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# Flow equations for gas and multiphase flow

The standard fluid flow equations require modification when the reservoir content is gas or multiphase flow is occurring. This article discusses those modifications and the equations to use for gas or multiphase reservoirs.

## Diffusitivity equation for gas flow

The diffusivity equation for liquids, Eq. 1, ....................(1)

was derived from three principles: conservation of mass, the equation of state for slightly compressible liquids, and Darcy’s law. This form of the diffusivity equation is linear, which makes solutions (such as theEi-function solution) much easier to find and which allows us to use superposition in time and space to develop solutions for complex flow geometries and for variable rate histories from simple, single-well solutions.

## Pseudopressure

Other forms of the equation for flow of gases must be developed because the equation of state for a slightly compressible liquid will not be applicable. First, introducing the real gas law, ....................(2)

to replace the slightly compressible equation of state results in a more complex, nonlinear partial differential equation. This equation can be partially linearized by introducing the pseudopressure transformation, ....................(3)

where p0 is an arbitrary "base" pressure, frequently chosen to be zero psia. The resulting form of the diffusivity equation is ....................(4)

Eq. 4 has the same form as the diffusivity equation for slightly compressible liquids, with pressure replaced by pseudopressure, pp. However, this equation is nonlinear because the product μct is a strong function of pressure. Fortunately, research has shown that the equation can be treated as linear, and theEi-function is valid for gases if μct is evaluated at the pressure at the beginning of a flow period until the time when boundaries begin to have a significant influence on the pressure drop at the well; that is, as long as the reservoir is infinite-acting.

## Pressure-squared and pressure approximations

By assuming that the product μz is constant, then, from Eq. 3, pseudopressure becomes ....................(4)

and the diffusivity equation becomes ....................(5)

The independent variable has become p2, and, in terms of this variable, theEi-function solution is valid when the assumption that μz is constant is valid. This is true (based on empirical evidence) even though Eq. 5 is nonlinear (pressure-dependent μct), but it is valid only for an infinite-acting reservoir.

Fig. 1 shows the range of validity of this assumption for a reservoir temperature of 200°F and several different gas gravities. The μz product is fairly constant at pressures below approximately 2,000 psia (the shaded area in the figure). Conclusions are similar at other temperatures from 100 to 300°F.

By assuming that the group p/μz is constant, from Eq. 3, pseudopressure becomes ....................(6)

and the diffusivity equation becomes ....................(7)

The independent variable has become p, and, in terms of pressure, theEi-function is valid (from empirical evidence) when the assumption that p/μz is constant is valid. This is true even though Eq. 7 is nonlinear (pressure-dependent μct) , but is valid only for an infinite-acting reservoir.

Fig. 2 shows the range of validity of this assumption (shaded area in the figure) for a reservoir temperature of 200°F and several different gas gravities. The group p/μz is fairly constant at pressures above approximately 3,000 psia as it is at other temperatures from 100 to 300°F.

The implication of these results is that the choice of variable for gas well-flow equations depends on the situation. The pressure-squared approximation is valid only for low pressures (p < 2,000 psia), the pressure approximation is valid only for high pressures (p > 3,000 psia), and the pseudopressure transformation is valid for all pressure ranges. For pressure transient test analysis using software, the pseudopressure is almost always the optimal variable to use. For hand analysis, only pressure or pressure-squared approaches are feasible.

## Pseudotime

Although the diffusivity equation written for gas flow has the same form as the diffusivity equation for slightly compressible liquids, with pressure replaced by pseudopressure, it is a nonlinear equation because the product, μct, is strongly pressure dependent. In some cases, the remaining nonlinearity cannot be ignored. To solve this problem, Agarwal introduced the pseudotime transformation to further linearize the diffusivity equation for gas. (The linearization is not rigorous, but is adequate for many practical purposes. ) The definition of pseudotime is ....................(8)

In terms of pseudotime, tap, the diffusivity equation becomes ....................(9)

Subsequent studies have shown that the pseudotime transformation is particularly useful for analysis of flow and buildup tests distorted by wellbore storage when using type curves designed to model flow of slightly compressible liquids.

Because the pressure in the integrand of Eq. 8 is a function of position in the reservoir, it is not obvious where the pressure is to be evaluated. Empirical observations indicate that the pressure should be evaluated at BHP during wellbore storage distortion for both buildup and flow tests. During the middle time region for buildup tests, it should be evaluated at BHP, and, for flow tests, at the average reservoir pressure at the start of the test. For flow tests in infinite-acting reservoirs, this is equivalent to using ordinary time as the independent variable.

## Normalized transformed variables

The pseudopressure and pseudotime transformations provide excellent results when used as part of the analysis procedure for gas well tests. However, they are inconvenient for two reasons: the values of both variables will often be in the range of 105 to 109, and they do not have units of actual pressure and time. Thus, the intuitive "feel" for the transformed variables is lost, and they may tend to be regarded as "black box" output—never helpful in test analysis. The use of pseudopressure and pseudotime require different test interpretation equations for oil wells than for gas wells.

These difficulties are overcome by normalizing pseudopressure and pseudotime by multiplying them by constants: ....................(10)

and ....................(11)

This normalization, or multiplication by appropriate constants, gives the new variables the same units—and similar ranges—as pressure and time, respectively. With these transformations, the equations for analysis of gas wells in terms of normalized pseudopressure and pseudotime, which are called adjusted pressure and adjusted time, are obtained from the equations for analysis of oil well tests by simple substitution. Of course, the transformations require the computer. Commercial well-test analysis software often provides these transformations.

Table 1 summarizes plotting methods and interpretation equations for oil well tests. It also presents information for gas well tests analyzed with ordinary pressure and time, adjusted pressure and time, pressure squared and time, and, finally, pseudopressure and time. The table includes a definition of pDMBH, a dimensionless pressure defined by Matthews, Brons, and Hazebroek that is useful in estimating current average drainage pressure. See this topic in Estimating average reservoir pressure from diagnostic plots.

In Table 1, the Horner time ratio (HTR) for gas well buildup tests is best estimated to be simply (tp + Δta)/Δta. This conclusion is based on the findings of Spivey and Lee.  Thus, when using adjusted pressure and time, the HTR is calculated using the actual producing time,tp.

## Non-Darcy flow

The flow equations shown to this point assume that Darcy’s law is an appropriate model for gas flow into wells. However, as the flow velocity and Reynolds number near the well increase, the result is a transition from laminar and turbulent flow and then to turbulent flow. This transitional (and possibly turbulent) flow is called non-Darcy (non-laminar) flow. The high velocities at which the flow is transitional occur in the immediate vicinity of the well, and the additional pressure drop caused by this transitional flow is similar to a zone of altered permeability that is characterized with a skin factor. In the case of non-Darcy flow, however, the additional "skin effect" caused by the deviations from Darcy’s law is rate dependent.

An adequate model for the apparent skin factor, s′, determined from a flow or buildup test is ....................(12)

In Eq. 12, s is the "true" skin because of damage or stimulation; D is a non-Darcy flow coefficient (assumed constant), with units of D/Mscf; and qg is the gas flow rate with units of Mscf/D. The absolute value of the gas rate is used because the contribution to the skin is positive regardless of whether the gas well is a producer or an injector.

The true skin for a gas well cannot be obtained from information in a single test conducted at constant rate (including a buildup test following constant-rate production). However, skin calculated from tests conducted at several different rates (for example, associated with a multipoint deliverability test on a well) can be used to determine the true skin and the non-Darcy flow coefficient. Fig. 3 illustrates the process for a well tested at three different rates, with an apparent skin factor determined at each rate.

The apparent skin factor extrapolated to zero rate is the true skin (in this case, 3.4), and the slope of the curve is the non-Darcy flow coefficient, D (in this case, 5.1×10–4 D/Mscf). When this method is used, take care to ensure that the permeabilities obtained from the different tests are the same; otherwise, the skin factors will be inconsistent and erroneous.

Often, only one test is available. In this case, the non-Darcy flow coefficient, D, can be estimated from ....................(13)

The turbulence parameter, β, can be estimated from ....................(14)

The correlation represented by Eq. 14 will provide only a crude estimate of the turbulence parameter, β. Further, the correlation assumes that the non-Darcy flow occurs in the formation near the wellbore rather than through the perforations. In a gravel-packed well, the most significant additional pressure drop caused by non-Darcy flow may occur in the perforation channels through the casing.

## Multiphase flow

The equations modeling flow in reservoirs can be modified to include multiphase flow. Perrine suggested simple and easily applied modifications and Martin gave them a theoretical basis. These modifications are based on the simplifying assumption that the saturation gradients in the drainage area of the tested well are small. Thus, as examples, the modifications may lead to reasonable approximations for solution-gas drive reservoirs and are inappropriate for water-drive reservoirs with a water bank (and saturation discontinuity) in the drainage area of the tested well. The Perrine-Martin modification for constant-rate flow in an infinite-acting reservoir is ....................(15)

and the Horner equation modeling a buildup test in an infinite-acting reservoir becomes ....................(16)

In Eqs. 15 and 16, qRt represents the total reservoir flow rate (RB/D) and is given by ....................(17)

and λt represents the total mobility, given by ....................(18)

The total mobility, λt, can be determined from a pressure buildup test run on a well that produces two or three phases simultaneously. Because Eq. 16 implies that λt is related to the slope, m, of a Horner plot of pws vs. log(tp+ Δt)/Δt by ....................(19)

The slope, m, of a plot of pwf vs. log(t) data from a constant-rate flow test has the same interpretation. Perrine also showed that the permeability to each phase flowing can be estimated from the relations ....................(20) ....................(21)

and ....................(22)

The quantity (qgqoRs/1,000)Bg in Eqs. 17 and 21 is the free-gas flow rate in the reservoir; that is, the difference in the total gas rate, qg, and the dissolved gas rate, qoRs/1,000. Skin factor for multiphase flow test analysis using semilog plots is calculated from ....................(23)

For analysis of tests using type curves, note that the pressure match point on a type curve is related to total and individual phase mobilities and rates by ....................(24)

and the time match point is related to the dimensionless storage coefficient by ....................(25)

The practical implication of Eqs. 24 and 25 is that total mobility and individual phase permeability are determined from the pressure-match point on a type-curve match. The dimensionless storage coefficient is determined from the time-match point resulting in the calculation of skin factor from ....................(26)

just as for single-phase flow. When the conditions for applicability of the Perrine-Martin approximations (small saturation gradients in the drainage area of the tested well) are not satisfied, use of a reservoir simulator for test analysis is an appropriate alternative.

## Nomenclature

 Bg = gas formation volume factor, RB/STB Bw = water formation volume factor, RB/STB ct = Soco + Swcw + Sgcg + cf = total compressibility, psi–1 CD = 0.8936 C/ϕcthrw2 , dimensionless wellbore storage coefficient D = non-Darcy flow constant, D/Mscf h = net formation thickness, ft k = matrix permeability, md ko = permeability to oil, md kg = permeability to gas, md kw = permeability to water, md M = Molecular weight of gas m = 162.2 qBμ/kh = slope of middle-time line, psi/cycle n = inverse slope of the line on a log-log plot of the change in pressure squared or pseudopressure vs. gas flow rate p = pressure, psi pi = original reservoir pressure, psi pp = pseudopressure, psia2/cp psc = standard-condition pressure, psia pwf = flowing BHP, psi pws = shut-in BHP, psi p0 = arbitrary reference or base pressure, psi qg = gas flow rate, Mscf/D qRt = total flow rate at reservoir conditions, RB/D qw = water flow rate, STB/D r = distance from the center of wellbore, ft rw = wellbore radius, ft Rs = dissolved GOR, scf/STB s = skin factor, dimensionless s′ = s + Dq = apparent skin factor, dimensionless t = elapsed time, hours tap = pseudotime, hours tp = pseudoproducing time, hours tD = 0.0002637kt/ϕμctrw2, dimensionless time T = reservoir temperature, °R Tsc = standard condition temperature, °R V = volume, bbl z = gas-law deviation factor, dimensionless Δt = time elapsed since start of test, hours β = turbulence factor λ = interporosity flow coefficient λt = , total mobility, md/cp μ = viscosity, cp μgwf = gas viscosity evaluated at pwf , cp μg = gas viscosity, cp μo = oil viscosity, cp μw = water viscosity, cp ϕ = porosity, dimensionless