Initial and Boundary Conditions: There are two boundary conditions and one initial condition required to complete the formulation and the solution of transient flow equation. The two boundary conditions are: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 = .

91

ContThe element has a width of dr and is located at a distance of r from the center of the well. The porous element has a differential volume of dV.

The rate of mass flow into an element minus the rate of mass flow out of the element during a differential time t must be equal to the mass rate of accumulation during that time interval92

ContMass entering the volume element during time interval t:

Where: = velocity of flowing fluid, ft/day = fluid density at (r + dr), lb/ft3A = Area at (r + dr)t = time interval, days

The area of element at the entering side is:

Combination of these two equation gives:Reservoir engineering II. Amin Azdarpour SEPTEMBER 201293

ContMass leaving the volume element:Adopting the same approach as that of the leaving mass gives:

Total Accumulation of Mass:The volume of some element with a radius of r is given by:

Differentiating the above equation with respect to r gives:

Total mass accumulation:Reservoir engineering II. Amin Azdarpour SEPTEMBER 201294

ContSubstituting for dV yields:

Replacing terms of general equation with those of the calculated relationships gives:

Dividing the above equation by (2rh)dr and simplifying, gives:Reservoir engineering II. Amin Azdarpour SEPTEMBER 201295

Total mass accumulation

ORWhere: = porosity = density, lb/ft3 = fluid velocity, ft/day(B)

ContThe previous equation (B) is called the continuity equation and it provides the principle of conservation of mass in radial coordinates.Darcys Law is essentially the basic motion equation, which states that the velocity is proportional to the pressure gradient (p/r):

Combining above equation with continuity equation gives:

Expanding the right-hand side by taking the indicated derivatives eliminates the porosity from the partial derivative term on the right-hand side:Reservoir engineering II. Amin Azdarpour SEPTEMBER 201296

ContPorosity is related to the formation compressibility by the following:

Applying the chain rule of differentiation to /t:

Substituting into previous equation gives:

Finally, substituting the above relation into previous equations gives:Reservoir engineering II. Amin Azdarpour SEPTEMBER 201297

Radial Flow of Slightly Compressible FluidsThe general partial differential equation used to describe the flow of any fluid flowing in a radial direction in porous media can be expressed by:

Assuming that the permeability and viscosity are constant over pressure, time, and distance ranges. This leads to:

Expanding the above equation gives:

Using the chain rule in the above relationship yields:Reservoir engineering II. Amin Azdarpour SEPTEMBER 201298

ContDividing the above expression by the fluid density gives:

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

Combining the above two equations gives:

The following term is very small and can be ignored:

Therefore:

Define total compressibility, ct, as: Reservoir engineering II. Amin Azdarpour SEPTEMBER 201299

ContCombining the above two equations and rearranging gives (t is expressed in days):

The above equation is called diffusivity equation and it is one of the most important equations in petroleum engineering. The equation is particularly used in analysis well testing data where the time t is commonly recorded in hours. The equation can be rewritten as:

Where: k = permeability, mdr = radial position, ftp = pressure, psiact = total compressibility, psi1t = time, hrs = porosity, fraction = viscosity, cpReservoir engineering II. Amin Azdarpour SEPTEMBER 2012100

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

where co, cw and cg refer to the compressibility of oil, water, and gas, respectively, while So, Sw, and Sg refer to the fractional saturation of these fluids.

The use of ct, simply accounts for the compressibility of any immobile fluids which may be in the reservoir with the fluid that is flowing.Diffusivity constant ( ) can be expressed as:

The diffusivity equation can then be written in a more convenient form as:Reservoir engineering II. Amin Azdarpour SEPTEMBER 2012101

ContThe diffusivity equation as represented by previous equation is essentially designed to determine the pressure as a function of time t and position r.

Before discussing and presenting the different solutions to the diffusivity equation, it is necessary to summarize the assumptions and limitations used in developing the diffusivity equation:1. Homogeneous and isotropic porous medium2. Uniform thickness3. Single phase flow4. Laminar flow5. Rock and fluid properties independent of pressure

Reservoir engineering II. Amin Azdarpour SEPTEMBER 2012102

Example 9Show that the radial form of Darcys equation is the solution to below equation.Reservoir engineering II. Amin Azdarpour SEPTEMBER 2012103

The applications of the pseudosteady-state flowAs soon as the pressure disturbance reaches all drainage boundaries, it ends the transient (unsteady-state) flow regime. A different flow regime begins that is called pseudosteady (semisteady) state flow.

Consider below figure, which shows a well in radial system that is producing at a constant rate for a long enough period that eventually affects the entiredrainage area. During this semisteady-stateflow, the change in pressure with timebecomes the same throughout the drainagearea. Section B in this figure shows thatthe pressure distributions become paralleled at successive time periods Reservoir engineering II. Amin Azdarpour SEPTEMBER 2012104

ContThe constant referred to in the above equation can be obtained from a simple material balance using the definition of the compressibility, thus:

Arranging:

Differentiating with respect to time t:

Expressing the pressure decline rate dp/dt in the above relation in psi/hr gives: Reservoir engineering II. Amin Azdarpour SEPTEMBER 2012105

Where: q = flow rate, bbl/dayQo = flow rate, STB/daydp/dt = pressure decline rate, psi/hrV = pore volume, bbl

ContFor a radial drainage system, the pore volume is given by (where A = drainage area, ft2):

Combining previous two equations give:

Examination of the above expression reveals the following important characteristics of the behavior of the pressure decline rate dp/dt during the semisteady-state flow:The reservoir pressure declines at a higher rate with an increase in the fluids production rateThe reservoir pressure declines at a slower rate for reservoirs with higher total compressibility coefficientsThe reservoir pressure declines at a lower rate for reservoirs with larger pore volumesReservoir engineering II. Amin Azdarpour SEPTEMBER 2012106

Example 10An oil well is producing at a constant oil flow rate of 1200 STB/day under a semisteady-state flow regime. Well testing data indicate that the pressure is declining at a constant rate of 4.655 psi/hr. The following additional data is available:

Reservoir engineering II. Amin Azdarpour SEPTEMBER 2012107h (ft)ct (psi-1) (%)Bo (bbl/STB)25151.3

Radial flow of slightly compressible fluidsThe diffusivity equation can be expressed by:

For the semisteady-state flow, the term (p/t) is constant and it can be expressed by:

Substituting previous equation into the diffusivity equation gives:Reservoir engineering II. Amin Azdarpour SEPTEMBER 2012108

OR

ContThe previous equation can be expressed as:

Integrating the above equation gives:

Where c1 is the constant of the integration and can be evaluated by imposing the outer no-flow boundary condition [i.e., (p/r)re = 0] on the above relation to give:

Combining the above two expressions gives:Integrating again:Reservoir engineering II. Amin Azdarpour SEPTEMBER 2012109

ContPerforming the above integration and assuming is negligible gives:

A more appropriate form of the above is to solve for the flow rate, to give:

The volumetric average reservoir pressure Pr is commonly used in calculating the liquid flow rate under the semisteady-state flowing condition. Introducing the Pr into previous equation gives: Reservoir engineering II. Amin Azdarpour SEPTEMBER 2012110

Where Q = flow rate, STB/dayB = formation volume factor, bbl/STBk = permeability, md

AND

DIFFUSIVITY EQUATIONis the diffusivity constant.

Compressibility in the equation is the saturation weighted compressibilityco oil compressibility (typically 10x10-6 psi-1)cw water compressibility (typically 3x10-6 psi-1)cf formation compressibility (typically 4x10-6 psi-1)So oil saturationSw connate waer saturation

It measures the speed of pressure disturbance propagation through reservoir so that equilibrium is attained.The larger the value, the more rapid the pressure equilibration.

Solutions of Diffusivity EquationIn order to solve the pde, we need to:Define initial conditions (specifying Pi at every point)Define boundary conditions (inner and outer boundary)

The solutions can be grouped according to the flow regime they represent:Steady StateSemi Steady StateTransient StateSystem with a flow across outer boundary, such that

System with a no flow outer boundary, such that

Infinite acting system such thatMost complex situation (represents the state during well testing)

CTR vs CTPBasic solutions can be divided into 2 types:Constant Terminal Rate (CTR)dq/dt constant, pressure varies with timeConstant Terminal Pressure (CTP)Constant bottomhole pressure, flowrate variesWill not be covered in the course

Solutions to be examined:

CTR Solution for radial model- Steady StateThere is a flow across the outer boundary to replace fluid produced at the wellPressure at outer boundary, Pe remains constant

In field units and with skin factor:

CTR Solution for radial modelSemi Steady StateThere is no influx across the outer boundary Constant rate of change of pressure (dp/dt) Sometimes called pseudo-steady or quasi-steady state.

In field units and with skin factor:

CTR Solution for radial model - Transient StateThe reservoir appears infinite acting applicable for short period (well testing hours) Pressure and rate of its changes are functions of distance and timeThe solution is usually referred as the line source constant terminal rate solution.

In field units:The reservoir pressure at radial position r at time t is given by:

Exponential Integral

If wellbore pressure is of interest:

Transient SolutionThe Ei(-x) function can be obtained from available graph.Provide quick evaluation of pressure drop expected when drilling a new well in producing fieldif x 0.01 , the Ei(-x) function can be approximated as

The line source solution is valid if the flowing time, t (hrs)

where

&

Dimensionless VariablesDimensionless Radius

Dimensionless TimeDimensionless Pressure

Skin FactorWhile drilling a well, must have higher wellbore pressure than reservoir pressureTo prevent inflow of reservoir fluid

Thus it is inevitable that some of the drilling fluid enters the formation which could cause:Plugging of pore spaces/perforationsClay swellingReduce permeability

This would create a damaged zone in the vicinity of the wellbore.

Can give significant effect, need to be quantifiedThis impairment (or enhancement) can be modeled through the concept of a mechanical skin.

Skin effect on Pressure Drawdown

rwreqPwfPe

P

ra

Pskin

If well is undamaged, pressure drawdown is lower than if the well were damaged (dashed line)Skin will increase pressure drawdown.

Definition by Van Everdingen;Where S is the mechanical skin factor (dimensionless).

Skin FactorAssumption of constant permeability around wellboreFormation damage during drilling and completion and during production causes alteration of permeability around wellbore.Extends up to a few feet from wellbore into reservoirIf reservoir fractured (naturally or by workover) permeability may be increasedEi function fails to account for these conditionsSkin zone defined as zone around wellbore with altered permeability

124

125

Skin Factor

Incorporating skin factor, the steady state inflow equation becomes:The effect of skin can be further quantified using Productivity Index (PI).

PI is a direct measure of well performance.Desirable to have PI as large as possible (small pressure drawdown)

Skin Factor

PI increases if S negative.S negativeS positive S=0Well stimulated (acidising, fracturing)Well damaged, consider well stimulation No damage in the well

S can be found from Pressure Buildup Test.

Skin effect on Pressure Drawdown

rwreqPwfPe

P

ra

S -veS +ve

Fluid flowIn summary, fluid flow depends on :reservoir geometryreservoir fluidsreservoir properties

Two methods to represent fluid flow:analytical solutions to diffusivity equationapproximation methods to diffusivity equation i.e. using finite difference/ finite element simulations

129

How important are computers?

New technologies allow us to gather better, more complete data and build clearer images of the reservoir Visualization of fluid flow

A well produces oil at a constant flowrate of 15 stock tank cubic metres per day (stm3/d). Use the following data to calculate the permeability in milliDarcys (mD).Dataporosity, f19%formation volume factor for oil, Bo 1.3rm3/stm3net thickness of formation, h, 40mviscosity of reservoir oil, m 22x10-3 Paswellbore radius, rw 0.15mexternal radius, re 350minitial reservoir pressure, Pi 98.0barbottomhole flowing pressure, Pwf93.5bar

qreservoir = qstock tank x Bo1bar = 105 Pa1 pas = 1000 cp

131

1

Unsteady State Flow RegimesDimensionless variables

normalised parametersdefine solution to diffusivity equation for dimensionless variablesdetermine solutioncalculate specific reservoir values from dimensionless solutiondimensionless radius, rD :

wDrrr= dimensionless time, tD :

2wDcrkttfm= dimensionless pressure, PD :

)P)(Pqkh2()t,(rPtr,iDDD-=mp (at a dimensionless radius and dimensionless time)

132

1

where r = radius in question rw = wellbore radius k = permeability t = time in question f = porosity m = viscosity c = compressibility h = thickness of the reservoir Pi = initial reservoir pressure Pr,t = pressure at the specified radius and time then the radial diffusivity equation becomes

DDDDDDDtPrPrrr1dddddd= (3.11) There are other definitions of dimensionless variables, such as dimensionless external radius

133

1

Unsteady State SolutionCTR solution obtained in several forms with different assumptions and mathematical analysesGeneral considerations

134

1

Wellbore pressure and flow rate responsePressure decline normally divided into 3 sections depending on the value of flowing time and reservoir geometry.

Initially, transient solution infinite acting reservoir case reservoir appears infinite in extent

Late transient boundaries start to affect the response

Semi-steady state or pseudo-steady state pressure perturbation affecting all parts of the reservoir no influx from aquifer

Hurst and van Everdingen SolutionCTR solution in 1949

Solved radial diffusivity using Laplace transform for both CTR and CTP

Solution describes pressure drop as function of time and radius for fixed values of re and rw rock and fluid properties.

Dimensionless variables and parameters:

PD = f(tD,rD,reD) where tD = dimensionless timerD = dimensionless radiusreD = re/rw = dimesionless external radius.

If the reservoir is fixed in size, i.e. reD is a particular value,

then the dimensionless pressure drop, PD, is a function of the dimensionless time, tD and dimensionless radius, rD.

The pressure in a particular reservoir case can then be calculated at any time and/or radius.

One of the most significant cases is at the wellbore since the pressure can be measured routinely during production operations and compared to the theoretical solutions.

The determination of a reservoir pressure at a location remote from a well may be required for reasons of technical interest, but unless a well is drilled at that location, the actual value cannot be measured.

At the wellbore radius, r=rw (or rD=1.0) PD = f(tD, reD) (3.13) i.e.

=--+-+=1mm21eDm212meDm21teD2eDDDD))()r(()r(e243lnrr2t)(tPJJJD2maaaaa (3.14) where am are the roots of 0)r()Y(J)()Yr(JeDm1m1m1eDm1=-aaaa J1 and Y1 are Bessel functions of the first and second kind This series has been evaluated for several values of dimensionless external radius, reD, over a wide range of values of dimensionless time, tD. The results are presented in the form of tables (from Chatas, AT, A Practical Treatment of non-steady state Flow Problems in Reservoir Systems, Pet. Eng. August 1953) in Well Testing by J Lee, SPE Textbook series, Vol 1. A summary of the use of the tables for constant terminal rate problems is as follows in Table 1.

Table Presents Valid for

2 i PD as a function of tD 101.6 hours

therefore ln approximation is not valid and the Ei function is used.

_1096992100.unknown

_1096992102.unknown

ii) (taking account of the conversion from stock tank to reservoir conditions via the formation volume factor for oil, Bo and also the fact that the radius, r, is now at 50m from the wellbore).

= 28703

= 0.254

P = 400x105 + 28703xEi(-0.254)

Ei(-0.254) = -1.032 (by linear interpolation of the values in Table 4)

P= 400x105 +28703x-1.032

= 400x105 -29622

= 39970378Pa

= 399.7bar

_1096992110.unknown

_1096992845.unknown

_1096992108.unknown

4) the pressure after 50 hours production at a radius of 50m from the wellbore

i) check ln approximation to Ei function

the ln approximation is valid if the time,

t > 365750s

t > 101.6 hours

therefore ln approximation is not valid and the Ei function is used.

_1096992113.unknown

_1096992115.unknown

ii) (taking account of the conversion from stock tank to reservoir conditions via the formation volume factor for oil, Bo and also the fact that the radius, r, is now at 50m from the wellbore and the time is now 50hours after start of production).

= 28703

= 0.020

P = 400x105 + 28703xEi(-0.020)

Ei(-0.020) = -3.355

P= 400x105 +28703x-3.355

= 400x105 -96300

= 39903700Pa

= 399.0bar

_1096992123.unknown

_1096992854.unknown

_1096992120.unknown

timeradiuspressure

(hours)(m)(bar)

0all400.0

40.15396.4

49.00398.8

450.00399.7

5050.00399.0