A COMPLEMENTARY CONVENTIONAL ANALYSIS FOR CHANNELIZED RESERVOIRS

Análisis convencional complementario para yacimientos alagados

Freddy-Humberto Escobar¹ and Matilde Montealegre-M.²

Many well pressure data coming from long and narrow reservoirs which result from either fluvial deposition of faulting cannot be completely interpreted by conventional analysis since some flow regimes are not conventionally recognized yet in the oil literature. This narrow geometry allows for the simultaneous development of two linear flow regimes coming from each one of the lateral sides of the system towards the well. This has been called dual linear flow regime. If the well is off-centered with regards to the two lateral boundaries, then, one of the linear flow regimes vanishes and, then, two possibilities can be presented. Firstly, if the closer lateral boundary is close to flow the unique linear flow persists along the longer lateral boundary. It has been called single linear flow. Following this, either steady or pseudosteady states will develop. Secondly, if a constant-pressure closer lateral boundary is dealt with, then parabolic flow develops along the longer lateral boundary. Steady state has to be developed once the disturbance reaches the farther boundary.

This study presents new equations for conventional analysis for the dual linear, linear and parabolic flow regimes recently introduced to the oil literature. The equations were validated by applying them to field and simulated examples.

Keywords: reservoir, linear flow, radial flow, parabolic flow.

Muchos datos de presión procedentes de yacimientos alargados y angostos, resultado de depósitos fluviales o callamiento, no pueden interpretarse por métodos convencionales puesto que existen algunos regímenes de flujo desconocidos en la literatura referente al método convencional. Esta geometría estrecha del yacimiento permite el desarrollo simultáneo de dos regimenes de flujo lineales, actuando a ambos lados alargados del yacimiento y dirigiéndose al pozo. Éste ha sido llamado flujo dual lineal. Cuando el pozo está descentrado con respecto a una de las dos fronteras laterales, uno de los flujos lineales desaparece y pueden presentarse dos posibilidades: primero, si la frontera más cercana es cerrada, un único flujo lineal persiste a lo largo de la prueba. Éste ha sido llamado flujo lineal único. Después de éste, se desarrollará el estado pseudoestable o estable. Sforndo, si la frontera cercana es de presión constante, entonces se desarrolla el flujo parabólico hacia el lado más largo del yacimiento. El estado estable deberá desarrollarse, una vez la perturbación haya alcanzado la frontera más lejana .

Este estudio presenta ecuaciones nuevas para análisis convencional para los regímenes de flujo dual lineal, lineal único y parabólico recientemente introducidos a la literatura petrolera. Las ecuaciones fueron validadas mediante su aplicación a ejemplos simulados y de campo.

Palabras clave: yacimientos, flujo lineal, flujo radial, flujo parabólico.

Very few researchers have focused their attention on long and narrow systems such as channel sands (Figure 1.a). Among them, we can name Ehlig-Economides and Economides (1985), Massonet, Norris, and Chalmette (1993), Mattar (1997), Nutakki and Mattar (1982) and Wong, Mothersele, Harrington, and Cinco-Ley (1986). A better description of channelized systems and a detailed interpretation technique was presented by Escobar, Saavedra, Hernández, C.M., Hernández, Y.A., Pilataxi, and Pinto (2004) who classified the linear flow regime into two closely related names: “dual linear flow” and “single linear flow” regimes as sketched in Figure 1.b. When the well is off-centered with respect to the lateral sides, firstly, we observe two linear flow regimes coming from both lateral sides of the well. They called this “dual linear flow”. Once the close-to-flow closer boundary is reached by the transient wave, a unique linear flow remains throughout the longer lateral boundary. Single linear flow can also be present when the well is located very close to one lateral boundary as shown in Figure 1.c. Escobar et al. (2004) called this “single linear flow” or simply, “linear flow”. However, if the closer boundary is at constant pressure, then “parabolic flow” develops and steady state follows this flow regime. The parabolic flow is recognized on the pressure derivative curve by a negative 0,5-slope straight line. The reader is referred to Escobar, Muñoz, Sepúlveda, Montealegre, and Hernández (2005a) and Escobar, Muñoz, Sepúlveda, J.A., and Montealegre (2005b) for a detailed explanation of the parabolic flow regime.

MATHEMATICAL FORMULATION

The dimensionless time quantities used by Escobar et al. (2004) were:

(1_2y3)

Dimensionless reservoir width and well position, as sketched in Figure 1.a, Escobar et al. (2004), are:

(4_5y6)

Dimensionless pressure is defined by Earlougher (1977) as:

(7)

Nutakki and Mattar (1982) presented an equation for the linear flow, as follows:

(8)

Escobar et al. (2004) and Escobar, Hernández, Y.A., and Hernández, C.M. (2007) have described the differences between the dual linear flow and the single linear flow occurring in elongated systems. Escobar et al. (2004) found out that Equation 8 was mistaken, so they proposed a new equation to describe dual linear, and single linear as well, flow regimes, such as:

(9y10)

Dual linear flow analysis

As depicted in Figure 1.b, this type of flow is formed by to linear flows occurring from each lateral side of the reservoir towards the well, so that they have an opposite direction to each other.

Replacing Equations 1 through Equation 7, into Equation 9 will yield:

(11.a)

For pressure buildup analysis, application of time superposition is required, therefore Equation 9 becomes:

(11.b)

Which implies that a cartesian plot of ∆P vs. either  or  will yield a straight line during dual linear flow behavior which slope, mDLF, and intercept, bDLF, are used to obtain reservoir width, YE, and dual linear skin factor, sDL.

Notice that Equation 13 is different to Equation 14 presented by Wong et al. (1986). However, our mathematical model agrees with those presented by Nutakki and Mattar (1982) and Ehlig-Economides and Economides (1985).

(14)

Single linear flow analysis

Regarding Figure 1.b, once the pressure disturbance reaches the first lateral boundary, one of the linear flow no longer exists, then a unique linear flow prevails which is called single linear flow regime. This flow also can take place when the well is very close to one of the lateral boundaries as sketched in Figure 1.c.

Replacement of Equation 1 through Equation 7 into Equation 10 will yield:

(15.a)

For pressure buildup analisis:

(15.b)

Equation 15.a and Equation 15.b indicates that a plot of ∆P vs. either  or  in cartesian coordinates will yield a straight line during linear flow behavior which slope, mLF, and intercept, bLF, are used to obtain reservoir width, YE, and linear skin factor, sL, respectively:

(16_17y18)

Parabolic flow analysis

The reader is referred to Escobar et al. (2005) for a better description of this type of flow. Escobar et al. (2004), Escobar et al. (2005a) and Escobar et al. (2005b) introduced the governing equation for pressure behavior dominated by parabolic flow, as follows:

(19)

Replacing of Equation 1 through Equation 7 into Equation 18 will yield:

(20.a)

For buildup analysis, superposition is required; then, the pressure equation is:

A cartesian plot of ∆P vs. either 1/ or  will yield a straight line during the parabolic dominated region which slope, mPB, and intercept, bPB, are used to obtain well position, bx, and parabolic skin factor, sPB, respectively, as:

(21)

(22y23)

Sui, Mou, Bi, Den, and Ehlig-Economides (2007) have also found the behavior depicted by a negative half-slope once linear flow regime has vanished in elongated reservoirs. However, they called it “dipolar flow”.

Simulated example

The synthetic pressure drawdown reported in Figure 2 was performed for a well inside a rectangular reservoir using the data given in the second column of Table 1.

The slope and intercept (mDLF = 1077 psi/h0,5, bDLF = 2792 psi) obtained from Figure 3 were applied into Equations 12 and Equation13 to yield an estimation of YE = 249,6 ft and sDL = 4,9, respectively. Also, from Figure 4 values of mLF = 1968,4 psi/h0,5 and bDLF = 4921 psi were read to be applied into Equation 16 to estimate a YE value of 242 ft and Equation 18 was used to estimate a sL of -8,6. A summary of the results are given in Table 2 . We observe a very good agreement between the calculated and the input YE value.

Field example

Escobar et al. (2004) presented an example of a pressure drawdown test run in a well in a channelized reservoir in the Eastern Planes basin in Colombia. Reservoir and well parameters are given in the third column of Table 1 and pressure and pressure derivative data are given in Figure 5.

As shown in Figure 5, radial, dual linear, parabolic flows and steady state are observed. Escobar et al. (2004) reported a permeability of 441 md and a mechanical skin factor of -4,9 determined using the TDS technique, Tiab (1993). As shown in Figure 6, the semilog slope is found to be 140 psi/cycle and P1h = 1158 psi. The conventional equation to estimate the mechanical skin factor is given by Earlougher (1977):