Tech Reservoir MBE Equations,Single Phase

PETE 4052 Well Testing Spring 2002Lecture 6 Equations for Single-Phase Porous Media Flow February 22

Equations for Single-Phase Porous Media FlowOutline0. Synopsis1. The Conservation Equation2. Use of Darcy’s Law in the Conservation Equation3. The Case of Small and Constant Compressibility4. The Linear Diffusivity Equation5. Dimensionless Variables6. Other Coordinate Systems and Notation7. Discussion

Related readingCraft et al., Sections 7.5-7.6, p. 231-234.Dake: Sections 5.1-5.4, p. 131-139.Earlougher: Sections 2.1-2.3, p. 4-6.Horne, p. 35.Matthews and Russel: Sections 2.1-2.3, p. 4-7.

6.0 Synopsis

The flow of a single, compressible fluid through porous, permeable rock can be described using a partial differential equation known as the diffusivity equation. Modified forms of the diffusivity equation can be used to describe gas flow. A similar equation can be derived for multiphase flow as well, and that equation is the basis for reservoir simulation. Clearly, the diffusivity equation is at the very heart of reservoir engineering and an intuitive understanding of this equation is essential to all who would do reservoir engineering.

Key Concepts Mass conservation Use of constitutive equations for velocity and density The form of the linearized diffusivity equation Assumptions in the linearized diffusivity equation Dimensionless variables Definitions of steady, pseudosteady, and transient flow Radius of investigation Diffusivity

6.1 The Conservation Equation

Many physical systems – ranging from solar collectors to river deltas to flow in reservoirs – can be analyzed using the principle of conservation. This principle is closely related to the idea of a control volume in thermodynamics; it is based on the idea that the amount of “stuff” (energy, mass, whatever) entering, leaving, created, and destroyed in a given volume must be balanced.

We will derive the conservation equation for a radial flow geometry, because this geometry is especially useful for well testing and inflow analysis. We could do it for any geometry we chose.

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Consider a cylindrical shell of radius r and thickness r (Figure 6.1).

The mass balance for this shell is (Flow in at r ) - (Flow out at r+Δr )= Accumulation between r and r+Δr

Let us consider each of the terms in Eqn. (6.1) in turn:

Flow in at r: Mass Inflow =Area×Superficial Vel .× Density × Time=(2 π rh ) u(r )ρΔt ...........(6.2a)

Flow out at r+r: [2 π (r+ Δr ) h ] u(r+Δr ) ρΔt ...............................................(6.2b)

Accumulation: (2 πrΔ rh φρ )t+ Δt−(2πrΔ rh φρ )t ............................................(6.2c)

Combining Eqns. (6.1) and (6.2), (2π rh ) u(r ) ρo Δt−2 π (r+Δr )hu (r+Δr ) ρo Δt=(2 πrΔ rh φρ)t +Δt−(2 πrΔrh φρ )t ........(6.3a)

Dividing by 2 πΔ rh Δt ,ru(r ) ρo(r )−(r+Δr )u(r+Δr ) ρo (r+Δr )

Δr=

(r φρ )t+Δt−(r φρ )tΔt ..................(6.3b)

If we take the limit as r and t go to zero, then we get,

− ∂∂r

(rρu )=r ∂φρ∂ t .......................................................(6.4)

Equation (6.4) is the conservation equation in radial coordinates. It states that the sum of the partial differential derivatives in r and t is zero. This is also known as a divergence equation; all conservation equations (for any quantity, in any coordinate system) can be expressed in a form very similar to Equation 6.4.

This equation must be manipulated further to be useful: it includes dependent variables , , and u, whereas we really want an equation in p only. We will use constitutive equations for these quantities to get the desired equation.

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Figure 6.1 – Radial control volume. The area of the outer face is 2π (r+ Δr ) h . The volume of the shell is

exactly π [ (r+Δr )2−r2] h . If r is small, this is approximately equal to 2 πrΔ rh . The pore volume can be computed simply by multiplying by the porosity.


6.2 Use of Darcy’s Law in the Conservation EquationWe will begin by using the differential form of Darcy’s Law in the conservation equation. In differential form, Darcy’s Law is

u=− kμ

∂ p∂r .............................................................(6.5)

Substituting Eqn. (6.5a) into the conservation equation [Eqn. (6.4)],

−∂∂ r [rρ(−k

μ∂ p∂r )]=r∂ φρ

∂ t

1r

∂∂r [rρ(kμ ∂ p

∂r )]=∂φρ∂ t ...............................................(6.6)

This gets us closer to the desired form, but the density and porosity still must be handled.

6.3 The Case of Small and Constant Compressibility

We will assume that the density and porosity are exponential functions of pressure:ρ( p )=ρRexp [ co( p−pR ] ................................................(6.7aφ ( p )=φR exp [c f ( p−pR ] ...............................................(6.7b)

where

co=1ρ

∂ ρ∂ p ...........................................................(6.8a)

and

c f =1φ

∂ φ∂ p ...........................................................(6.8b)

Substituting Eqns. (6.7) into Eqn. (6.6),1r

∂∂ r [rρR exp [co ( p− pR ]( k

μ∂ p∂ r )]= ∂

∂ t {ρR exp [co ( p−pR ] φR exp [c f ( p− pR ] }......(6.9a)

Expanding the left-hand side of Eqn (6.9a),

∂∂ r [rρRexp [ co( p− pR ](k

μ∂ p∂ r )]

¿(r kμ

∂ p∂ r )∂∂ r {ρR exp [co ( p−pR ]}+ρRexp [ co( p−pR ]∂∂r (r k

μ∂ p∂r )

¿ ρR coexp [co ( p− pR ]∂ p∂ r (r k

μ∂ p∂ r )+ρR exp [co( p− pR ]∂∂ r (r k

μ∂ p∂ r )

¿ ρ {co(r kμ )(∂ p

∂r )2+∂∂ r (r k

μ∂ p∂ r )}

............(6.9b)Similarly, expanding the right-hand side of Eqn. (6.9a),

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∂∂ t { ρR exp [co( p− pR ] φR exp [c f ( p−pR ]}=

ρR φR {(co+c f )exp [co ( p− pR ]exp [ c f ( p−pR ] }∂ p∂ t

¿φρ(c0+c f )∂ p∂ t ............................(6.9c)

Recombining Eqns. (6.9b) and (6.9c),

ρr {co (r k

μ )(∂ p∂r )

2+∂∂r (r k

μ∂ p∂ r )}=φρ(c0+c f )

∂ p∂ t

co (kμ )(∂ p∂ r )

2+1

r∂∂r (r k

μ∂ p∂r )=φ(c0+c f )

∂ p∂ t

co (kμ )(∂ p∂ r )

2

+1r

∂∂r (r k

μ∂ p∂r )=φct

∂ p∂ t ..........................(6.10)

where c t=(c0+c f ) . For systems with connate water and/or immobile gas, the definition of total compressibility can be extended to be

c t=c f+So co+Sw cw+Sg c g ............................................(6.11)Notice that the only dependent variable is now pressure; however, the porosity is still a function of pressure.

6.4 The Linearized Diffusivity EquationAlthough Equation (6.10) is in pressure, it is nonlinear. It is very difficult to solve nonlinear partial differential equations, and we therefore seek a simplified, linear form to work with.

The nonlinearity comes from two different sources.

The gradient-squared term: co(r k

μ )(∂ p∂ r )

2

The porosity term: φc t

∂ p∂ t

In many systems, the porosity varies slowly and it can be replaced by it value at the average pressure. This is usually adequate.

The gradient-squared term is often small because (1) the gradient is small and (2) the oil compressibility was assumed to be small. If these assumptions are made (see discussion), then the linearized diffusivity equation can be written as

1r

∂∂ r (r k

μ∂ p∂r )=φct

∂ p∂ t ...............................................(6.12a)

If k/ is constant,1r

∂∂ r (r ∂ p

∂r )=φμ c t

k∂ p∂ t ...............................................(6.12b)

This equation is very important! It will form the basis of our study of inflow and pressure transient analysis (or well testing).

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6.5 Dimensionless VariablesWe used the concept of a dimensionless variable when we discussed the skin factor. We will extend that discussion now to better understand the linearized diffusivity equation. It seems sensible to make radius dimensionless on the wellbore radius,

r D=rrw

r=r Dr w

∂∂ r

=drD

dr∂∂ r D

=1r w

∂∂ rD .................................................(6.13)

With this substitution, the diffusivity equation is1r w r D

1rw

∂∂r D (r w r D

kμ

1rw

∂ p∂r D )=φc t

∂ p∂ t

1r D

∂∂r D (rD

kμ

∂ p∂ rD )=φc t r w

2 ∂ p∂ t ....................................(6.14)

This definition needs no unit conversions; it works in consistent and field units. If we assume k and are constant,

1rD

∂∂ r D (rD

∂ p∂r D )=φμ ct rw

2

k∂ p∂ t ...........................................(6.15)

As for our case with skin, we will define the dimensionless pressure as

pD= 2π khqμB

( pi−p )..................................................(6.16a)

in consistent units or

pD=0 . 00708 khqμB

( pi−p)..............................................(6.16b)

in field units. Now the differential equation is1rD

∂∂ r D

(rD

∂ pD

∂ r D)=φμ ct rw

2

k∂ pD

∂ t ..........................................(6.17)(We can choose the definition of dimensionless pressure pretty freely because p appears in all terms in the equation; the reasons for this choice should be clear from the form of Darcy’s Law and we will discuss it more later in the course). Because all other terms are dimensionless, the definition of dimensionless time can be deduced from Eqn. (6.17):

tD= ktφμ c t rw

2........................................................(6.18a)

in consistent units or

tD=0.0002637 ktφμ c t r w

2....................................................(6.18b)

in field units, t is given in hours – if it were in days, the conversion factor is 24 times larger.

Our final, dimensionless form of the equation is1

rD

∂∂ r D

(rD∂ pD

∂ r D)=∂ pD

∂ tD ................................................(6.19)

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This form of the equation is simpler than earlier forms, because there are only variables and no parameters in it. The most important thing about dimensionless variables is the way they scale the solution. Regardless of the values of parameters such as k, , , ct, q, and so on….the dimensionless pressure at the same (rD,tD) is the same for all infinite-acting radial flow problems.

More generally, if the appropriate scaling is used in bounded or non-radial problems, one can define dimensionless pressures that have this same property (for example, for a fractured well or a well in the center of a square). We need not compute solutions for all different combinations of parameters – the dimensionless solution is the scaled solution that can be used for all of these cases.

6.6 Other Coordinate Systems and NotationIn linear coordinates, the diffusivity equation is

∂2 p∂ x2=

φμ c t

k∂ p∂ t .......................................................(6.20)

For any coordinate system, one can write the diffusivity equation in terms of the divergence, which some of you may be familiar with

∇2 p=φμ c t

k∂ p∂ t .......................................................(6.21)

DiscussionAssumptionsThe steps and assumptions used to derive the linearized diffusivity equation are summarized in Table 6.1, below:

The assumptions are very important to be familiar with. Study this table! In particular, consider the following: The compressibility of gas may not be small or constant. Thus, this form of the diffusivity

equation may not be accurate. We will consider an alternative form, in real gas potential (or pseudopressure) later in the course.

The assumption that the product of the compressibility times the square of the gradient term is small may not be valid (1) for gas or (2) near the well, where the gradients are high. These problems can also be addressed using real gas potential.

Unconsolidated rocks may have very high pore compressibilities, cf. This can make the

equation nonlinear, because both and

∂ p∂ t depend on pressure. This problem is not solved

using pseudopressure, but may be reduced (but NOT eliminated) using pseudotime.

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Eqn. Step Assumptions6.4 Conservation Equation h is constant; radial geometry6.6 Constitutive Equation for velocity Darcy's Law holds

6.11 Constitutive Equation for density; nonlinear diffusivity equation

Porosity and fluid density are exponential functions of pressure; the pore and fluid compressibility are constant

6.12a Diffusivity Eqn with variable k, The product fluid compressibility times the square of the gradient of pressure is small

6.12b Linearized diffusivity equation k, and are constant

Table 6.1 Derivation of the Linearized Diffusivity Equation


The mobility k/ may not be constant, especially for gas or highly compressible rocks. This can introduce an additional nonlinearity. Pseudopressure corrects for the viscosity variation, and can be generalized to correct for permeability varying with pressure.

The Steady-State Equation in Dimensionless FormWe previously wrote Darcy’s Law as

q= 2π kh

μB ln ( rr w

)( p(r )−p(rw ) )

We can rearrange this to get

ln ( rr w

)=2 π khqμB [( pi−p(r ))−( p i−p(r w ))]

and introduce the definitions of rD (Eqn. 6.13) and pD (Eqn. 6.16) to getln (rD )=pD(r D)−pD(1 )

If we use a special definition of pD that is sensible for steady-state systems (where pw is constant),

pD=2π khqμB

( p−pw )

Then the dimensionless steady-state equation is simplypD(r D)=ln(r D) .......................................................(6.22)

The dimensionless form is simpler for steady state systems, as well. As we will see later, we can get this equation by solving the linearized diffusivity equation.

Diffusivity

The group

kφμ ct appears in the diffusivity equation in all coordinate systems. It is known as the

diffusivity and is usually represented with the symbol or “eta”:

η= kφμ ct ...........................................................(6.23a)

or, in field units,

η=0 .0002637 kφμ ct

[ft2

hr ].............................................(6.23b)

The diffusivity is important because it controls the speed with which pressure information travels through the reservoir. In general, higher values of mean more rapid transmission of information; you can see this because the diffusivity is multiplied by t (and divided by the square of the wellbore radius) to get the dimensionless time. Thus, at same dimensional time a reservoir with higher diffusivity will have a greater dimensionless time – things happen faster. Increasing permeability has the same effect as decreasing any of the other three parameter (, , ct). Consider why Higher mobility, k/ causes more rapid pressure transmission Higher porosity and compressibility cause slower transmission (ct is also known as the

storativity).

Some Basic CasesSeveral reservoir systems of interest can be classified based on the value of the right-hand side of Eqn. 6.19:

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∂ pD

∂ t D=0

: Steady state, the pressure is not changing with time

∂ pD

∂ t D=constant

: Pseudosteady state, usually occurs when rate is changing slowly and compressibility is nearly constant.

∂ pD

∂ t D=f (rD , tD )

: Transient flow.

Radius of Investigation

In transient flow, the radius of investigation is a useful concept (Horne, p. 35). As we will see later, that the solution of the partial differential equation predicts that there is an infinitesimally small pressure change everywhere, instantly. However, from a physical and practical point of view, the pressure changes propagate more slowly. One definition of the radius of investigation is that, at any time, it is the closest boundary that would have had a detectable effect at the well. For an infinite-acting well, this gives

rinv=0 . 03√ktφμ c t

¿0 .03√ηt ....................................................(6.24a)With our previous definition of tD, Eqn. (6.18b),

rinv=0 .03√kφμ c t

φμ ct rw2 t D

0 . 0002637 kr Dinv=1.8√t D .........................................(6.24b)

Sometimes an alternative definition of dimensionless time is used, based on area rather than wellbore radius

tDA=0 . 0002637 kt

φμ c t A ...................................................(6.25a)For a well in the center of a circle,

tDA=0 . 0002637 kt

φμ c t πre2

=8 .938×10−5 ktφμ c t re

2..................................(6.25b)

As we will see later, boundaries affect this system when tDA>0 . 1 so

0 .1=8 .938×10−5 ktφμ ct rinv

2..............................................(6.25c)

Solving for rinv,

rinv=√8 . 938×10−5

0 . 10ktφμ c t r inv

2

¿0 .02897√ktφμ ct ..............................................(6.26)

Eqn. (6.26) is about the same as Eqn. (6.24a), which was obtained by a much more complicated analysis using derivatives of Green’s functions.

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A couple of things are worth noting: We see (Eqn 6.24b) that diffusivity controls the radius of investigation. Radius of investigation increases with the square-root of time; this behavior is characteristic

of solutions to the diffusivity equation. The time to investigate near-well regions is typically too small to be well-represented in data

(see below), whereas in low-permeability reservoirs it may take a long time to detect boundaries (see below).

Example 1: Radius investigated in one minute in a good-quality reservoir.Assuming k = 100 md, = 0.25, ct = 1010-6/psi, = 0.8 cp, and t = 1/60 hr, rinv is

rinv=0 .03√100 (1/60)(0.25 )(0 .8 )(10×10−6)

=27 ft. Thus, even after 1 minute, the part of the reservoir

being investigated is far beyond the skin region. Wellbore weirdness (storage and temperature transients on the pressure gauges) will usually obscure these data.

Example 2: Radius investigated in 1 week in a low-permeability, low-pressure gas reservoir.Assuming k = 0.010 md, = 0.05, ct = 20010-6/psi, = 0.05 cp, and t = 168 hr, rinv is

rinv=0 .03√ 0 .010(168 )(0.05 )(0 .05 )(200×10−6 )

=55 ft. So after one week, only 55 feet have been

investigated. Because such tight gas reservoirs are usually completed on large, it is difficult to detect boundaries 100’s or 1000’s of feet away.

The difference in these two time scales is solely due to the differences in diffusivity, . Example 1 is high-diffusivity, example 2 is low-diffusivity.

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Tech Reservoir MBE Equations,Single Phase

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