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Porous media like hydrocarbon reservoirs may be composed of a wide variety of rocks with different porosity and permeability. Our study shows in algorithms and in synthetic numerical simulations that the flow pattern of any particular porous medium, assuming constant fluid properties and standardized boundary and initial conditions, is not affected by any spatial porosity changes but will vary only according to spatial permeability changes. In contrast, the time of flight along the streamline will be affected by both the permeability and porosity, albeit in opposite directions. A theoretical framework is presented with evidence from flow visualizations. A series of strategically chosen streamline simulations, including systematic spatial variations of porosity and permeability, visualizes the respective effects on the flight path and time of flight. Two practical rules are formulated.

Flow analysis in porous media is at the macroscopic scale governed by Darcy’s law. The local flux and velocity are controlled by reservoir parameters such as permeability and porosity and by fluid properties such as density and viscosity. However, permeability and porosity are not directly connected, neither by their formal definition nor by dimensional units. In fact, porosity [fraction of void space] is a static material property of a porous medium. Permeability [m^{2}] is an independent dynamic scaling parameter in Darcy’s law that relates the fluid transmission flux [m^{−3} s^{1}] through a unit area [m^{2}] by the ratio of the applied pressure gradient [Pa m^{−1}] and the transmitted fluid’s dynamic viscosity [Pa s].

Although physically independent, there often exists an empirical relationship between the permeability and porosity for a given porous medium [

(a) SEM image of porosity structure for diatomite (from Erdőbénye, Hungary, see [

Darcy’s law features permeability, but porosity does not directly appear. However, the equation of motion includes both permeability and porosity, which are important parameters for predicting, respectively, the flow paths (FP) and time of flight (TOF) of fluids transporting in porous media [

Although for any particular rock type a higher permeability generally correlates with higher porosity (Figure

There is considerable risk for drawing overly quick conclusions in practical situations. Consider a simple doublet composed of one injector and one producer (Figure

Example 1: Illustration of possible misinterpretation of faster water breakthrough. Streamlines are tracked in blue and red curves are the time of flight contours spaced for 0.25 years. Water breakthrough takes 1 year in a priori model (a) but the posterior observation shows breakthrough occurs already after half a year (b).

A priori model

Posterior observation

In another example, consider a localized zone of faster flow, such as a longitudinal hydraulic fracture connecting two vertical wells in Figure

Example 2: Streamline trajectories passing through a hydraulic fracture zone. Flow plane is perpendicular to wells and fracture surface.

Our study intends to deliver a practical set of rules for the relationships highlighted in Figures

The basic flow visualization model uses 5 injectors and 5 producers in a direct line drive arrangement (Figure

Well constellation used in this study assumes one horizontal injection well with water injection rates controlled by 5 inlet control valves (ICVs). Flooding occurs by direct line drive between the water ICVs and 5 vertical producer wells where pressure can be monitored with installed bottomhole assemblies (BHAs; after [

Previous theoretical work and field experiments have demonstrated that the relationship between injected fluid and production is not always simple [

The porosity of porous media could be divided into connected porosity and unconnected porosity. Connected porosity could be defined as the ratio of the volume of fluid that can flow into the rock, while fluids cannot access unconnected pores. In this study, we assume the porosity is connected porosity. In our present study, we avoid any transient effects by imposing stationary, creeping flow and assume incompressible fluids. Under this assumption, the flow rates of injectors and producers will be kept constant.

In a porous medium, the material which is homogeneous at the scale of 1 millimeter might present heterogeneities at the scale of 100 meters. Unconventional reservoirs require dealing with multiscale problems due to the very low permeability in shale gas and oil reservoirs. On the microscopic scale, the fluid may perform as non-Darcy flow where classical Darcy’s law will no longer stand. To better model non-Darcy flow, different tools and flow equations have been proposed to simulate the atomic level of the diffusion phenomena; for example, the fractional diffusion equations were proposed to replace the classical diffusion equation [

While new tools might be useful for some unconventional reservoir plays, currently, for most popular models and simulators in petroleum industry, the conventional flow equations such as the stationary Stokes flow including Darcy’s law are still widely used. These equations are generally built on continuum mechanics, which is based on the assumption that bulk properties such as permeability, density, and porosity, all, vary continuously over the continuum on the macroscopic scale (usually between millimeter to kilometer). In continuum mechanics, at each point of the continuum, each specific bulk property is represented by the average over a representative elementary volume [

We will show below the theoretical proof of our contention that

We will first deliver explicit theoretical proof for the above rules. The physical parameters that control the streamline trajectories and the time of flight are solely due to reservoir properties and initial conditions. First, consider the following Stokes equation for stationary, creeping flow of an incompressible fluid [

If next we multiply

To show Rule 1 more directly, the relation between permeability/porosity and TOF is derived next. It is known that the time of flight,

An alternative expression relates the velocity field to the stream function; in the 2D case, we have the following corresponding relation:

Combining expressions (

After presenting and demonstrating the theoretical development in the previous paragraphs, next we will use numerical experiments to validate the inferred relationships between streamline trajectory patterns and permeability [see (

Our flow visualization experiments use a numerical streamline tracing algorithm, which allows comparison of time of flight as well as streamline patterns for the advancing flood front as it becomes affected by variations in reservoir properties. The base data used for our experiments are listed in Table

Numerical model parameters.

Parameters | Dimensions | Number of grids | Porosity | Permeability | Injectors | Producers |
---|---|---|---|---|---|---|

Value | 250 × 200 × 1 | 250 × 200 × 1 | 20 | 100 | Water, 1000 | Oil, 1000 |

| | | | | | |

The numerical streamline algorithm calculates the TOF information by using the methodology described in Datta-Gupta and King [

For simple reservoir models especially if it is homogeneous porous medium and simple injectors and producers placement as shown in Figure

The analytical streamline method and the numerical streamline algorithm have been cross-validated in our previous studies [

Verification of the numerical streamline method with the analytical streamline method (after [

The aim of our study is to investigate the effect of permeability and porosity on FP and TOF by some systematically designed numerical experiments. From the synthetic simulations, we can check our assertion that an increase in permeability decreases the time of flight, whereas an increase in porosity increases the time of flight

The first series of experiments is comprised of heterogeneous reservoirs including isotropic domains with both permeability and porosity contrasts.

The second series of experiments is comprised of streamline tracing cases which include

This section presents results of flow visualizations including domains with both permeability and porosity contrast. First a homogeneous base case is presented, with reservoir and design parameters specified in Table

The flow visualization of Figure

Top view of flow in horizontal reservoir with finite thickness but free flow between top and bottom boundaries (resp., above and below the image plane). Horizontal spacing between wells is 6 ft and vertical separation of injector and producer well arrays is 30 ft. Streamlines (red curves) and flood flight times (rainbow colors) are based on ECLIPSE pressure data augmented with proprietary velocity calculations based on the flux balance of adjacent cell nodes. Scale bar gives TOF in days and is valid for all three cases (a–c). Water front advance is outlined by the yellow leading edge as the waterflood moves away from the 5 injectors (lower row). (a) Base case permeability 100 mD and porosity 10%, TOF = 0.055 days. (b) Increasing porosity to 20% results in linear increase of TOF = 0.11 days. (c) Further increase of porosity to 40% results in linear increase of TOF = 0.22 days.

Figure

We next keep the permeability constant throughout the reservoir but the porosity is assumed to be heterogeneous, made up of two distinct domains that have a porosity, respectively, lower and higher than the ambient reservoir. The domain of lower porosity results in a local shortening of TOF (Figure

Matrix and domains of heterogeneous porosity both have base case permeability of 100 mD. Matrix has 40% porosity, except for domains of the yellow box (5% porosity) and blue box (80% porosity).

The porosity is held constant throughout the reservoir except for one low permeability domain (rectangular region) between the injector and producer wells where the porosity is systematically but uniformly varied to show the additional impact on TOF in such low permeability domains. For example, a high permeability zone will diverge streamlines such that more streamlines cross the higher permeability domain (Figure

Streamlines (red curves) and flood flight times (rainbow colors). (a) Domains of lower permeability result in local increase in TOF. Matrix permeability is 100 mD; rectangular central zone between injectors and producers has permeability of 20 mD. Porosity is equal in both domains (20%). (b) Increase of TOF in low permeability domain is undone when porosity in low permeability domain is adjusted from 20% to 5%. (c) Reversely, TOF can be increased still further than in (a) when porosity is adjusted from 20% to 80%.

Figures

Streamlines (red curves) and flood flight times (rainbow colors). (a) Domains of higher permeability result in local decrease in TOF. Matrix permeability is 100 mD; rectangular central zone between injectors and producers has permeability of 1000 mD. Porosity is equal in both domains (20%). (b) Decrease of TOF in high permeability domain is undone when porosity in low permeability domain is adjusted from 20% to 80%. (c) Reversely, TOF can be decreased still further as compared to (a) when porosity is adjusted from 20% to 5%.

This section considers results for reservoirs with directional permeability anisotropy while keeping porosity constant throughout the reservoir.

It is well known from the early work of Muskat [

Permeability anisotropy studies.

Formation location | Horizontal permeability (nD) | Vertical permeability (nD) | Ratio (horizontal/vertical) | Measuring method | Reference |
---|---|---|---|---|---|

Barnett | 96.3 | 2.3 | 45 | Pulse decay method | Bhandari et al., 2015 [ |

Eagle Ford | 50–210 | 50–80 | 4 | Pulse decay method | Mokhtari and Tutuncu, 2015 [ |

Wilcox | 11800 | 500 | 23 | Transient pulse method | Kwon et al., 2004 [ |

Longmaxi | 199.3–201.3 | 8 | 25 | Pulse decay method | Pan et al., 2015 [ |

Krechba | 12–58501.4 | 0.6–264 | 10–100 | Transient pulse method | Armitage et al., 2011 [ |

Impact of permeability anisotropy on FP and TOF. Streamlines (red curves) and flood flight times (rainbow colors). (a) Permeability in the horizontal

Local permeability anisotropy may occur in many situations. In clastic sediments, it may occur along fractures and fault zones, which may either increase permeability due to dilational effects, or reduce permeability, for example, due to clay smear. For example, carbonate-bearing fault zones can be interpreted as damage zones with the permeability increasing 2 orders of magnitude relative to that of the undamaged protolith [

Streamlines (red curves) and flood flight times (rainbow colors). (a) Domains of lower permeability result in local increase in TOF. Matrix permeability is 100 mD; upright to bottom left slanted zone has permeability of 10 mD. Porosity is equal in both domains (20%). (b) Matrix permeability is 100 mD; upright to bottom left slanted zone has increased permeability of 1000 mD. Porosity is constant and equal in all domains (20%).

Our final set of simulations is shown in Figures

Two central clay or shale walls obstruct flow, except for sand filled fractures that leave a high permeability conduit for the flood advance across the two central low permeability walls. Low permeability walls also occur above and below the well arrays. (a) There are two gaps on the second clay wall from the top. (b) There is only one gap on the second clay wall from the top. (c) The streamlines (red curves) and flood flight times (rainbow colors for TOF scaled in unit of days) for case (a), where sand permeability is 100 mD and clay is 100 nD; porosity everywhere equals 20%. (d) The streamlines and flood flight times for case (b). (e) Pressure map for case (a) (rainbow colors scaled for pressure in psi units). (f) Pressure map for case (b). Flow is in steady state, due to which pressures remain steady.

The corresponding pressure maps for the models of Figures

The compartments may themselves have more or less uniform pressure distributions. Figure

The effect of differential pressures is separately graphed in Figures

Pressure profiles along lines AB, CD, EF, and GH, where labels 1, 2, and 3 correspond to the pressure of the three compartments mentioned in Figure

Case H uses the same layered reservoir as for Case G but highlights the effect of heterogeneous porosity distribution on FP and TOF. The reservoir is still divided into layered zones, gray areas (Figure

The numerical streamline workflow is applied for both reservoir settings illustrated in Figures

Two central clay or shale walls obstruct flow, except for sand filled fractures that leave a high permeability conduit for the flood advance across the two central low permeability walls. Low permeability walls also occur above and below the well arrays. (a) The streamlines (red curves) and flood flight times (rainbow colors for TOF scaled in unit of days) for case of Figure

In this study, the relationships between the time of flight and flow path in porous media with spatial changes in permeability and porosity are derived and illustrated, both theoretically and numerically. Two fundamental rules are formulated to capture the essence of the relationships.

A principal conclusion of our study, based on careful derivations and systematically designed streamline simulations, is that permeability distribution alone governs the spatial distribution of streamlines. Porosity does have an influence on the time of flight but does not affect the streamline path. Those are not trivia conclusions, and although many practitioners may have empirical understanding of these relationships, we know of no systematic treatise with an explicit theoretical proof and substantiated numerical experimental validation for the following rules.

While these two rules are generic and applicable to any porous medium, the motivation of our study mainly comes from the surging interest in shale oil and gas production, in particular when studying the effect of anisotropy fabrics on reservoir flow, to which our two rules apply. For example, the emergence of oil and gas recovery from shale formations has led to renewed interest in laboratory measurements of anisotropic permeability properties of shale samples from major shale plays (e.g., Eagle Ford, Barnett in the US, and emerging plays elsewhere, e.g., Longmaxi formation, Sichuan Basin, China, and Vaca Muerta, Argentina). Numerous studies have detailed the permeability properties (e.g., [

Our simulations of Cases E and F used permeability anisotropy ratios of 5 and 10, respectively, which are conservative anisotropy factors compared to the range of values given in Table

Our simulation of flow across sand compartments separated by shale walls (Figures

A related finding from the Appendix is that, assuming constant injector rates in the reservoir, the pressure increases with viscosity of the produced fluids. Pressure will also increase due to any decrease(s) in local permeability and/or porosity.

Further insight obtained from our flow simulations is that larger pore space increases residence time of fluid passing through each pore (slowing TOF in such high porosity zones). Our results also shed light on residence time of acid injectates. For example, acid stimulation will more effectively react with high porosity zones. A common conjecture is that a higher effect of acidization is due to the larger volume of acid or larger contact surface area available for the acid simulation. But with the result of our study a higher effect of acid could also be a consequence of longer residence time in the high porosity spaces. Also, acid injection may reach producer wells faster via the low porosity zones and the injected acid reaches the high porosity space much slower (Figure

The rules highlighted in our study also apply to streamline studies for well productivity optimization and history matching. Several studies have been completed by us applying streamline solutions to a variety of geothermal and hydrocarbon production systems, including the optimization of well locations and well production rates in order to obtain maximum water sweep across oil drainage volumes [

High fidelity reservoir simulations that include complex reservoir geometry may obscure some of the systematic effects highlighted in our systematic study. Our conclusion is that permeability and porosity both need to be accounted for in well productivity models and in history matching process for the reservoir property parameters, for the following reasons:

Spatial permeability changes affect the streamline pattern (or particle flight paths if flow is transient).

Spatial porosity changes do not affect the streamline pattern.

Porosity changes for otherwise unchanged permeability distributions will affect the time of flight, but not the streamline pattern.

When the permeability is anisotropic and/or heterogeneous, the streamline patterns are affected. However, any spatial porosity changes will affect the time of flight without changing the streamline patterns.

Any temporal permeability changes due to reservoir compaction as production proceeds will effectuate streamline migration over the life cycle of a field.

In addition to permeability and porosity gradients of otherwise isotropic domains, spatial anisotropy of permeability can independently affect well productivity. Anisotropy and permeability both control the flow (or flight) path, leading to dispersion or channeling (Figure

Here we demonstrate that when the fluid flux of the injectors and producers should remain constant, the reservoir pressure changes inversely proportional to the changes of permeability and proportionally to the changes in viscosity. We performed two sets of experiments. In the first set of experiments, we varied the permeability value from 100 mD to 50 mD and 25 mD. In the second set, the viscosity contrast of water and oil was kept constant but absolute values were changed from 0.5 cp/2 cp to 1 cp/4 cp and 2 cp/8 cp. The resulting pressure changes are summarized in Table

Pressure changes according to different permeability and viscosity.

Porosity (%) | Permeability (mD) | Viscosity (water/oil) (cp) | Pressure distribution (psi) |
---|---|---|---|

0.2 | 100 | 0.5/2 | 362–7361 |

0.2 | 50 | 0.5/2 | 705–14590 |

0.2 | 25 | 0.5/2 | 1140–27691 |

0.2 | 100 | 1/4 | 705–14590 |

0.2 | 100 | 2/8 | 1140–27691 |

The pressure distributions for different permeability and viscosity. The corresponding pressure range is given in Table

100 md; 0.5 cp/2 cp

50 md; 0.5 cp/2 cp

100 md; 1 cp/4 cp

25 md; 0.5 cp/2 cp

100 md; 2 cp/8 cp

Porosity

Velocity vector

Velocity in the

Velocity in the

Velocity in the

Density

Pressure

Total compressibility

Dynamic viscosity

Permeability

Component of permeability tensor

Gravity in the

Volumetric flux in the

Initial gas flow rate

Stream function in 2D

Time of flight

Spatial distance along streamline.

= m

=

= Pa·s

= kPa

=

= mD.

The authors declare that they have no conflicts of interest.

Lihua Zuo gratefully acknowledges financial support from members of the Texas A&M University Joint Industry Project, MCERI (Model Calibration and Efficient Reservoir Imaging). This study was sponsored by start-up funds from the Texas A&M Engineering Experiment Station (TEES) provided to the corresponding author.

_{2}storage project at the Krechba Field, Algeria