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Chapter 10

Material Balance Equation of Multilayer Gas Reservoir

Abstract

In recent years, more and more multilayer gas reservoirs have been discovered worldwide. Dynamic reserve estimation of multilayer producers is a major challenge for reservoir engineering. The material balance equation (p/Z plot) is one of the fundamental methods for reserve estimation. It is usually used in single-layer gas reservoir.

In Section 10.1, we study the applicability of the multilayer gas reservoir material balance equation for reserve estimation by single well numerical simulation. There are several parameters that are implicated in the reserve estimation of multilayer gas reservoirs, such as the connectivity between layers of the gas reservoir, the difference between formation pressures of the layers, and the difference between permeability of the layers.

In Section 10.2, we present a new method to estimate the single well dynamic reserve and individual reserve recovery for two-layer commingled gas reservoirs, which provides a foundation for the reasonable and efficient development of multilayered gas reservoir.

In Section 10.3, we establish a material balance equation of gas reservoir with a supplying region and performance forecast method by the semipermeable wall model. The results on real cases indicate that the proposed model can improve the accuracy of early performance forecast.

Keywords

Multilayer gas reservoir; Material balance equation; Dynamic reserve; Supplying region; Performance forecast; Semipermeable wall model

10.1 Material Balance of Two-Layer Gas Reservoir

The dynamic reserve estimation is one of the major tasks in gas field development. The dynamic reserve is the produced original gas in place (OGIP) that is estimated with a dynamic method when formation pressure reduces to zero. Calculation methods include the material balance method (Sun, 2011a), elastic binomial, extended drawdown test, production decline analysis method, numerical simulation, etc.

The material balance method is the main method for reserve estimation in a gas field. One of the basic assumptions of the method is that a gas reservoir consists of a single and homogeneous gas tank. The recovery percent of reserve should be higher than 5% as the method applies to reserve estimation. Several successful analyses have been performed on single-layer reservoirs. However, the classic material balance equation is unable to accurately forecast production performance of the multilayer gas reservoir.

The multilayer gas reservoir can be divided into two types depending on the existence of crossflow between layers. It is a commingled system if the layers have been isolated by a nonpermeable layer, while it is a crossflow system if there is connectivity between layers.

The production performances of multilayer reservoirs with or without crossflow are different. For a multilayer reservoir without crossflow, the individual layer rate is proportional to the formation flow capacity (kh) when the skin factor is the same for all the layers in the early period; the individual layer rate is proportional to the individual layer reserve in the long time period (PSSF).

For a multilayer reservoir with crossflow, flow is more complicated. For a circled closed multilayer reservoir with crossflow, the individual layer rate is relevant to skin factor, formation flow capacity (kh), and communication between zones. When the skin factors are the same, the individual layer rate is proportional to the formation flow capacity (kh); when the skin factors are different, the smaller the skin factor, the larger the individual layer rate is.

10.1.1 Physical Model

Assuming the gas reservoir is divided into two interconnected relatively homogeneous blocks, as shown in Fig. 10.1, we can concentrate the plane flow resistance to the flow at the wall between the blocks and assume the flow resistance is zero within the block. Under this scenario, the pressure within the block is only time-related. Semipermeable wall resistance on the flow can be calculated by the equivalent flow resistance method, as shown in Eq. (10.1). The semipermeable wall model can be used any time to ensure that the pressure is balanced within the block, so each district can meet the conditions of traditional material balance. In this model, the heterogeneity and the flow resistance of the stratum are considered, which makes the material balance much closer to the practical cases.

f10-01-9780128128534 — Fig. 10.1 Physical model of the material balance of the two-layer gas reservoir with crossflow.

$R = \frac{μ}{A} (\frac{h_{1} / 2}{{k_{1}}_{v}} + \frac{h_{2} / 2}{k_{2 v}} + \frac{h}{k_{v}})$ $R = \frac{μ}{A} (\frac{h_{1} / 2}{{k_{1}}_{v}} + \frac{h_{2} / 2}{k_{2 v}} + \frac{h}{k_{v}})$

si1_e (10.1)

The following assumptions are used in this section. The multilayer reservoir is homogeneous, isotropic, horizontal, and infinite. There are homogeneous, low permeability, and thin shales between layers; the storage of the shale is negligible compared to that of the layers. The reservoir is filled with a slightly compressible fluid of small and constant compressibility, and the flow obeys Darcy's law; the gravity can be neglected; a well completely penetrates the reservoir; the skin of the well is infinitely thin; and the well produces with a constant rate q from t=0. The multilayer reservoir is modeled by the semipermeable model.

10.1.2 Mathematical Model of a Two-Layer Gas Reservoir Without Crossflow

The material balance for gas is described by the following pair of equations:

$G_{1} \frac{d ρ_{1}}{d t} = - ρ_{S C} q_{1}$ $G_{1} \frac{d ρ_{1}}{d t} = - ρ_{S C} q_{1}$

(10.2)

and

$G_{2} \frac{d ρ_{2}}{d t} = - ρ_{S C} q_{2}$ $G_{2} \frac{d ρ_{2}}{d t} = - ρ_{S C} q_{2}$

(10.3)

For a multilayer reservoir without crossflow, the individual layer rate is

$\frac{q_{1}}{q_{2}} = \{\begin{cases} \frac{{(k h)}_{1}}{{(k h)}_{2}} = \frac{γ_{1}}{γ_{2}} early time \\ \frac{G_{1}}{G_{2}} = \frac{α_{1}}{α_{2}} PSSF \end{cases} q_{1} + q_{2} = q_{g}$ $\frac{q_{1}}{q_{2}} = \{\begin{cases} \frac{{(k h)}_{1}}{{(k h)}_{2}} = \frac{γ_{1}}{γ_{2}} early time \\ \frac{G_{1}}{G_{2}} = \frac{α_{1}}{α_{2}} PSSF \end{cases} q_{1} + q_{2} = q_{g}$

si4_e (10.4)

The individual layer pressure is

${(\frac{p}{Z})}_{j} = \{\begin{cases} {(\frac{p}{Z})}_{i} (1 - \frac{γ_{j} G_{p}}{G_{j}}), early time γ_{j} = \frac{{(k h)}_{j}}{\sum_{j = 1}^{2} {(k h)}_{j}}, j = 1, 2 \\ {(\frac{p}{Z})}_{i} (1 - \frac{α_{j} G_{p}}{G_{j}}) = {(\frac{p}{Z})}_{i} (1 - \frac{G_{p}}{\sum_{j = 1}^{2} G_{j}}), PSSF α_{j} = \frac{G_{j}}{\sum_{j = 1}^{2} G_{j}}, j = 1, 2 \end{cases}$ ${(\frac{p}{Z})}_{j} = \{\begin{cases} {(\frac{p}{Z})}_{i} (1 - \frac{γ_{j} G_{p}}{G_{j}}), early time γ_{j} = \frac{{(k h)}_{j}}{\sum_{j = 1}^{2} {(k h)}_{j}}, j = 1, 2 \\ {(\frac{p}{Z})}_{i} (1 - \frac{α_{j} G_{p}}{G_{j}}) = {(\frac{p}{Z})}_{i} (1 - \frac{G_{p}}{\sum_{j = 1}^{2} G_{j}}), PSSF α_{j} = \frac{G_{j}}{\sum_{j = 1}^{2} G_{j}}, j = 1, 2 \end{cases}$

si5_e (10.5)

where

$G_{p} = q_{g} t$ $G_{p} = q_{g} t$

(10.6)

10.1.3 Mathematical Model of a Two-Layer Gas Reservoir with Crossflow

The material balance for gas is described by the following pair of equations:

$G_{1} \frac{d ρ_{1}}{d t} = - ρ_{S C} q_{1} + \frac{k^{*} A}{μ} ρ \frac{\partial p}{\partial x}$ $G_{1} \frac{d ρ_{1}}{d t} = - ρ_{S C} q_{1} + \frac{k^{*} A}{μ} ρ \frac{\partial p}{\partial x}$

si7_e (10.7)

$G_{2} \frac{d ρ_{2}}{d t} = - ρ_{S C} q_{2} - \frac{k^{*} A}{μ} ρ \frac{\partial p}{\partial x}$ $G_{2} \frac{d ρ_{2}}{d t} = - ρ_{S C} q_{2} - \frac{k^{*} A}{μ} ρ \frac{\partial p}{\partial x}$

si8_e (10.8)

For a multilayer reservoir with crossflow, the individual layer rate is

$\frac{q_{1}}{q_{2}} = \frac{{(k h)}_{1}}{{(k h)}_{2}}, q_{1} + q_{2} = q_{g}$ $\frac{q_{1}}{q_{2}} = \frac{{(k h)}_{1}}{{(k h)}_{2}}, q_{1} + q_{2} = q_{g}$

si9_e (10.9)

where k* is average permeability:

$k^{*} = \frac{h + \frac{h_{1} + h_{2}}{2}}{\frac{h_{1}}{2 k_{1}} + \frac{h}{k} + \frac{h_{2}}{2 k_{2}}}$ $k^{*} = \frac{h + \frac{h_{1} + h_{2}}{2}}{\frac{h_{1}}{2 k_{1}} + \frac{h}{k} + \frac{h_{2}}{2 k_{2}}}$

si10_e (10.10)

Initial condition:

$ρ_{2} (t = 0) = ρ_{1} (t = 0) = ρ_{0}$ $ρ_{2} (t = 0) = ρ_{1} (t = 0) = ρ_{0}$

(10.11)

The individual layer pressure is

${(\frac{p}{Z})}_{1} = {(\frac{p}{Z})}_{i} \{1 - \frac{q_{g} t}{G_{1} + G_{2}} - \frac{q_{g}}{β {(1 + α)}^{2}} (\frac{γ_{1}}{G_{1}} - \frac{γ_{2}}{G_{2}}) [1 - e^{- β (1 + α) t}]\}$ ${(\frac{p}{Z})}_{1} = {(\frac{p}{Z})}_{i} \{1 - \frac{q_{g} t}{G_{1} + G_{2}} - \frac{q_{g}}{β {(1 + α)}^{2}} (\frac{γ_{1}}{G_{1}} - \frac{γ_{2}}{G_{2}}) [1 - e^{- β (1 + α) t}]\}$

si12_e (10.12)

${(\frac{p}{Z})}_{2} = {(\frac{p}{Z})}_{i} \{1 - \frac{q_{g} t}{G_{1} + G_{2}} - \frac{α q_{g}}{β {(1 + α)}^{2}} (\frac{γ_{1}}{G_{1}} - \frac{γ_{2}}{G_{2}}) [1 - e^{- β (1 + α) t}]\}$ ${(\frac{p}{Z})}_{2} = {(\frac{p}{Z})}_{i} \{1 - \frac{q_{g} t}{G_{1} + G_{2}} - \frac{α q_{g}}{β {(1 + α)}^{2}} (\frac{γ_{1}}{G_{1}} - \frac{γ_{2}}{G_{2}}) [1 - e^{- β (1 + α) t}]\}$

si13_e (10.13)

where

$α = \frac{G_{1}}{G_{2}}, β = \frac{k^{*} A}{G_{1} μ C_{g} h}, γ_{1} = \frac{{(k h)}_{1}}{\sum_{i = 1}^{2} k h}, γ_{2} = \frac{{(k h)}_{2}}{\sum_{i = 1}^{2} k h}, G_{p} = q_{g} t$ $α = \frac{G_{1}}{G_{2}}, β = \frac{k^{*} A}{G_{1} μ C_{g} h}, γ_{1} = \frac{{(k h)}_{1}}{\sum_{i = 1}^{2} k h}, γ_{2} = \frac{{(k h)}_{2}}{\sum_{i = 1}^{2} k h}, G_{p} = q_{g} t$

si14_e (10.14)

10.1.4 p/Z Curve of a Two-Layer Gas Reservoir

10.1.4.1 p/Z Curve of a Two-Layer Gas Reservoir Without Crossflow

There are two lines for a p/Z curve of a two-layer gas reservoir without crossflow. The slopes are $- {(\frac{P}{Z})}_{i} \frac{γ_{j}}{G_{j}}$ $- {(\frac{P}{Z})}_{i} \frac{γ_{j}}{G_{j}}$ and $- {(\frac{P}{Z})}_{i} \frac{α_{j}}{G_{j}}$ $- {(\frac{P}{Z})}_{i} \frac{α_{j}}{G_{j}}$ respectively. The OGIP is $\sum_{j = 1}^{2} G_{j}$ $\sum_{j = 1}^{2} G_{j}$ when $α_{j} = γ_{j}$ $α_{j} = γ_{j}$ , as shown in Fig. 10.2.

f10-02-9780128128534 — Fig. 10.2 (p/Z)_j curve of a two-layer gas reservoir without crossflow.

10.1.4.2 p/Z Curve of a Two-Layer Gas Reservoir with Crossflow

Early Time Period

In the early time, $e^{- β (1 + α) t} \to 1$ $e^{- β (1 + α) t} \to 1$ , the derivatives of Eqs. (10.12), (10.13) are

$d {(p / Z)}_{1} / d (q_{g} t) = - {(p / Z)}_{i} \frac{γ_{1}}{G_{1}}$ $d {(p / Z)}_{1} / d (q_{g} t) = - {(p / Z)}_{i} \frac{γ_{1}}{G_{1}}$

(10.15)

$d {(p / Z)}_{2} / d (q_{g} t) = - {(p / Z)}_{i} \frac{γ_{2}}{G_{2}}$ $d {(p / Z)}_{2} / d (q_{g} t) = - {(p / Z)}_{i} \frac{γ_{2}}{G_{2}}$

(10.16)

In the early time, the slope is identical to the case where there is no crossflow. The pressure difference of the two layers is so small that the crossflow rate can be neglected.

Long Time Period

In the PSSF, $e^{- β (1 + α) t} \to 0$ $e^{- β (1 + α) t} \to 0$ , the derivatives of Eqs. (10.12), (10.13) are

$\frac{d {(\frac{p}{Z})}_{2}}{d (q_{g} t)} = \frac{d {(\frac{p}{Z})}_{1}}{d (q_{g} t)} = {(\frac{p}{Z})}_{i} \{- \frac{1}{1 + α} (\frac{α γ_{1}}{G_{1}} + \frac{γ_{2}}{G_{2}})\} = - {(\frac{p}{Z})}_{i} \frac{1}{G_{1} + G_{2}}$ $\frac{d {(\frac{p}{Z})}_{2}}{d (q_{g} t)} = \frac{d {(\frac{p}{Z})}_{1}}{d (q_{g} t)} = {(\frac{p}{Z})}_{i} \{- \frac{1}{1 + α} (\frac{α γ_{1}}{G_{1}} + \frac{γ_{2}}{G_{2}})\} = - {(\frac{p}{Z})}_{i} \frac{1}{G_{1} + G_{2}}$

si23_e (10.17)

There are also two lines for the p/Z curve of a two-layer gas reservoir with crossflow. The OGIP is $\sum_{j = 1}^{2} G_{j}$ $\sum_{j = 1}^{2} G_{j}$ when $α_{1} / G_{1} = α_{2} / G_{2}$ $α_{1} / G_{1} = α_{2} / G_{2}$ , as shown in Fig. 10.3. In the long time, OGIP is $\sum_{j = 1}^{2} G_{j}$ $\sum_{j = 1}^{2} G_{j}$ as a single layer.

f10-03-9780128128534 — Fig. 10.3 (p/Z)_j curve of a two-layer gas reservoir with crossflow.

10.1.5 Influencing Factor Analysis of Reserve Estimation

Parameters used in simulation are as follows: the well is drilled with two zones, the supply radius is 750 m, and the vertical permeability is the same for the two zones. The wellbore diameter is 0.1 m, the thickness is 20 m, and permeability is 0.1 md for each zone; the formation temperature is 80 °C, original formation pressure is 20 MPa, relative density of natural gas is 0.6, pseudocritical pressure is 4.67 MPa, and pseudocritical temperature is 195.7 K. Suppose a single well is put into production with an output of 5×10⁴ m³/d, the production cycle is 300 d, and the time of shut-in is 30 d. Six cycles are simulated and accumulative gas production is 0.9×10⁸ m³. The highest shut-in pressure is used for the average formation pressure for reserve estimation.

10.1.5.1 The Effect of an Interlayer Heterogeneity on Dynamic Reserve Estimation

The following assumptions are made: the two zones have the same thickness of 10 m and the sum of the permeability-thickness product is 100 md.m. Dynamic reserve estimation on different permeability-thickness product is shown in the Fig. 10.4. The result of the dynamic reserve decreases with the increased heterogeneity between layers. When the permeability-thickness product of the two zones is 1:99, the reserve estimation is 2.85×10⁸ m³, which is 48.5% of the reserve (5.88×10⁸ m³)when the permeability-thickness product of the two zones is equal to each other.

f10-04-9780128128534 — Fig. 10.4 The effect of permeability-thickness product on reserve estimation. (After Sun et al. (2008). SPE 113665, Permission to publish by the SPE, Copyright SPE.)

The average formation pressure is determined by the shut-in pressure buildup. The measured average pressure by this way is not to represent the average pressure of the multilayer formation, but it is closer to the average pressure of the weak zone. The simulation result indicated that the more heterogeneity the layer is, the more the pressure declines, and the lower the measured formation pressure is, as shown in Fig. 10.5. Therefore, if the apparent formation pressure, p/Z, is less than expected, the reserve estimation would be lower.

f10-05-9780128128534 — Fig. 10.5 The effect permeability-thickness product on formation pressure. (After Sun et al. (2008). SPE 113665, Permission to publish by the SPE, Copyright SPE.)

10.1.5.2 The Effect of Crossflow on Dynamic Reserve Estimation

If there is crossflow between two adjacent zones, the flow could be considered as linear flow. On the basis of the linear flow equation of Darcy's law, the interlayer crossflow could be calculated by the following formula:

$q_{c i} = \frac{2.714 \times 10^{- 5} T_{s c} {\bar{k}}_{v i} A_{i}}{μ_{g} Z p_{s c} x_{i}} (p_{i}^{2} - p_{i + 1}^{2})$ $q_{c i} = \frac{2.714 \times 10^{- 5} T_{s c} {\bar{k}}_{v i} A_{i}}{μ_{g} Z p_{s c} x_{i}} (p_{i}^{2} - p_{i + 1}^{2})$

si27_e

where the flow distance is half of the thickness of two adjacent formations:

$x_{i} = \frac{h_{i} + h_{i + 1}}{2}$ $x_{i} = \frac{h_{i} + h_{i + 1}}{2}$

The vertical permeability of the two adjacent zones is

${\bar{k}}_{v i} = \frac{h_{i} + h_{i + 1}}{\frac{h_{i}}{k_{v i}} + \frac{h_{i + 1}}{k_{v i + 1}}}$ ${\bar{k}}_{v i} = \frac{h_{i} + h_{i + 1}}{\frac{h_{i}}{k_{v i}} + \frac{h_{i + 1}}{k_{v i + 1}}}$

si29_e

Assuming that the total of permeability-thickness product of the two zones is 100 md.m, the simulation result indicates that the vertical permeability has a great impact on the reserve estimation, as shown in Fig. 10.6.

f10-06-9780128128534 — Fig. 10.6 The effect of vertical permeability on reserve estimation. (After Sun et al. (2008). SPE 113665, Permission to publish by the SPE, Copyright SPE.)

When the vertical permeability is smaller than 1E−4 md, the more heterogeneity is, the less dynamic reserve declines are. When the permeability is 1:4, the reserve estimation is nearly 90% of the OGIP although the vertical permeability is only 1E−7 md. However, when the permeability is 1:19, the reserve estimation is only 60% of the OGIP.

When the vertical permeability value exceeds 1E−4 md, the heterogeneity has little impact on the reserve estimation and the dynamic reserve is close to 5.88×10⁸ m³.

The above results indicate that the crossflow between the two zones is under the control of interzonal permeability. Although the vertical permeability is a small value, the exchange volume in the whole contact area is large. When the vertical permeability is above 1E−4 md, it plays an important role in interzonal pressure balance.

10.1.5.3 The Effect of the Permeability on Reserve Estimation

Suppose that the permeability-thickness products are 100, 200, and 300 md.m respectively and the thickness of each zone is 10 m. Dynamic reserve estimation under different permeability is shown in Fig. 10.7. The permeability and permeability-thickness product both influence the reserve estimation. With the same permeability-thickness product, the higher the permeability value, the larger the reserve estimation is.

f10-07-9780128128534 — Fig. 10.7 The effect of the permeability on reserve estimation. (After Sun et al. (2008). SPE 113665, Permission to publish by the SPE, Copyright SPE.)

10.1.6 Method of Improving the Ratio of Dynamic Reserve to OGIP

10.1.6.1 Well Stimulation for Formation with Low Permeability

Suppose that the permeability-thickness products are 10 and 90 md.m, respectively, the thickness of each zone is 10 m, and the skin factor of the high permeable zone was 0. Dynamic reserve estimation under different skin factors is shown in Fig. 10.8. If the skin factor of each zone is zero, the ratio of dynamic reserve to OGIP is only 73.5%. If the layer with low permeability is stimulated and when the skin factor is less than −5, the ratio of dynamic reserve to OGIP is above 91.7%.

f10-08-9780128128534 — Fig. 10.8 The effect of well stimulation for a low-permeability reservoir on reserve estimation. (After Sun et al. (2008). SPE 113665, Permission to publish by the SPE, Copyright SPE.)

10.1.6.2 Reduce the Production Output

Suppose that the permeability-thickness products are 10 and 90 md.m respectively, the thickness of each zone is 10 m, and the skin coefficient is zero for each zone. Firstly, the production is implemented with an output of 5×10⁴ m³/d, and an adjustment is made at a later stage. The dynamic reserve estimation result is shown in Fig. 10.9. If the skin factor of each zone is zero, the ratio of dynamic reserve to OGIP is only 73.5%. If a certain adjustment of production output is made at a later stage of development, the producing reserve could be improved. When the production output is adjusted at 3×10⁴ m³/d, the ratio of dynamic reserve to OGIP would be as high as 84.5%.

f10-09-9780128128534 — Fig. 10.9 The effect of production adjustment on reserve estimation at a late period of development.

10.1.6.3 Single-Layer Production for Layer with Low Permeability

The interlayer replacement could be adopted and layer with low permeability may produce independently to improve the accuracy of dynamic reserve.

10.2 Reserve Estimation of a Two-Layer Commingled Gas Well

Multilayer reservoirs can be divided into two types according to the crossflow between layers, that is, crossflow reservoirs and no-crossflow (commingled) reservoirs. The commingled production is an effective way to develop the multilayer gas reservoir without crossflow. Performance of commingling wells are different due to the differences in average pressures, the permeabilities, and OGIP.

10.2.1 Physical Model and Mathematical Model

Assume the gas reservoir consists of two layers, there is no interlayer fluid exchange, and each layer is in line with the traditional material balance conditions, as shown in Fig. 10.10. In addition, (1) the reserves of the layers are G₁, G₂ respectively, and the reserve in barrier is ignored; (2) the single well production is q (the layer productions are q₁, q₂, respectively); (3) the single well cumulative production is G_p (the layer cumulative productions are G_p1, G_p2 respectively); (4) the water effect is negligible; and (5) the initial formation pressure is the same.

f10-10-9780128128534 — Fig. 10.10 Two-layer material balance physical model.

The layered material balance equation is

${(\frac{P}{Z})}_{1} = {(\frac{P}{Z})}_{i} (1 - \frac{G_{P 1}}{G_{1}})$ ${(\frac{P}{Z})}_{1} = {(\frac{P}{Z})}_{i} (1 - \frac{G_{P 1}}{G_{1}})$

si30_e (10.18)

${(\frac{P}{Z})}_{2} = {(\frac{P}{Z})}_{i} (1 - \frac{G_{P 2}}{G_{2}})$ ${(\frac{P}{Z})}_{2} = {(\frac{P}{Z})}_{i} (1 - \frac{G_{P 2}}{G_{2}})$

si31_e (10.19)

$G_{p} = G_{P 1} + G_{P 2}$ $G_{p} = G_{P 1} + G_{P 2}$

(10.20)

According to Eqs. (10.18)–(10.20) can be derived as

$G_{1} \frac{{(\frac{P}{Z})}_{i} - {(\frac{P}{Z})}_{1}}{{(\frac{P}{Z})}_{i} - {(\frac{P}{Z})}_{2}} + G_{2} = G_{p} \frac{{(\frac{P}{Z})}_{i}}{{(\frac{P}{Z})}_{i} - {(\frac{P}{Z})}_{2}}$ $G_{1} \frac{{(\frac{P}{Z})}_{i} - {(\frac{P}{Z})}_{1}}{{(\frac{P}{Z})}_{i} - {(\frac{P}{Z})}_{2}} + G_{2} = G_{p} \frac{{(\frac{P}{Z})}_{i}}{{(\frac{P}{Z})}_{i} - {(\frac{P}{Z})}_{2}}$

si33_e (10.21)

Assume

$Y = \frac{G_{p} {(\frac{P}{Z})}_{i}}{{(\frac{P}{Z})}_{i} - {(\frac{P}{Z})}_{2}}; X = \frac{{(\frac{P}{Z})}_{i} - {(\frac{P}{Z})}_{1}}{{(\frac{P}{Z})}_{i} - {(\frac{P}{Z})}_{2}}$ $Y = \frac{G_{p} {(\frac{P}{Z})}_{i}}{{(\frac{P}{Z})}_{i} - {(\frac{P}{Z})}_{2}}; X = \frac{{(\frac{P}{Z})}_{i} - {(\frac{P}{Z})}_{1}}{{(\frac{P}{Z})}_{i} - {(\frac{P}{Z})}_{2}}$

si34_e

Then

$Y = G_{1} X + G_{2}$ $Y = G_{1} X + G_{2}$

(10.22)

It can be seen that, X~Y is a linear relationship. Thus, the individual reserve parameters can be obtained by the measured point's data regression, and the individual recovery percent of reserve can be established. This calculation requires at least two shut-in pressure calculations with one run-in-hole.

10.2.2 Examples

Well parameters are as follows. Two layers are drilled, and the supply radius is about 1000 m. The wellbore radius is 0.1 m, and the total thickness of formation is 20 m; the porosity of each layer is equal to 0.1, the formation temperature is 80 °C, the initial formation pressure is 30 MPa, the natural gas relative density is 0.6, the pseudocritical pressure is 4.67 MPa, and the pseudocritical temperature is 195.7 K. The OGIP is 7.65×10⁸ m³, respectively; permeability-thickness product of the first layer, (kh)₁, is about 30 md.m, and permeability-thickness product of the second layer, (kh)₂, is about 10 md.m. The well produces for 2500 d at a 10 MPa constant bottomhole flowing pressure. Cumulative gas production of the first layer is 3.80×10⁸ m³, and the cumulative gas production of the second layer is 1.9×10⁸ m³.

Fig. 10.11 is a data match curve in Table 10.2. The individual dynamic reserves are 7.72×10⁸ and 7.62×10⁸ m³ respectively, for which the errors compared with the original reserves are 0.9% and 0.4% respectively, and the error compared with OGIP is 0.26%.

f10-11-9780128128534 — Fig. 10.11 Match curve of X and Y in Table 10.2.

Table 10.1

Two-Layer Gas Reservoir Stimulation Results at a Constant Pressure

Production Time (d)	Total Production (10⁴ m³)	First Layer		Second Layer
Production Time (d)	Total Production (10⁴ m³)	p/Z (MPa)	G_P1 (10⁴ m³)	p/Z (MPa)	G_P2 (10⁴ m³)
0	0	30.83	0	30.83	0
10	509	30.68	374	30.78	135
20	912	30.56	671	30.73	241
30	1340	30.43	985	30.69	355
40	1765	30.31	1297	30.64	468
50	2188	30.18	1607	30.60	581
60	2609	30.06	1915	30.55	694
70	3027	29.93	2221	30.50	806
80	3442	29.81	2524	30.46	918
90	3855	29.69	2826	30.42	1029
100	4265	29.57	3125	30.37	1140
200	8232	28.41	6005	29.93	2227
300	11,960	27.33	8684	29.51	3276
400	15,461	26.33	11,174	29.10	4287
500	18,746	25.40	13,484	28.71	5262
1000	32,344	21.68	22,709	26.95	9635
1500	42,282	19.14	29,006	25.48	13,276
2000	50,000	17.26	33,671	24.25	16,329
2500	56,939	15.52	38,000	23.20	18,939

t0010

Table 10.2

Two-Layer Gas Reservoir Dynamic Reserve Calculation (Adjacent Point)

Production Time (d)	Total Production (10⁴ m³)	Y (10⁸ m³)	X	G_P1 (10⁸ m³)	G_P2 (10⁸ m³)	G_P1 Error (%)	G_P2 Error (%)	G Error (%)
0	0	0	0
10	509	28.7810	2.7713
20	912	28.8220	2.7726	30.388	55.434
30	1340	28.8165	2.7703	2.341	22.330	69.4	−191.9	−61.2
40	1765	28.7964	2.7668	5.802	12.742	24.2	−66.6	−21.2
50	2188	28.7703	2.7629	6.657	10.379	13.0	−35.7	−11.3
60	2609	28.7412	2.7588	7.019	9.378	8.3	−22.6	−7.2
70	3027	28.7103	2.7545	7.210	8.850	5.7	−15.7	−5.0
80	3442	28.6783	2.7501	7.325	8.534	4.3	−11.6	−3.7
90	3855	28.6456	2.7457	7.399	8.330	3.3	−8.9	−2.8
100	4265	28.6124	2.7412	7.450	8.189	2.6	−7.0	−2.2
200	8232	28.2708	2.6960	7.561	7.886	1.2	−3.1	−1.0
300	11,960	27.9270	2.6509	7.620	7.727	0.4	−1.0	−0.3
400	15,461	27.5865	2.6063	7.635	7.688	0.2	−0.5	−0.1
500	18,746	27.2510	2.5624	7.641	7.672	0.1	−0.3	−0.1
1000	32,344	25.6792	2.3569	7.646	7.658	0.1	−0.1	0.0
1500	42,282	24.3630	2.1848	7.649	7.653	0.0	0.0	0.0
2000	50,000	23.4236	2.0620	7.649	7.652	0.0	0.0	0.0
2500	56,939	22.9990	2.0064	7.649	7.652	0.0	0.0	0.0

t0015

It can be seen from Table 10.2 that the individual reserve is calculated with two adjacent points: G₁ is smaller than the theoretical values and G₂ is larger than the theoretical values. In order to ensure that the error of dynamic reserves is <5.0%, the recovery percent of reserve must be >2.5%; in order to ensure that the error of the individual reserve is <5.0%, the recovery percent of reserve must be >5.0%.

10.3 Material Balance Equation of Gas Reservoir with a Supplying Region

The traditional method of material balance takes the gas reservoir as containers while the heterogeneity and the flow resistance of the stratum are neglected. Therefore, the results tend to be optimistic when using laminar-inertial turbulent flow analysis and the material balance equation to predict performance on heterogeneous gas reservoirs with low permeability. To improve the accuracy of performance forecast, a semipermeable wall model is proposed. According to the law of conservation of mass, a gas material balance equation with a supplying region and performance forecast method are established. Assuming the gas reservoir with low permeability is divided into two interconnected strips that are relatively homogeneous blocks, we can concentrate the plane flow resistance to flow at the wall between the blocks and let the flow resistance be zero within block. Hence, the pressure within the block is only time-related. Semipermeable wall resistance on the flow can be calculated by the equivalent flow resistance method. The semipermeable wall model can be used any time to ensure that the pressure balances within the block, so each district can meet the conditions of traditional material balance. In this model, the heterogeneity and the flow resistance of the stratum are considered, which makes the material balance much closer to the practical cases. In addition, the performance of well can be easily and simply forecast. The changing rules of pressure-time and rate-time are explained. The results on real cases indicate that the proposed model can improve the accuracy of the early performance forecast. This study can direct the rational and effective development of the similar gas reservoirs.

10.3.1 Introduction

Gas reservoir production performance forecast is the core task of gas field development management throughout the lifecycle of gas field development. In the beginning stages of the development of the gas field, the material balance method and the numerical simulation method were the main methods for performance forecast. The traditional material balance method is very simple. That is, the gas field is assumed as a homogeneous container, and gas reservoir heterogeneity is ignored completely. The result of forecasting always has a tendency for optimism. On the contrary, numerical simulation is too complex to be applied in general gas reservoir engineering. It requires the detailed formation parameter distribution, which is difficult to obtain in the early period of gas field development. Its application is limited when a large amount of work is required.

It is urgent to develop a method for gas reservoir production performance forecast in the early time of gas field development, which not only considers the regional heterogeneity, but also forecast the gas field performance quickly and easily. In recent years, the multitank material balance method is studied by several studies. Its result is reliable compared with the numerical simulation.

10.3.2 Material Balance Equation of Gas Reservoir with a Supplying Region

10.3.2.1 Physical Model

Assume the gas reservoir is divided into two interconnected relatively homogeneous blocks, as shown in Fig. 10.12. We can concentrate the plane flow resistance to flow at the wall between the blocks and let the flow resistance be zero within the block. Hence, the pressure within the block is only time-related. Semipermeable wall resistance on the flow can be calculated by the equivalent flow resistance method, as shown in Eq. (10.23). The semipermeable wall model can be used any time to ensure that the pressure balances within the block, so each district can meet the conditions of traditional material balance. In this model, the heterogeneity and the flow resistance of the stratum are considered, which makes the material balance much closer to the practical cases.

f10-12-9780128128534 — Fig. 10.12 Physical model of material balance of a two-tank region gas reservoir (Sun, 2011a).

$R = \frac{μ}{h} (\frac{L_{1} / 2}{k_{1}} + \frac{L_{2} / 2}{k_{2}} + \frac{W}{k})$ $R = \frac{μ}{h} (\frac{L_{1} / 2}{k_{1}} + \frac{L_{2} / 2}{k_{2}} + \frac{W}{k})$

si36_e (10.23)

10.3.2.2 Mathematical Model

In the near-wellbore tank region, the material balance is

$G_{1} \frac{d ρ_{1}}{d t} = - ρ_{S C} q_{g} + \frac{k^{*} L h}{μ} ρ \frac{\partial p}{\partial x}$ $G_{1} \frac{d ρ_{1}}{d t} = - ρ_{S C} q_{g} + \frac{k^{*} L h}{μ} ρ \frac{\partial p}{\partial x}$

si37_e (10.24)

In the outer tank region, the material balance is

$G_{2} \frac{d ρ_{2}}{d t} = - \frac{k^{*} L h}{μ} ρ \frac{\partial p}{\partial x}$ $G_{2} \frac{d ρ_{2}}{d t} = - \frac{k^{*} L h}{μ} ρ \frac{\partial p}{\partial x}$

si38_e (10.25)

where k* is average permeability of the flow control region

$k^{*} = \frac{W + \frac{L_{1} + L_{2}}{2}}{\frac{L_{1}}{2 k_{1}} + \frac{W}{k} + \frac{L_{2}}{2 k_{2}}}$ $k^{*} = \frac{W + \frac{L_{1} + L_{2}}{2}}{\frac{L_{1}}{2 k_{1}} + \frac{W}{k} + \frac{L_{2}}{2 k_{2}}}$

si39_e (10.26)

The initial condition is

$ρ_{2} (t = 0) = ρ_{1} (t = 0) = ρ_{0}$ $ρ_{2} (t = 0) = ρ_{1} (t = 0) = ρ_{0}$

(10.27)

Individual region pressure is

${(\frac{p}{Z})}_{1} = {(\frac{p}{Z})}_{i} \{1 - \frac{q_{g} t}{G_{1} + G_{2}} - \frac{q_{g}}{G_{1} β {(1 + α)}^{2}} [1 - e^{- β (1 + α) t}]\}$ ${(\frac{p}{Z})}_{1} = {(\frac{p}{Z})}_{i} \{1 - \frac{q_{g} t}{G_{1} + G_{2}} - \frac{q_{g}}{G_{1} β {(1 + α)}^{2}} [1 - e^{- β (1 + α) t}]\}$

si41_e (10.28)

${(\frac{p}{Z})}_{2} = {(\frac{p}{Z})}_{i} \{1 - \frac{q_{g} t}{G_{1} + G_{2}} + \frac{q_{g}}{G_{2} β {(1 + α)}^{2}} [1 - e^{- β (1 + α) t}]\}$ ${(\frac{p}{Z})}_{2} = {(\frac{p}{Z})}_{i} \{1 - \frac{q_{g} t}{G_{1} + G_{2}} + \frac{q_{g}}{G_{2} β {(1 + α)}^{2}} [1 - e^{- β (1 + α) t}]\}$

si42_e (10.29)

where

$α = \frac{G_{1}}{G_{2}}, β = \frac{k^{*} L h}{G_{1} μ C_{g} W}$ $α = \frac{G_{1}}{G_{2}}, β = \frac{k^{*} L h}{G_{1} μ C_{g} W}$

si43_e

For variable flow rate, according to Duhamel's principle, we have

${(\frac{p}{Z})}_{1, j} = {(\frac{p}{Z})}_{i} \{\begin{array}{l} 1 - \frac{1}{G_{1} + G_{2}} \sum_{k = 1}^{j} (q_{k} - q_{k - 1}) \times (t_{j} - t_{k - 1}) \\ - \frac{1}{G_{1} {(1 + α)}^{2}} \sum_{k = 1}^{j} \frac{(q_{k} - q_{k - 1})}{β_{k}} [1 - e^{- β_{k} (1 + α) (t_{j} - t_{k})}] \end{array}\}$ ${(\frac{p}{Z})}_{1, j} = {(\frac{p}{Z})}_{i} \{\begin{array}{l} 1 - \frac{1}{G_{1} + G_{2}} \sum_{k = 1}^{j} (q_{k} - q_{k - 1}) \times (t_{j} - t_{k - 1}) \\ - \frac{1}{G_{1} {(1 + α)}^{2}} \sum_{k = 1}^{j} \frac{(q_{k} - q_{k - 1})}{β_{k}} [1 - e^{- β_{k} (1 + α) (t_{j} - t_{k})}] \end{array}\}$

si44_e (10.30)

${(\frac{p}{Z})}_{2, j} = {(\frac{p}{Z})}_{i} \{\begin{array}{l} 1 - \frac{1}{G_{1} + G_{2}} \sum_{k = 1}^{j} (q_{k} - q_{k - 1}) \times (t_{j} - t_{k - 1}) \\ + \frac{1}{G_{2} {(1 + α)}^{2}} \sum_{k = 1}^{j} \frac{(q_{k} - q_{k - 1})}{β_{k}} [1 - e^{- β_{k} (1 + α) (t_{j} - t_{k})}] \end{array}\}$ ${(\frac{p}{Z})}_{2, j} = {(\frac{p}{Z})}_{i} \{\begin{array}{l} 1 - \frac{1}{G_{1} + G_{2}} \sum_{k = 1}^{j} (q_{k} - q_{k - 1}) \times (t_{j} - t_{k - 1}) \\ + \frac{1}{G_{2} {(1 + α)}^{2}} \sum_{k = 1}^{j} \frac{(q_{k} - q_{k - 1})}{β_{k}} [1 - e^{- β_{k} (1 + α) (t_{j} - t_{k})}] \end{array}\}$

si45_e (10.31)

10.3.2.3 Deliverability Equation and Recharge Equation

There is only one production well in the reservoir, and the deliverability equation is as follows:

$Δ p^{2} = A q + B q^{2}$ $Δ p^{2} = A q + B q^{2}$

(10.32)

The recharge flow rate is proportional to the pressure difference in two regions:

$q_{c} = \frac{Δ p^{2}}{R}$ $q_{c} = \frac{Δ p^{2}}{R}$

si47_e (10.33)

We can forecast the gas reservoir production performance by Eqs. (10.28)–(10.31).

10.3.3 Example and Discussion

10.3.3.1 Example

Parameters used in the example are as follows. The thickness is 10 m, as shown in Fig. 10.13. The near-wellbore tank region permeability is 10 md, the outer tank region permeability is 5 md, the low-permeability region permeability is 0.1 md, the absolute open flow potential is 59×10⁴ m³/d, the formation temperature is 80 °C, the original formation pressure is 30 MPa, and the relative density of natural gas is 0.6. The deliverability equation is

f10-13-9780128128534 — Fig. 10.13 Real model of a two-tank region gas reservoir.

$Δ p^{2} = 7.74344 q_{g} + 0.127014 q_{g}^{2}$ $Δ p^{2} = 7.74344 q_{g} + 0.127014 q_{g}^{2}$

10.3.3.2 p/Z Curve

The well is producing at a constant rate of 10.0×10⁴ m³; the relationship of the apparent formation pressure of the near-wellbore tank region and the produced gas volume is shown in Fig. 10.14. There is an obvious breakpoint.

f10-14-9780128128534 — Fig. 10.14 p/Z curve of a two-tank region gas reservoir.

In the early stages of well development, the pressure difference between the regions is very small, so the recharge rate can be neglected. The relationship of (p/Z)₁ and G_p is a line, and the slope is $- {(\frac{p}{Z})}_{i} \frac{1}{G_{1}}$ $- {(\frac{p}{Z})}_{i} \frac{1}{G_{1}}$ . The derivative of Eq. (10.28) can be written as

$\frac{d {(\frac{p}{Z})}_{1}}{d (q_{g} t)} = - {(\frac{p}{Z})}_{i} \frac{1}{G_{1} + G_{2}} [1 + \frac{G_{2}}{G_{1}} e^{- β (1 + α) t}]$ $\frac{d {(\frac{p}{Z})}_{1}}{d (q_{g} t)} = - {(\frac{p}{Z})}_{i} \frac{1}{G_{1} + G_{2}} [1 + \frac{G_{2}}{G_{1}} e^{- β (1 + α) t}]$

si50_e (10.34)

In the long time, the exponential term of Eq. (10.34) can be neglected, the second line appears, and the slope is $- {(\frac{p}{Z})}_{i} \frac{1}{G_{1} + G_{2}}$ $- {(\frac{p}{Z})}_{i} \frac{1}{G_{1} + G_{2}}$ , as shown in Fig. 10.13.

The reserve derived from the multisection (p/Z)₁ curve is lower than that of the conventional (p/Z) curve, and the reserve difference ΔG can be expressed as

$Δ G = \frac{μ c_{g} W}{k^{*} (L h)} (\frac{G_{2}^{2}}{G_{1} + G_{2}}) q_{g}$ $Δ G = \frac{μ c_{g} W}{k^{*} (L h)} (\frac{G_{2}^{2}}{G_{1} + G_{2}}) q_{g}$

si52_e (10.35)

The above formula indicates that when the multiregion reserve keeps constant, ΔG depends on not only the characteristic parameters of the gas reservoir, but also the initial production associated with the production area. The residual reserve decreases with the increase of gas reservoir permeability and the contact area between the blocks. The residual reserve also decreases with the decrease of the stripe widths of the two intervals. The residual reserve increases with the increase of the production rate, which is consistent with the laboratory experiment result.

10.3.3.3 Performance Forecast

The well is producing at a constant rate of 10.0×10⁴ m³, and the result comparison of the conventional and multitank model is shown in Fig. 10.15. There is a difference about the period of stabilized production. It is 0.85a for the multitank model method and approximately 1.32a for the conventional method. The result of gas reservoir production performance forecast is shown in Fig. 10.16.

f10-15-9780128128534 — Fig. 10.15 The forecast result comparison for the conventional and multitank method.

f10-16-9780128128534 — Fig. 10.16 The forecast result of the multitank method (Sun, 2011a).

The numerical simulation method shows that the stable production period is 0.90 years if the well produces with a constant rate of 10.0×10⁴m³/d. The pressure field is shown in Fig. 10.17. The result of the multitank model is close to the result of the numerical simulation method.

f10-17-9780128128534 — Fig. 10.17 Distribution of pressure field for numerical simulation method (the end of the period of stable production).

10.4 Summary

(1) For a multilayer gas reservoir, if the individual thickness is the same for the two zones and there is no crossflow, the reserve estimation result is mainly influenced by interlayer heterogeneity and the permeability. Dynamic reserve declines as heterogeneity between layers increases. For the same permeability-thickness product of the formation, the higher the permeability is, the larger the individual reserve is.

(2) For a multilayer gas reservoir, if the individual thickness is the same for the two zones and there is crossflow, the reserve estimation result is mainly influenced by vertical permeability and heterogeneity between layers.
When vertical permeability is smaller than 1E−4 md and the permeability ratio of the two zones is above 1/4, the reserve estimation can be calculated by the material balance method. Moreover, the estimation error of the reserve is within 10%.
When vertical permeability is above 1E−4 md, the effect of heterogeneity on reserve estimation is little, and the estimation result is fairly accurate.

(3) The ratio of dynamic reserve to OGIP can be improved by means of well stimulation, optimization of production, replacement of different layers, etc.

(4) The recovery percent of the reserve can be analyzed by the layered material balance equation. The new calculation method is simple and practical; the individual reserve error is <5% when the recovery percent of the reserve is greater than 5.0%.

(5) For multiple wells with the same producing positions, the total gas reservoir shut-in approach should be applied to acquire the layer average formation pressure, and then the above method can be applied to estimate the dynamic reserve and evaluate the individual recovery percent of the reserve.

(6) For the case of more than two layers, firstly, all layers should be divided into two sets of layer groups according to geological knowledge; then the above method can be applied to estimate the dynamic reserve.

(7) The two-tank material balance model can be easily extended to the N-region model; it can be used to forecast the performance of gas reservoir with serious heterogeneity.

(8) The two-tank model is based on the condition that the deliverability equation is reliable. For low-permeability or tight gas reservoir, the deliverability equation is not reliable, so the result of forecast will make a great difference.

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Table of Contents for Chapter 10: Material Balance Equation of Multilayer Gas Reservoir

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10.1 Material Balance of Two-Layer Gas Reservoir

10.1.1 Physical Model

10.1.2 Mathematical Model of a Two-Layer Gas Reservoir Without Crossflow

10.1.3 Mathematical Model of a Two-Layer Gas Reservoir with Crossflow

10.1.4 p/Z Curve of a Two-Layer Gas Reservoir

10.1.4.1 p/Z Curve of a Two-Layer Gas Reservoir Without Crossflow

10.1.4.2 p/Z Curve of a Two-Layer Gas Reservoir with Crossflow

Early Time Period

Long Time Period

10.1.5 Influencing Factor Analysis of Reserve Estimation

10.1.5.1 The Effect of an Interlayer Heterogeneity on Dynamic Reserve Estimation

10.1.5.2 The Effect of Crossflow on Dynamic Reserve Estimation

10.1.5.3 The Effect of the Permeability on Reserve Estimation

10.1.6 Method of Improving the Ratio of Dynamic Reserve to OGIP

10.1.6.1 Well Stimulation for Formation with Low Permeability

10.1.6.2 Reduce the Production Output

10.1.6.3 Single-Layer Production for Layer with Low Permeability

10.2 Reserve Estimation of a Two-Layer Commingled Gas Well

10.2.1 Physical Model and Mathematical Model

10.2.2 Examples

10.3 Material Balance Equation of Gas Reservoir with a Supplying Region

10.3.1 Introduction

10.3.2 Material Balance Equation of Gas Reservoir with a Supplying Region

10.3.2.1 Physical Model

10.3.2.2 Mathematical Model

10.3.2.3 Deliverability Equation and Recharge Equation

10.3.3 Example and Discussion

10.3.3.1 Example

10.3.3.2 p/Z Curve

10.3.3.3 Performance Forecast

10.4 Summary

Table of Contents for
Chapter 10: Material Balance Equation of Multilayer Gas Reservoir