As already noted, to study the long-term effect of increased effective stresses on the reservoir permeability, the core samples were taken from a terrigenous oil productive object (Fig. 1). The average reservoir depth is 1489 m, the initial reservoir pressure is 14.5 MPa, and the gas saturation pressure is 8.5 MPa. Standard core samples (length and diameter 30 mm) were drilled from the original core material, which were then extracted and dried (Fig. 1). Then, for all prepared samples, using the PIK-PP laboratory equipment, the nitrogen permeability was determined at the minimum pressure (0.5 MPa) of all-round core samples compression.

The average porosity of the samples was 22.4%, and the permeability was 656 mD. For further long-term experiments, several standard samples were selected, having the highest permeability, without natural fractures, and represented only by sandstone (with no clay inclusions). In the selected samples, residual water saturation was created using a reservoir water model, oil saturation was simulated with kerosene.

For long-term studies of samples, the UIK-5VG laboratory equipment was used, which allows measuring the permeability of the rocks during fluid filtration, while it is possible to vary the pore pressure and core sample compression pressure. At the UIK-5VG, the end faces of the sample are rigidly fixed, and compressive pressure are created on the side surface of the sample. Below are the equations for determining the required values of the reservoir and compressive pressure in accordance with the given reservoir conditions.

The value of vertical stresses was determined from the following equation:(1) σv=ρgH.

The following equation also can be used:(2) σv=γH.

For the considered oil field conditions the parameter γ was set equal to 0.022 MN/m³. In this case, for an average core sampling depth of 1489 m, the vertical stress is 32.8 MPa. Effective vertical stress is defined as:(3) σveff=σv−αp0.

The initial vertical effective stress is 21.9 MPa. The effective horizontal stress can be found in the equation:(4) σheff=σveffv1−v.

As follows from Eq. (4), the initial horizontal effective stress is 10.3 MPa. In the experimental research, the average effective stress should correspond to the average effective stress in the reservoir:(5) σaveff=σveff+2σheff3 and equal to 14.2 MPa.

The constant conditions of the experiment are usually the initial reservoir pressure and the gas saturation pressure. At the same time, if the reservoir pressure falls below the saturation pressure, it should also be taken into account. For the considered reservoir the saturation pressure is 8.5 MPa, however, the reservoir pressure has already decreased to 7.5 MPa, in this regard it is advisable to take the limit value of the reservoir pressure of 5 MPa.

Taking into account that at the UIK-5VG the ends of the samples are rigidly fixed and using the equations of the elasticity theory, the effective stresses on the ends is determined by the equation:(6) σaxialeff=2vσconfeff.

Thus, the average stress in the sample can be determined from the following equation, taking into account the fact that for the standard sample, the side surface area is 2 times larger than the end area:(7) σavcoreeff=(2σconfeff+σaxialeff)/2.

After transformation:(8) σavcoreeff=σconfeff(1+v).

The effective stress from the compression pressure at the UIK-5VG is determined by the following equation:(9) σconfeff=σconf−p.

As a result, we obtain the ratio based on which it is possible to determine the compression pressure in the experiment:(10) σaveff=σavcoreeff=σconfeff(1+v).

And the final equation is following:(11) σconf=σaveff1+v+p.

Based on this equation, it is possible to determine the compression pressure during testing of core samples, knowing the effective stress and reservoir pressure under-conditions. Table 1 shows the program of the experiments, obtained according to the above equations. As can be seen from Table 1, the total time for each experiment was 39 h. It should be noted that the research program included three loading cycles for 11.5 h at three different values of effective stresses. The first loading cycle corresponds to the initial reservoir conditions (effective stress 14.2 MPa) and is required to reach the initial value of the core samples’ permeability. Further, this permeability value will be used to determine the nature of the relationship between permeability and increasing effective stresses.

The next loading cycle at constant effective stresses corresponded to the decrease of reservoir pressure by 3 MPa (effective stress is 17.2 MPa), and the last loading cycle simulated the ultimate decrease in reservoir pressure to 5 MPa, i.e., by 9.5 MPa (effective stress is 23.7 MPa).

As can be seen from the research program, in each loading cycle, 5 permeability measurements were made, while the holding step of loading was gradually increased from 0.5 to 5 h. This is due to the fact that at the constant effective stress, the change in permeability has a damped character, and a more discrete change in this characteristic does not make sense.

In total, long-term studies were carried out on the three different core samples. For example, Fig. 2 shows the results of such an experiment for the one sample, which shows the changes in permeability depending on the effective stresses and the experiment duration. As can be seen from Fig. 2, with an increase in effective stresses to the initial reservoir conditions value (14.2 MPa), an intensive permeability decrease from 745 to 563 mD, or by 24.4% is observed.

Then, in the process of holding the sample under the constant effective stress of 14.2 MPa (initial reservoir conditions), the permeability of the sample also decreases from 563 to 530 mD or by 5.9%, which is caused by viscoplastic rock deformations. With the further effective stresses increasing, the permeability continues to decrease, including under constant effective stresses of 17.2 and 23.7 MPa, which corresponds to the decrease in reservoir pressure by 3 and 9.5 MPa, respectively. As can be seen from Fig. 2b, in each holding cycle under the constant effective stress, the change in permeability has the damping character, which indicates that the viscoplastic rock deformations tend to the limit value. It also should be noted that when the effective stresses (reservoir pressure recovery) were restored to the initial value of 0.5 MPa, the permeability did not recover to its initial value and has a value of 642 mD, which is 86.2% of the initial value of 745 mD. Thus, we can establish that the viscoplastic strains of the sample are 13.8%.

In Fig. 3, the blue circles show the averaged relative permeability values for all three tested samples, which were determined from the last results in each core sample holding cycle under the constant effective stress. The permeability value was related to its limiting value during the first holding cycle, which corresponds to the initial reservoir conditions. As can be seen from Fig. 3, the maximum decrease in permeability with a decrease in reservoir pressure from 14.5 to 5 MPa, on average, amounted to 20.6%, which is a very significant value and, undoubtedly, should affect the well’s productivity. It should be noted that in this case, the approximating function should be built from the effective stresses corresponding to the initial reservoir conditions (14.2 MPa).

In scientific publications, various equations that describe the change in permeability depending on effective stresses are used [16–19]. In various studies quite often the changes in the porosity of samples are first determined from effective stresses or reservoir pressure, and then the changes in permeability are determined from the porosity-permeability relationships [1,4,17,20]. It was shown in [15] that the experimental relationships of the decrease in the relative permeability of terrigenous reservoirs with an increase in effective stresses (a decrease in fluid pressure) are quite accurately approximated by the following analytical function:(12) krel=1−a(σaveff−σ0eff)b+(σaveff−σ0eff).

After statistical processing of the laboratory experiments results, the following analytical equation of the change in the relative permeability of the reservoir rock under the influence of effective stresses was obtained (Fig. 3):(13) krel=1−0.35(σaveff−14.2)6.58+(σaveff−14.2).

This equation was later used in the numerical modeling of the near-wellbore zone stress state during the process of reservoir and bottomhole pressure drawdown.

At present, poroelasticity equations are often used to determine the stress state of reservoir rocks in the oil and gas fields [1,4]. In this case, the mutual influence of rock stresses and the inhomogeneous distribution of the reservoir are taken into account. In this work, for numerical calculations, the ANSYS finite element modeling software package was used, which has proven itself well in solving geomechanical problems associated with the development of hydrocarbon fields [21,22].

The fluid-containing porous medium was modeled by representing it as a two-phase material (liquid and rock) and applying the basic principles of Biot theory. At the same time, the fluid flow was considered single-phase. It was assumed that the porous medium is completely saturated by fluid. The basic equations for solving the problem of consolidating the Biot theory implemented in the ANSYS software package, are given below:(14){ ∇•σ+f=0ИлИ∇•(σ′-αpI)+f=0αε˙V+1Kmp˙+∇•q=S

The relationship between the effective stress tensor and elastic deformations of the rock skeleton:(15) σ′=Dεe.

The relationship between fluid flow and pore pressure is described based on Darcy’s law:(16) q=−k∇p.

This study assumed that after the decrease in reservoir or bottomhole pressure, a sufficient amount of time passed for the fluid filtration mode to be established, in this case, at each calculation step, the initial conditions corresponded to the boundary conditions. The poroelastic model implemented in ANSYS allows for the setting of two types of boundary conditions for two simulated media (liquid and solid phase). For the liquid, either the liquid filtration rate or its pressure at the model boundaries can be set. For the solid phase (reservoir rock), either displacements or stresses at the boundaries of the model can be specified. It is also assumed in the calculations that the liquid is incompressible.

The simultaneous equations are derived from the element formulations which are based on the principle of virtual work [23]:(17) ∫V⁡σδedV=∫V⁡fBδudV+∫Г⁡fГδudГ

The left side of Eq. (17) is the virtual work. Taking into account Eqs. (14)–(16), (17), after discretization of differential equations by the finite element method, the connection of the adjoint problem can be expressed as a system of equations [24]:(18) [KLLTE][ΔuΔp]=[FR]

The matrix E can be defined as the difference: T – D.(19) R=Q−TP.

The system of Eq. (18) is divided into two components:–

stress equations:(20) KΔu=F−LΔp,

The connection between the equations is carried out through the matrix L.

For the numerical calculations, the four-node quadrangular finite element CPT212 was used, according to the ANSYS classification. This element is used to solve poroelasticity problems.

The sensitivity of the finite element grid to the calculation results was tested for the case when the reservoir pressure is equal to the initial value (14.5 MPa) and the bottomhole pressure drawdown is equal to the maximum value (9.5 MPa). In this case, the maximum pressure drop occurs in the bottomhole zone of the well, which is the most sensitive factor for numerical calculation. The remaining characteristics were taken in accordance with Table 3. For the calculations, the finite element scheme shown in Fig. 4 was used. For comparison, the filtration rate characteristic on the well’s wall was used, on the basis of which the well productivity was subsequently determined.

Table 2 shows the calculation results. The number of finite elements varied from 10 to 100 with a step of 10. As can be seen from this table, starting from 40 finite elements, there is no difference in the filtration rate. Thus, the optimal number of finite elements in the design scheme was chosen, equal to 40.

Since it is not possible to analytically determine the fluid filtration rate on the well’s wall, taking into account the changing effective stresses and rock permeability, the comparison of the results was carried out on a model with a constant permeability equal to the initial value (530 mD), while, as in Section 3.3, the maximum bottomhole pressure drawdown was set equal to 9.5 MPa. Analytically, the filtration rate was determined based on the Dupuis equation for the flow rate:(22) Qw=2πkavhΔpμln(rk/rw).

Dividing the well flow rate by the area of the lateral surface of the well (2πr_wh), an analytical relationship for determining the fluid filtration rate on the well’s wall was obtained:(23) Vw=kavΔprwμln(rk/rw).

Substituting into expression (23) the above values of permeability and bottomhole pressure drawdown, the radius of the well and the radial extent from Table 3, obtained that the value of the fluid flow rate will be 0.5791 10⁻¹ m/s. Using similar initial data, the fluid flow rate calculated in ANSYS amounted to 0.6019 10⁻¹ m/s. The difference in the analytical and numerical calculations is 3.8%, which is quite acceptable. Apparently, the difference in the calculations is due to the fact that in numerical calculations, the pressure distribution in the near-wellbore zone is affected not only by drawdown pressure, but also by the change in effective stresses. Thus, a comparison of calculations shows that the chosen method of numerical simulation can be used to calculate the fluid flow filtration rate and well productivity.

Table 3 shows the initial data used for numerical simulation. Fig. 4 shows the finite element axisymmetric design scheme used to model the near-wellbore zone. The scheme was created in such a way that near the wall there was a maximum concentration of finite elements, since in this area there is the greatest change in most of the calculated parameters: reservoir pressure, effective stresses and reservoir permeability. As the boundary conditions of the model, the pressure on the well’s wall and on the radial extent and the vertical stress were set. On the lower boundary of the model, zero displacements along the normal to the surface were set.

Using the developed finite element scheme, multivariant modeling of the near-wellbore zone was performed. In the calculations, the bottomhole pressure drawdown value and the pressure at a distance from the well were varied, which simulated a decrease in reservoir pressure. Figs. 5 and 6 present the main results of the simulation: the change in the relative permeability of the near-wellbore zone and the relative productivity ratio of the production well depending on the bottomhole pressure drawdown. Also in the calculations, the reservoir pressure varied. Relative permeability was defined as the current permeability divided by the permeability at initial reservoir pressure (calculated maximum value). The relative productivity ratio was determined in a similar way-the current calculated value of the productivity ratio was divided by its maximum value, which in this case corresponded to the initial reservoir pressure and the minimum bottomhole pressure drawdown.

As can be seen from Fig. 5, an intense decrease in permeability is observed near the well, caused by a decrease in bottomhole pressure and the formation of a depression funnel. The magnitude of the decrease in permeability increases with an increase in bottomhole pressure drawdown, which is caused by the effective stresses increasing. At the initial reservoir pressure, depending on the bottomhole pressure drawdown, the permeability decreases from 6% (drawdown 4 MPa) to 17% (drawdown 9.5 MPa).

Reducing the reservoir pressure correspondingly reduces the permeability in the entire near-wellbore zone. So, with a decrease in reservoir pressure from 14.5 to 8.5 MPa with zero bottomhole pressure drawdown, the permeability in the entire near-wellbore area decreases by 8% (Fig. 5b). When bottomhole pressure drawdown increased, the permeability decreases by an additional 8.7%.

The most objective indicator of well exploration in conditions of changing reservoir and bottomhole pressure is the productivity ratio, which was determined as the well flow rate divided by the bottomhole pressure drawdown. Next, the relative productivity ratio was calculated (Fig. 6) as the current productivity ratio divided by the maximum value of this parameter.

As can be seen from Fig. 6, at the initial reservoir pressure of 14.5 MPa and maximum bottomhole pressure drawdown of 9.5 MPa, the productivity ratio decreases by 10%, which is caused by the compaction of the reservoir rock under the increasing effective stresses. With a decrease in reservoir pressure, a decrease in well productivity is observed. Thus, with a minimum modeling reservoir pressure of 8.5 MPa and a maximum bottomhole pressure drawdown of 3.5 MPa, the relative productivity ratio will be 87% of the maximum value, i.e., will decrease by 13%.