Unsteady-State and Pseudosteady-State Flow

Unsteady-State and Pseudosteady-State FlowBY :NARUMI NURUL ANNISAYUMAS CITRA RESMIWINDYA PUTRIRIKA BUDI NOVIAWATICHYNTIA BILQISHI KADEK GERARDAVESYAIFUL AZIZGUSRIANDIMAS IRFANA

DefinitionUnsteady- state flow is flow that occurs while the rates and/or the pressure are changing with time. Consequently, it covers all reservoir flow, except for the specific situation when the rates and pressure do not change with time. A complete understanding of unsteady- state flow is necessary for any competent reservoir engineer. It is the backbone of such useful techniques as pressure-buildup analysis, drawdown analysis, interference tests, reservoir boundary delineation, prediction of water encroachment, prediction of disposal- well behavior , gas-well tests, and drilldtem-test (DST) analysis

Physical DescriptionThe time effect of the production and the fact that it takes some period of time before the entire reservoir is affected are one of the concepts with which engineers have the most trouble. Consequently, we will try to examine this concept in different ways to gain a physical understanding of the phenomena.Physically, this procss requires time so the pressure effect can move across the reservoir. Note that as the pressure effect moves out into the reservoir, it continues to have a smaller effect on the pressure of each subsequent reservoir segment as the segment raii increase. The increase in radius causes an increase in the segment size, and thus, a larger amount of fluid withdrawal is required to obtain the same pressure drop. Also,note that as the radius increases, the crossectional area, 2rh or A in the darcy equation, increases and the pressure gradient decreases correspondingly.

Radial Diffusivity EquationWhereas steady-state flow is governed by very simple equations, unsteady-state flow is best described by a partial- differential equation known as the radial diffusivity equation:

When written in a finite- difference form, the equation is much more meaningful to most engineers:

Constant- terminal-rate solutionThe constant- terminal rate solution is by far the most useful of the two general solutions to the radial diffusivity equation. It is an integral part of pressure- buildup analyses, interference tests, drawdown analyses, isochronal tests of gas wells, and many other important reservoir analysis techniques.Pseudosteady-state flowPseudosteady-state flow is actually the finite acting portion of the constan- rate solution to the radial diffusivity equation. Thus, it canbe analyzed by the same pressure- function solution described for the constant- rate solution. However, this case of the constant-rate solution lends itself to much simpler equations and metods so it is importanat that we look at it as a special case. Although this flow is recognized as a special case, it appears that most reservoirs spend more of their history in the pseudosteady state than in any other flow regime.In the constant-rate solution the rate is known to be constant at some part of the reservoir, and the pressure calculated throughout the reservoir. Conversely, in the constant-terminal-pressure solution the pressure is known to be constant at some point in the reservoir and the cumulative flow at any particular time across the subject radius can be evaluated.The constant-pressure solution takes the form of a relationship between the reduced time and the reduced cumulative flow, Q.

Constant-Terminal-Pressure Solution

Recognizing that for a constant P the rate,q, varies. We could define the cumulative flow at any time as ;

Thus, comparing Eqs 3.51 and 3.55, we see that :

Remember that Eqs 3.56 must be applied by superposition. Consequently, The relationship is not as simple as it may appear.

Constant-Terminal-Pressure SolutionThe application of the constant-pressure solution is similar to that of constant-rate data. The reduced time is based on the radius at which the cumulative flow is desired and at which the constant pressure is known. The reduced time values also incorporates the time which the cumulative flow is desired.

Constant-Terminal-Pressure SolutionCompressibility has previously referred to as simply water, oil, or gas compressibility.By referring to the derivation of the radial diffusivity equation, we see that value required for the compressibility of a system is the amount of fluid forced from a unit pore volume for 1.0 psi of pressure drop. If the pore volume is occupied 100% by one fluid and remains constant, we would evaluate only the compressibility of the one fluid, which would be the effective compressibility.

Effective CompressibilityAs the reservoir pressure is depleted by production, the rock pressure or overburden pressure remains unchanged. Although the overburden is supported almost entirely by the mechanical strength of the rock, it is also supported to a small extent by the pressure in the pores

Effective CompressibilityThe formation compressibility should not be confused with the change in pore volume that often accompanies the production from a reservoir with an initial reservoir pressure similar to the rock pressure. The pore volume change usually equal to the production. This phenomena usually is observed in reservoir gas.Effective CompressibilityIf we were limited in our practical application of the constant-rate and constant-pressure solutions to situations in which we simply had one well producing from a reservoir at a constant pressure, the two solutions to the radial diffusivity equation would hardly be worthwhile. However, using the concept of superposition, we can account for the effects of producing from more than one well and for the effects of rate and pressure changes. We can even extend the use of infinite-acting solutions by artificially setting up boundary conditions. The concept of superposition is not the product of reservoir engineering. The same technique is applied to many equations governing energy mass transfer.

Superposition

SuperpositionLinear Unsteady-State FlowLinear unsteady flow fundamentals are seldom needed in petroleum reservoir engineering. Common practice is to produce low permeability gas reservoirs using a very large artificial fracture treatmen known as MHF. The fracture provides a combination of circumtances that leads to linear unsteady state flow when the well is first to put on production or is shut in.

Spherical or hemispherical flow often prevails a long enough period so flow geometry characteristics may be discornible and in many cases useful. Spherical or hemispherical flow occurs when only a small portion of the reservoir thickness is perforated. This situations may result when botton water or gas cap gas is to be avoided. Use of a sidewall fluid sample that involves one perforation may also result in spherical flow.Spherical Unsteady State Flow