New Understanding of Transient Pressure Response in the Transition Zone of Oil-Water and Gas-Water Systems

Well test analysis requires a preselected model, which relies on the context input and the diagnostic result through the pressure logarithmic derivative curve. Transient pressure outer boundary response heavily impacts on the selection of such a model. Traditional boundary-type curves used for such diagnostic purpose are only suitable for single-phase flow in a homogeneous reservoir, while practical situations are often much more complicated. This is particularly true when transient pressure is derived during the field development phase, for example, from permanent down-hole gauge (PDG), where outer boundary condition such as an active aquifer with a transition zone above it plays a big role in dominating the late time pressure response. In this case, capillary pressure and the total mobility in the transition zone have significant effect on the pressure response. This effect is distinctly different for oil-water system and gas water system, which will result in the pressure logarithmic derivatives remarkably different from the traditional boundary-type curves. This paper presents study results derived through theoretical and numerical well testing approaches to solve this problem. The outcome of this study can help in understanding the reservoir behavior and guiding the management of mature field. According to the theoretical development by Thompson, a new approach was derived according to Darcy’s law, which shows that pressure response in the transition zone is a function of total effective mobility. For oil-water system, the total effective mobility increases with an increase in the radius of transition zone, while for gas-water system, the effect is opposite.


Introduction
With the increased reliability and technology enhancement, more and more PDG have been installed in oilfields around the world. Long-term data from PDG have the potential to provide more information about a reservoir than data from traditional pressure transient tests [1][2][3]. In particular, during the oil field development stage, the influence of multiphase flow due to long-term nature of PDG data is significant [4][5][6].
Some researchers proposed different scale models to study different types of reservoirs, such as fractured tight reservoirs [7,8], preferential flow path with non-Darcy flow reservoirs [9], mature waterflooding reservoirs [10], and shale gas reservoirs [11,12]. It is well known that the pressure logarithmic derivative curve has been used to identify true boundary model, but for multiphase flow well testing [13,14], it is much more complicated, there is a great influence on pressure derivative due to the changes of fluid properties in the transition zone [15][16][17]. For example, the logarithmic derivative of bottom-hole pressure goes up and then goes down in the late time for gas-water reservoir systems with an active aquifer underneath. If using a traditional boundary-type curve for the model diagnostic, the wrong conclusions may be obtained.
In the literature, some articles have been published on multiphase flow, including those from Ramakrishnan and Wilkinson [18], Thompson (1997), and Roadifer [19]. Ramakrishnan and Wilkinson took account of Buckley-Leverett theory to describe the saturation profile and developed the radial Buckley-Leverett model [18]. Thompson examined the behavior of transient pressure for singlephase flow and multiphase flow in heterogeneous reservoirs. He used mass conservation equations and Darcy's law to derive the pressure derivatives, which can be used to interpret well test from single-phase and multiphase flow. For singlephase gas reservoir, the pressure derivative is a function of the changing of rates with time. But for multiphase flow, Thompson focused on gas-condensate reservoir and water injection in an oil reservoir. He found that multiphase flow drawdown is heavily influenced by the mobility in the region, where the mobility is changing most rapidly with time. But for both oil-water and gas-water systems, Thompson did not consider the influence of transition zone [13].
Roadifer [19] examined the pressure behavior in a multiphase reservoir with a constant pressure boundary. Buckley-Leverett theory was used to analyze multiphase well test at the same time. Some laws on water saturation front were derived numerically. Roadifer used Thompson equation to interpret the pressure derivative of oil-water and oil-gas systems with constant pressure boundary. But he did not consider the influence of capillary pressure [19].
In fact, for multiphase flow system, capillary pressure cannot be ignored [20,21]. In this work, based on Thompson equation, but also considering capillary pressure, a new theoretical equation is derived from Darcy's law, which shows that the pressure response in the transition zone is a function of total effective mobility and capillary pressure.
In order to study the rules of multiphase flow in transition zone, a 3D model was set up using Eclipse to simulate the pressure responses due to multiphase flow. The oilwater and gas-water systems were studied. Capillary pressure was considered under closed reservoir or constant pressure outer boundaries. Numerical well testing results can be interpreted qualitatively by this new theoretical equation for different multiphase reservoir systems. It is particularly useful for analyzing the transient pressure outer boundary response.

Theory
According to the law of conservation of matter for black oil system, isothermal multiphase flow in radial homogeneous reservoir is described by a partial differential equation [22], that is, diffusivity equation. When the conservation of mass equation in each flowing phase is satisfied, Darcy's law, which describes pressure losses of system, is also satisfied.
First, in an oil-water system, capillary pressure is considered and gravity effect is ignored. The production rate can be written at any radial location as [23][24][25] where A r is the cross-sectional flow area, ft 2 , that is, Equations (1), (2), and (3) can be rearranged as where λ t r, t is the total effective mobility, that is, λ t r, t = λ o r, t + λ w r, t = K ro S w /μ 0 + K rw S w /μ w Equation (4) can be applied to bounded reservoirs (outer boundary condition: ∂P e /∂r| r=r e = 0, t > 0, where r e is the radial extent of reservoir and P e is the pressure of outer boundary) or infinite-acting reservoirs (outer boundary condition: lim r→∞ P r, t = P i , where P i is the initial reservoir pressure).
Considering finite-acting reservoirs, separating variables of (4) and taking integral along a radial direction: The drawdown solution can be obtained from (5): Finally, according to the results of laboratory experiments, capillary pressure is a function of water saturation and water saturation is a function of radial distance and elapsed time. With respect to the natural logarithm of elapsed time, (6) can be differentiated, and the pressure derivative is given by Equation (7) is the general equation, which applies to a radial flow system producing under defined wellbore boundary conditions. From this equation, it can be seen that the pressure logarithmic derivative is a function of the total effective mobility, capillary pressure, liquid rate, and water saturation. Because this equation is highly nonlinear, an analytical solution cannot be obtained. So the numerical well testing approach was used to study multiphase flow rules, which considered phase effective permeability, capillary pressure, saturation gradient, and complex formation properties. The numerical results can be analyzed qualitatively by (7).

Numerical Well Testing
In this section, a 3D model was built to simulate multiphase flow in order to derive numerical solution. At the same time, (7) was used to verify qualitatively the accuracy of the numerical solution.

Numerical Model.
In the constructed numerical model, the basic simulation grid consisted of 51 cells in the Idirection and 51 cells in the J-direction, with 6 layers (in both the anticline model and the flat model) (Figure 1). For the anticline model, the angle of formation bedding (α) is about 2.8°and much smaller, so the gravity of liquid can be neglected. Two systems, an oil-water model and a gas-water model, with different properties were used in the simulations. Data for the simulation model are summarized in Table 1. 3.1.1. Grid Description. Corner point geometry grids were used with approximate dimension of 200 × 200 × 30 ft. Since bottom-hole pressure is sensitive to the size of grid, a nested grid technique was used in the model ( Figure 2). This not only makes the wellbore better connected to the grid thereby avoiding numerical dispersion but also improves the simulation speed. 3.1.3. Relative Permeability Modeling. Two sets of relative permeability data to oil-water model ( Figure 3) and gaswater model ( Figure 4) were used, which show that when K ro is equal to K rw , the corresponding water saturation is more than 0.5, which denotes that the rock has water wetting property.
3.2. Case Study 1: Oil-Water System. The studies were designed to test the impact of reservoir and fluid properties on the pressure response. All cases are the start of a single producer at a constant flow rate. These key parameters are listed in Table 2. 3.2.1. Single-Phase Flow. The first case considered singlephase flow. As shown in Figure 5, these are simulation results of the drawdown (DD) tests for closed system. The pressure derivatives of the two cases from flat and anticline models are the same. It means that the shape of model does not affect the numerical solutions of models.
For single-phase flow in these closed systems, mobility does not change and capillary pressure is not present, that is, ∂λ t r, t /∂t = 0 and P o = P w , P c = 0; therefore, (7) reduces to

Geofluids
Hence, the pressure derivative is a function of the flow rate changes with time in the reservoir.
Before pseudo-steady state is reached, the flow rate in the reservoir will continue to increase, for example, if t < t pss , where t pss is the time to reach the pseudo-steady state; we can get According to the result of (9), (8) shows that the pressure derivative will remain positive until the pseudo-steady state is reached, that is, At late time, when the pressure disturbance reaches a closed boundary, the formation pressure decreases with time, until the drawdown at wellbore is equal to the drawdown at boundary, from then on, the pseudo-steady state flow will start.
At the same time, the formation flow rate at any location with elapsed time will approach to some constant value and the increment of changes for formation flow rate at any location will approach to zero, that is, After the pseudo-steady state flow, the increment of changes for formation pressure at any location will approach to constant.
Equation (13) indicates that in late time, the slope of derivative in log-log plot is unit 1. Figure 5 shows that the numerical result is identical to that from (13).
All of these results are consistent with single-phase flow solutions in closed system. No P c Anticline 4700 Center Closed system 10 Closed system: the no-flow condition implies zero superficial velocity at the outer boundary and hence the local pressure gradient must also be zero, that is, ∂P e /∂r| r=r e = 0, t > 0; aquifer: the well is situated in the center of a drainage area with a constant pressure, equal to the initial pressure, P i , maintained along the outer boundary, P r e , t = P i , t > 0.

Two-Phase Flow.
For two-phase flow in homogeneous reservoir, the capillary pressure was not considered at the beginning, that is, P o = P w , P c = 0. According to Buckley-Leverett theory, in oil-water transition zone, there exists saturation front, and the reservoir can be divided into three regions ( Figure 6). The first region is between the wellbore and the downstream side of saturation front, r w < r < r − f ; the second region is between the downstream side of saturation front and the upstream side of saturation front, r − f < r < r + f ; and the third region is between the upstream side of saturation front and outer boundary of the reservoir, r + f < r < r e .
After the pressure reached the boundary of closed system and the pseudo-steady state occurred, the increment of change for volumetric flow rate at any location in late time will approach to zero, that is, Δq t r ′ , t → 0, t > t pss or ∂Δq t r′, t /∂t → 0, t > t pss ; the general pressure derivative equation (7) can be simplified as Ahead of saturation front, the flow is single-phase (oil) and the total effective mobility is unchanged, so dλ t t /dt = 0, r w < r < r − j ; from upstream of saturation front to outer boundary, the total effective mobility increases and finally becomes constant, so at late time, dλ t t /dt → 0, r + f < r < r e , then (14) can be reduced to Equation (15) is the generalized pressure derivative equation for transition zone of oil-water or gas-water systems. Since (15) is a highly nonlinear equation and cannot obtain analytical solution, it can be used to qualitatively validate the numerical solution and interpret the pressure behavior in transient zone.
According to the interpretation results of numerical solution, the pressure derivative curve, the total mobility curve of saturation front, the fluid rate of saturation front, and the total mobility derivative curve of saturation front were obtained. As shown in Figure 7, the pressure derivative starts to go down at 48.16 hours, and then goes up. How can this phenomenon be explained? Figure 8 shows that when the well produces by depletion, at downstream saturation front, with oil produced, the combined water did not flow and the oil mobility was decreased, so the total mobility was decreased until to the point of 48.16 hours. At upstream saturation front, the combined water starts to flow, water encroachment will occur, and the total mobility starts to increase rapidly.
After 48.16 hours, the λ t r, t of upstream saturation front increases, so ∂λ t r, t /∂t increases with time and ∂λ t r, t /∂t > 0 (Figure 9). According to (15), dΔP o t /d ln t is negative and decreasing with time.  Figure 6: According to Buckley-Leverett theory, using saturation front, reservoir can be divided into three regions. The first region is between the wellbore and the downstream side of saturation front, r w < r < r − f ; the second region is between the downstream side of saturation front and the upstream side of saturation front, r − f < r < r + f ; and the third region is between the upstream side of saturation front and outer boundary of the reservoir, r + f < r < r e .

Sensitivity Studies
(1) Effect of Different Oil-Water Contacts (OWC). Under different OWC conditions (as listed in Table 2, the depths of OWC for case OW-1 and OW-9 are 4735 ft and 4700 ft, resp.), there is a large variation in the pressure derivative. Figure 10 shows that the pressure derivatives go down overall at the transition zone, but for high oil-water contact, because the saturation front is much higher, the pressure disturbance reaches the saturation front quickly. Hence, the pressure derivative goes down earlier and much further.
(2) Effect of Capillary Pressure. In practice, capillary pressure exists everywhere in reservoirs with multiphase flow as long as oil saturation is different from water saturation [26][27][28]. This is reservoir's inherent nature. According to analysis above, considering the capillary pressure of saturation front in transition zone, (7) can be rearranged for transition zone as For λ equivalent r ′ , t = λ w r ′ , t / λ t r ′ , t = λ w r ′ , t / λ w r′, t + λ o r′, t = 1/ 1 + λ o r′, t /λ w r′, t , from the downstream saturation to the upstream saturation front  Figure 8: (a) The total mobility curve and (b) the volumetric of water curve just downstream of the saturation front (r → r − f ), before 48.16 hr, the combined water does not flow, but because the oil is moving, the total mobility decreases. (c) The total mobility curve and (d) the volumetric of water curve at upstream saturation front (r → r + f ), from 48.16 hr, the water encroachment will occur, the total mobility starts to increase rapidly. 7 Geofluids decreases with time, so λ o r ′ , t /λ w r ′ , t decreases with time. Hence, λ equivalent r ′ , t increases and ∂λ equivalent r ′ , t /∂t > 0. But for ∂P c r′, t /∂S w , it is negative as shown in Figure 3. However, Figure 11 indicates that ∂S w r′, t /∂r′ is positive, so < 0 17 Figure 12 shows that the transition zone pressure derivative of the reservoir with capillary pressure goes down more than that without capillary pressure.
(3) Effect of Constant Pressure Outer Boundary. As shown in Figure 13, with and without capillary pressure, the pressure derivative followed nearly the same trend in reservoir with constant pressure at the outer boundary.
(4) Effect of Oil Viscosity. In order to study the pressure behavior of a transition zone caused by varying oil viscosities, four cases were designed, in which oil viscosity is 1 cp, 5 cp, 10 cp, and 25 cp (the oil/water viscosity ratio is 2, 10, 20, and 50), respectively.
According to the radius of investigation equation,     Figure 12: The log-log plot of drawdown of a closed system. It shows that the transition zone pressure derivative of reservoir with capillary pressure goes down more than that without capillary pressure. 8 Geofluids If the oil/water viscosity ratio is higher, the time of the pressure reaching transition zone is longer. This is consistent with simulation results shown in Figure 14.
Meanwhile, according to numerical simulation results, under low oil/water viscosity ratio condition (such as the oil/water viscosity ratio is 2), the pressure derivative did not decrease in log-log plot. As shown in Figure 15, although the total mobility has changed, the changes of total mobility are very small, especially in late time, ∂λ t /∂t → constant; hence, the total effective mobility did not change quickly at saturation front of transition zone under low oil/water viscosity ratio condition, so the phenomenon of the pressure derivative going down at transition zone will not occur.

Case Study 2: Gas-Water System
3.3.1. Two-Phase Flow. For a gas-water reservoir system, there are two boundary system conditions: the closed boundary and the constant pressure boundary. Sensitivity studies were designed for these systems. The key parameters are listed in Table 3.
As with the studies of the oil-water system, capillary pressure was not considered at first. As shown in Figure 16, the pressure derivative of the wellbore pressure starts to go up at 10.10 hours. This is due to the rapid decrease in total mobility ( Figure 17). According to (7), at 10.10 hours, the λ t r, t decreased, so the absolute value of ∂λ t r, t /∂t increased and ∂λ t r, t /∂t < 0, but dΔP o t /d ln t is positive and increasing with time.

Sensitivity Studies
(1) Effect of Capillary Pressure. According to (18), because − q t r′, t /r′λ 2 t r′, t ∂λ t r′, t /∂t > 0, ∂λ equivalent r′, t /∂t ∂P c r ′ , t /∂S w ∂S w r ′ , t /∂r ′ > 0, so dΔP o t / d ln t > 0. Figure 18 shows that the transition zone pressure derivative of a reservoir with capillary pressure goes up more than that without capillary pressure.
(2) Effect of Constant Pressure Boundary. Considering constant pressure boundary, Figure 19 shows the simulation results of drawdown (DD) test and buildup (BU) test under constant pressure boundary conditions. The pressure derivatives of DD and BU both go up when the pressure reached transition zone. Figure 20 shows that the transition zone pressure derivative in a reservoir with capillary pressure goes up more than that without capillary pressure.
(3) Distance of Movement for Saturation Front. In order to simulate the saturation front movement, PDG data were generated by simulation ( Figure 21). The first DD test and the last DD test were selected for analysis. As shown in Figure 22, for the first DD, at 2.123 hours, the pressure reached the saturation front, but for the last DD, the time for pressure reaching the saturation front is about 1.190 hours. Using (18), for the first DD, the radius investigation of R 1 is calculated as about 1712.237 ft; for the last DD, the radius investigation of R 2 is calculated as about 2282.869 ft. The angle of formation bedding (α) is about 2.8°, H = sin α × R 2 − R 1 ; the height of saturation front can be obtained as about 27.87 ft. Figure 23 shows that the height of saturation front is about 27.7 ft. In comparison with the well test and simulation results, the error is only 0.6%.

Field Application
In order to apply the pressure behavior of the study in practice, two field examples are studied: a gas well test and an oil well test.

Well A: Gas Well
Test. This well is located between two faults. The distances from the well to the faults are 2100 ft and 3600 ft (Figure 24). The average permeability of the gas 9 Geofluids reservoir is 1.8 mD, and the porosity is 11.5%. In order to test well productivity, a well test was performed ( Figure 25). From the test data, the last build up data was analyzed.
Because geologist engineers thought that the gas well did not encounter a water layer during the drilling, this gas reservoir is assumed to be a dry gas reservoir. Therefore, a single-phase theory was used to interpret the well test.  Figure 15: (a) The total mobility curve and (b) the total mobility derivative curve at upstream saturation front with oil viscosity at 1 cp. Although the total mobility has some changes, the total mobility derivative is very small and close to constant in late time.    Figure 17: The total mobility curve at upstream saturation front (r → r + f ), from 10.10 hours. The total mobility starts to decrease rapidly. 10 Geofluids From the log-log plot (Figure 26), at late time, the pressure derivative went up, using one fault model to match this test response; at 670 ft away from well A, there is a no-flow boundary. Combining the geological knowledge, reservoir engineers believed this no-flow boundary may be a subseismic fault. But after this well was put into production for two months, the water broke through. The lesson learnt from this case was that the use of singlephase theory to interpret the test is inappropriate. Although well test data had some ambiguities, if the impact of multiphase flow was not taken into account, especially at transition zone, which can affect the late time pressure response, the analysis result could lead to a completely wrong decision. In fact, in this well test log-log plot, the pressure derivative goes up at late time, purely due to the changes of total mobility in transition zone, and nothing to do with the reservoir outer boundary.

Well B: Oil Well
Test. This reservoir is an anticline reservoir ( Figure 27). The average buried depth of this reservoir is 1650 ft, and the oil viscosity is 120 cp. But the average permeability of the formation is 3 D, and the porosity is 0.27. Hence, although oil property is very bad, the formation property is very good. Using conventional method to produce, the single well productivity is very high. The key of the field development is the evaluation of formation energy. So well Elapsed time (hr) ∆m(p)′ GW-1-∆m(p)′ without P c GW-2-∆m(p)′ with P c Figure 18: The log-log plot of the pressure response in gas-water reservoir with P c and closed boundary. The pressure derivative with capillary pressure goes up more than that without capillary pressure.   testing was used ( Figure 28). From the test data, the last DD and the last BU were selected for the analysis. The DD and BU plots show that at late time, the pressure derivatives all go down (Figures 29 and 30), which was a sign of the pressure reaching aquifer, and it means that the reservoir has higher energy.     According to the well testing result, at early development phase, the natural depletion mechanism development was advised. But after 6-month production, some wells in this reservoir could not continue production, including well B.
In fact, for this reservoir, if multiphase flow theory was used to interpret the well test, in the log-log plot of DD, the pressure derivative decreased, purely as a response to the change of the total mobility in the transition zone. Again, this is nothing to do with the outer boundary.

Conclusions
(1) Based on a theoretical development by Thompson, a new expression can be derived from Darcy's law. According to this expression and the numerical well test results, we reasonably interpret the pressure behavior of transition zone in oil-water and gaswater reservoirs. The results presented in this work are generally applicable to multiphase reservoir, which have either an infinite-acting or a constant pressure outer boundary.
(2) Capillary pressure in the transition zone has an insignificant impact on pressure response.