- The fracture height is equal to the pay zone thickness.
- The reservoir is single-phase state.
- The reservoir is volumetric, i.e., there is neither water encroachment nor water production from the reservoir.
- Isothermal condition.
- Production is assumed to be solely due to change in volume of fluid and rock.

Hydraulic fracture nodes were used to simulate the length, aperture and the conductivity of the existing fracture. The poroelastic model is verified against analytical solutions (Detournay and Cheng 1988) and then it has been extended to predict the oil production rates from 2009 to 2020 for the studied field.

Figure 10 shows the comparison between the oil production rates from the finite difference and finite element method. The matching between the results from the two different methods is good, which is an indication that LGRs method was succeeded in simulating the hydraulic fracture.

**Fig. 10** Prediction of oil rates from two different methods.

## Fracture design

In this section, we discussed building the rock mechanics model to design the fracture and how to optimize the fracture design.

### Building the rock mechanic model

The rock mechanic properties have been calculated using the open hole logs (sonic, density and neutron porosity logs). Although only compression velocity is available from the sonic log, the shear velocity has been calculated using the following correlations for both the sand and shale layers.

Shear velocity prediction Results from this limited depth range study show that shear velocities vary linearly with compressional velocities for both sands and shale. A linear relationship between Vp and Vs has also been observed by Castagna et al. (1985) and Williams (1990). The equations for sand and shale shear velocity predictions are given by

Vs=0.7149Vp−2367.1,Sand

Vs=0.6522Vp−1902.2,Shale

The rock mechanic properties (Poisson’s ratio and Young’s modulus) have been calculated using the following correlations:

The closure pressure for M AR/G and the stress profile for the layers have been calculated using the following correlation:

The fracture toughness has been calculated using the following correlation

_{IC}=0.313+0.027*E

Another method for calculating the Poisson’s ratio has been used, using Fig. 11 which shows the relationship between porosity, sonic transit time and Poisson’s ratio. This method is considered less accurate compared with the method discussed above.

**Fig. 11** The relationship between porosity, sonic transit time and Poisson’s ratio.

Table 4 shows the summary of the rock mechanics calculations for the M AR/G and the layers above and below it. This table also shows good alignment between the calculated Poisson’s ratio using the sonic data and the porosity charts for the Shale layers but for the sand layer there is large difference. In order to cover these uncertainties around the Poisson’s ratio, different scenarios were used in the stress profile calculations. Appendix 1 presents the different stress profile scenarios.

**Table 4** The rock mechanics calculations.

## Optimize the fracture design and pump schedule

Several hydraulic fracture designs were evaluated to determine the optimum fracture design given the uncertainty of the stress profile when compared to the simulation results. Appendix 2 summarizes the results of the different designs comparison.

After this sensitivity analysis we found that massive hydraulic fracturing treatment is required for Fig 16 in order to achieve long half-length and high fracture conductivity. This massive frac will increase the history production rate by 4–5 times. The fracture half-length that could be achieved using the proposed design length ranging between, 600 and 800 ft. The fracture conductivity achieved by using the proposed model, ranging between 10,000 and 20,000 ft (md) depends on the proppant type used. Coarse proppant type is preferred (12–18); medium strength will give high conductivity. The amount of proppant required to achieve this design is ~360,000 Lb. In most of the cases the frac height will propagate downward toward the carbonate layer. In order to avoid propagating the frac toward the carbonate layer, very small frac design should be used (~40,000 Lb of proppant). This small frac size will not increase the productivity from well as required and the production rate is expected to drop rapidly after the fracture. Proppant flow pack additive should be used to avoid proppant flow back after the treatment and avoid damaging the sucker rod pump.

### LGR concept and design

In many problems we need a higher resolution (finer grid) than our grid permits. An example is where we model gas coning near the horizontal well. With a high resolution as in Fig. 12, we can track the gas front accurately, and give a good estimate for time and position of the gas breakthrough in the well. Also, the cells are sufficiently small that they can be classified as either gas or oil filled.

**Fig. 12** Gas cone near a horizontal well, fine grid.

When the same problem is modeled on a coarse grid, we see that the shape of the cone is completely lost, and the front is no longer clearly defined (Fig. 13).

**Fig. 13** Gas cone near a horizontal well, coarse grid.

Using the resolution of Fig. 12 on the entire grid is typically not possible due to memory limitations and computing time. One possibility is to extend the fine grid in all directions with coarser cells, as shown in Fig. 12. This is, however, not recommended solution, since the resulting long and narrow cells are sources of computational errors, especially when the size difference between large and small cells in the grid becomes too large.

In such situations it is much better to use local grid refinement (LGR). As the name implies, this means that part of the existing grid is replaced by a finer one, and that the replacement is done locally.

The LGRs which will be discussed in this section are regular Cartesian. The appropriate keyword is then CARFIN (Cartesian refinement). Basically a box in the grid is replaced by another box with more cells. The keyword is followed by one line of data, terminated by a slash. Note that only one LGR can be defined in one CARFIN keyword. The keyword must be repeated for each new LGR. Keyword ENDFIN terminates current CARFIN. The syntax is then,

CARFIN

Cartesian Local Grid Refinement

‘Name’ I1 I2 J1 J2 K1 K2 NX NY NZ NWMAX Name_of_parent_LGR/

ENDFIN

If we want an LGR on a volume that is not a regular BOX, this can be done by amalgamating several local grids into one. Each of the LGRs must be defined in the standard manner, and hence be on a regular BOX. There is no limit to the number of LGRs that can be amalgamated.

The LGR is the process of dividing one or several grids in the reservoir model into smaller sized grids allowing enhanced grid definition, which is essential for modeling hydraulic fractures using permeability multipliers. Because the fracture does not have a perfect rectangular shape, in fact it usually takes an elliptical shape; it is required to give different ratios in the refinement of each layer which cannot be achieved using only one LGR. This limitation leaves you with the choice of either to use amalgamated LGRs or to choose simplicity and sacrifice modeling the actual geometry; instead you will need to use a rectangular shaped fracture which does not accurately mimic the actual fracture geometry.

In this model several LGRs were needed in order to accurately describe the fracture geometry. LGRs were used for each well, one for every layer. Each layer was divided separately with different ratios to allow sensitivity on the fracture length and height.

This LGR resulted in increase of the grid cells number to be 77,076 cells, which is greater than the original number of cells by 4,836 cells (6.7 %). This increase in the cell number resulted in longer computing time by about 30 % simulated on the same machine which is actually an achievement compared to the normal cartesian LGR; a normal LGR would have increased the number of cells to 116,856 cells (61.8 %) increasing the computing time by up to more than five times the original computing time, and the model required a more powerful workstation to simulate the results (Fig. 14).

**Fig. 14** The dynamic without LGR.

The hydraulic fracture is implemented in this simulation study by choosing different cells inside LGR region and multiply its original permeability by a factor in order to represent the hydraulic conductivity of the created fracture as Fig. 15.

**Fig. 15** The dynamic with LGR.

The actual field production history was received after the implementation of the optimum frac scenario and was compared to the simulated results to show a match with great accuracy with the predicted rates and pressures. This match validates that the technique used in modeling this frac actually is capable of mimicking the real reservoir performance.

The keywords used in this process were:

LGR It is the first keyword used in the LGR creation, found in the Runspec section, this keyword is used to introduce the number of LGRs present and their main specifications.

CARFIN It is presented in the grid section, and it defines the cells included in the LGR and the number of their subdivisions.

HXFIN &HYFIN It is the keyword responsible for defining the ratios by which the cell should be divided; if not included the cell will be divided equally. This is keyword should be included in the CARFIN keyword (i.e., Before ENDFIN)

AMALGAM It is the keyword responsible for combining the separate LGRs in one group. Without this keyword LGR cannot be introduced in two adjacent cells.

WELSPECL Used instead of WELSPEC keyword for the wells located in the LGR, it used to introduce the wells that are present in the LGR cells.

COMPDATL Used instead of COMPDAT keyword for the wells located in the LGR, it is used to introduce the information about the completion for the wells included in the LGR cells.

## Conclusion

- Most of the previous publications simulate the hydraulic fracture in a finite difference simulator by using the ordinary technique of grids refinement. The resulting long and narrow cells lead to huge computing errors. This paper presents the concept of LGRs method and its implementation in a finite difference simulator.
- The modeling of hydraulic fractures could be achieved using amalgam LGR as previously shown with small effect on the computing times and will yield reliable and accurate results as concluded from the comparison of the postfrac production to the simulated rates.
- Hydraulic fracturing stimulation is expected to increase the production rate from Abrar-1 well by 2.5–3 times.
- Long half-length (above 600 ft) and high fracture conductivity (above 10,000 ft MD) are required and can be achieved in order to maximize the production rate and ultimate recovery from the reservoir.
- Massive hydraulic fracturing treatment will be required in order to achieve the required objective.

The key risks associated with the fracture treatment are the propagation of the fracture toward the carbonate interval below M.AR/G reservoir. Bending on the permeability of this carbonate layer, water production may be seen after the frac treatment.