A CHEMICAL COMPACTION MODEL FOR SANDSTONE

Abstract

A method for modeling compaction in a reservoir including obtaining a profile of vertical burial depth, a profile of effective stress against the vertical burial depth, and approximating a plurality of rock grains with a hexagonal closed-packed arrangement of spheres to estimate compacted mechanical and chemical porosity profiles.

Description

BACKGROUND

Compaction is a diagenetic process that can alter rock properties of interest in reservoir characterization, the process of modeling a reservoir to facilitate optimizing production of hydrocarbons from the reservoir. Compaction can cause the decrease in volume of a mass of sediment that occurs upon and during burial of the sediment to form rock. Compaction tends to result in a reservoir porosity trend in depth, which in turn influences extraction of hydrocarbons from the reservoir. Therefore, modeling compaction is useful for improving hydrocarbon production operations.

Compaction is mainly caused by overburden loading and involves changes in the packing density of framework grains and loss of intergranular pore space. Compaction generally includes both physical and chemical compaction. Physical compaction, also termed mechanical compaction, mainly involves grain reorientation and repacking accompanied by water expulsion from porous sediments, or even fracturing and cleavage of brittle grains and plastic deformation of ductile grains with additional overburden loading, which will significantly reduce pore space in sandstones. Chemical compaction, sometimes termed pressure solution, in sandstones mainly refers to grain dissolution, diffusion, and precipitation at the grain-to-grain contacts, which is also widespread in sandstones. Chemical compaction reduces the porosity and is primarily dependent on vertical stress, temperature, diffusive flow, and pore fluid chemistry.

In modeling chemical compaction, typically grains are treated as spheres in a simple cubic packing arrangement defined by a cubic unit cell. The simplicity of the cubic closed-packed cell arrangement results in simplicity of calculations with these models. However, these models tend to over-estimate the effects of compaction, resulting in a final porosity after burial that is unrealistically low in comparison to observed, real-world compaction trends, for example as determined by well logging. While existing mechanical compaction models can be sufficiently accurate, inaccuracies in chemical compaction models tend to cause difficulties in properly modeling overall compaction, which is made up of both mechanical and chemical compaction. Accordingly, there exists a need for improved compaction models.

SUMMARY

This summary is provided to introduce a selection of concepts that are further described below in the detailed description. This summary is not intended to identify key or essential features of the claimed subject matter, nor is it intended to be used as an aid in limiting the scope of the claimed subject matter.

In one aspect, embodiments disclosed herein relate to a method of modeling compaction in a reservoir including a profile of vertical burial depth against geological time, a profile of effective stress against burial depth, and an approximation of rock grains based on a hexagonal closed-packed arrangement to estimate mechanical and chemical compaction profiles against vertical burial depth in a subterranean region.

Other aspects and advantages of the claimed subject matter will be apparent from the following description and the appended claims.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 shows a flow chart of a method according to one or more embodiments.

FIGS. 2A, 2B, 2C show a hexagonal closed-packed arrangement, a hexagonal closed-packed unit cell, and a hexagonal geometric form, respectively, according to one or more embodiments.

FIG. 3 shows the direction of stresses effective vertical stress σ_V and effective normal stress σ′, in a hexagonal unit cell in accordance with one or more embodiments.

FIG. 4 shows an exemplary burial depth history of a hypothetical field in accordance with one or more embodiments.

FIG. 5 shows an exemplary thermal history of a hypothetical field according to one or more embodiments.

FIG. 6 shows variations of porosity with depth change during the burial history for an exemplary combined chemical and mechanical compaction model (Model 3) in accordance with one or more embodiments, a comparative cubic close-packed chemical model (Model 1), and a comparative mechanical model (Model 2).

FIG. 7 shows a system in accordance with one or more embodiments.

DETAILED DESCRIPTION

Embodiments disclosed herein are related to a method of modeling compaction that incorporates a model of chemical compaction based on modeling grain packing with a hexagonal closed-packed arrangement of grains. Use of the hexagonal closed-packed arrangement provides an improved estimate of chemical compaction in comparison to use of a cubic close-packed arrangement.

In one or more embodiments, an initial porosity of a formation stone and a compacted porosity of the formation stone are determined from a model of packing of grains of the formation stone. The formation stone may be sandstone. In one or more embodiments, grains of a formation stone are modeled as identical spheres. Spheres in different layers may be modeled as identical in terms of size. In one or more embodiments, the packing arrangement of the grains is modeled as hexagonal close packing. If the grains are not reshaped, the hexagonal closed-packed arrangement is the tightest arrangement.

FIG. 1 shows a flow diagram of an application of the chemical compaction model. In FIG. 1, the process 100 begins by obtaining data to use in the model. This includes two different plots, one, at block 102, of burial depth against geological time, and a second plot, at block 104, of effective stress against burial depth. Following this, at block 106, the radius of rock grains with a hexagonal closed-packed arrangement of spheres is approximated. With this data, at block 108, the porosity from mechanical compaction can be estimated for varying times and effective stress values until a threshold is reached. Once the threshold is reached, at block 110, the porosity from chemical compaction can be estimated using a model based on varying times, effective stress values, and the arrangement of rock grains. Using this data, a complete compaction model is created from burial depth, porosity from mechanical compaction, and porosity from chemical compaction. The hexagonal closed-packed model accounts for mechanical and chemical compaction simultaneously using complex equations to more accurately align with real world data than a cubic closed-packed model.

Thus, in one aspect, embodiments disclosed herein relate to a method 100 of modeling compaction that includes at block 102, obtaining a profile of vertical burial depth against geological time within a subterranean region, at block 104, obtaining a profile of effective stress against the burial depth within the subterranean region; at block 106, approximating a plurality of rock grains in the subterraneous region with a hexagonal closed-packed arrangement of spheres having an approximated radius; at block 108, estimating a compacted porosity profile against the vertical burial depth for porosity greater than or equal to a threshold, using a mechanical compaction model incorporating the effective stress; and at block 110, estimating the compacted porosity profile against the vertical burial depth for porosity less than the threshold, using a chemical compaction model incorporating the geological time, the effective stress, and the arrangement of rock grains.

FIG. 2A shows a hexagonal closed-packed arrangement of identical spheres. In one or more embodiments, the hexagonal closed-packed arrangement is considered as repeats of a unit cell. The unit cell is the unit cell for the grain packing. FIG. 2B shows a hexagonal closed-packed unit cell for identical spheres. FIG. 2C shows a representation of a hexagonal prism for identical spheres. The hexagonal prism shown in FIG. 2C is a geometric form of the hexagonal closed-packed unit cell shown in FIG. 2B. The base edge a of the hexagonal prism is shown in FIG. 2C and the height h of the hexagonal prism is shown in FIG. 2C. The black dots in FIG. 2C represent sphere centers.

In one or more embodiments, the hexagonal closed-packed unit cell can be used to estimate a porosity. To estimate porosity, first a grain volume is determined based on the hexagonal closed-packed unit cell. The grain volume is determined for a hexagonal closed-packed unit cell and is the total volume of all grains within a single hexagonal closed-packed unit cell. The grain volume depends on the shape of the grains. The shape may be modeled as a sphere, where the radius of the sphere is determined by the grain size, for example half the average grain cross-sectional size. The average grain cross-sectional size may be measured by counting on thin sections. In other embodiments, the average grain cross-sectional size may be measured using a sieve analysis or estimating grain size in the field through a visual comparison with grain size charts. In one or more embodiments, for the unit cell shown in FIG. 2B, the grain volume V_s is determined based on equation 1.

$\begin{array}{cc}{V}_s=6\times \frac{4}{3}\pi R^3& \left(1\right)\end{array}$

The value 6 is from six equivalent grains (spheres) in the unit cell. R is the radius of the grain.

In one or more embodiments, a geometric volume is determined based on the unit cell. The geometric volume is a volume of a geometric form for the unit cell. The geometric volume may be determined for a hexagonal prism. The geometric volume depends on the shape of the grains. As for the unit cell, the shape may be modeled as a sphere. For the geometric form shown in FIG. 2C, the geometric volume V_h is determined based on equation 2.

$\begin{array}{cc}{V}_h=\frac{3\sqrt{3}}{2}{a}^{2}h& \left(2\right)\end{array}$

The dimension a is the base edge of the hexagonal prism and h is the height of the hexagonal prism. The base edge a of the hexagonal prism equals 2R (R is the radius of the sphere). The height h of the hexagonal prism equal

$4\sqrt{\frac{2}{3}}R.$

In one or more embodiments, an initial porosity is determined from the grain volume and the geometric volume. The initial porosity is the porosity before compaction. The general relationship between the initial porosity Ø, the grain volume V_s, and the geometric volume V_h, is given by equation 3.

$\begin{array}{cc}\varnothing =\left(1-\frac{V_s}{V_h}\right)\times 100\%& \left(3\right)\end{array}$

The initial porosity is determined for the unit cell as shown in FIG. 2B and the geometric form as shown in FIG. 2C. Thus, equations 1, 2, and 3 may be combined, giving the initial porosity Ø as shown in equation 4.

$\begin{array}{cc}\varnothing =\left(1-\frac{V_s}{V_h}\right)\times 100\%=\left(1-\frac{6*\frac{4}{3}\pi R^3}{\frac{3\sqrt{3}}{2}\times {\left(2R\right)}^2\times \left(4\sqrt{\frac{2}{3}}R\right)}\right)\times 100\%=C.& \left(4\right)\end{array}$

Based on equation 4, the initial porosity Ø for a hexagonal closed-packed arrangement is a constant value C. The constant value C from equation C is about 25.95%. The constant value C is radius-independent. When estimating chemical compaction, different values can be used for the constant value C. For example, in the example below a threshold value of 24.5% is used for estimating chemical compaction to avoid stress singularities. Stress singularities occur when the normal stress is equal to an infinite value, which occurs when the radius of each contact surface is zero at C equal to 25.95%.

FIG. 3 shows a hexagonal closed-packed unit cell with an effective vertical stress (σ_v) and an effective normal stress (σ′) between grains demonstrated visually. The effective normal stress is the component of the effective vertical stress that is perpendicular to the particle contact area. The effective vertical stress is oriented perpendicular to the horizontal plane and is equal to the overburden stress. The effective vertical stress relates to multiple aspects of the compaction model described herein including the approximated radius of rock grains, the removed thickness, the radius of each contact surface, and the effective normal stress. The effective vertical stress also relates, in a separate relationship to strain rate, diffusivity, film thickness, and stress coefficient of solubility. In application, the effective vertical stress is calculated in the first steps of the model using the burial depth, to then calculate the porosity from mechanical compaction.

In one or more embodiments, the removed thickness δ is determined from a relationship between the effective vertical stress and the effective normal stress for the unit cell. The removed thickness is the removed thickness due to dissolution at grain contacts. For the unit cell as shown in FIG. 3, the relationship between the vertical stress σ_V and normal stress σ′ is provided in equation 5.

$\begin{array}{cc}{\sigma }^{\prime }=\frac{6\sqrt{3}{\left(R-\delta \right)}^2{\sigma }_V}{9\sqrt{6}\pi r^2}=\frac{\sqrt{2}{\left(R-\delta \right)}^2{\sigma }_V}{3\pi r^2}& \left(5\right)\end{array}$

where δ is the removed thickness due to dissolution at grain contacts and r is the radius of each contact surface. The radius is given by r=[(R+y)²−(R−δ)²]^1/2, where y is the overgrowth thickness due to reprecipitation). Reprecipitation is the process of dissolving at grain contacts and reprecipitating as grain overgrowth at the free grain surface.

In one of more embodiments, the overgrowth thickness y is calculated from a conservation of volume relationship assuming that dissolution volume from grain contacts will precipitate at adjacent grain surfaces, with pressure solution volume equaling precipitation volume. Equation 6 demonstrates this conservation of volume, assuming pressure solution takes place at each contact and precipitation occurs evenly at the grain surface.

$\begin{array}{cc}6*\frac{4}{3}\pi R^3=6*\frac{4}{3}{\pi \left(R+y\right)}^3-72*\frac{1}{3}{\pi \left(\delta +y\right)}^2\left(3+2y-\delta \right)& \left(6\right)\end{array}$

In one or more embodiments, the relationship between strain rate {dot over (δ)} and the effective vertical stress σ_V and effective normal stress σ′ is provided in equation 7.

$\begin{array}{cc}\stackrel{.}{\delta }=\frac{8\mathrm{Dwb}{\sigma }^{\prime }}{r^2}=\frac{8\sqrt{2}{\mathrm{Dwb}\left(R-\delta \right)}^2{\sigma }_V}{3\pi r^4}& \left(7\right)\end{array}$

where D is the diffusivity of solute in solution film, w is the film thickness, and b is the stress coefficient of solubility. The diffusivity is given by D=D₀e^(−H/RT), where D₀ is the diffusivity extrapolated to infinite temperature, H is an activation energy, R is the universal gas constant equal to 8.3145 J*mol⁻¹*K⁻¹, and T is the absolute temperature. Further, in one or more embodiments, there is a relationship between strain rate {dot over (δ)} and removed thickness provided in equation 8.

$\begin{array}{cc}\delta =\stackrel{.}{\delta }·\nabla T& \left(8\right)\end{array}$

where ∇T is the time period.

In one or more embodiments, a compacted volume is determined for a compacted form derived from the unit cell and geometric form. The compacted volume is a volume of the compacted form. In one or more embodiments, the compacted volume is determined from compaction of a hexagonal prism. In one or more embodiments, the compacted volume depends on the shape of the grains. In one or more embodiments, for a compacted hexagonal prism compacted from the geometric form shown in FIG. 2C, the compacted volume V_c is determined based on equation 9.

$\begin{array}{cc}{V}_c=\frac{3\sqrt{3}}{2}\times 2^2\times 4\sqrt{\frac{2}{3}}\times {\left(R-\delta \right)}^3.& \left(9\right)\end{array}$

δ is the removed thickness and R is the radius of the sphere, as shown above, without compaction).

In one or more embodiments, a relationship between compacted porosity and removed thickness is determined using an assumption that total volume is conserved. With the assumption of conserved total volume, the relationship between the compacted porosity Ø_c and the grain volume V_s and a compacted volume V_c is then given by equation 10.

FIG. 2C. Thus, equations 1, 9, and 10 may be combined, giving the compacted porosity as shown in equation 11.

$\begin{array}{cc}{\varnothing }_c=\left(1-\frac{6*\frac{4}{3}\pi R^3}{\frac{3\sqrt{3}}{2}\times 2^2\times 4\sqrt{\frac{2}{3}}\times {\left(R-\delta \right)}^3}\right)\times 100\%=\left(1-\frac{\pi R^3}{3\sqrt{2}{\left(R-\delta \right)}^3}\right)\times 100\%& \left(11\right)\end{array}$

δ is the removed thickness due to dissolution at grain contacts. The compacted porosity is then determined by obtaining the removed thickness.

FIG. 7 shows a system in accordance with one or more embodiments. The computer system (702) is used to provide computational functionalities associated with described algorithms, methods, functions, processes, flows, and procedures as described in the present disclosure, according to one or more embodiments. The illustrated computer (702) is intended to encompass any computing device such as a server, desktop computer, laptop/notebook computer, wireless data port, smart phone, personal data assistant (PDA), tablet computing device, one or more processors within these devices, or any other suitable processing device, including both physical or virtual instances (or both) of the computing device. Additionally, the computer (702) may include a computer that includes an input device, such as a keypad, keyboard, touch screen, or other device that can accept user information, and an output device that conveys information associated with the operation of the computer (702), including digital data, visual, or audio information (or a combination of information), or a graphical user interface (GUI).

The computer (702) can serve in a role as a client, network component, a server, a database or other persistency, or any other component (or a combination of roles) of a computer system for performing the subject matter described in the instant disclosure. The illustrated computer (702) is communicably coupled with a network (730). In some implementations, one or more components of the computer (702) may be configured to operate within environments, including cloud-computing-based, local, global, or other environment (or a combination of environments).

At a high level, the computer (702) is an electronic computing device operable to receive, transmit, process, store, or manage data and information associated with the described subject matter. According to some implementations, the computer (702) may also include or be communicably coupled with an application server, e-mail server, web server, caching server, streaming data server, business intelligence (BI) server, or other server (or a combination of servers). The computer (702) can receive requests over network (730) from a client application, for example, executing on another computer (702) and responding to the received requests by processing the said requests in an appropriate software application.

The computer (702) includes an interface (704). Although illustrated as a single interface (704) in FIG. 7, two or more interfaces (704) may be used according to particular needs, desires, or particular implementations of the computer (702). The interface (704) is used by the computer (702) for communicating with other systems in a distributed environment that are connected to the network (730). Generally, the interface (704) includes logic encoded in software or hardware (or a combination of software and hardware) and operable to communicate with the network (730). More specifically, the interface (704) may include software supporting one or more communication protocols associated with communications such that the network (730) or interface's hardware is operable to communicate physical signals within and outside of the illustrated computer (702).

The computer (702) also includes at least one computer processor (705). Although illustrated as a single computer processor (705) in FIG. 7, two or more processors may be used according to particular needs, desires, or particular implementations of the computer (702). Generally, the computer processor (705) executes instructions and manipulates data to perform the operations of the computer (702) and any algorithms, methods, functions, processes, flows, and procedures as described in the instant disclosure.

The computer (702) further includes a memory (706) that holds data for the computer (702) or other components (or a combination of both) that can be connected to the network (730). For example, memory (706) can be a database storing data consistent with this disclosure. Although illustrated as a single memory (706) in FIG. 7, two or more memories may be used according to particular needs, desires, or particular implementations of the computer (702) and the described functionality. While memory (706) is illustrated as an integral component of the computer (702), in alternative implementations, memory (706) can be external to the computer (702).

The application (707) is an algorithmic software engine providing functionality according to particular needs, desires, or particular implementations of the computer (702), particularly with respect to functionality described in this disclosure. For example, application (707) can serve as one or more components, modules, applications, etc. Further, although illustrated as a single application (707), the application (707) may be implemented as multiple applications (707) on the computer (702). In addition, although illustrated as integral to the computer (702), in alternative implementations, the application (707) can be external to the computer (702).

Each of the components of the computer (702) can communicate using a system bus (703). In some implementations, any or all of the components of the computer (702), both hardware or software (or a combination of hardware and software), may interface with each other or the interface (704) (or a combination of both) over the system bus (703) using an application programming interface (API) (712) or a service layer (713) or a combination of the API (712) and service layer (713). The API (712) may include specifications for routines, data structures, and object classes. The API (712) may be either computer-language independent or dependent and refer to a complete interface, a single function, or even a set of APIs.

The service layer (713) provides software services to the computer (702) or other components (whether illustrated or not) that are communicably coupled to the computer (702). The functionality of the computer (702) may be accessible for all service consumers using this service layer. Software services, such as those provided by the service layer (713), provide reusable, defined business functionalities through a defined interface. For example, the interface may be software written in JAVA, C++, or other suitable language providing data in extensible markup language (XML) format or another suitable format. While illustrated as an integrated component of the computer (702), alternative implementations may illustrate the API (712) or the service layer (713) as stand-alone components in relation to other components of the computer (702) or other components (whether or not illustrated) that are communicably coupled to the computer (702). Moreover, any or all parts of the API (712) or the service layer (713) may be implemented as child or sub-modules of another software module, enterprise application, or hardware module without departing from the scope of this disclosure.

Example

This example illustrates the methodology shown in FIG. 1. The methodology is applied in an exemplary work-flow to sandstone in a hypothetical field with an assumed burial depth, thermal history, quartz grain average diameter, and grain radius, and an assumption there is no sedimentary matrix, and for quartz overgrowths no other cements. It is assumed only mechanical compaction happens until it is as tight as hexagonal closed-packed, then the chemical compaction starts. The example assumes an initial porosity of 40.2%.

The following stages of a workflow can be conducted to model the compaction process.

Stage 1: Obtain necessary data including burial depth vs. geological time and effective stress vs. burial depth. FIG. 4 shows a plot of burial depth against geological age and FIG. 5 shows a plot of temperature against geological age. Both of these plots demonstrate hypothetical data utilized in the beginning of the example. The burial depth data is used in the calculations of effective normal stress throughout the compaction process. The temperatures at varying geological ages are used in the model during the calculations of diffusivity, which is temperature-dependent. Diffusivity is utilized to calculate strain rate, which leads to finding the additional removed thickness and then the porosity due to chemical compaction in the example. In application, there are multiple methods for measuring the profile of temperature against the geological time including fission track analysis, fluid inclusion analysis, organic maturity analysis, vitrinite reflectance analysis, and apatite and zircon (U−Th)/He thermochronometry. This model can be used with historical data, as was done in this example.

Stage 2: Calculate the effective stress, σ_v (MPa) based on equation 12 (assuming no overpressure) using the data from Stage 1.

$\begin{array}{cc}{\sigma }_v=\rho \mathrm{gh}& \left(12\right)\end{array}$

σ_v is the effective vertical stress; ρ is the burial loading density and is assumed to be 2.3 g/cm³; g is the gravitational acceleration (9.81 m/sec²); and h is the vertical burial depth selected from the graph of the data in FIG. 4.

Stage 3: Model the mechanical compaction with equation 11 as the mechanical model, meanwhile, setting a threshold for chemical compaction.

$\begin{array}{cc}\mathrm{IGV}={\mathrm{IGV}}_f+\left({\varnothing }_0+m_0-{\mathrm{IGV}}_f\right)e^{-{\mathrm{\beta \sigma }}_v}& \left(13\right)\end{array}$

IGV is the intergranular volume (sum of matrix, cements and intergranular porosity) and represents the porosity after mechanical compaction. IGV_f is the porosity of a stable packing configuration (minimum compacted IGV value due to mechanical compaction) and is assumed to be 0.23. ϕ₀ is the depositional porosity and is assumed to be 0.4. IGV, IGV_f, and ϕ₀ are all expressed as fractions. β is the exponential rate of porosity decline with effective stress (1/MPa) and is assumed to be 0.06.

Stage 4: Evaluate a threshold condition for porosity. The threshold condition is that the porosity is less than or equals a threshold value to avoid stress singularities at grain contacts. Once a threshold condition for porosity is satisfied in Stage 3, go to Stage 5 for chemical compaction. For this example, the threshold is assumed to be 0.245 in practical applications to avoid stress singularities caused by using 0.2595, which sets the radius of each contact surface to zero. By assuming the threshold is 0.245, stress singularities are avoided.

Stage 5: Calculate the current removed thickness δ according to equation 11 (i.e. computing δ based on setting the known mechanical compaction porosity IGV to Ø_c and grain radius). The grain radius is assumed to be 0.018 cm. Then, calculate the strain rate {dot over (δ)} based on equation 7. It is assumed that the film thickness w equals 10⁻⁷ cm and the stress coefficient b equals 6*10⁻¹⁰ cm³ quartz/cm³ solution. Diffusivity (D in the equation 7) can be calculated based on the equation 14.

$\begin{array}{cc}D=D_0{e}^{\left(-H/\stackrel{\_}{R}T\right)}& \left(14\right)\end{array}$

D₀ is the diffusivity extrapolated to infinite temperature, H is an activation energy, R is the universal gas constant, and T is absolute temperature. In the example, D₀ equals 3.488*10⁻⁹ cm²/sec, H equals 6 kcal/mol, and R equals 8.3145 J*mol⁻¹*K⁻¹.

Stage 6: Compute the additional removed thickness using the equation 8 during the current period. By adding the removed thickness δ (calculated in Stage 5) and the additional removed thickness, one can obtain the total removed thickness by the end of the current geological time period. Then, the porosity at the end this period can be computed by equation 11.

Stage 7: Next is to repeat Stage 5 and Stage 6 to model chemical compaction in the next period until the time reaches the present.

FIG. 6 is a plot comparing three different compaction models. As discussed previously, the simple cubic packing arrangement unit cell is represented by Model 1. Model 2 is a model with mechanical compaction as described above but no chemical compaction. Model 1 is based on chemical compaction only from a cubic packing arrangement unit cell. Model 3 is based on the present method, that is based on a mechanical compaction model combined with the chemical compaction model based on a hexagonal closed-packed arrangement. The data of FIG. 7 demonstrates that Model 3 acts as an accurate representation of compaction at both deep burial depths and shallow burial depths, when chemical compaction and mechanical compaction each dominate, respectively.

FIG. 6 demonstrates, for Model 3, the porosity loss by both mechanical and chemical compaction shows that the loss caused by chemical compaction is approximately 1.9%. In this example, the data indicates that mechanical compaction is dominated in shallow burial depth whereas chemical compaction plays a significant role in deep burial depth, as is consistent with observed real-world compaction. The intensity and occurrence of chemical compaction can vany significantly depending on the geological and environmental factors involved. In some cases, chemical compaction may not occur at all, while in others, it may play a significant role in the compaction process. In cases where mechanical compaction has ceased due to tight grain packing, chemical compaction can become the dominant process if it has occurred in the specific case.

FIG. 6 demonstrates, for Model 2, the porosity loss by mechanical compaction alone, shows that this porosity loss is approximately 16.7%. The data of FIG. 7 demonstrates that Model 2 ignores the effects of chemical compaction in deep burial depth, where chemical compaction plays a significant role.

FIG. 6 demonstrates, for Model 1, based on the simple cubic arrangement, the model shows a rapid porosity loss due to chemical compaction, confirming that existing models inaccurately overestimate the influence of chemical compaction. Model 1 shows a post-compaction porosity of 5% at 7000 feet and then a constant porosity at depths beyond 7000 feet, which is unrealistic. Although it does not include mechanical compaction. Model 1 is shown because it is the conventional model used for chemical compaction.

Embodiments of the present disclosure may provide at least one of the following advantages. The present improved method of chemical compaction modeling is based on using a hexagonal closed-packed arrangement to construct a unit cell for grain packing providing a more realistic compaction modeling as compared to mechanical compaction modeling alone and as compared to chemical compaction modeling based on using a cubic unit cell. Chemical compaction modeling based on a cubic unit cell overestimates chemical compaction and thus results in a lower porosity than is realistic.

Although only a few example embodiments have been described in detail above, those skilled in the art will readily appreciate that many modifications are possible in the example embodiments without materially departing from this invention. Accordingly, all such modifications are intended to be included within the scope of this disclosure as defined in the following claims.

Claims

1. A method of modeling compaction in a reservoir, comprising:

obtaining a profile of vertical burial depth against geological time within a subterranean region;

obtaining a profile of effective stress against the burial depth within the subterranean region;

approximating a plurality of rock grains in the subterraneous region with a hexagonal closed-packed arrangement of spheres having an approximated radius;

estimating a compacted porosity profile against the vertical burial depth for porosity greater than or equal to a threshold, using a mechanical compaction model incorporating the effective stress; and

estimating the compacted porosity profile against the vertical burial depth for porosity less than the threshold, using a chemical compaction model incorporating the geological time, the effective stress, and the arrangement of rock grains.

2. The method of claim 1, wherein obtaining the effective stress comprises obtaining an average burial loading density for the subterranean region and determining an effective stress based on, at least in part, the vertical burial depth, a constant gravitational acceleration, and the average burial loading density.

3. The method of claim 1, wherein the mechanical compaction model incorporates a depositional porosity, an exponential rate of porosity decline with effective stress, an initial proportion of matrix material, and a porosity of a stable packing configuration.

4. The method of claim 1, wherein approximating the plurality of rock grains comprises obtaining the approximated radius.

5. The method of claim 4, wherein obtaining the approximated radius comprises measuring an average size of core rock grains in a core sample from a well in the subterranean region.

6. The method of claim 1, wherein the chemical compaction model incorporates calculating a removed thickness based on, at least in part, the threshold porosity and the approximated radius.

7. The method of claim 6, wherein the chemical compaction model further incorporates iterating determining an additional removed thickness based on, at least in part a time differential, the effective stress, and the approximated radius; adding the additional removed thickness to the removed thickness to obtain an updated removed thickness; and determining a value of compacted porosity based, at least in part, on the updated removed thickness.

8. The method of claim 7, wherein determining the additional removed thickness comprises determining a radius of a grain contact surface, a removed thickness due to dissolution at the grain contact surface, and an overgrowth thickness due to precipitation.

9. The method of claim 8, wherein the overgrowth thickness due to precipitation is based on the approximated radius and the removed thickness according to a volume conservation relationship.

10. The method of claim 7, wherein the additional removed thickness is further based on, at least in part, a strain rate, based, at least in part on the effective stress and the approximated radius.

11. The method of claim 10, wherein the strain rate is further based on, at least in part, the removed thickness, the radius of a grain contact surface, the radius of a grain within the rock, and a diffusivity.

12. The method of claim 11, wherein the diffusivity is determined based on a diffusivity constant extrapolated to infinite temperature, an activation energy, a universal gas constant, and an absolute temperature.

13. The method of claim 12, further comprising obtaining profile of temperature against the geological time and determining the absolute temperature from the profile of temperature.

14. A method of extracting petroleum from a reservoir, comprising:

drilling a well in the reservoir;

modeling compaction in the reservoir according to claim 1 to obtain the compacted porosity profile; and

adjusting a parameter of producing the petroleum from the well based on the compacted porosity profile.