SPE 124484

New Technique to Determine Biot Coefficient for Stress Sensitive Dual Porosity Reservoirs H.H. Abass and A.M. Tahini, Saudi Aramco; Y.N. Abousleiman, Oklahoma University; and Mirajuddin Khan, Saudi Aramco

Copyright 2009, Society of Petroleum Engineers This paper was prepared for presentation at the 2009 SPE Annual Technical Conference and Exhibition held in New Orleans, Louisiana, USA, 47 October 2009. This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.

Abstract Poroelasticity plays a vital role in many applications of rock mechanics in petroleum engineering. To model dual porosity stress sensitive reservoirs such as tight gas, it is necessary to quantify the stress-dependant hydraulic conductivities of their porous components, the matrix and natural fractures. The role of effective stress concept near the wellbore is more pronounced as the reservoir pressure declines rather rapidly approaching the bottom hole flowing pressure. The porous media changes when reservoir pressure is altered and therefore the elastic coefficient () becomes a function of reservoir pressure for both components; matrix and fractures. Rock samples from tight formation were tested to determine the stress-dependant matrix permeability. Various combinations of net effective stresses were applied and the corresponding permeabilities were measured at each stress level. Applying the definition of effective stress with an iterative assumption of Biots coefficient enabled the determination of () as a function of pressure. Additionally, the Biot coefficient was determined for a natural fracture. A tensile fracture was simulated by splitting a whole core following a Brazilian test procedure. The stress dependent permeability was evaluated under varied effective stresses simulating a reservoir depletion scenario. The concept of effective stress within the natural fracture was totally different than that of rock matrix; therefore, the effective stress function for both matrix and natural fractures were evaluated and representative functions for both media were obtained. These stress-dependent permeability functions can then be used in any rock mechanics application in petroleum engineering. The developed experimental procedure is fairly simple and is discussed in details, and selective relavant applications are presented. The new technique for determining the Biot coefficient is based on the application it is being used for; the effect of effective stress on matrix and natural fracture permeabilities. This paper shows that two poroelastic coefficients, and not one, should be used in any dual porosity system to obtain reasonable prediction of reservoir performance. Varying the coefficient as a function of pore pressure allowed for determining a pressure dependant function for a given porous medium. Introduction Tight reservoirs behave as dual porosity/permeability systems in which the rock matrix, natural fractures network and created hydraulic fractures, if these reservoirs are hydraulically fractured, contribute to the hydrocarbon transport in a very complex manner. Permeability loss due to increasing effective stress as a result of reservoir depletion can result in substantial cumulative recovery loss1. An increased effective stress, which is a combined effect of stress and pore-pressure, may decrease reservoir permeability slightly or considerably depending on its initial permeability. The permeability of tight reservoirs has been postulated to be highly sensitive to changing effective stresses. This permeability sensitivity to changing stresses is most pronounced in tight; over pressured, naturally fractured reservoirs where the apertures of natural fractures are very sensitive to applied closure stress resulted from reservoir depletion. A working model consistent in both fluid flow and geomechanical considerations is required to link various fluid-rock information (e.g., flow/storage properties, rock mechanical properties, reservoir fluid pressure and stress level) measured by

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different techniques and to forecast reservoir performance2. Natural fractures impact on production has been studied by various authors whom have proposed that these fissures if kept open can contribute substantially to production.3-9

Transvers anisotropy of Biot coefficient in the vertical and horizontal directions was experimently determined10. Understanding the hydraulic characteristics of the matrix frame, natural fractures, and created hydraulic fractures as a function of effective confining stress is vital to design optimum stimulation treatments, to predict reservoir performance via reservoir simulation. The objective of this study was to investigate permeability reduction characteristics of natural fractures in highly stressed reservoirs with lowered pore pressure. A pore pressure versus permeability relationship is essential for optimizing field development scenarios. This matrix-fracture flow transfer function has been used to simulate fluid flow through fractured porous medium. A shape factor is usually imbedded within the transfer function. Considering the work presented in this paper, a new perspective for the transfer function including the shape factor should be considered to include the stress dependant fracture aperture and its permeability, in addition to the stress dependant matrix permeability. Effective Stress Concept The effective stress concept suggests that pore pressure helps counteract the mechanical stress carried by the grain-to-grain contact. The efficiency of reservoir pressure, pr, in supporting the earth stresses is measured by the poroelastic factor , and the relationship is11,12:

pa= .... (1)

Where , is the effective stress, and is the total stress. The coefficient, , is given by:

= 1ccma

b, 0 1 .. (2)

With the bulk compressibility, cb, given by:

c3(1 2 )

Eb=

(3)

If the rock has low porosity as it is the case in TGS, the matrix compressibility, cma, may be close to cb, and therefore approaches zero. Conversely, with high porosity, the matrix compressibility is much smaller than the bulk compressibility; therefore, becomes closer to one. The role of effective stress concept near the wellbore is more pronounced as the reservoir pressure declines rather rapidly, approaching the bottomhole flowing pressure (BHFP). Understanding the effective stress concept in TGS is critical to explain many of the unique observations encountered in producing these reservoirs. Lets consider a given pressure drawdown profile of a single well. The reservoir pressure as a function of radial distance from a given well can be simplified as follows:

w

e

wwew r

rrrPPPrp )()( +=

... (4)

For example, if pe = 12,000 psi, pw = 2,500 psi, re = 4,000 ft, rw = 0.5 ft, =15,000 psi, and = 1.0 or 0.3, then the pressure and effect stress gradients around the flowing wellbore for the given example is shown in Figure 1.

SPE 124484 3

0

4,000

8,000

12,000

16,000

0 1,000 2,000 3,000 4,000Distance (ft)

Pres

sure

or st

ress

, psi

reservoir pressureeffective stress (Alfa=1)effective stress (Alfa=0.3)total stress

Figure 1: Reservoir pressure and effective stress gradients around a wellbore.

The greatest pressure drop occurs within a short distance from the wellbore. Therefore, the effective stress will be the highest near the wellbore, causing permeability reduction in addition to that caused by radial flow convergence and skin. Although is assumed to be constant, it is really a function of pressure. The near wellbore permeability in specific, and the reservoir permeability in general, changes with pressure, thus affecting our well test analysis, single-well reservoir simulation studies, and the reservoir management strategies as a whole. Therefore, modeling this mechanism is critical to how these petroleum engineering tools are used for TGS reservoir evaluation and performance prediction. Changes in fracture permeability caused by changes in effective stress, caused by changes in the pore pressure, have been observed at both laboratory and field studies. Although the terminology Stress-sensitive reservoirs has been widely used in the literature, all reservoirs exhibit stress-sensitive permeability. The effective stress magnitude and resulting deformation which produce permeability changes, differs from one reservoir to another. The effective stress is a function of pore pressure, total stress, and Biots coefficient, and the deformation is a function of the rock elastic and plastic characteristics. Therefore the resulting permeability change is a very complex function for an analytical equation and should be experimentally evaluated for a given rock formation at in-situ conditions. Experimental Simulation An experimental procedure was designed to simulate the reservoir permeability (matrix, natural fractures and induced fractures) reduction as a function of increasing effective stress. Whole core samples were used with dimensions of 4-in. diameter and various lengths (4 - 8 in). The sample is then tested for matrix conductivity, as it is positioned inside the rock mechanics loading frame where confining pressure is applied around the sample and a linear flow is established at a given pore pressure to determine the permeability as a function of various combinations of confining and pore pressures. Carbonate Sample Selected samples such that they do not appear to have microfractures were tested to determine the stress-dependant matrix permeability. Various combinations of net effective stresses were applied and the permeability measured at each stress level. Table 1 presents all combinations of applied confining stresses and the pore pressure levels and gradients for a given flow test. Recalling the definition of effective stress as given in equation 1, it is required to assume a value for the Biots coefficient (). Assuming is 1, and plotting matrix permeability as a function of effective stress, we obtain Figure 2. Close examination of Figure 2 suggests that for a given effective stress multiple values of permeabilities are measured. This is not an experimental error rather the assumption of being one is not valid. The next step is to change and replot the stress-dependant permeability function until a meaningful trend is obtained. Since is function of stress, then varying it within the constraints from the first step would produce the stress-dependant permeability presented in Figure 3 with estimated function (p).

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Table 1: applied confining stresses for various flow tests

Cp Pout Pin Pav P Q K psi psi psi psi psi mlcc/m D 2488 2004 1000 1502 1004 7.644 0.313

4519 4010 3000 3505 1010 6.359 0.2588

4506 2013 1000 1507 1013 4.224 0.1714

8555 8006 7000 7503 1006 5.124 0.2093

6495 4002 3000 3501 1002 3.616 0.1484

6487 1999 1000 1499 999 3.178 0.1308

8511 4006 3000 3503 1006 2.857 0.1167

8511 2005 1000 1502 1005 2.595 0.1061

10012 2005 1000 1502 1005 2.521 0.1032

Stress-dependant matrix permeability

0

0.1

0.2

0.3

0.4

0 2,000 4,000 6,000 8,000 10,000

Effective confining stress, psi

Mat

rix p

erm

eabi

lity,

Mic

ro-D

arcy

Stress-dependant matrix permeability

y = 1.9989x-0.1236

y = 21.548x-0.5937

0.0

0.1

0.2

0.3

0.4

0.5

0 2,000 4,000 6,000 8,000 10,000Effective confining stress, psi

Mat

rix p

erm

, Mic

ro-D

arcy

0.5

0.6

0.7

0.8

0.9

Bio

t Coe

ffici

ent

Perm

Biot Coeff.

Figure 2: Stress-dependant matrix permeability ( =1). Figure 3: Stress-dependant matrix perm with variable (p). Tight Sandstone Reservoir 1 In this case a very tight carbonate formation was tested to determine the absolute permeability as a function of effective stress. As explained earlier, combinations of hydrostatic stress and pore pressure were designed to provide enough data to determine meaningful trends such that the poroelastic coefficient, , can be logically unfold for a given rock formation. The benefit of determining, , this way is to ensure that the process used to yield, , is the same for which, , is being applied to. In other words, if we are evaluating the fluid flow through porous media under a given stress path, then, , should be determined based on its effect on the conductivity of that specific porous medium. Examining the k-Pc relationship evaluation plotted in Fig. 4, as is changed such that it is 0, 0.25, 0.65, and 1, an immediate awareness of a reasonable trend appears when =0.65. Detailed modeling of the data set may be further performed to refine as a function of Pc, then an iteration computation is required, however, an explicit prediction is necessary. One possibility is to use the area enclosed within the higher and lower boundaries enclosing the data points. The objective is then to determine the dependent variable when the defined area is minimum.

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Figure 4: From left to right; permeability changes as function of confining pressure assuming = 0; permeability changes as function of confining pressure assuming = 0.25; permeability changes as function of confining pressure assuming = 0.65; permeability changes as function of confining pressure assuming = 1.0. Tight Sandstone Reservoir 2 Similar measurements and analysis are performed for a tighter sandstone cores from Reservoir 2. As it can be noticed from Fig. 5 that the iterations of has resulted in reasonable trend appears when =0.35.

Figure 5: From left to right; permeability changes as function of confining pressure assuming = 0; permeability changes as function of confining pressure assuming = 0.35; permeability changes as function of confining pressure assuming = 0.6; permeability changes as function of confining pressure assuming = 1.0. Tensile Fracture When the reservoir pressure decreases, the elastic displacement in response to the increase in effective stress will cause natural fractures to close leading to a decline in reservoir productivity. The matrix medium feed the natural tensile fractures and the latter conduct the fluids to the wellbore. The decline in conductivity with increasing effective stress should follow a logical declining rate to support a given production rate.

Figure 6: Simulated tensile fracture shows the failed whole core (left) and the resulting fracture surface (right).

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The elastic closure response occurs when the net effective horizontal stress increases as a result of reservoir depletion. The elastic response to close the fracture follows Hookes law of elasticity and it is controlled by Youngs modulus of the formation:

Ee

= .. (5)

The aperture of the fracture will decrease causing a corresponding reduction of fracture conductivity. If we assume 50 ft of the rock perpendicular to the fracture will contribute to fracture closure, then for a Youngs modulus of 3 x 106 psi, the decrease in fracture width corresponding to a decrease in reservoir pressure from 7,000 to 4,000 psi will be 0.05 inches. The fracture will not close by 0.05 inches rather the contact points (asperities) will carry the applied stress to prevent fracture closure if they are strong enough to withstand the stress. The compressive strength of the asperities will determine the final fracture permeability. The reduction in conductivity is due to a combined effect of elastic response and compressive failure of the asperities. Compressive failure also generates rock particles and fines that will further reduce fracture conductivity. The fracture asperities in tensile and shear fractures differ considerably, as the first one is not accompanied with formation shifting while the rock in the latter experience formation shifting which generate a higher conductivity. As shown in Fig. 6, a 4-in sample was failed in tension following a Brazilian test. The induced failure plain represents a tensile fracture. The flow testing through the fracture was performed and separated from the total permeability using the following equation:

mamafftt AKAKAK += ...... (6) By following the same analysis provided earlier in Fig. 7, one can show that = 0.95 provides the best relation for the K-Pc. A final dominating fracture has caused the resulting permeabilities to be in the range of milidarcy rather than microdacry observed in the previous samples. This indicates that the fracture is main conduit for the fluid transmission, and therefore should be only assigned to the fracture and not to the whole sample. Although it has not attempted in this paper, a new transfer function between the matrix and fractures maybe derived as a function of the poroelastic coefficients in the two media.

Figure 7: From left to right; permeability changes as function of confining pressure assuming = 0; permeability changes as function of confining pressure assuming = 0.25; permeability changes as function of confining pressure assuming = 0.5; permeability changes as function of confining pressure assuming = 0.95. Applications 1) Pore Volume Compressibilities Pore volume compressibility (PVC) plays a critical role as a driving meganisim for fluid flow through porous media and the recovery of oil and gas. The PVC is an important parameter to match the recovery performance of a given reservoir. Its role becomes even more critical in naturally fractured reservoirs as it is necessary to assign two PVCs for the matrix and fractures in a dual porosity system. It has been documented that quantifying fracture compressibility is a major problem when simulating naturally fractured reservoirs13. The following equation has been suggested to determine fracture compressibility14:

=

h

kf

PP

ClnP

1-

k

(7)

SPE 124484 7

Where Pk is the net stress applied on the fracture and Ph is the fracture healing stress. For example, if we assume Pk is 4400 psi and Ph is 20000 psi, then Cf is calculated to be 0.00015 psi-1. To understand how the technique presented in this paper can provide an estimate of fracture compressibility, the following equation is assumed to describe the sample with one fracture:

f

ma

cc1 =f .. (8)

Equation 8 assumes that the bulk compressibility is controlled by the fracture compressibility. If we assume a typical matrix compressibility of 2.5 x 10-7 psi-1, then must approach 1 to get the fracture compressibility of 0.00015 psi-1. This is in agreement with the fracture test where is close to 1.0 for the collected data to follow a reasonable trend. Additionally a fracture acts as a piston like in counteracting the external strength which supports a value of 1.0 to be assigned to the fractures in a dual porosity system. For the non-fractured blocks, the new technique suggests testing the rock formation that has no obvious fractures to determine the appropriate . If we apply the following equation with same value of matrix compressibility, it is observed from Fig. 8 that the resulting from the test can provide an estimated value of the bulk compressibility and if natural fractures or microfracures exist in the rock formation.

b

ma

cc1 =b (9)

0.000000

0.000002

0.000004

0.000006

0 0.2 0.4 0.6 0.8 1Biot Coef f icient

Figure 8: Bulk compressibility as a function of Biot coefficient

2) Reservoir Simulation A mathematical model15 was used to compare stress-dependant permeability vs. constant permeability. The reservoir properties used the simulation were; a gas reservoir with a net pay of 46 feet, reservoir pressure of 7620 psi, permeability is 0.5 md at initial conditions. A single porosity reservoir was considered with constant matrix permeability and with stress-dependant matrix permeability. Fig. 8 shows the results from these two cases labeled as w/o stress and w/stress. In the w/o stress case, the initial matrix permeability was kept constant as the reservoir pressure decreases, while in the w/stress case the initial matrix permeability decreases as the reservoir pressure decreases. The stress effect is responsible for 50% loss of the PI.

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0

0.5

1

1.5

2

2.5

0 50 100 150 200 250Time (days)

PI, M

scf/d

/psi

w/o stressw/ stress

Figure 8: PI as a function of time for single porosity with and without the effect of stress on matrix permeability.

3) Tight Gas Reservoirs The effect of stress-dependant permeability becomes more exagurated in tight gas reservoirs. It is therefore, imperative to consider the stress-dependant in fractures as well as in the matrix in tight gas reservoirs. The microfractures are not necessarily characterized by a unit poroelastic coefficient as it depends on their apperature and surface rouphness. To get relastic reservoir simulation results in tight gas reservoirs, extensive experimental testing should be performed on matrix and micro-fractured rock samples to determine the Biot poroelastic coefficients in both media. There is a mjor difference between naturally fracture reservoirs and naturally fractured tight gas reservoirs. The natural fractures in tight gas reservoirs are basically microfractures especially in deep tight gas sand. The fundamental difference between low gas permeability (0.1 md - 1 md) and tight gas permeability (0.001 md - 0.1 md) appears to be the difference in the values of the matrix and fractures poroelastic coefficients. In tight gas san reservoirs, filling the existing microfractures with small mesh proppant will lead to more production than depleting the reservoir as a naturally fractured reservoir. This conclusion does not carry a general consensus rather it depends on the stress level and mechanical characteristics of the reservoir formation. In shallow reservoirs, the natural fractures may behave as infinite conductivity fractures and filling them with proppant will only reduce their contribution to the overall reservoir flow efficiency. However, in a reservoir with micofractures, the limited initial conductivity may vanish when reservoir pressure is decreased. 4) Laboratory measurements of Several methods have been introduced in the literature to evaluate the poroelastic factor, , required for various applications in petroleum engineering16 - 19. The determination of from fluid flow through rock samples under confining stress provides the data that reflects the actual process of fluid recovery under stress. Such direct technique to determine based on rock permeability under stress has been attempted in the literature20; however constant value of was reported. The current techniqure is based on the stress dependant permeability under the effect of pore pressure and confining stress. It is important to simulate the natural process of flowing the reservoir at high pressure initially and then as the reservoir is depleted the, pore pressure decreases causing the effect stress to increase. The new method uses combined effects of the pore pressure and confining stress and not to perform flow testing at constant pore pressure gradient at increasing levels of confining stress.

Conclusions 1. A new laboratory procedure to determine the poroelastic coefficients for the matrix and fractures media is presented.

The coefficients should be used in a dual porosity reservoir simulation study to obtain reasonable prediction of reservoir performance.

SPE 124484 9

2. This study has uncovered an important phenomenon related to the stress dependant poroelastic effect during production of naturally fractured reservoirs. The poroelastic coefficient in the matrix domain is considerably different than that of the natural fractures system.

3. In tight gas sand reservoirs or stress-sensitive reservoirs where permeability loss is substantial, keeping the natural fractures open should be the primary objective. Propping these fractures with small proppant mesh at early time should be considered as an effective reservoir management strategy for these reservoirs.

4. Many wells in naturally fractured reservoirs are initially good producers but after a short period of time a sharp decline in productions is observed. This is frequently interpreted as a flush production which is a rapid drainage of the fracture network, whereas fluid bleed-off from the lower permeability matrix rock occurs at much lower rates. This study suggests a new explanation related to unsynchronized permeability reduction rate in the matrix and fissures media due to different poroelastic coefficients in these media.

5. The contribution of matrix, natural fractures and microfractures to the overall reservoir productivity follow different stress-dependant permeability functions. The permeability functions of these porous components should be carefully determined for any reservoir simulation study.

Nomenclature A : Fracture area E : Youngs modulus K : Bulk modulus, psi K : Permeability, md Pe : External pressure, psi Ph : Fracture healing stress, psi Pk : Net stress on fracture, psi Pw : Wellbore pressure, psi PI : Productivity Index, Mscf/d/psi P : Pressure drawdown (Pe-Pw), psi re : External radius, inch rw : Wellbore radius, inch w : Fracture displacement during width development : Biots coefficient e : Displacement due to elastic response : Effective grain-to-grain stress.

t : Total minimum horizontal stress Subscript b : bulk e : external e : elastic f : fracture ma : matrix p : Pore t : total w : well References 1. Kasap, E. Schlumberger and Bush, E. S. Occidental Petroleum Corporation; Estimating a Relationship Between Pore Pressure and

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9. Hynes, B., Abdelmawla, A., and Strongberg, S., 2008. Impact of Pore Volume Compressibility on Recovery from Depletion and Miscible Gas Injection in South Oman. Paper SPE 115274 presented at the Annual Technical Conference & Exhibition, Denver, Colorado, 21-24 September.

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