line source solution

Well Test Analysis, © UTP JAN 2012

Line Source Solution


Infinite Cylindrical Reservoir With Line-Source Well

Assume that

(1) a well produces at a constant rate, qB

(2) the well has zero radius

(3) the reservoir is at Uniform pressure, Pi Before production

begins; and

(4) the well drains an infinite area (p → pi ; as r→ ∞

Under those conditions, the solution to the diffusivity equation is.

Where P is the pressure at a distance r from the well and t.

and the Ei function or exponential integral.


The Ei-function solution is an accurate approximation to the more

exact solution for time

The times less than ( left hand side), the assumption of zero well size( assuming the well line source or sink) limits the accuracy of the equation.

At times greater than ( right hand side), the reservoir's boundaries begin to affect the pressure distribution in the reservoir, so that the reservoir is no longer infinite acting.

A further simplification of the solution to the flow equation is possible: for X<0.02, can be approximated with an error less than 0.6% by

To evaluate Ei, table is used for 0.02<X≤10.9

for X≤0.02, natural logarithm approximation is used

For X>10.9 , Ei function can be considered zero for application in well testing


In practice, we find that most wells have reduced permeability (damage) near the wellbore resulting from drilling or completion operations. Many other wellsare stimulated by acidization or hydraulic. So theequation for calculating (P) fails to model such wellsproperly; its derivation holds the explicit assumptionof uniform permeability throughout the drainage areaof the well up to the wellbore. if the damaged orstimulated zone is considered equivalent to an altered

zone of uniform permeability (ks) and outer radius (rs), the

additional pressure drop across this zone ∆ps can bemodeled by steady state radial flow equation. According to the following figure :


This equation simply states that the pressure drop in the

altered zone is inversely proportional to ks rather

than to k and that a correction to the pressure drop in

this region (which assumed the same permeability, k,

as in the rest of the reservoir) must be made.

The total pressure drop at the well-bore can calculated

by combining the last two equations.


If a well is damaged (ks < k) , s will be positive, and the greater the contrast between ks, and k and the deeper into the formation the damage extends, the larger thenumerical value of s. There is no upper limit for s.Some newly drilled wells will not flow at all before stimulation; for these wells, ks=0 and s→∞. If a well is stimulated (ks >k), s will be negative, and the deeperthe stimulation, the greater the numerical value of s. when ks=k then s=0.An altered zone near a particular well affects only the pressure near that well - i.e., the pressure in the unaltered formation away from the well is not affected by the existence of the altered zone. Said another way, we use Eq. with skin to calculate pressure at the sand face of a

pressures at the sand-face of a


well with an altered zone, but we use Eq. with no skin tocalculate pressures beyond the altered zone in theformation surrounding the well. Example: - Calculation of Pressures Beyond the Wellbore Using the Ei-Function SolutionA well and reservoir have the following characteristics:The well is producing only oil; it is producing at aconstant rate of 20 STB/D. Data describing the well andformation areμ = 0.72 cp,k = 0.1 md,ct = 1 . 5 x 1 0E-5 psi-1

pi = 3,OOOpsi,re = 3,00Oft,rw, = 0.5 ft,B, = 1.475 RB/STB,h = 150 ft,Φ = 0.23, ands = 0.


Calculate the reservoir pressure at a radius of 1 ft after 3 hours of production; then, calculate the pressure at radii of 10 and 100 ft after 3 hours of production.

Solution: The Ei function is not an accurate solution

to flow equations until t >3.79 x 105 Φμctrw2/k

Then: = 2.35 hrs which is less than (t=3hrs)

So Ei solution could be used with satisfactory accuracy if the reservoir is still infinite acting at this time. The reservoir will act as an infinite reservoir until t<948 ,here is=211,900, thus for any time less than 211900 the reservoir is still infinite acting.Now the pressure at r=1ft is:


=3000+(100)Ei(-0.007849)

P=3000+100ln(1.781x0.007849)=2573psi

At a radius of 10 ft,

P=3000+100

=3000+100E(-0.7849)=3000+100(-0.318)=2968psi

In this calculation, we find the value of the Ei

function from a table . Note, as indicated in the

table, that it is a negative quantity.

At a radius of 100 ft,

=3000+100

p=3000+100Ei(-78.48) = 3000


Values of exponential integral -Ei (-x)


Pseudo-steady-State Solution

Actually, this solution (the pseudo-steady-state) is simply a limiting

form solution of diffusivity equation requires that we specify two

Boundary conditions and an initial condition. A realistic and

practical solution is obtained if we assume that

1-a well produces at constant rate, qB, into the wellbore

(q refers to flow rate in STB/D at surface conditions, and

B is the formation volume factor in RB/STB);

(2) the well, with wellbore radius rw, is centered in a cylindrical

reservoir of radius re, and that there is no flow across this outer

boundary; and

(3) before production begins, the reservoir is at uniform pressure, p i.

The most useful form of the desired solution relates flowing

pressure, pwf, at the Sand-face to time and to reservoir rock and

Fluid Properties. The solution is:


which describes pressure behavior with time for a well centered in a cylindrical reservoir of radius re.

The limiting form of interest is that which is valid for large times, so that the summation involving exponentials and Bessel functions is negligible; after this

t>948ɸµcre2/k

Or


Note that during this time period we find, by differentiating the last equation with respect to time we get

Since the liquid-filled pore volume of the reservoir, Vp (cubic feet), is

then

Thus, during this time period, the rate of pressure decline is inversely proportional to the liquid-filled pore volume Vp. This result leads to a form of well testing sometimes called reservoir limits testing, which seeksto determine reservoir size from the rate of pressure decline in awellbore with time.


This solution is useful also for some other applications.

It involves replacing original reservoir pressure, p i, with average pressure, within the drainage volume of the well. The volumetric average pressure within the drainage volume of thewell can be found from material balance.

The pressure decrease (pi- ) resulting from removal of qB (RB/D )of fluid for t hours.A total volume removed of 5.615 qB(t/24) cu) is equal to

Substituting in the main equation yields

P

P


Or

The main equation and the final equation after

counting for skin becomeOr

and


Further we can define average permeability(kJ)

such that:

This average permeability, kJ proves to have considerable value in

well test analysis, as we shall see later. Note that for a damaged well,

the average permeability kJ is lower than the true, bulk formation

permeability k in fact, these quantities are equal only when the skin

factor is zero. Since we sometimes estimate the permeability of

a well from productivity-index (PI) measurements, and since the

productivity index J (STB/D/psi), of an oil well is

defined as


this method does not necessarily provide a good estimate of formation permeability, k. Thus, there Is a need for a more complete means of characterizing a

producing well than exclusive use of PI information.Example:A well produces 100 STB/D oil at a measured flowing

bottom-hole pressure (BHP) of 1,500 psi. A recent pressure survey showed that average reservoir pressure is 2,000 psi. Logs indicate

a net sand thickness of 10 ft. The well drains an areawith drainage radius, re, of 1,000 ft; the boreholeradius is 0.25 ft. Fluid samples indicate that, at current reservoirpressure, oil viscosity is 0.5 cp and formationvolume factor is 1.5 RB/STB.


1. Estimate the productivity index for the tested well.

2. Estimate formation permeability from these data.

3. Core data from the well indicate an effective

permeability to oil of 50 md. Does this imply that

the well is either damaged or stimulated? What

is the apparent skin factor?

Solution:


3-Core data frequently provide a better estimate

of formation permeability than do permeabilities

derived from the productivity index, particularly

for a well that is badly damaged. Since cores

indicate a permeability of 50 md, we conclude

that this well is damaged. so


Radius of investigation:The radius-of-investigation concept is of both quantitative and qualitative value in well test design and analysis. By radius of investigation, ri, we mean the distance that a pressure transient has moved into a formation following a ratechange in a well.We will show that this distance is related to formation rock and fluid properties and time elapsed since the rate change. If we consider a well producing from a formation originally at 2000 psi.Both, well and formation have the following characteristics:


And calculate the pressure distribution at different radius at production time of ( 0.1,1,10,100) hrs as its shown in the above figure.Two observations are particularly important:1. The pressure in the wellbore (at r=rw)decreases steadily with increasing flow time; likewise,pressures at other fixed values of r also decrease withincreasing time.


2. The pressure disturbance (or pressure transient) caused by producing the well moves further into the reservoir as flow time increases. For the range offlow times shown, there is always a point beyondwhich the drawdown in pressure from the originalvalue is negligible.Now consider a well into which we instantaneouslyinject a volume of liquid. This injection introduces apressure disturbance into the formation; thedisturbance at radius ri will reach its maximum attime tm , after introduction of the fluid volume. Weseek the relationship between ri and tm.


Stated another way, in time t, a pressure disturbance reaches a distance ri, which we shall call radius of investigation, as given by the equation

The radius of investigation given by the aboveequation proves to be the distance a significant

pressure disturbance is propagated by production or injection at a constant rate. For example, for the formation with pressure distributions shown in the above figure, the application of the above equation yields the following results.


Comparison of these results with the pressure distributions plotted

shows that ri as calculated from the equation is near the point at

which the drawdown in reservoir pressure caused by producing the

well becomes negligible. We also use the same equation to calculate

the radius of investigation achieved at any time after any rate change

in a well. This is significant because the distance a transient has

moved into a formation is approximately the distance from the well

at which formation properties are being investigated at a particular

time in a well test. The radius of investigation has several uses in

pressure transient test analysis and design. A qualitative use is to

help explain the shape of a pressure buildup or pressure drawdown

curve. For example, a buildup curve may have a difficult-to interpret

shape or slope at earliest times when the radius of investigation


is in the zone of altered permeability, ks nearest the well bore. Or more commonly, a pressure buildup curve may change shape at long times when the radius of investigation reaches thegeneral vicinity of a reservoir boundary, or somemassive reservoir heterogeneity. In practice, we find that a heterogeneity orboundary influences pressure response in a wellwhen the calculated radius of investigation is of the order of twice the distance to theheterogeneity.The radius-of-investigation concept provides a


guide for well test design. For example, we may want to sample reservoir properties at least 500 ft from a tested well. How long a test shall be run?Six hours? Twenty-four hours? We are not forcedto guess – or to run a test for an arbitrary lengthof time that could be either too short or too long.Instead, we can use the radius-of-investigationconcept to estimate the time required to test tothe desired depth in the formation. The radius ofinvestigation equation also provides a means ofestimating the length of time required to achieve

"stabilized" flow (i.e., the time required for


a pressure transient to reach the boundaries of a tested reservoir).

For example, if a well is centered in a cylindrical drainage area of Radius re, then, setting ri=re, the time required for

stabilization, ts, is found to be

Example:We wish to run a flow test on an exploratorywell for sufficiently long to ensure that the well willdrain a cylinder of more than 1,000-ft radius.Preliminary well and fluid data analysis suggest thatk= 100 md, Φ=0.2, ct =2x 10-5 psi, and μ=0.5 cp. What length flow test appears advisable?What flow rate do you suggest?


Solution:

The minimum length flow test would propagate a pressure transient approximately 2,000 ft from

the well (twice the minimum radius of

Investigation for safety). Time required is

line source solution

Documents